Brandon’s story about his experience with the “2,3,5,_” series IQ tests and conversation that ensued prompted me to share some questions from a Dupage County Regional Office of Education Physics test. An old version of the test is available in a MS thesis from Northern Illinois University. The motives for creating this test are laid out in that thesis and are, worthy. The MS thesis examines student answers to the test and discusses some statistical properties of the answers to most, though not all, of the questions.
I’ve been tutoring physics students and like to know the topics, depth and level students are tested on. So I try to obtain any testing materials presented students. Naturally, I was interested in the questions themselves and I’ve been looking them over. I generally like to create ‘similar to’ questions and add them to a large database of questions I have been compiling.
I was looking at this question this morning, and thought of Brandon:
I know many of you will immediately solve this question correctly. But having tutored, I know that some high school students will have problems with it. Naturally, some people will say, “Of course, that’s because they don’t know physics!”.
Perhaps. Or perhaps not.
As a tutor, I know why some students who know physics blow this question. After reading a question like this, some students will look puzzled and ask: “Is the wagon on a hill? Is there friction?” Questions who ask these sorts of questions when presented with multiple choice questions tend to do badly on multiple choice tests. But these questions aren’t evidence of not understanding physics; they are evidence of understanding it well enough to detect ambiguity in the question as posed.
Bear in mind: This particular question is presented to students who have been doing all sorts of questions involving finding the acceleration of blocks on inclines with or without friction.
In my view, a “good test taker” who grasps high school physics pretty well will very quickly answer this question. A “good test taker” is one who has internalized the following “rule of high school tests” and applies it so reflexively they don’t even know they did it:
If the question says nothing about something that might affect the solution, and “not enough information” is not one of the options, assume the question-writer wants the test-taker to interpret the ambiguity to imply “the question writer wants us to assume the effect of that thing is zero”.
For this question, application of the rule means the kid is pushing on a level surface and frictional effects are absent.
When that rule is applied to this question, the answer is
v=(50 N/10 kg) 3 s = 15 m/s. Which is “C”.
Otherwise, if the wagon might be on an incline of unknown angle and/or there is friction, and that kids pushing force is the total force he exerts on the wagon, the truly correct answer would “Not enough information.” That answer is not among the options.
What to do?
What to do depends on what options one has.
If someone is in the business of “writing tests”, part of the solution is to have several people who share Brandon’s pickiness and who knows some physics goes over the questions and flag those he finds ambiguous and explains why.
Assuming the student needs to be ‘fixed’ the solution is more challenging. If a student’s high school teacher asks a lot of this sort of question, one’s concern is to help the student get a decent grade while not screwing up their understanding of physics. The fix needs to involve getting the student to learn “the rules of multiple-choice questions” while also managing to help them retain the level of understanding that caused them to ask the questions they asked while also being able to being able to take multiple choice tests quickly.
The fact that students may need to learn to apply “the rule” does explain why some parents do enroll kids in “test prep” services whose function can be to help students learn and apply “the rules of multiple choice tests”. One hopes this does not create other problems along the way.
Can the question be improved? Oddly, I’m not sure. I know it can be if the students are to answer questions at leisure. I’ve been trying to write questions that test understanding at a similar level to those they might experience in class. Where I see ambiguity, I try to eliminate it. I’m not sure I succeeded, but here’s a try:
Note the downside: being very specific makes the question longer. That’s a disadvantage on timed tests. When taking the ROE Physics tests, the teachers think students should be budgeted 50 minutes for 90 questions at this level or roughly 50*60/90 = 33 s. The questions are not difficult; most can be answered without use of a calculator. But they need to be answered quickly. Knowing time is an issue, one needs to give some thought to the order of any clauses in the sentence so as to create a question that scans well. I admit to never being quite sure I’ve achieve that. (Also: yes, it would be nice my answers were sorted in either ascending of descending order.)
Challenge:
I’m going to ask people to propose their theories of why honors physics students performed worse on this problem than non-honors:
I have my theory which involves knowing the answer to Question 3.
Some people who see that already know my theory for why honors students might blow Q 7 at a higher rate than non-honors students. Of course my theory could be incorrect. You tell me yours!
I used to enjoy finding unorthodox ways of reasoning answers to these questions, and one way was to spot which answers would involve ambiguity (so choose the other). From a chem exam:
“A balloon filled with H2S will rise in air. T or F?”
Well, if H2S was lighter than air, they would have to say something about the weight of the balloon. So F.
or
“Mammals native to Australia are always marsupial”
Well, humans have been in Australia for many millennia. It’s not what they meant, but if the question were T, then that would be an ambiguity. So F.
(They wanted students to think of some native rats).
My favorite (because I was otherwise flummoxed) was:
“Xylem tubes in monocotyledons are
a) Exarch and open
b) Endarch and closed
c) Exarch and closed”
Well, I still don’t know the facts, and probably have mis-remembered the choices. But they clearly want to test whether you know about ex/end and open/closed. And if the answer were b, you’d only have to know it was endarch. Or a, you’d only have to know it was open. Hence c.
Nick,
Clearly, they needed a “d” on that last one. I used to be a good guesser too, and I suspect I used tricks like that.
My mom took a teacher ed class at IIT when I was a highschool freshman. For some reason, I’d gone to campus with her and then stopped by the classroom. The professor (Dr. Shub or something like that) was discussing some test the ACS liked high school chemistry kids take, and the average grades which was something like 60 or so. He then suggested everyone take it. As I was there, he suggested I took it too so we could learn a baseline.
Before I took it he asked me if I knew any Chemistry, and I told him no– except that we’d discussed the periodic table in 8th grade. (We took biology as freshman.) Anyway, I did ok for a high school student. Not stellar but better than average. So, Dr. Shub suggested it might be because I heard chemistry at home and I said no, I pretty much guessed. He and the teachers all wanted me to explain how came up with answers. So, I said I guessed them all. They then asked how I’d guessed, and I explained my system. It included things like you’ve described. But I no longer remember them all.
Also… the room did contain a periodic chart, so that helped with questions like “which of the following is not a _____ ” where ____ is some category. I had no idea what those categories were, but I could tell what was in the same general region of the periodic chart. As in: Three choices are on the far left, 1 on the right. The one on the right is ‘different’ and so a good guess. And with names “three share an ending vs. the ones that doesn’t”.
It’s sort of amazing this sort of reasoning improves guessability on multiple choice. Obviously, the trick is going to work better on some tests than others. But learning to guess on multiple choice is a skill kids in public school can certainly benefit from.
The rule on multiple choice is that if you can eliminate even one answer as obviously false, then guess. I remember hearing that someone who had never studied Greek, took an SAT or something on Greek and achieved a reasonable score.
On math, there’s usually at least one answer that looks reasonable at first glance, but is wrong. Those aren’t that hard to eliminate either. Test prep is mostly about learning tricks like that.
DeWitt,
I was admitted to U of I undergrad but ultimately didn’t go there. But we didn’t know my decision until after the date for the CLEP tests. So mom signed me up for the maximum number we could take. (It was one of those things like $X for 1 test, $ 1.8*X for two and so on. 4 tests cost the same as three or something like that.)
I took the CLEPS, at one of the breaks I was tired and tried to persuade my mom we should skip the last one because I knew absolutely nothing about any of those. I took some sort of test in “Art”. As in paintings and such. When we got our results back I was stunned to discover I’d CLEPed out of ART.
But really, a lot of questions were things like “Which if the following is not an impressionist?” Then there were three artists who I knew lived in the same century and one from a much, much earlier century. I knew I’d gotten some right for reasons like that, but at the time I never imagined that would be enough to get 3 hours of college credit. I didn’t know anything much about art and still don’t!
A test in art? Never saw one of those. I suppose you could legitimately ask questions like: Which of the following artists like to paint nude women.
Lucia,
I was not an honors student in physics. I’m more representative of the “passed with a C” demographic. I’d like to take a stab at answering your question, about why #7 is harder for honor students, but before I can do that I’m curious if I’ve even got it right.
Negative velocity, positive acceleration. It’s slowing down, right?
mm. Velocity has direction, speed does not. Since acceleration is change in velocity, acceleration also has a direction. Acceleration is changing in #3. Is this (changing acceleration) the same as ‘accelerating’? I guess.
What do honor students say the answer to #7 is?
Oh. Maybe I see. Problem 3 highlights that acceleration can be a change in direction, not speed. What if this was the case in p7? The cAR wouldn’t be slowing down.
Lucia,
The answer to #7 would appear to be c, and to #3 the answer would appear to be c, at least if you consider acceleration as r*omega^2. I’m not seeing any obvious connection between the two. The earlier example (pushing on a cart) is a nutty question because it does not say “horizontal surface” and “frictionless”, so someone who understands more will see abiguity in the question…. but still,,the ‘best’ answer is pretty obvious. I am all for reducing ambiguity in these kinds of questions; you never want more understanding to lead to fewer correct answers
Nick Stokes,
A better question would specify a weightless balloon, of course. Poorly framed questions, like that one, fail on multiple counts: it confuses the student who really understands, and it fails to accurately determine if the student grasps important concepts and important information: 1) the molecular weight of a compound is the sum of the atomic weights of the constituent atoms, 2) the density of a gas is proportional to its molecular or atomic weight, 3) the approximate composition of air, and the resulting “average” molecular weight of air based on its composition, 4) the molecular weight of sulfur and hydrogen, and 5) lower density gas “rises” (or floats) in a higher density gas.
lucia:
Personally, I think that goes beyond being a good test taker. You could always worry, “But what if the wagon is resting against a wall?” Or, “What if there is another child pushing in the opposite direction?” Or, “What if there is a giant magent behind the child applying a force of X to the wagon in the opposite direction?” Or you could just assume when answering questions, all relevant information is provided.
Then again, I once took a test which instructed students to assume all relevant information was provided. A classmate of mine said, “But it doesn’t tell us what the force of gravity is!” So yeah, I’m not sure there’s a way to word things to avoid ambiguity >.<
If the “right” answer is a, then I’m not sure. I would expect honors students to pick that more often than non-honors students because you can have acceleration in the opposite direction of the direction you are traveling, which could be called “positive” even though it would mean you’re slowing down.
If the “right” answer is b, then I’d guess the reason is because non-honors students see “positive acceleration” and think “speeding up.” Which is kind of fair, I guess…? You can define acceleration’s directional frame of reference to match whichever direction you are traveling, in which case “positive acceleration” will always be “speeding up.” That seems weird to me though. I don’t know why you would have your coordinate system change rather than your acceleration term’s directional coefficient.
Or maybe I’m missing something else all together.
Actually, quick addendum to my last comment. Constantly redefining your frame of reference makes sense when you’re driving a car because your frame of reference is constantly changing since you’re moving. What I don’t get is why one would use that as an analytical approach for an object you’re examining, such as when picking the “right” answer for a test talking about some car’s movement.
Brandon,
A frame of reference can be moving relative to some other reference frame, but still not “changing”. Constant velocity of a reference frame is completely invisible within that reference frame.
SteveF:
I agree, but I’m not sure what your point is. Could you clarify?
SteveF,
“A better question would specify a weightless balloon, of course….
it confuses the student who really understands”
Well, no. A student who understands will do the reasoning you describe, find H2S is heavier than air, and so the balloon weight doesn’t matter. And the examiners, expecting that, didn’t bother to specify. That’s how you can guess.
It’s all what ifs. The bohemian artist Whistler came from a military family, but flunked out of West Point and went to Paris. Whence he used to remark
“If silicon had been a gas, I would have been a general.”
Brandon,
Note that it doesn’t sayaccleration is in the “opposite” the direction of velocity. It merely says “positive”.
To answer the question correctly you assume this is a 1-D motion problem in cartesian coordinates. So, in that case, positive and negative do mean acceleration is in the ‘opposite’ direction. But if — possibly because you answered #3 moments ago, your mind considers the possibility of circular motion, you could hesitate and think, “Well.. if it’s changing direction acceleration is not zero. And if it’s circular motion, that acceleration could be “positive” for some choice of coordinate system.
The degree of complication violate the “rules of multiple choice”. But I also suspect that the honors students are more likely to worry that the car might be traveling at constant speed but changing direction. In contrast others who forgot all about acceleration springing from change in direction will at least start off with the assumption of “it’s moving in a straight line.)
Given that 30 s are allowed, they might end up picking… whatever. (At what rate? dunno. Many will sort it out.)
One way to test the theory this was the problem is to do split testing and give some students the question worded to start with “A car traveling on a straight track….” and one that omits that– as Q7 does. See if the bias against honors students persists.
Mark
Do you see what my theory had to do with Q3 which they’d just answered. Q7 doesn’t actually say “straight line”. So yes, if it was turning, aceleration could be positive– but not in the direction of motion. It would be positive in some other direction.
I don’t know if this is what flashed through the minds of some honors students while answering. But it is the sort of thing that would be more likely to flash through the minds of better students. OTOH: extremely skilled ones would then discount it as “not what they meant” and still pick the right answer. All in 30 s.
NickS,
But the difficulty is if the question writer actually thought their goal was to have students figure out whether the H2S molecule is heavier than air based on their understanding of molecular weights and the ideal gas law, the question didn’t test that. It tested something else.
What it tested was not a bad thing– but it might not be what the test writer intended to test.
Brandon,
But has “all relevant information” been provided in Q30? Is saying absolutely nothing about friction the same as providing the information that the coefficient of friction is zero? (Which it never is?) I don’t think that’s quite right.
I think it’s more accurate to say that if it occurs to you that the magnitude of the coefficient of friction matters and they haven’t provided it, you remember that often– in highschool– it is assumed zero, and you go ahead and assume that.
In fact there is nothing in the question to indicate whether μ=0 is a good assumption. The question doesn’t say ‘very smooth” or “slippery” surface. It doesn’t say “rough surface”. And it’s not as if no-friction is a rule that works for ‘most’ problems involving sliding things along surfaces in real life. Nearly everyone knows they can lean on a couch and it won’t move– same for boxes, cars, sleds and so on. Friction is often not-negligible.
My view is relevant information is not provided. So the real thing is the student needs to learn: If they say nothing about friction, assume there is none. If they say nothing about incline, assume level. (Unless one of the options is “not enough information provided, in which case, that’s the answer!)
Lucia,
Makes sense to me. I’d bet your theory is correct.
The use of the qualifiers ‘positive’ and ‘negative’ implied to me that the acceleration opposed the velocity. But I’m not really sure that was warranted.
:> I see now why I was a ‘C’ type of physics student. 30 s? It took me over 10 minutes to notice what #3 could indicate for #7. And I was looking for it, too.
Brandon
The right answer was “a” and honors students picked it less often than non-honors students.
If you automatically think ‘rectilinear motion” (or “straightline” as more hs students would) (a) is obviously the right answer. BUT, if that was assumed to be a detail left out of the problem statement, then why is one of the options “changing direction”? The reason “changing direction is wrong is… well…to some extent because to get the answer right, you assume that the car is moving an a straight line. And more over, if you thought it was changing directions, you have no bench mark to decide if the acceleration perpendicular to the direction of motion is “positive” or “negative”– which is ambiguous.
The questions would be unambiguous if it specified the car was moving in a straightline.
Mark,
It’s been years since most of us took multiple choice tests– and taking them is a skill. A good test taker would have purged the memory of Q3 from their mind by the time they hit Q7. Nevertheless. students who are expected to know the answer to Q3 and who can’t fall into the “zen” of test taking might be forced to pause on Q7. Or not.
Those who don’t remember turning at constant speed results in acceleration won’t have that to deal with when reading Q7. They just think about “speeding up” or “slowing down” and– quite likely– get the right answer.
lucia:
Of course not providing the information friction is zero isn’t the same as providing the information friction is zero >.<
But if you assume all relevant information has been provided, and no information about friction has been provided, then the assumption is that no information about friction is relevant. That is, friction can be disregarded in its entirety. You could assume it's because the coefficient of friction is small enough its effect would vanish due to rounding, or you could just say, "Thought problems don't necessarily reflect reality so they may not factor in some things real world problems would actually factor in."
I know you’re making this point for a different reason, but I actually did make that exact point myself 😛
So if they think the car is traveling at a constant speed but changing direction, what answer would they pick? a and b are out since they don’t give constant speeds, c is out because the velocity is changing due to the angular acceleration, and d is… I don’t even know what. It’s ungrammatical, so I can’t tell if it would be wrong. I guess depending on how you changed it to fix the grammatical problem, it might fit your idea…?
(That is why I didn’t consider this interpretation when giving answers. I don’t see an answer anyone could reasonably pick if one chose to interpret the question as referring to circular motion.)
Who knows what a nervous test taker might pick? But “instantaneously changes direction” can correspond to “constant speed but changing direction”. Some cognitive dissonance could occur. What someone will ultimately put in the bubble– no one knows.
The motion doesn’t need to be circular for to have non-zero acceleration without change in speed. The students who take this test have been doing all sorts of “roller coaster” problems and so on. The roller coaster tracks aren’t circular– and yet acceleration is often ‘positive’ or ‘negative’ relative to ‘up’ or ‘down’ in these problems.
I’m not sure “ungrammatical” means “d” can’t be the answer. On the other hand, I suspect that is often a feature or “wrong answers”. The test writer took pains with the correct on and then winged it on the ‘wrong’ ones.
So it’s changing direction: whose to say that non-zero acceleration is not “positive”. It’s not “positive in the coordinate aligned with the velocity”. But “positive in ‘y'” is positive even if it’s not “positive in ‘x'”. So “instantaneously changing direction” could potentially be attractive.
(The fact is:it’s the honors students who blew this one relative to the other students. Mind you: it might just be ‘statistical outlier’. But. Still.)
lucia:
I knew as soon as I typed “circular” I should have typed “angular.” I had typed angular before, but I was in a rush and didn’t feel like going back to change that.
I guess…? Again, this just brings me back to my view you can always worry about all sorts of things. The car could be a Transformer in disguise, sent to save us from giant robots out to kill us all and steal our resources. I’m not sure that would actually change the problem any, but I don’t see why I’d bother taking the time to ponder if it would when trying to answer a simple question on a test.
I mean, if it’s for fun, sure. I get that. But if honors students are doing just as part of their normal test taking practice, I think they have bad teachers. I’d be inclined to teach them a lesson by failing them on a very easy test then explain to them, “I never said the test was in base 10; it was in base 8.” Because hey, technically questions don’t specify these things.
And then I could tell them there is no answer to Question 7 because it asks, “What is happening to the car?” and slowing down isn’t happening to the car; it’s what the car is doing. And hey, Question 3 doesn’t specify the speedometer is accurate…
I never thought much of honors programs. My experience with schooling was the more time I spent with it, the dumber I felt. I honestly don’t think I learned a thing in any of my main classes. I initially thought you’d have to be able to think a little to know how to apply different formulas, but really, schools have you do problems so many times it can just become rote.
The only classes that actually made me think were electives. And maybe one AP history class, because my teacher actually encouraged thinking there. I think she was the first person growing up who didn’t insult me when I said the Civil War wasn’t “because of slavery.” Which was funny because I was taking two US history classes that semester, one honors, and one AP (don’t ask). The honors teacher kicked me out of class for arguing that point with him. The AP teacher encouraged me to look at the many different contributing factors.
Brandon
If a Transformer in disguise is moving in a straight line with negative velocity and positive acceleration it is slowing down. If it has constant speed but is turning, its acceleration is positive in the direction pointing to the center of a circle that describes the radius of curvature.
I’m discussing things that are relevant (whether it is going in a straight line or not) and which the students have just been taught. With 30s to answer the question, that matters.
It says that is the speed; nothing said about measurement. There’s no ambiguity there.
Depends. The teacher from kids at Hinsdale I tutored has a teacher who seemed to like the “rote” method. The ones in Naperville (different system) wasn’t into ‘rote’. Sister Abels, my high-school teacher, wasn’t into “rote”. But I think many are.
I hated my highschool– but it did have some strengths. Not valuing ‘rote’ was one of them.
Lucia,
If the coordinate system is allowed to be different between the “velocity” and the “acceleration,” then the answer could be anything at all.
Given that the wording was specifically chosen to include “positive” and “negative,” and given that there are only 4 choices, at some point even a physics honor student is going to have to stop thinking about all the ways in which he or she can interpret the question in a smarter way than the test writers and just pick a damned answer!
Nick,
If the hypothetical balloon is specified as zero weight, then knowledge of the listed concepts is required to answer correctly. If a specific weight is given, but no volume of gas is given, then the problem is poorly framed because (among other things) there is insufficient information, and it allows the kind of guess you deacribe. Same thing with no mention at all of weight of the balloon. There are lots of ways to ask a poorly framed question which can defeat the purpose of the question.
Oliver
Of course. Given the testing rate, they have 30 seconds. And there is a best simplest answer and good test takers need to figure that out quickly even if the question is contains an ambiguity.
I’m just posing the question of whether we have theories of why one group– honors did worse than the other group not honors under the circumstances.
In a minute, I’ll show you another question for comment.
Brandon,
You wrote:” Constantly redefining your frame of reference makes sense when you’re driving a car because your frame of reference is constantly changing since you’re moving. ”
.
That is what I was referring to. A frame of reference is not changing unless it is undergoing acceleration. Linear movement (relative to things outside that frame of reference) makes no difference, so there is no need to “constantly redefine” the reference frame, absent acceleration.
Oliver, here’s another one.
Students seem to have done fine on this question. That said: I think the formatting is unfortunate and could have cause students to spend more time than necessary sorting out the format. The three formatting issues:
1) The text should not be obscured behind arrows or the box.
2) Text should be more closely linked to the corresponding arrows.
3) Either spacing or a line would make the division between the question easier to see quickly.
Others might immediately see that the word “gravity” corresponds to the downward arrow on option (a). But at first glance, I thought the word “normal” did. I sorted it out and I suspect students did too– but I do think the formatting is poor and likely consumes an extra 5 seconds or so.
Does this matter? Well… the time allotted per question is 30 seconds. Perhaps this seems unimportant, but if every question was like that, the time ‘robbed’ by ambiguity and/or poor formatting corresponds to 7-8 questions out of 90 given. The issue will be worse for “poor test takers” than for “good test takers” and it’s orthogonal to “knowing physics.”
Remember: this is a county wide test distributed to many schools. It is not a test written by one harried teacher given to one set of students. Evidently, some teachers use it as their final exam. (Though I’m really not sure how many do that and they are permitted to mix the question into their exam. I suspect many mix the mechanics questions into the winter term exam and the EM questions into their fall term.)
This doesn’t really pertain directly, but it was nagging at me. The initial problem with the child accelerating the wagon would seem to require a child of unusual athletic talent. Kid needs to be in contact with the wagon for the three seconds to be applying the force, so at the end that kid is cooking along at 15 m/s too.
.
Adults don’t really sprint faster than 12 m/s.
.
But there’s no ambiguity here. If you don’t accept the numbers, you can’t do anything with the problem at all. Still, it could be that by trying to ‘sanity check’ the results of some problems that were never meant to be ‘sane’ in the first place, students might be driven to question stuff that they’ve done correctly.
I don’t know.
SteveF,
I suspect Brandon meant ” constantly changing since you’re moving [ in the way cars typically move]. â€
On the road, cars typically accelerate, decelerate, turn, go up hills, down hills, stop, start. In parking lots, they often back up. If you center your coordinate system on the car you’ll have a non-inertial reference frame.
That said: I don’t think the issue of “non-inertial reference frame” would give any high school test taker any pause because I think none would really have thought about that.
In my theory (which could be wrong) it’s merely the issue of a student who knows turning results in acceleration having to reread and decide and some do so incorrectly, while those who have forgotten that will be able to answer more quickly. Did that matter? I don’t know. The thesis itself points out the difference in performance and it’s a reason that occurred to me.
Lucia,
The poor formatting/printing of the diagrams is a horrible waste of time during a timed test…. unless the test is trying to determine ability to read partially obstructed words. The rest of the question seems perfectly OK.
Lucia,
“That said: I don’t think the issue of “non-inertial reference frame†would give any high school test taker any pause because I think none would really have thought about that.”
.
Really? That surprises me. Without the fundamental concept of an inertial reference frame, I don’t see how curvilinear acceleration or Newton’s Laws make any sense at all. The students you tutor don’t understand reference frames?
Mark Bofill
This sort of problem does cause some students problems both because:
a) They have been told that they should learn to reality check their answers as that helps them find errors and
(b) Some of them seem to have already internalized that habit even before they take physics.
So some who have internalized the rule of reality checking and know 15 m/s is over 30 mph will pause to double check. Others might not. And some will recognize that the advice to ‘reality check’ is often more useful in real life than on multiple choice tests!
At least 1/2 my students will pause if the answer doesn’t make any sense.
Notice I changed the child into a ninja. A teenage mutant ninja turtle would permit an even broader range of ‘realistic’ answers as they are fictional.
SteveF,
Reference frames are discussed very early on and they do know the difference between distance traveled by a passenger on a bus relative to his seat on the bus and relative to the earth.
As far as I can tell, the issue isn’t tested much — except in very focused questions in kinematics. But they have seen “reference frame” as a constant.
What I haven’t seen any questions on tests like “Regents”, or high school worksheets around the web, and so on that test them on understanding the importance of “non-inertial reference frame” itself. So they can pretty well gloss over that and little time is spent and my impression is the average high school student whose taken a first contact physics class (not AP) will not even think about “non-inertial reference frame” issues when solving a problems.
Perhaps I’m wrong, but that’s my sense.
Licia,
Do the tests still have questions about pulleys ( frictionless, of course), thermal expansion, and projectile motion?
The Dupage County test had nothing about pulleys. The New York Regents also normally doesn’t. Here’s all the NY Regents tests:
http://www.nysedregents.org/physics/
I’m trying to remember now, but of students I tutored, I think only 1 was in a curriculum that covered pulleys. None covered thermal expansion. All covered projectile motion.
None covered the first law of thermodynamics, but all covered work energy. All covered conservation of momentum. One system covered torque, the others did not. Some systems did “modern physics”, some did not. Some did more depth; others more “conceptual”.
With respect to what’s on this test: The DuPage County ROE test was a teacher motivated project (and a very good project generally). They decided the test would cover only those topics that were common to all curricula. So projectile motion is on that test. So are Newton 1,2,3. Universal gravitation is not; torque is not.
I suspect, but don’t know, that the choice of emphasis in the first contact physics class is influenced somewhat by whether the school is more focused on having students pass the “AP Physics C” tests or the “AP Physics I” tests. The former require calculus covers ‘mechanics’ and “EM”. The latter don’t require calculus and cover a broader range of topics.
If you are going into Engineering schools will generally not grant credit toward graduation for Physics I because a student who takes that will not have the pre-requisite depth of knowledge in mechanics or EM to continue on to the engineering courses and the many topics are at insufficient depth to ‘count’. To get college credit based on an AP an engineering student needs to take Physics C. By the same token: in non-engineering majors a school might grant a larger number of hours for Physics I. So if you are majoring in Econ or French, you might prefer to take Physics I. I don’t know how schools decide which tests prepare their students for, but I suspect the demographics of the parents in the suburb affect the choice.
SteveF:
BTW, one of the ‘questions’ presented a student that most annoyed me was a True/False one:
“The orbit of the moon is circular.”
The eccentricity of the moons orbit is, evidently 0.055.
The eccentricity of Mars orbit is 0.0934. This is sufficiently eccentric for Kepler to detect lack of circularity, formulate his laws and have them named for him. Moreover, one of the famous things Newton did was show that his laws could be used to derive Kepler’s laws.
If they were get credit, the student had to answer True. The answer of “False” was “wrong”.
Now, I get that some people might think that eccentricity of 0.055 is “close enough” to a circle to make “True” a ‘better’ answer than False, but I think that could be debated. My view is that shouldn’t be a “T/F” question.
For what it’s worth, the eccentricity of the earth’s orbit around the sun is 0.015 which is distinctly less than the eccentricity of the moon’s orbit. People discuss epihelion and perihelion for the earth particularly with respect to Milankovich cycles. So lots of knowledgable people would say the earth’s orbit is ‘not circular’ and that the effects of non-circularity may be enough to impact climate.
Fortunately, the DuPage ROE test did not have questions with debatable answers. I agree with all their answers are the ‘best’ one. I just think some of the questions on the test do have the feature test questions sometimes have. The questions are sometimes a bit ambiguous and that can sometimes force students to interpret using “the rules of multiple choice tests”.
Ideally, a test is proof-read by multiple people to eliminate these issues– but as a practical matter, they often aren’t. (Heck, I’m not even saying these issues mean the test is bad or that they cause great harm. But I am saying kids do need to learn some “test taking strategies”.)
Re: lucia (Comment #138433)
That’s a terrible problem as printed. Interpreting the text to mean that the arrow in the top figure points in the direction of the velocity (and hence the direction the rope must be pulling), I think I can figure it out by blocking out the labels in the answer choices entirely…
But why is “Rope” on the left side of the figure?
oliver,
I think the word “rope” on the left hand side is yet another poor formatting choice. I think one might say “inexcusably poor”.
I am very puzzled that a question formatted this badly was on the test. I understand that one of the teachers might somehow not grasp the fact this question is poor. But I do not understand who the other teachers involved in the project didn’t say “this formatting sucks. Could you fix it? ” If it could not be fixed (which it can) the question should be omitted.
There’s another question that could be improved by better formatting, but it’s not quite as bad.
The question would be improved if the 14N was near the arrow in the question statement. Ideally the 6N’s should be aligned with their arrows.
I think students did ok on this one so presumably they sorted it out. But the formatting still strikes me as sub-optimal.
Lucia,
“I’m trying to remember now, but of students I tutored, I think only 1 was in a curriculum that covered pulleys. None covered thermal expansion. All covered projectile motion.
None covered the first law of thermodynamics, but all covered work energy. All covered conservation of momentum. One system covered torque, the others did not. ”
.
Seems like pretty thin gruel.
In the physics achievement test I took in 1968, I remember multiple questions on pulleys, including complicated pulleys pulling on another pulley’s rope, simple machines like wedges, levers, and screws that provide mechanical advantage, and multiple questions on thermal expansion. The one I remember most clearly went something like: “Imagine a horizontal metal rod of uniform initial temperature, a little over 3 meters long and 1 cm in diameter, supported on its left end by a fixed structure, and supported near its other end, 3 meters from the fixed end, by a 0.1 cm diameter rod with its long axis perpendicular to the long axis of the metal rod, and which is in turn supported by a flat horizontal surface. (a diagram was given) If the coefficient of thermal expansion of the metal rod is 1 * 10^(-5) per degree C, and the 3 meter long rod warms by 20 C, and assuming there is no slippage between the two rods, which of the following statements is most correct:
a) The supporting rod will turn clockwise by approximately XX degrees
b) The supporting rod will turn counterclockwise by approximately YY degrees
c) The supporting rod will turn clockwise by approximately ZZ degrees
d) The supporting rod will turn couterclockwise by approximately QQ degrees
e) The supporting rod will not turn”
.
What I remember most about that test is that many of the questions required knowledge of multiple physical concepts and an reasonable competence in algebra and geometry to answer correctly.
SteveF
My opinion is that is not the case for the DuPage ROE physics test and I would say it is not the case for the NY Regents with the caveat that the Regents does a contain free response section — but still my impression is each question is focused on 1 principle.
However, the AP Physics tests do require student to apply multiple physical concepts and display competence in algebra and geometry but the concepts are limited to those covered. So: one might do use work energy, Newtons 1st law, kinematics in 1 question. But heat transfer will not appear. This isn’t much different from college tests where you won’t suddenly have a circuits question spring up on a statics test.
For what it’s worth: the analysis of the question for the DuPage ROE test deemed them “easy”; that was based on properties of the student responses. I’d have to show graphs they used to determine “level of difficulty” which– if I understand correctly has to do with the location of an inflection point on a particular sort of plot. (Note: results would depend on who took the test, so it’s really “level of difficulty relative to subjects tested.).
The MS thesis analyzing the results also discussed the functions of “easy” vs. “hard” tests and the function of having a mix of problems difficulties. (It was an interesting read.)
On what students actually do take: Those high school students who continue to AP generally take either the “AP C” flavors or they take AP Phys 1 (and possibly 2). As far as I can tell, none include thermal expansion.
That said: AP Physics 1 test does not appear contain thermal expansion– it’s just mechanics and EM — I diagnose that by looking at the formula sheet:
https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap-physics-1-equations-table.pdf
AP physics 2 must cover some conduction because they list Fourier’s law, but there is no coefficient of thermal expansion.
https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap-physics-2-equations-table.pdf
AP C does not cover the 2st law of thermo, fluids, heat transfer or the like. It is (a) Mechanics and (b) EM.
SteveF,
It looks like the test your recall is now called
“SAT Subject Test in Physics”. It used to be called the ‘achievement test’.
https://en.wikipedia.org/wiki/SAT_Subject_Test_in_Physics
I took this test. I don’t remember what was on it. At. All.
By the way, if you want to see the copy of the exam analyzed it’s available in appendix C of this thesis:
http://search.proquest.com/docview/1554346073
Lucia,
Yes, that is the test. The format was a little different, but most of the subject matter looks to be the same; I don’t think there was any thermo, just basic concepts in heat and heat transfer. I didn’t know any thermo to speak of when I was 18, but I was familiar with everything else. I do recall that you had to be able to do trig, algebra, and geometry without any outside information, prompting, or formulas, or you were pretty much screwed.
help
Maybe I’m thinking about driving too much, but can’t get 7, at all.
assuming forward is positive, backwards negative..
negative velocity has the car going backwards.
positive acceleration, means you are accelerating in a forward gear.
which means you’ve trashed your gearbox, or are in a forward gear sliding backwards on ice..
I really don’t get 7.
Barry,
Strangely, this didn’t trouble me. I thought of the signs as arbitrary, maybe referring to some external coordinate system. Like, north is positive and south is negative, and the car was facing south.
This actually did cross my mind; is the car braking or is it in gear?
I think braking qualifies as an acceleration that opposes the velocity. Remember, acceleration is defined as a change in velocity. But the ‘in gear’ case didn’t bug me too much either; I figured maybe it was a MythBusters episode or something and the car was going to get trashed in the experiment.
[Edit: Cars can brake while in gear. I didn’t identify the cases properly above. I meant, is the acceleration due to braking or due to the forward drive system opposing the velocity.]
Barry,
Put the gar in reverse, hit the accelerator. You are now going backwards. Next: Hit the brakes. Your
velocityacceleration is now positive, but you are still going backwards (until you stop.) You are slowing down.Being able to think through the real physical analog is a skill teachers actually are trying to teach. Braking could easily have been discussed as an example in class.
As long as someone doesn’t over complicate it and reads backwards/forwards as “in the same line”, the problem is easy, there is no physical problem and I suspect can be done in <20 second and the kid moves on.
(That's why, for example oliver and brandon are right that you can't 'complicate' the issue when taking a test. Which is why I say that when you encounter this, you have to figure out how to get a kid to be able to take tests. In life, a person can take the extra 30 second to confirm, "you mean it's moving in a straight line.... right? Or answer, "If it's moving in a straight line, then."
Interesting that I had to double think the problems because 7 was presented before 3 . answers a for 7 and c for 3 then.
reading the replies I found I was confusing the problem 3 answer for problem 7 answer due to having the order the way they were originally presented.
Also made me think of breaking and slowing when entering a roundabout, accelerating when in the curve and continuing to accelerate when coming out to get back to the original speed.
Kid running at 15m/s was interesting, but even more interesting was the fact that the calculation was too difficult come up so ended up just guessing in my mind the feasible 6m/s. I’m a physics major type person but not have been doing any Fs/m for years and years.
I think multiple choice questions are just for teachers who can’t differentiate right answer from wrong one.
The same guess effect happened to me a while ago. I was asked how much is .60 times .50. I misread and chose .33 from options available! Being wrong had nothing to do with my capability, but with time limit, font and potentially misleading options. Put a hard time limit, you get quick and dirty answers.
A misleading a-b-c-d question is an efficient way of making differences where none really exist.
Barry,
It doesn’t need to be ice. You’d get a lot of smoke, but you wouldn’t trash the transmission. My avocation is being a volunteer turn marshal (flagger) at sports car road races, I’ve seen this exact situation more than once.
From an earlier comment: the sign of the acceleration from the reference frame of a car traveling in a circle at constant speed depends on whether the car is traveling clockwise or counterclockwise. Lateral acceleration is the product of the speed and angular velocity. The sign of the angular velocity depends on whether one is turning right or left.
I’ve played with inertial navigation using the three axis accelerometer and three axis angular velocity sensor on a smart phone. The problem is that the output of the sensors is noisy and you have to double integrate to get position. Hence, the calculated position drifts rapidly from the real position. So you need a reference to correct the drift. That’s obviously GPS. One could use just GPS, but most don’t update rapidly enough. But the GPS data is Earth frame and the other data is in the car frame, so one needs to transform between the two frames. That involves either quaternions or cosine direction matrices. Fun.
Drones have this built in. You can even buy all the pieces and build your own navigation and flight control system for not very much. The software is open source and free. With a helicopter drone, you need a magnetometer as well because GPS doesn’t give you heading information unless you’re moving. Most smart phones have magnetometers too.
There’s a lower-speed alternative, relative to having your ends swapped, for forward acceleration in gear while moving backward. A lot of people back out of a parking spot by giving a small “push” in reverse — just enough to roll the car back. You push in the clutch almost immediately, then when you are just about where you want to be, you shift into 1st and start driving away. It’s all very low speed, so no gearboxes get trashed, but to an outside observer the car is clearly “accelerating” forward while backward, smoothly through zero and into going forward.
Also, normal GPS gives you course information when you’re moving, which is a bit different from heading. To get heading you still need a compass or a GPS with differential antennas.
oliver,
Then you’re slipping the clutch rather than the tires.
Unless you’re into drifting, yaw on a car is small and isn’t easy to measure. So course and heading are the same to all intents and purposes. To measure yaw with GPS, you need not only at least two antennas, but also a receiver capable of centimeter level accuracy, which is quite expensive. A local fixed reference station may also be necessary. IIRC, that sort of system is called RTK for Real Time Kinematics. Professional racing teams can afford that sort of thing.
DeWitt,
I think averaging over time is also used to get higher accuracy out of GPS in some cases. Link here.
Mark,
That works for a stationary receiver, not so much for one that’s moving. For a moving receiver, you can use inertial data to improve precision with a good IMU. A high precision GPS receiver uses doppler shift in the satellite broadcast frequency as well as the GPS data in the signal for position and velocity determination.
Re: DeWitt Payne (Comment #138458)
Yup, you sure are.
However, in the example of a helicopter drone, yaw is non-negligible.
Also, if you’re sideways in your car and there’s lots of smoke, yaw can again be non-negligible!
What you really need is (sub-)centimeter relative precision between the antennas, which is a bit different problem for GPS than centimeter level absolute accuracy.
Short attention span theater here.
From the comment that started the whole discourse on the difference between heading and course. Not to mention that it’s probably not possible to space the antennas on a consumer level drone far enough apart to determine direction when stationary disregarding that the cost of the two antenna GPS would far exceed the cost of the rest of the drone.
DeWitt, Oliver,
If you are talking about GPS based compasses, these are gradually becoming cheaper. See Panbo.com, for example. Still expensive, of course.
DeWitt,
🙂 I loved this. This describes me to a — oh look! Something shiny.
Re: DeWitt Payne (Comment #138462)
Didn’t you specifically bring up heading and magnetometers in the context of helicopter drones?
Come to think of it, didn’t you use the example of a sports car with non-negligible yaw to show that positive acceleration during negative velocity is possible?
This is a branching from the the theme of this thread, but I think there is enough similarity that it is not completely off topic.
Tom Fuller has posted an essay on “Mathematical Decay” at his Lukewarmer’s blog I think speaks to innumeracy and how fallacies can take root. And possibly ultimately get the fallacies weeded out.
https://thelukewarmersway.wordpress.com/2015/08/22/climate-change-predictions-and-mathematical-decay/
The math insights of the community at this site are among the best available. Comments and critiques of “Mathematical Decay” would be very interesting.
SteveF,
The Furono SC-30, for example, goes for ~$2,200 and weighs 2.5kg. A magnetometer chip weighs on the order of a gram and probably uses a lot less power too. You can get a 9 degree of freedom breakout board including digital motion processing that weighs on the order of grams for $35. Add another $50 or so for GPS.
Oliver,
No.
The comment on drifting was in reply to your nitpick.
I was also talking about what’s available for vehicle data logging on a smart phone.
These types of questions used to infuriate me as a student. The answers are not physically possible; just how the hell can a child power a trolley at a velocity of 15m/s? I know what children are, i know what childrens toys are and know what 15m/s is, so obviously I have the wrong answer; could a kid push a trolley at 1.5 m/s? Well yes, that must be the answer and I have forgotten a factor of 10 somewhere
DocMartyn,
I still don’t think a kid could manage it, but I’ve been obsessing about it for no particularly good reason. How fast could a kid get going, reasonably, and could he finish the acceleration with a shove?
Say he can somehow spend last full half second shoving the cart. That’s still 2.5 seconds of keeping up with it; that’s 12.5 m/s, which beats wikipedia’s claim for the adult sprint record.
I guess if we put the kid on rollerblades or give him some other tool it becomes reasonable, or make him a Teenage Mutant Ninja Turtle or something like Lucia suggested.
I was on the verge of setting up an experiment with my eight year old yesterday before I realized I was going off the deep end and let it go.
DeWitt,
Yes the autopilot on my boat uses a multi-axis magnetometer, combined with a three axis accelerometer. It gives heading pitch and yaw updates each 0.1 second, accurate to well under 1degree. The whole thing (sensors and cimputer) is the size of a baseball and weights 30 or 40 grams.
I’ve got one of these puppies sitting around waiting for me to find a project that needs it. On board 3 axis accelerometer, decent microcontroller, 15 bucks. No magnetometer laying around although sparkfun sells one.
Re: DeWitt Payne (Comment #138468)
I commented on the difference between heading and course in reply to what you said:
“With a helicopter drone, you need a magnetometer as well because GPS doesn’t give you heading information unless you’re moving.”
To a helicopter drone, heading and course really are different things, so it wasn’t just idle nitpicking.
Doc Martyn,
In many cases, the ‘ideal’ multiple choice question:
1) Does not require a calculator to solve or at least avoids it.
2) Poses a question in a context where one can judge whether the result is physically realistic.
3) Give a physically realistic result, all values “given” in the problem are physically realisitic, all intermediate values one might compute are physically realistic.
4) Attractive ‘distractors’ (wrong answers) one would get if one made a common error is still physically realistic. (For example: if the answer involves taking a square root, the wrong answer obtained by forgetting to take the square root is still fairly realistic.)
5) I worded unambiguously. (Or at least all aspects that matter are specified.)
6) The text is scannable and doesn’t required the student to try to write a sentence diagram to figure it out.
7) None of the “wrong” answers are even arguably correct while at least some are not physically impossible.
8) The quickest way to find the right answer applies application of the content you are planning to test. (Nick Stokes gave examples where the ‘right answer’ could be obtained by applying ‘test taking’ principles even where the student knew no content. Those are not good questions.)
There are probably a bunch of other ideal features. But when you want at least some problems to have numerical answers it’s hard to achieve all of them.
There may be other features. The child pushing a wagon didn’t need a calculator. But the ‘correct’ answer is physically unrealistic. I often find myself writing questions where “superman” does something or the planet is “vulcan”, “krypton” or “planet X”. The hope is to at least alert the student they shouldn’t check the answer against “physically unrealistic”.
“Superhuman ninja” in place of “child” might at least prevent students who know no kid can run at 15m/s from wasting time double and triple checking that.” But some teachers seem to dislike those. (My impression is they want the problem to teach other side information that might prepare a student to win Jeopardy! )
FWIW: One problem a student presented me irritated me a lot. It was a conservation of momentum problem where, given the known in the problem statement, mechanical energy increased during the collision. The students were not asked to find how much mechanical energy was dissipated. Good thing. . .
There is an Australian Maths test for high school students in their last 3 years. Zero if a question is not answered .
Minus a point if answered incorrectly.
5 MC responses , 4 points for a correct answer.
timed test with increasing level of difficulty, last 10 quite severe.
It was obvious after seeing a few that 2 answers out of 5 were quite “wrong”
Left a choice of 3.
In all such cases the middle answer seems to have a much higher chance of being correct.
I advised my sons to do the easy problems to best of their ability and follow this tactic, answer all questions after eliminating the dross for the hard questions.
Alas they all opted to not answer the hard questions thus reducing their potential score.
Wish I had been young enough to do these tests.
My boys had superb maths skills but no understanding of probability and risk at that age 15-17.
oliver,
It was a nitpick.
An inexpensive GPS as in a smart phone, for example, does not give directional information unless it’s moving. It’s irrelevant whether that information is course or heading. It isn’t there and therefore a magnetometer is necessary, as I stated. Your response, on further consideration of what you actually meant, that a magnetometer is necessary anyway, seems to me to be intended to be argumentative rather than informative. I’ll keep that in mind for the future.
Re: DeWitt Payne (Comment #138476)
You are, of course, welcome to your own interpretation. I remind you, however, that it was you who brought up the helicopter application, in which case it is clearly relevant whether it’s course or heading.
What’s ironic is that I agreed with you that a magnetometer or other compass is necessary, but you interpret that as proof that I was being argumentative rather than informative.
Might not be a bad idea. If you post stuff that’s wrong, it’s possible that people might point it out. If in turn you respond by being trying to be snarky instead of simply acknowledging that you did, of course, mean the correct thing, then what could have been a minor, informative moment might instead turn argumentative.
Re: lucia (Comment #138474)
The use of calculators is an interesting question. On the face of it, the use or non-use of calculators doesn’t seem that relevant to testing of physics understanding. However, given that time is scarce in these tests, the use of the calculator seems like a needless distraction from what’s (ostensibly) being tested. Most of the relevant test problems I can think of wouldn’t seem to be be materially different with round numbers or reasonably simple fractions.
Of course, now I have to ask: What was this strange problem?
lucia:
I can’t help it if a course fails to teach students about how alien technology may impact their lives via robots disguised as automobiles. All I can say is if a student reads a question which does nothing to remotely suggest a factor should be considered but decides it should be considered anyway simply because the question doesn’t explicitly say otherwise, they’re bad at taking tests. Quite possibly, because they’ve had bad teachers.
Say what? We may talk about students not reading these questions in the intended way, but Question 3 clearly doesn’t say anything “is the speed.” It clearly talks “about measurement.” It says:
There isn’t a word about the speed itself. All the question refers to is what the “speedometer reads,” which is a measurement. If we’re going to worry about unstated technicalities and unhinted possibilities, inaccurate measurements is a possibility we should consider.
SteveF:
Sorry for the slow response; I went out of town for a short trip this weekend. I think lucia cleared this up for you, but just to make sure, I agree with what you’re saying. Cars just undergo lots of acceleration as you drive around. You speed up, slow down and turn (angular acceleration) all the time. Because of that, you wind up constantly redefining your frame of reference when driving around.
Which is fine. You wouldn’t want to have to picture the road around you as a 2-d map with a bunch of cars on it to drive. It’s far easier to just look around and judge your situation. Intuitive approaches like that are also often good for giving directions. They’re just usually not very good for analytical problems.
Brandon,
I agree a student should not read in feature that are not remotely in the problem. I disagree that the notion of turning is not remotely in the problem; meanwhile, transformers clearly are not. Among other things, the answers include the possibility of “turning”. If the test writer really believed that their wording of the question itself implied “the car is not turning”, then that answer is wrong because the question itself says “the car is not turning”. The fact is: the question does not say that.
And if the car is turning, it is accelerating. That acceleration could be positive– just on in the direction of motion.
Beyond that, the turning could make a difference to the answer; the transformers would not.
It’s also worth mentioning that if we look at similar questions on the NY Regents– which has a much larger budget for proof reading– that sort of question is much more carefully worded.
The example I gave is not a horrible question. And of course students who take multiple choice questions need to learn the rule of “keeping it simple as possible”, but that doesn’t mean that those who understand more might not take longer to answer questions whose wording permits possible interpretations that force greater amount of thought before deciding on the correct one. And those same students might punt, guess and move on at a greater rate.
Oh– you’re right that 3 says “speedometer”. My mistake. But I do think certain types of ambiguities do throw bad test takers, and the possibility of inaccurate speedometer isn’t one of them. If it was, I’d criticize that question for that.
No one is advocating worrying about things. I said students who have trouble with these things need to learn the rules of multiple choice– which is to read the problem as simply as possible.
That some need to learn this is not the same as saying the ambiguities do not exist. They do– that’s why they need to learn to deal with them.
Oliver
Sometimes they are. Students want to turn everything into numbers as soon as possible, and depending on the problem done, that can mean picking up and setting aside the calculator several times. OTOH, this strategy sometimes simplifies the algebra as they don’t need to shuffle too many variables.
In many cases they aren’t materially different. I was just pre-calculator in high school. All angles were 30, 45 or 60! I suspect many books took pains to see to it that t2 ended up being something like 4,9,16…, 81 so students didn’t waste time taking the square root. In such cases, the test writer might fail to remember that kids don’t run 30m/s.
Of course the test writer might forget to check this anyway.
Billiard balls colliding. It specified the masses, initial velocities and angles, and you had to find final velocities. Some angles aren’t possible. When I looked that the problem statement… I thought… ehrmmm…. So I checked afterwards.
I suspect I know how this problem arose. It involves “moodle” and “calculated multiple choice”. https://docs.moodle.org/23/en/Calculated_multichoice_question_type
I love moodle and calculated multiple choice. But unless the test writer carefully attends to numbers, they can write problems where the ‘g’ force on a passenger on a roller coaster will kill him, or the car will drop off the rails when it goes around the loop-de-loop and so on. You sometimes need write a script on the side to make sure the questions aren’t screwy. Of course, you have to think of this in advance which you may not when quickly entering questions that seem simple.
Hugh
I don’t think that’s the reason why they are used. I think the main reason they are used is that they take less time to grade.
Depending on what they are testing….
Hypothetically, if the purpose of the test is to test readability of a font, then people blowing questions because of the font teaches the test writer something. In that case, the test should be used to grade the testtakers.
In physics, test certainly shouldn’t be testing “ability to read fonts”.
I doubt this is the reason high school physics teachers write ambiguous question or have physically unrealistic ‘right’ answers. I think it’s sometimes difficult to write some that are 100% clear especially if you know what you mean. Often, the problems can’t be detected by a copy editor because the ambiguity isn’t a grammar issue. It’s information that is left out.
Consider these:
1. “Superman flies up to a meteor and hurls it at 30 m/s. What is his recoil speed?”
“Superman flies up to a meteor, stops. Grabs the meteor, hurls it at 30 m/s. What is his recoil speed?”
“Superman is holding a meteor. The two are motionless, he hurls the meteor at 30m/s. What is his recoil speed?”
In 1 a student might ask: What was Superman’s velocity when he bashed into the meteor? But a test writer might think it’s “obvious” you mean Superman stopped — because that’s what was in the writer’s head when they wrote it.
The rule of ” when ambiguity exists, interpret any multiple choice question to be as simple as possible” suggests the student should assume that superman is motionless relative to the meteor before he hurls it and 30m/s is relative to their initial position and velocity.
But the fact that one should assume their initial velocity is 0m/s doesn’t really mean that’s what the problem says. Unless “there is not enough information” is an option, a student who thinks they detect ambiguity should very quickly use 0 m/s. If that guess is wrong then there is no answer.
(Note: in all, the problem does need to give Superman’s mass and the mass of the meteor! Otherwise… there really is no answer!)
Brandon,
I wouldn’t put it quite that way. The reference frame of the car doesn’t change. The roll, pitch and yaw axes don’t change with respect to the longitudinal, lateral and vertical axes of the vehicle. Acceleration always pushes you back in the seat. What changes is the relationship of the car frame of reference to the Earth frame of reference. The change can be calculated by integrating the accelerations twice. Once to get velocity and again for position. Or you can just look out the window.
lucia:
I think you may have just succeeded in writing the most confusing paragraph I’ve read all week, and I spent an hour this morning tracking down why Mark Steyn has a section in his book which argues Michael Mann fiddled with the modern instrumental record in his reconstruction of past temperatures, all ultimately based on the fact the northern hemisphere warms faster than the southern one. (Obviously, it was not explained that way.)
So… yeah. I’m just going to let you reword that if you want. I’m sure I could probably figure out what you meant, but my head already hurts without trying to figure out things like why the test writer is answering any question, much less why a question itself says something that it doesn’t say…?
The key here is I have little sympathy for bad test takers.* I don’t approach this situation looking for ways to help them. If people want to do it, they can. I’m not going to try to stop them. I’m just not going to be particularly helpful. In fact, I’ll probably mock the people for being bad at taking tests.
And that’s what I meant by worrying about things. It’s not that I’m advocating worrying about things. It’s that if bad test takers are going to make me worry about trivial points, I’m just going to roll with it. I’m confident in the end, I can out-nitpick anyone, no matter how bad they may be at taking tests.
*I’ve known people who are bad at tests purely because of anxiety. They would have no problem with questions like these without the pressure of a “test.” I have sympathy for that. It’s the refusal to consider simple interpretations that gets on my nerves.
DeWitt Payne:
Huh? How do you figure acceleration always pushes you back in the seat? Acceleration has at least 180 degrees it can push you in, depending on turning and if you’re speeding up or slowing down. I say “at least” because if you want to consider hills/elevation change, acceleration can be a 3-d issue.
Regardless, I don’t know why you say the roll, pitch and yaw axes don’t change. It’s not actually true, but I don’t think anyone interpreted what I said as suggesting the car itself changed.
But I’d like to think everyone understood that’s what I meant when I said you constantly “redefine” your frame of reference while driving. You define your frame of reference in relation to the Earth’s frame of reference, so as you drive, you have to redefine it in relation to the Earth’s frame of reference. Or in some circumstances, you might choose to redefine it in terms of some other frame of reference.
Brandon,
I should have been more specific. What I meant was that when you push down on the accelerator pedal, you’re pushed back in the seat. Lineal acceleration is always along the vehicle longitudinal axis, even when the longitudinal axis isn’t aligned with the direction of travel, i.e. yaw isn’t zero. Obviously the vector sum of lateral and longitudinal acceleration can have any direction on the horizontal plane defined by the lateral and longitudinal axes of the car. You can even have vertical acceleration if you’re going up a hill or driving around a banked track.
I’m also coming at this from the point of view of measuring the components of acceleration and angular velocity using three perpendicular axis sensors.
Brandon:
I don’t see what’s confusing:
This means: I think you think a student should not read in a feature that is not remotely in the problem. And I also think they shouldn’t.
I thought that was responsive to this statement you posted
I then wrote:
To clarify: I took your general position to be that the notion of “turning” is not in Q 7. Perhaps I mistook your position.
However, if that is your position, I think turning is suggested by the problem. Among other things: “changing directions” is contained in answer “d”. Turning involves changing directions. This is certainly likely to suggest a student should consider what happens if the car is “turning”.
In that comment, I also referred back to your transformer example. I agree that transformers are not remotely mentioned. So: turning touches on the question; transformers does not.
Someone wrote question 7. In my view, if that question writer took the position that the problem statement as worded meant to communicate “assume the car is not turning (or changing direction)”, they would not include answer “d”. So, in fact, when the question is taken as a whole, the question writer wants the student to consider “turning” (i.e. changing direction).
I have inferred that your position is that someone thinking “what if the car is turning” when reading that question is akin to someone wondering “what if a transformer is driving”. Perhaps I am mistaken in your point. But if you think that, I disagree with you.
I hope that clarifies.
lucia, I normally wouldn’t comment on this, but I thought it was funny I had a moment of confusion because you edited your latest comment and that caused me to get lost when switching between applications which had your comment open. Your edit to clarify things caused me confusion 😀
You can only come up with this interpretation if you leave out part of that quote of yours. You’ve somehow left out the part of the answer I’ve drawn attention to before, the part which makes the answer ungrammatical. The full answer is, “Instantaneously changes direction.”
When a car turns, it does not instantaneously change directions. It changes directions over time. If you want to get hyper technical, you could perhaps argue the car changes instantaneously changes directions an infinite number of times in infinitesimal increments (or some finite value approaching this if you wish to avoid Zeno’s paradox and the like), but that requires a special level of obnoxiousness from the test taker. And the answer still won’t fit the question as the question asks what is “happening,” not what “happens.”
I still find your original wording awkward, but apparently the primary part of my problem was just in failing to realize you think a turning car can be fairly described as “[i]nstantaneously changes direction.” I suspect you might be able to figure out why that could cause a person some confusion.
I believe you have inferred my position quite accurately. I believe I have inferred your position quite accurately as well.
It appears we are in disagreement because you think an answer which refers to an instantaneous change in direction means the test taker should consider turning to be a possibility whereas I think the phrase should, if anything, make them rule out the possibility of turning as the word instantaneous cannot apply to turning in any sense we’d normally use with a car but can only apply to the simpler systems of movement this question was intended for.
DeWitt Payne, I think I would agree with you about that, but… your approach just seems needlessly complicated. I can’t see the point of worrying about most of that when looking at how you’d define things while driving a car. You can, of course, but it seems like overkill.
Brandon
No idea why you think the answer being ungrammatical matters. Yes, it says “Instantaenously changes directions”. We both agree on that.
Yes it does.
adjective
1.occurring, done, or completed in an instant :
an instantaneous response.
2.existing at or pertaining to a particular instant :
the instantaneous position of the rocket.
I think I am allowed to read the word ‘instantaneous’ following definition (2) especially as that reading is very common in physics and is even discussed at some length in high school and college texts. In the instant the car is turning, its instantaneous acceleration in the direction perpendicular to the instantaneous velocity is not zero.
By “hyper-technical”, you seem to mean “If you want to discuss this they way people in physics do…” Or “If you want to lay the foundations of calculus like Newton did….” or something like that? I’m not sure why this would be “hyper technical” on a physics test.
You are really loosing me here. “Happening” implies “this instant”: as ‘instantaneous’ using definition (2) above.
Yes. I think turning a car does involve “instantaneously changing directions”. I guess it causes you confusion because you are unaware of the range of use of the word “instantaneous”.
So it seems you rule out my interpretation of (d) as possible because you either are not aware that instantaneous can mean “existing at or pertaining to a particular instant :
the instantaneous position of the rocket., and that that is how it is used and discussed in physics texts or you reject that meaning as acceptable.
Brandoon.
Are you serious? Because there is an answer that says ‘instantaneously changes direction’ you think this will cause test takers to rule out a turn?
I’m having a hard time swallowing that somehow.
~shrug~
Might have seemed I was asking rhetorical questions, I wasn’t. At least I didn’t mean them to be.
.
In question #1 I’m wondering if Brandon is arguing to explore if the position is arguable, or for some other purpose, rather than out of a personal belief in the position. I’m asking him to let me know if this is the case. Perfectly all right in my book, just helps me sort out how much attention I want to pay.
.
Question #2 restates my understanding of his position as a question to verify that I understand him properly, because it astonishes me that he’d believe this. Maybe it shouldn’t, but it does.
Thanks.
Brandon,
So we’re supposed to understand exactly what you mean when you phrase it ambiguously, but you’re free to (willfully) misinterpret anyone else’s imprecise statements? I think not. Are you channeling Joshua? Apparently.
This one time, I had an English test that I had not studied for at all. I knew essentially none of the content, and was guaranteed to fail.
One section was about events in English history matched with the corresponding dates. I noticed that the teacher had listed the events themselves in order, but randomized the order of the dates. Thus, I could answer them all correctly by simply knowing that 1776 came after 1066.
This prompted me to carefully examine the other sections of the test, and likewise “reverse engineer” them. From knowing almost nothing, I could get a near perfect score by simple reverse engineering the test.
The test was multiple #2 pencil bubble choice test graded by a machine. As was usual, the teacher chose a student at random to take the forms to the central office to get graded. This time, she chose me. I had a deer-in-the-headlights sort of panic attack at that point — because it relied upon the teacher trusting me that I wouldn’t cheat, changing my own sheet now that I had the master sheet in hand. But of course, I had already “cheated” after a fashion.
Watching the forms get graded by the machine, I watched as my fears were confirmed. I had gotten the best grade in the class by a wide margin. If the teacher questioned me, to see if I actually knew the material, it would be obvious that I didn’t, thus she could prove that I’d “cheated”.
As it turns out, the teacher never questioned me about this.
But anyway, my point is this: teachers have limitations on the questions they can ask. Considering things from their point of a view solves a lot of problems. You know what If “x^5 + y^4 = 113”, then you know what the answer is because you know that the teacher is going to choose integers, and that only 2 and 3 are small enough to get something as low as 113. I recently helped my niece with math were she struggled with the concept that with “powers”, the test was only going have to the numbers 2 through 5, because otherwise the numbers got too big.
OT:
Sea ice this year is different. Antarctic sea ice area in the satellite era set a record low for the day on August 21. It’s been decreasing when it should be increasing. Meanwhile, Arctic sea ice area has been increasing recently even though the extent has been decreasing. The record low Antarctic area and only fourth from the bottom Arctic area caused global area to also set a record low August 20.
DeWitt Payne.
“The record low Antarctic area and only fourth from the bottom Arctic area caused global area to also set a record low August 20.”
Interesting observation and thanks for putting it up.
In the interest of accuracy/balance it might have been fair to note,
-that less than a year ago the global sea ice area was at a record high for possibly a number of weeks.
– that at -0.617 the Antarctic sea ice might be at an anomalous low anomaly for this time of year but has been much lower in the past at other times of the year, down to – 1.75 million in 1980 before the sustained and persistent rise in Antarctic ice for the last 305 years and in particular the last 2 years where it has been > 2SD for over 2 years.
-That PIOMAS is increasing overall in the Arctic for the last 3 years.
-The Arctic is very interesting at the moment as it has a ring shaped structure of disappearing ice towards Canada.
Thus is very similar to the ring shaped structure in the Antarctic in MacMurdo Sound last Southern Summer.
I feel the ring which has already broken will completely disappear as still 2 weeks of melting to go.
But, if it persists, due to hopefully thicker volume, for 2 weeks the resulting refreeze should cause a very steep rise in Arctic sea ice area and extent in the following 4 weeks. Temps above 80 latitude already below zero should help.
DeWitt Payne:
Say what? I think what I said was fairly clear, but where have I suggested people aren’t allowed to misunderstand me? If people don’t understand something I say, they’re free to ask questions. If they misunderstand something I say, then I’ll try to correct their understanding. It’s not like I’ve insulted anyone for misunderstanding me in this thread. The only people I’ve made fun of are bad test takers.
As for the other portion of your comment, whose statements have I misinterpreted, willfully or otherwise? I don’t recall anyone complaining I’ve misinterpreted or misrepresented them. I might be forgetting a case, but communication is fraught with difficulties. I try not to worry much about misunderstandings as long as they can be resolved.
So… yeah. I have no idea what you’re talking about. Rather than just making vague accusations, could you try being specific? It would help people judge whether what you’re saying is reasonable or not.
Mark Bofill:
To answer the question, I am serious.
Your answer is a bit off. You miss the context that I don’t think a test taker should be considering the possibility of a turn in the first place as the question does nothing to suggest the possibility of a turn. lucia’s position was a test taker would consider the possibility of a turn because answer d suggests a turn. My response was twofold: 1) Answer d is ungrammatical so I don’t why a test taker would read anything into what it says as it cannot be a correct answer; 2) what it says does not describe a turn so even if the test taker did read anything into it, it shouldn’t be that the car is turning.
Personally, I don’t think answer d should be used to judge the question’s meaning at all because it is ungrammatical, so it is clearly not a right answer. But, and this is a big but, if we are going to focus entirely on it, we cannot cut words out of it. An instantaneous change in direction cannot happen in any sort of curved motion (save the hyper technical sense I remarked upon earlier) because curved motion is ongoing, not instantaneous.
Which again goes back to the fact the answer is ungrammatical. The question clearly asks about an ongoing activity. Instantaneous changes in direction are not ongoing activities. The answer simply cannot fit the question.
lucia, I’d like you to consider these two statements:
And remember the meaning of a word depends on how the word is used, including the grammatical structure of the sentence it is in. A word’s meaning can change simply by changing other words around it. While you think, or don’t think, on that, I’m going to leave this discussion because it annoys me and I don’t care to deal with it.
I’m sure you’ll probably form some negative opinion of me in regard to this situation, but the reality is your entire response to me is based solely upon flouting the grammatical structure of the question and answer you discuss then turning around and saying you don’t know why I’d care about the grammar.
The reality is I am perfectly aware of the meanings you describe. They just don’t fit if you read what was written as whole due to the grammatical structure. But hey, why should I care about that? It’s not like grammar matters when talking about how to interpret things.
Oh wait, it does. Maybe I should care.
Sorry, I know I said I was going to walk away, and I meant it. I was just lying down to go to bed when I realized something I had somehow failed to notice. It seemed to relevant to not point out, and since nobody had responded yet, I thought I’d add it as a final thought.
The idea of “instantaneous” change in direction is silly even if one assumes we’re looking at a single instant instead of a period of time, such as lucia suggests. If you define your problem in terms of individual instants, then by definition, all changes must be instantaneous. Answer d’s inclusion would have been purely superfluous, and you could have added the same word to answers a-b.
So… yeah. I’d say you don’t instantaneously change directions with curved motion, but if you do, you also instantaneously speed up and instantaneously slow down. That wasn’t mentioned in options a-b for some reason. My theory is because answer d was ungrammatical and should never have been interpreted anything like is being suggested, but others are entitled to their own theories.
But yeah, feel free to ignore me, mock me or whatever. I’m going to go instantaneously accelerate and decelerate myself a few infinite times to get myself to bed. Hopefully I don’t get mixed up with all the instantaneous changes in directions it takes to get from my bed to my chair. I hear that 1,098,803,583,857,293,219,047,276th left is a tricky one.
The concept of an impulse is commonly used in dynamics (mechanics) to model idealized rigid body collisions. Impulses usually involve instantaneous changes of velocity.
Brandon,
Thanks for your response.
My post was a bit snippy. Sorry, I was having a bad day yesterday. If I didn’t respect your brains, it wouldn’t have astonished me that you took a position I think is off, FWIW.
oneuniverse,
Yes. Impulse is the integral of force over time and used in a variety of problems. In these problems one generally is only concerned with the change in momentum over some period and not the details inside the period. But in truth, the time during which changes occur is not zero.
Brandon,
Yes. They could have left the word ‘instantaneously’ out. They could have put it in the other answers. The entire question relates to what is happening in that instant, so the word is superfluous.
Hi Lucia,
My understanding is that when modelling rigid body collisions (eg. in the way described in the linked note), the period during which the change of momentum occurs is considered to be of zero length ie. an instantaneous change – the period t_0 to t_0, where t_0 is the time of collision (or, per the note, the width of the interval of the Dirac delta function where it is not 0).
The question doesn’t specify the vector in reference to which the velocity is negative and the acceleration positive. Perhaps it’s meant to suggest that they’re negative with respect to each other ie. their scalar product is negative?
[Edit: I agree, in general, impulse is the intergral of force over time]
Mark Bofill:
For what it’s worth, it being snippy didn’t bother me. I have no idea why you think my position is off, but that wouldn’t cause the remark you responded to. People are free to think I’m wrong, even in remarkably dumb ways. That’s not going to offend me. It’s just going to make me curious why they think I got something so wrong.
Brandon,
Excellent. I will take this to mean that I did not give lasting offense despite my somewhat offensive tone. Since you express curiosity about why I think you’ve got this wrong, I will explain my reasoning.
.
We’re talking about students taking tests. Students come to the table in a distribution; some will have an excellent grasp of grammar. Some will not. Some will care if an answer is grammatically incorrect, and others will be completely unaware that a grammatical problem existed. In any event, they will be under pressure to select the answer deemed ‘correct’ under a time constraint.
.
I don’t have any evidence to support the following idea, I base this on personal experience. Perhaps I base this to some extent on vague recollections of conversations with other students after taking tests. I don’t think whether or not an answer is grammatical really ranks among the top, say, 5 reasons a student would exclude an answer on a physics / math / science test. I think some students might notice a grammatical error, but I don’t think the presence or absence of a grammatical error would be indicative to them that the answer isn’t the one the test giver is looking for. Why? People occasionally make grammatical errors, this is well known / assumed. The test taker is motivated to find the answer the test giver deems correct (generally speaking). It’s reasonable (to me) to assume that a student observing a grammatically incorrect answer that nonetheless would apparently be the correct answer except for the grammatical error would select the answer, because they are motivated to try to obtain as high a score as possible on the test, because they know unintentional grammatical errors occur, and because it seems reasonable to assume that the presence of a grammatical error in an answer
is[edit: is likely] due to test writer error rather than being present due to intent to disqualify the answer (unless the test is a grammar test)..
Part of the impatience apparent in my earlier query was due to the fact that it seems tedious to explain this in detail and the notion I have (apparently without basis) that this is fairly intuitively obvious. Live and learn I guess! 🙂
one universe,
Of course. I’m not sure what point you are trying to make. I did say that people can model the time as zero– but that doesn’t make it so.
I’ll elaborate: “modelling” doesn’t mean “describing exactly what happened in a way that is correct in all details. And “considered to be of zero length” doesn’t mean that the time really has zero length.
In any case, even in a collision problem, the time it need not be “considered 0”. In fact, many impulse/momentum problem worksheets assigned to high school students ask them things like:
The change in momentum during a car crash was dp.
(a) What was the impulse.
(b) The time period was dt, what is the average force during the collision?
(c) Car bumpers were added and increased the time to dt_2. What is the average force?
(d) Explain why car bumpers are useful.
So: Yes, when doing part (a) the students don’t worry about the time and for all practical purposes, the time is in some sense treated as 0. But students are expected to know that it is not actually zero.
It isn’t really 0 and never can be. People’s modeling decisions — however useful they might be to determining the solution they are aiming for– cannot actually change reality. Reality is: a moving car cannot change its direction a finite amount in zero time.
oneuniverse
Yes. The student is clearly supposed to assume the acceleration and velocity are positive in opposite directions. Most students who know physics would sort that out. The only issue is: Who might take longer to do the question, and who might hesitate and do it wrong. And who might spend to much time on the question.
Lucia,
I was providing an example where instantaneous changes of velocity occur in the formulation of a model commonly used to solve a certain class of mechanical physics problems. It was among the subjects I was taught at school, and I thought it might also apply to the US as well ie. a student seeing the words “instantaneously changes direction” might think of rigid body collisions, as I did.
I specifically said, it’s used to “model idealized rigid body collisions”. I didn’t say the time of collision in real life is zero, and I’m not sure why you would think that I think that. Anyone reading the short notes I linked to would see that they state “This is clearly idealised: real bodies deform; the forces they exert are not instantaneous [..]”. I thought this was fairly obvious, and didn’t need restating.
oneuniverse:
That’s fine. In other words, it seems you meant to agree and provide further info. I wasn’t quite sure. That’s why I said I wasn’t sure.
One problem with blog comments is that a person’s point isn’t quite clear to others. It’s not a matter of grammar or style– it’s just that tone is lost. Also: it’s generally not safe to assume that material at a link will clarify what you mean — or to think that people who visit the link will “know” what point you are making. It just doesn’t work. (This is an empirical observation– not a value judgement about whether someone reading the material at the link should or should not be able to grasp the main point the person leaving the link intended to make. It’s simply a fact that they often don’t.)
Yes. They might. I hadn’t thought of that possibility.
Mark Bofill:
Lasting offense? You didn’t give any offense. I wasn’t bothered by how you responded at all. I was curious, as you gathered, but unfortunately, I can’t really respond to explain why I think you’re wrong as I said I’d walk away from the discussion. To try to walk the fine line of explaining without discussing, I’ll just say I would agree with the point you make, save that it misses why this particular grammatical mistake matters.
There’s a difference between saying, “You misplaced a comma, so that answer is wrong” and saying, “The grammatical failure of that answer renders it incoherent, so it is wrong.” I wouldn’t expect a test taker to think the former; I would expect one to think the latter.
Brandon,
Ok. This is reasonable to me in principle, I’ll chalk up our difference to the idea that in your view, in this specific instance the grammatical failure nullified the answer. I’m happy to accept that this is the case, and it works out well that you do not elect to discuss it, as the details do not seem particularly pertinent to me.
.
Thank you.
Brandon,
I agree with Mark on the grammar issue. I also think that a teacher had made a grammatical mistake of that nature on an option the teacher thought correct, and a student advanced your theory about grammar, they would be unlikely to give that student or the class points. I think it unlikely the students could get an administrator to overrule them on a physics test.
That said: If one is taking wild guesses, grammar mistakes can suggest an option is likely wrong. The reason is the test writer takes care to write the question and the correct answer. They think of a few wrong answers they imagine tempting and often their level of care declines as they write the distractors. So ‘wrong’ answers then to contain larger numbers of grammar and spelling errors.
In this case, the answer is wrong. So it really doesn’t matter what theory the student had for deeming it wrong.
Lucia,
Thanks for pointing this out, this did not occur to me.
Mark,
I’ve been entering questions for students to practice on. I definitely enter the question and correct answer first. Then I need to think of “good” wrong answers. For cases with answers that are text, you sometimes just want to get it over and move on. But really…. you can’t just type in “Leprechauns!” for option ‘d’.
Calculated multiple choice isn’t so bad. With ‘calculated multiple choice’ by moodle, I usually code in common blunders (like losing the 1/2 on kinetic energy, forgetting to take a square root and so on.) But sometimes you run out of good ones. So if the right answer is “X” you code in stuff
(1+error)*X where error is a wild card which can take on a range of values.
I do wish calculated multiple choice had a “sort ascending” choice instead of “sort” though. That would make the answers more scannable.
A big reason I have not posted for the past year or so is that I have been teaching introductory physics courses, so I have to worry about this sort of thing more or less constantly…
That said: I have no idea why honors students would do worse than non-honors students on Lucia’s question 7, and in fact I find it hard to believe. I’m tempted to test it in my course. If I do, I will be asking the question in the (implied) context of motion in one dimension, that is to say, before we start covering circular motion, so if Lucia’s theory is right fewer of the smarter students would be likely to be confused by two-dimensional complications…
Lucia,
I agree, my first comment needed some explanatory text . (I did in fact attempt to edit the comment at the time to remedy that, but the pop-up window came up blank in both IE and Chrome, and I left it at that. The edit function worked with my second comment, however.)
Robert (#138495) – good story, thank you.
julio–
I think that means your theory is “false positive”. It may have been.
lucia:
I’m pretty sure I’d have no trouble convincing an administrator answer d is wrong, even if it were the intended answer, but really, I don’t want to keep talking about this because I’m trying very hard not to insult you for being worse than the bad test takers I was mocking before. Because honestly, you are being way worse.
Mark Bofill:
Alright.
Brandon,
(d) is wrong for reasons other than any theory about grammar. No one disputes it’s wrong. My position is that if the question was otherwise right, this “grammar” theory would be unconvincing.
I can live with that being your opinion. I also note that people aren’t exactly falling all over endorsing your theory about grammar.
lucia:
Yes, please make sure to point out I did not specify “if the question was otherwise right.” I understand the more relevant qualifier of, “if the answer was otherwise right” was clearly too obvious for me to need to say it, so there was no reason to point out I didn’t specify it 😀
Sure. And people aren’t exactly falling all over endorsing your theory about anything either. There’s what, two people who have said anything on the issue? I believe that was my count, and one of them doesn’t even want to look at the details at all.
If you’re going to appeal to popularity, you should try having some actual popularity first.
Though if I really wanted to argue the point, I’d say the count is about even. I don’t though, so I’ll just leave that thought there.
Brandon,
I haven’t said people are endorsing my theory. I also haven’t suggested that I should mock you or anyone else for not endorsing it.
Lucia-I don’t know, I think it’s intriguing. I will probably test it, maybe as early as next week (I give short in-class quizzes every day and that looks like a good question). I have 87 Honors students and 252 regular ones, which is not a bad size sample. (Now you see what keeps me so busy…)
I fully expect the honors students to do better than the others, but, as I explained above, such a result would be compatible with your theory, since we will not really be covering motion in two dimensions for at least another month or so.
I could try to ask the question again, in the same words, after covering two-dimensional motion, but it might feel too awkward at that point–
Lucia,
I think you’d mention moodle before, but I didn’t google it till now. Sounds like an interesting system. 🙂
Julio,
It would be interesting to see if it repeat with a different batch of students. It could be a false positive. I haven’t sorted out if the statistically significant results were diagnosed using Bonferroni. But there were very few individual tests that poked out with any bias toward any of the groups tested — no where near 1/20 one would expect if the diagnosis did not use Bonferroni. Lots of questions were tested for several potential baises so we do expect some false positives.
Even if you repeat, that won’t tell us if my theory is correct. But it would show a repeat, which would be interesting.
Dang Lucia. I’d never even heard of Bonferroni. Thanks!
Do you tutor in statistics too? How much do you charge? 😉
Mark,
Moodle has good things and bad things. It’s free, so I can hardly bitch. The main thing is I can set up diagnostic tests, figure out where the students are confirmed lost or back on track. Of course that assumes the tutees will take tests. As they are busy, sometimes swamped and I don’t actually hold a grade over them that is hit and miss.
But what I can do is have practice problems set up to discuss and have them work on during tutoring. I have modified the ROE’s and coded at least 1 version of each of the mechanics problems to use for practice. (You can see a modification– I replaced the kid with a ‘ninja’, tweaked to specify the floor is horizontal and said mentioned the floor was frictionless. Then I used the “calculated multiple choice”.
After your comment I turned it into a “super human ninja” because I do want round numbers and I want 10 versions of the numbers. “super human ninjas” can run faster, jump higher and push with greater force than children.
Mark
Statistics? Only entry level stuff. I charge $40/hour. Which is either cheap or expensive depending on how you look at it. Parents will pay more but… it just doesn’t seem right. I like tutoring, so that’s about what seems fair to me.
Fall has started so I’m focusing on physics. Also: I only take fairly nearby.
lucia,
So superhuman ninja’s can have traction to push something on a frictionless floor? I’m impressed.
Dewitt,
If Ironman can use thrusters, I’m sure a superhuman ninja can too! Or the floor is metal and they have magnetic shoes. Or something. But yeah…. obviously, there is an issue there.
Things are constantly getting pushed or pulled on frictionless floors in physics.
lucia:
See, I knew you would say this, and I rolled my eyes before you even did. Because the obvious answer is, I didn’t say you said people are endorsing your theory. It’s as true as your statement. And if you respond to say I implied it, then the obvious response is:
Implied that somehow the issue of popularity favored you. I hadn’t said anything about anyone having endorsed my position, but you decided to bring up the lack of endorsement as though it made a point.
Or if you don’t respond that way, then we can be like, “Oh, okay. You didn’t say that either.” In that case, we just wound up having accomplished nothing via a fruitless exchange of trivially true statements.
Though at that point, I feel I’d be better off just going back and discussing grammar. At least then I’d get the point of these comments. And heck, the reason I stopped is I figured other people wouldn’t want me to keep talking about it, but when I tried to stop, people responded by trying to get me to not stop, so maybe I was wrong.
Brandon
Nope. That’s not what I was implying.
I was responding to
I meant that when one person has their own pet theory and suggests they are holding themselves back from mocking people for not adopting it, I don’t worry about their temptation to resort to mockery or insult. This has nothing to do with whether my position is more popular and doesn’t insinuate mine is so.
The fact is: I have no only not mocked anyone for not subscribing to my idea– I invited people to share their theories for why it happened. Julio appears to subscribe to false positive–and he may be right. I even suggested that testing would be required and it sounds like he might tweak the question and give it a try.
So given my position with respect to my theory — as being a tentative one that is considered as a possibility that would require testing– I really don’t think it’s a problem that other people might be dubious of my theory. After all: perhaps it’s wrong and observed event is just a false positive. Not a big deal for me.
No one is trying to get you to not stop.
lucia,
You could have the same conversation with Doug Cotton with nearly zero changes.
BTW, you are now officially a Renaissance Woman. with your disclosure regarding tutoring.
lucia, if you think:
Is in any way implied by:
I think you are sorely mistaken. But then, I haven’t suggested I’m wanting to mock you for not adopting my “pet theory,” and you are way off-base in saying things like:
Since this disagreement between us on grammar wasn’t about why people answered the question on the test wrong, but on a different issue arising from that discussion so… yeah. I just won’t pursue this chain of response any further.
I said I was going to stop. You immediately asked me questions which I couldn’t answer if I did stop. Now, maybe you weren’t trying to get me not to stop, but that sort of behavior encourages me not to. And because of things like that, I’m going to just scratch the idea all together and start over. And as part of starting over, I will bite my tongue and not say a single hostile word after this comment despite them being well deserved.
So this is all very simple. Question 3 says:
As most people understand, when you’re going around a curve, if your “speedometer reads a constant 30 mph,” then it must read “30 mph” for some period of time. That means you spent some period of time going around the curve. In the same mindset, when Question 7 asks:
This car would be spending some unspecified amount of time traveling with a negative velocity and positive acceleration. This means the car is, as answer a says, “Slowing down.” It’s all very simple. In these questions, the car is going around a curve or slowing down for some unspecified period of time. The test doesn’t bother to say how long that period of time is because it just doesn’t matter.
Or at least, that’s how I thought everyone would understand it. The previous discussion broke down before I realized there’s apparently another view. In that view, the car isn’t going around the curve for five seconds. The car isn’t slowing down for five seconds.
It’s doing it for only a single instant. That is, answer a doesn’t have to say, “Slowing down.” It could instead be reworded as, “Instantly slowing down.” Because in physics:
Under this view, when Question 7 says “A car is traveling with a negative velocity,” it doesn’t mean the car is traveling with a negative velocity for some unspecified amount of time. It means the car has a negative velocity for a split second, infinitesimal amount of time we freeze things at to study.
And when the car is “slowing down,” it isn’t “slowing down” over some period of time,” it is… slowing down in that split second, infinitesimal amount of time we freeze things at to study. If the test said a person is walking down the street, it would not mean the person is actually taking steps. It would mean the person is frozen mid-stride, caught in some freeze frame moment so we can study him for our problem.
Is that interpretation sensible? I don’t think so. It is nothing like how the questions would be interpreted in normal conversation, so I referred to it as being okay in a “hyper technical” sense. lucia responded:
But what about the test remotely suggests her interpretation makes any sense? This entire issue arose because answer d to Question 7 is the ungrammatical, “Instantaneously changes direction.” That doesn’t fit the question asked. I pointed this out a number of times. lucia’s position is its ungrammatical, but that doesn’t matter because you could just remove “instantaneously” and get an answer that fits (save for grammar). Or you could add the word instantly in front of the other answers and they’d still fit as well.
In other words, she says “instantaneously” is completely superfluous there. So not only is the answer ungrammatical, it includes a word it doesn’t need, a word that could just be thrown into the other answers. How needlessly complicated is that? How much more sense would it make to just say, “No, the question is meant like a normal person would mean it, a car is traveling down the road”? I think a whole lot more.
Especially since Question 3 doesn’t make any sense under lucia’s interpretation. Question 3 explicitly says the car’s speedometer “reads a constant 30 mph.” You can only read a constant value over an interval of time. If these questions really were intended as being in reference only to instantaneous, split second moments, the wording of Question 3 would be nonsensical.
So where does lucia’s interpretation come from? The only basis for it seems to be the word “Instantaneously.” There is nothing else which remotely hints at it. But even there, there is a far more obvious explanation. The user oneuniverse says:
And later clarifies:
Anyone familiar with high school physics should already know this topic. As will anyone who has ever had to think about the angles to line up a shot in a pool game. Doing that involves the exact kind of problem oneuniverse describes. You know when your cue ball hits another ball, it will go off in a direction based on the angle you hit it at. But you also know your cue ball will change directions based on that angle too, so you need to make it doesn’t go off in the wrong direction.
Now technically, the cue ball doesn’t instantaneously change directions. It actually deforms a bit as it transfers momentum to the ball it hit. There are a bunch of details and factors involved, but for the most part, they get ignored. When playing pool, and when doing high school physics, you mostly just think, “The cue ball hits the other ball and immediately bounces off.”
.
This post came about because of discussions of how to interpret things. So in that light, I ask a sincere and genuine question. When an average person, or high school student if you think the difference matters, reads a phrase like, “A car is traveling down a road,” how should he interpret it?
1) A car is traveling down a road for some unspecified amount of time which we’ll look at.
2) A car was traveling down a road, and we’ve taken an instantaneous, nanosecond slice of time out of it to look at.
Similarly, when he reads a car is “speeding up,” should he imagine a person in a car pressing his foot down on the gas pedal, or should he imagine some frozen stillcapture of a moment in time with an invisible acceleration vector on it?
Now, I had never even considered Option 2. I think it sounds ludicrous, especially given the context of Question 3. I have done plenty of problems within frameworks where one examines things at instantaneous intervals like in Option 2, but in each case, it was made clear so the reader knew. Sometimes it was specified, but more often, it was simply clear via context as that was the only way the problem being presented made sense.
In casual conversations, I’ve sometimes had to discuss things in a framework like in Option 2. In each case, it had to be specified so people knew. That’s because the people involved normally didn’t think in that manner. Normally, if you told them something was “happening,” they’d think it was an ongoing, continuous thing. That’s simply how the language is normally used in the social circles I’ve been in.
Maybe it’s different somewhere else. I don’t think so. I’ve never seen any evidence it is. I’ve read at least a couple thousand books. I’ve never seen anyone use framework 2 as their default. Nobody’s ever told me that’s how they normally use words before. But hey, I’d still be willing to consider that it could maybe be a legitimate thing. Maybe it’s just me not being aware of something.
Only, there’s a huge problem. Nobody here ever suggested the inverse. Nobody here ever suggested maybe there’s just two entirely valid ways of looking at things. No. The response I got was, there is only one way; it must be this way. I was told I am wrong to read sentences like, “A car is going around a curve” and imagine a car actually driving around a curve. That is why I was hostile.
Because I think the interpretation for Question 7 given by lucia is ludicrous, but I am certain the interpretation I have for Question 7 is fine. And if anyone wants to tell me I am wrong “because physics,” well… I promised not to say anything hostile, so I’ll just stop there.
Lucia,
I knew I should have stayed with that contract in Milwaukee! :p
No I’m joking. I’ll just go on scrounging the free crumbs that fall around here every so often and learn stuff that way. 🙂
Mark,
Milwaukee is well outside of my tutoring range. 🙂
“Anyone familiar with high school physics should already know this topic. As will anyone who has ever had to think about the angles to line up a shot in a pool game. Doing that involves the exact kind of problem one universe describes. You know when your cue ball hits another ball, it will go off in a direction based on the angle you hit it at. But you also know your cue ball will change directions based on that angle too, so you need to make it doesn’t go off in the wrong direction.
Now technically, the cue ball doesn’t instantaneously change directions. It actually deforms a bit as it transfers momentum to the ball it hit. There are a bunch of details and factors involved, but for the most part, they get ignored”.
Problems.
1. you seem to be drifting between the directions of the cue ball and the object ball on and after making contact. It would seem better to discuss each instance separately.
in which case
problem
2. Given all reasonable assumptions the object ball can do 2 things.
It can continue in the straight line of the cue ball hitting it [is zero degrees an angle?]
or it can go off in a direction [angle] as you have described.
3. The cue ball? now we have problems.
First if it hits it straight on it will of course, all other things being equal continue in a straight line [a] or go off in a direction[b] or stop if the shot has reverse spin enough to stop the ball[c] or go off in a curve or parabola if there is enough side spin [d] or spin backwards [e] or a curve backwards [f] if you get my drift.
Thinking about it further I feel I have never played on a level pool table surface so 2. is unreasonable
Technically, deforming included, there will be an “instantaneous” change of direction.
It would not involve all of the ball at first as electromagnetic etc forces are not instantaneous but as each part of the impact develops in their instantaneous time slots the cue ball under the narrow conditions specified, All normal motion, flat table, no spin and some angle must have when any force is registered enough to register as a force, a change in direction in that part of the ball which is now in possession of that force such that the whole ball must be changing in direction. It is merely the repository of that force and that force has a vector or direction.
g. I sometimes miscue or hit the cue ball so hard it jumps off the table.
just kidding.
[…]
No-one ever takes an instantaneous view of motion as their framework? Uh. Mazing.
angech,
Apparently you’ve either never taken a calculus course or you’ve forgotten it.
Also, many problems in physics courses deal with idealized objects, i.e. perfectly elastic billiard balls on a frictionless surface. We used air troughs in my college physics lab to approximate no friction.
Brandon,
If I understand you properly, if you feel that I have not shown your point of view sufficient respect and/or due consideration, then I also apologize for that. I hope this helps.
.
If this does not help, could you suggest something that would? I’m at something of a loss to figure out how to rectify the situation.
DeWitt
I lost a lot of money playing poker at Uni but not at Billiards. I was so bad at Billiards I could not play anyone for money.
I did calculus somewhere many years ago.
This was more a fun foray into physics and semantics.
Sorry to offend.
No offense taken. It’s just that instantaneous is sort of a term of art in math and physics. Calculus, for example, is all about finding the limit of a function as Δt → 0 Zeno’s paradox of Achilles and the tortoise, is not really a paradox. It has a solution.
The notion of “instantaneous motion” has been known to create logical difficulties ever since the ancient Greeks. DeWitt mentioned the “Achilles and the tortoise” paradox. I like the “arrow” paradox myself.
On the other hand, ever since Newton developed calculus, physicists have felt free to talk about “instantaneous velocity” or “instantaneous acceleration.” In classical physics, we can define these things precisely, mathematically, and we can even build devices that measure these idealized concepts approximately, to a degree of accuracy that is only limited, in principle, by quantum mechanics (that is, the uncertainty principle).
Right now my students are in a lab using (relatively crude) motion sensors that will generate for them curves for the position, velocity and acceleration of objects as functions of time. So it makes perfect sense for them to talk about the velocity, or the acceleration, of an object at this instant.
Yes, it’s an approximation. But so is almost everything in physics, and we can make it (in principle) so precise, that is to say, so close to the mathematical, idealized limit, that you would never know the difference (not without extremely sensitive, specialized equipment, anyway). That makes all these things useful concepts, and all physics students are told about them.
The breakdown of these concepts at the quantum level is what I find, personally, most interesting. Nobody talks about “velocity” in quantum mechanics, for instance: we can define operators for position and momentum, but not velocity (let alone acceleration). This may be a reflection of the fact that, on a very fundamental level, there were problems with the concept itself from the beginning. So Zeno may have been right, after all, in pointing this out…
Feynman on Zeno:
” The Greeks were confused by such problems, being helped, of course, by some very confusing Greeks.”
I don’t think mathematicians (including von Neumann) were very happy with the Dirac delta function either, until further developments in functional analysis put it on a more rigorous footing.
RB–
Funny. Of course, most kids might ask: “Why not do what a dog or cat would do? Get a bead on the tortoise’s trajectory and go there?”
That solves the whole “infinite series” thingie, but replaces it with “How do you know its trajectory?”
Feynman has an entire section on this idea of instantaneous speed in Chapter 8 of the Lectures, with the example of a hypothetical lady who protests her speeding ticket. His discussion of Zeno’s paradox concludes with
Calculus placed the idea of speed on a firmer footing, but to my understanding, it was still not considered mathematically rigorous until further developments were made in measure theory .
The rigorous basis for the use of limits was actually established by Cauchy, 150 years after the invention of calculus. More here .
RB thanks for both these tips.
Have only read the first and wish I knew calculus.
The second starts with the student asking about a car doing 50 mph what does it mean. Very apt. Perhaps I can find time tonight Aus time.
DeWitt, is it possible to have a black swan event, say a 5 sigma deviation up or down in ice area without invoking CO2 as a driver in up or volcanoes in down?
Could you envisage in any way that current events are just natural variability?
angech, the first link is very dense – I just linked to it to indicate that the foundations of calculus, as with many contributions of physicists to math, were not very rigorous by the standards of fundamental math. The second link is more accessible and relevant for the purposes of discussing speeds.
To come back to some of the earlier discussion by Brandon S:
This view is correct. As explained by Feynman discussing the hypothetical lady caught speeding at 60 mph (Chapter 8, Section 8.2),
Re: RB (Comment #138559)
People who don’t spend much time trying to develop mathematical techniques to solve actual physics problems have much more time to worry about “fundamentals.” 😉
I spent the last 18 or so hours bedridden from being sick, and I’m reading this from the bottom up, so my responses will be out of order. I was going to wait until I had read more, but I just saw this and had to respond:
While that remark is true, nothing about calculus requires the use of limits, so it’s debatable how much significance that point has. Indeed, many earlier mathematicians showed awareness of things the over emphasis on limits managed to paper over for quite some time, effectively causing those earlier ones to predict “discoveries” made in the 1900s a thousand or more years in advance.
Okay, I quit. I effing quit. lucia, there is no way you could possibly read this:
See the word “default,” and think:
Is a reasonable response. I repeatedly made the point in my comment that there are multiple frameworks one can use, and that which framework one uses depends on the circumstance. I explained why the context of this particular circumstance leads me to believe one framework was clearly intended. I then said in my experience, the default framework people use is one particular framework.
There is nothing, absolutely nothing, in that, which could possibly make you think I meant nobody ever uses the other framework. Hell, I explicitly said I’ve used the other framework!
I explicitly said I’ve said used the framework, yet somehow, you turn around and derisively respond to portray my comment as though I said nobody ever uses it? My comment was nothing like you suggest, and there isn’t a way in hell you could have read it and thought it was.
You aren’t trying to listen to a word I say. You aren’t trying to have anything resembling an honest discussion. You’re being a dishonest hack just so you can insult me because… I don’t know why. All I know is it’s about as offensive as anything can be, and I’m not going to stay here and let you lie about what I say.
Even Joshua wouldn’t stoop as low this low.
Mark Bofill:
Again, you haven’t bothered me. I think you haven’t really looked into the situation, but you also haven’t argued about it to tell me I’m wrong, so that’s fine.
You’re not like lucia who just responded to my comment where I put a significant and sincere effort into explaining the disagreement and my position by misrepresenting it in the most blatany way imaginable just so she could make a derisive and dismissive remark. That’s the sort of behavior which has offended me. It’s not a behavior you’ve engaged in. You haven’t read me saying:
Done plenty of problems within framework 2. And:
Had to discuss thigns in a framework like 2, but:
Never seen anyone use framework 2 as their default. I’m sure you’d have no problem understanding that because it’s quite clear. Even if you didn’t know or care what framework 2 is, you’d understand I’ve used framework 2 plenty of times, but it’s just never been my or anyone else I’ve known’s default framework. Because of that, you would never respond to me by saying:
Because obviously, sometimes people do take framework 2 as their framework. I do it myself, at times. I don’t use it as my default framework, but if it is the one appropriate for the situation at hand, I use it.
So no, I’m not upset with you. You’re not the one blatantly misrepresenting what I say or treating me rudely. You’re not the one responding to my sincere attempts to resolve disagreements and get past any ill will by slapping me in the face. So please, don’t keep apologizing. You have no reason to.
RB,
Yes. And interestingly, that is an example in which Feynman points out that the cop is using the notion of “instantaneous speed” as his default and it would be obvious to all listeners that he was the one who was using the term in a way that is common to most people.
Hi RB, thanks for those articles.
Following on from your first link, Euler: The Master of Us All contains some examples of Euler’s amazing “non-rigorous” proofs. I found it an insipirational and well-written book that felt like it transmitted some of the joy Euler might have felt in exploring mathematics. (It has a $0.00 price on the MAA website, so I assume it’s ok to read the PDF copy).
( Edit to #138570 : “Following on from your second link [..]” )
Re: Brandon Shollenberger (Comment #138564)
Would you mind explaining what you mean by the above?
The N1 question.
A balloon filled with helium floats in air. What happens if the ballon is instead filled with SF6 (Eli does not want to be in the same ZIP code with a balloon filled with H2S).
It’s OK Eli. I believe the balloon idea won’t fly.
DeWitt
“The record low Antarctic area and only fourth from the bottom Arctic area caused global area to also set a record low August 20.”
That broken ring of ice in the Beaufort sea is fading fast unfortunately. Still, is getting colder there. If some stays expect a rapid refreeze
oliver:
Sure. I’m fighting off a migraine though so I’m going to be brief and not provide the details/documentation I’d like to though.
Axioms are things we assume to be true without any proof so that we can build systems of logic upon them. Because they’re assumptions, they are arbitrary, and there’s no inherent reason we must pick one over another. People generally just pick whichever ones seem to work best. Limits arise from an axiom which says (non-zero) infinitesimals do not exist. Infinitesimals are infinitely small numbers, so that axiom means when you get infinitely close to something, you’ve reached it.
That axiom is an old one. It’s also one which was violated by many of the people who claimed to follow it. A number of people who endorsed it created mathematical proofs which relied on infinitesimals to work. Those proofs were dismissed as incomplete and other people sought out different proofs which didn’t rely on infinitesimals. Over time, mathematics formalized calculus largely around the idea of limits and solidified the axiom forbidding the existence of infinitesimals as the “right” way of doing math.
In more recent times, there’s been an upsurge of what’s known as “non-standard calculus,” mathematical frameworks which don’t rely on the previously formalized approaches to things. A number of these do not have a restriction on the existence of infintesimals, meaning they do not require the existence or use of limits (though oddly enough, at least one allows both infinitesimals and limits at the same time). This allows one to do calculus without relying on the approaches Cauchy formalized.
In effect, calculus has a class of frameworks you can use to solve problems. Cauchy formalized one set of them which was already hugely popular. That helped make it be largely viewed as “calculus” rather than just one type of calculus. The result was non-standard calculus was largely dismissed as a joke, viewed as something which could never be a real type of mathematics. That’s true even though we now know non-standard calculus can be formalized every bit as well as standard calculus, and in fact, it is better at solving some problems than standard calculus could ever be.
So ultimately, if those people who had shown an intuitive grasp of the non-standard approach to things by relying on infinitesimals in their proofs had gone with it instead of going with the “right” answer which said (non-zero) infinitesimals don’t exist, we could have had non-standard calculus develop right along with standard calculus, which would have been far healthier. I don’t have the energy to do justice to what I mean by the “healthier” part right now, but the reality is a lot of people intuitively think more in line with non-standard calculus than in standard calculus, but they are taught they’re just wrong to do so because that’s not how math works. A great example can be seen in the debate over whether or not .9 repeating equals 1, which I encourage you to read my post about here:
http://www.hi-izuru.org/wp_blog/2014/01/0-999-does-not-equal-1/
What it comes down to is mathematics doesn’t have one “right” way of doing things. It lets you pick whatever set of axioms you want to use to solve the problem at hand. A long time ago, a number of people chose to use infinitesimals to solve some problems, but at the same time, they decided infinitesimals can’t exist because they messed up some other problems people were trying to solve. If they wouldn’t have made that decision, and would have instead embraced the possibility of using different frameworks for different problems, mathematics could have developed very differently.
(Of course, even once people started embracing the idea of using different frameworks more widely, many still ridiculed the idea of non-standard analysis, so maybe it wouldn’t have changed things that much. It’s hard to tell. Quite a bit in math happened or didn’t happen just because of who or what was popular.)
angech, go over to Neven’s blog for an update.
Brandon, If 0.99999 repeating gets to be 1.0 what does 0.55555 repeating get to be? I believe nature has a valid reason that some numbers can only be determined to an accuracy relative to the amount of work of observation. I believe this is related to the same principle driving Heisenberg uncertainty.
Human brains routinely simplify the universe for the purpose of broader understanding. We take short cuts. Most of the time this pays off. Once in a while we will get fooled by them, which forms the basis for the magician’s profession.
Sorry if I added to your migraine.
Ron Graf:
You can believe what you want as the answer to your question depends on the number system you choose to use. You can choose to use the real number system, as taught in high schools across the country, and believe 0.9 repeating does not equal one, in which case 0.5 repeating just equals 0.5 repeating. Or you could choose to use a different number system.
For instance, you could choose to use the integer system. In that system, 0.5 repeating wouldn’t exist because there is no such thing as non-whole numbers. You could have 0 or 1 but nothing in-between. That’s a genuine number system taught to children everywhere. There is no rule of mathematics or logic which makes it any less valid or rational than any other number system. The The choice to “believe” in it is every bit as valid as the choice to “believe” in any other.
And therein lies the problem with your remark. You say you “believe nature has a valid reason that some numbers can only be determined to an accuracy relative to the amount of work of observation,” but many numbers in math aren’t determined relative to an accuracy related to nature. Many numbers can be determined to an absolute level of accuracy, simply by the nature of the number systems we create. Nature has nothing to do with it, save in that we can often find those same numbers reflected in nature.
To answer your question though, I have done problems where I wound up writing 0.555…0, 0.555…5 and 0.555…9, each as a different potential answer to your question. Indeed, I’ve played games where determining which value between those (or others) was the answer to a problem could determine which play was the winning move. Coming up with the wrong answer could result in me losing the game. It normally wasn’t with repeating 5s (3s and 9s were far more common) because 5s convert to 0s more easily than most numbers, but it did happen from time to time if things aligned properly.
No problem. I couldn’t sleep, so I was just playing a Flash game anyway. It’s a tower defense, if you know what those are. Interestingly, this discussion reminded me of project I’ve been working on for them. I had the idea of creating generalized equations to represent tower defense games. It obviously works better with some games than others due to how well their mechanics can be generalized into equations, and how effective it is depends on how good a job you can do, but I thought it’d be cool to have a tower defense game in the form of mathematical equations.
Anyway, the reason I bring it up is for some reason I find doing math soothing so talking about math and working on this project some actually helps make things better, not worse. I had completely forgotten about the project, so if I hadn’t been discussing math while playing this game, I’d probably not have thought about it at all. So hey, you helped!
Re: Brandon Shollenberger (Comment #138588)
Brandon, I wasn’t sure which “debate” you were referring to, so I clicked over to your post… which starts off with the statement:
Now, the fact that such “proofs” are indeed offered in textbooks and in analysis classes the world over doesn’t prove that you’re wrong, but it does raise the obvious question: why do you think you’ve hit upon some unique insight in of math that almost everyone else doesn’t ‘get’?
(As an aside, if you were shown the statement 1/3 = 0.3333… would that be untrue/ambiguous as well?)
oliver:
Did you bother to read past that statement? If so you should have found the answers to your questions. What I said in that post isn’t remarkable in any way. It’s just a discussion of basic principles of math. You have to read more than the introductory sentence to understand it, but… that’s sort of to be expected in any discussion of math.
Eli, I go there nearly every day but Neven fired me ages ago.
He has great graphs but does not update them when the ice grows so is stuck in Layla land 2012 with some.
Anthony also has a great sea ice page with the advantage of being up to date.
Currently much hullabaloo about Northwest passages possibly opening and miracles of melting which will make this the third or fourth worst Arctic extent in the 45 years of record.
DeWitt did not comment when sea ice area globally at the start of the year was the highest on record but felt an urge to mention when it dropped to the lowest for one day earlier this week ( now rapidly rebounding.
The best one could say is that when a measure exceeds known boundaries on both sides within a year that none of the current movements would seem to have any bearing on whether the world is currently warming or not.
But you know better.
Ron Graf:
As this was unanswered : in the frameworks where 0.99999 recurring is equal to 1, 0.55555 recurring is equal to the rational number 5/9 .
As described here , what Brandon states is true, but the alternative is a convention with no apparent use.
RB:
That’s not true. You’ve somewhat misrepresented what that guy said, and what that guy said doesn’t even seem to be accurate. I mean, unless when he says “invent strange new objects,” he means you need to invent new things in the same sense all mathematical systems require you invent new things, in which case his wording is quite peculiar.
But then, maybe I’m misunderstanding him. A single sentence quoted without context in the middle of a Wikipedia article isn’t exactly what I’d consider a good source for understanding things.
angech,
We never bet here on the sea ice maximum for either pole. We did bet in previous years about the Arctic sea ice minimum extent as measured by JAXA using a seven day moving average. Not coincidentally, the Antarctic maximum occurs at nearly the same time as the Arctic minimum. The Antarctic has been at or near record highs for several years now. Another record high wasn’t worth commenting about in my opinion. I thought it was much more interesting that the Antarctic was at a record low. So shoot me.
I’m not Neven or any of his fanboys. I have several bones to pick with him, including his claim that he invented using Cryosphere Today area and JAXA extent to calculate Arctic sea ice concentration.
I was never very good at taking tests. I would over think the simple questions or not RTFQ very carefully (oh, answer only two of the four essay questions).
While I was reading the above comments, i could not help but think that a lot of the comprehension problems being discussed could eaily apply to some government tax forms i have attempted to complete in the past.
Jeff Norman,
Tax forms (outside the basics) are often written in a fashion which, while allowing correct completion if you follow instructions to the l-e-t-t-e-r, are essentially incomprehensible. You have turn off all thinking, or you will mess it up for sure. The very best are so complex that the form just says (paraphrasing), ” this is so complicated that you have no chance of doing it right, so you should just let us calculate it for you”.
DeWitt, my chip on my shoulder, apologies.I’m not Brandon but I can be quite annoying and obtuse at times.
A mixture of trying to understand in a fairly rigid belief framework.
Trying to improve and open my mind.
Sometimes the best stuff I write is the stuff that does not send due to I-pad/location/timing problems.
Invariably when I have time to think about my response I become less aggressive and more considerate of other people’s feelings.
I modify or eliminate the snark that so easily comes to mind with mindless anger at what I needlessly read into other people’s comments.
You are right, it was an interesting event and not one I was hoping for pre September.
It was good that Lucia gave credit to Brandon in the opening of this blog.
I never knew an olive twig contained so much energy.
Re: Brandon Shollenberger (Comment #138601)
Yes, I did.
You said there was a “debate,†so I was looking for evidence of such a thing. Now you are saying that the content of the post “isn’t remarkable in any way,†which I can agree with, (other than the above statement). I did read past the introductory sentence, which is how I decided that I agree. Incidentally, I also didn’t find answers to any other questions that I currently have.
But to go back to something you said previously:
Re: Brandon Shollenberger (Comment #138595)
It doesn’t have much at all to with “belief.†Proofs “0.999… = 1†are shown all the time, by professional mathematicians who do, in fact, understand math. The proof is explicitly understood to be a consequence of axioms, which is how mathematics works. Nor is there any real danger of confusion when you see x = 0.999… of thinking, oh, perhaps x is really meant to be an integer, and therefore 0.999 != 1.
Wait a minute, 0.9 repeating equals one in the real number system, as taught in high schools across the country and 0.5 repeating equals 0.5 repeating as well as the rational number 5/9 within the same system.
Re: RB (Comment #138620)
I think most everyone was able to filter the minor typo and get the gist of Brandon’s sentence…
Re: oliver (Comment #138621)
Fair enough. Upstream, there was a paraphrase of what Gowers said in his book
Perhaps, he was referring to things such as dispensing with some of our familiar rules of subtraction wherein in an alternative system two different numbers can have a zero difference , the distributive law breaks etc.
oliver:
There’s been a huge debate over whether or not .9 repeating equals 1 for years. My post doesn’t provide evidence of the debate because I didn’t think anyone would doubt it since it’s so common, and it’s all over the internet. There are thousands of topics and posts on the internet talking about it. It’s the entire reason you can find so many “proofs” online.
Leaving aside my personal belief many professional mathematicians don’t understand math better than the average person, it is highly misleading to say the “proof is explicitly understood to be a consequence of axioms.” If you look at most of the proofs given online, almost none say a word about axioms. If you ask most people to explain the answer, almost none will say a word about axioms. That’s because most people have no idea there is a connection between the answer and the axioms you choose to use.
Of course, some mathematicians might. That has nothing to do with the proofs commonly offered though. The proofs commonly offered are offered to people who don’t even realize there is a choice between the real number system and any other system. Those people are then presented proofs which assume you use the real number system but don’t actually state that assumption. Because of that, the proofs arise purely from an unstated assumption. It’s a classic case of begging the question.
A person who intuitively thinks .9 repeating does not equal 1 is not making the assumption (non-zero) infinitesimals do not exist. Despite that, they’re being presented proofs which only work if you make the assumption (non-zero) infinitesimals do not exist. That is wrong.
I speak as someone with experience. I’ve always loved math, but nobody had ever suggested you could use different axioms at different times to me. I didn’t realize the answer to this question depended on your choice of axiom until college, despite talking about the question with over half a dozen different math teachers. I even wound up talking to two professors in college who had PhDs in math, neither of whom could understand why .9 repeating would ever not equal 1.
RB:
I’m not certain what that piece means when it refers to “pseudo-reals” offhand but I don’t see why allowing for the existence of infinitesimals should break the distributive law. I think the only reason it does in what he describes is the strange role those “pseudo-reals” fill. It sounds like he’s saying they can be smaller than another number an infinitely small amount even though infinitely small numbers don’t exist. I’m not entirely certain.
What I am certain of is it is easy to construct a number system which allows for the existence of infinitesimals which allows for the distributive law. I’ll demonstrate by subtracting a number from another number which has an infinitesimal component. I’ll label the infinitesimal x.
You have (2 + x) and 2. You subtract them: 2 + x – 2. You now have x.
I wonder if there is going to be a chance to discuss the latest tactic of the cliamte obsessed to explain the pause:
Adjust it away and pretend it never happened.
hunter,
The main thing about the pause is it may finally be ending. I haven’t had a chance to look at the most interesting paper– which has Zeke as a coauthor. So of the others are just sort of “whatever”.
The announcements surrounding the paper suggests ‘the pause’ has become very important to “them”. But to me, the important issue is discrepancy between models and projections. (And as the AR5 is now out, we need a little time to let climate happen before testing is worthwhile.)
I’m not sure that mathematicians offering different proofs of the same theorem is evidence there is anything controversial about a particular result, such as 0.9 repeating exactly equally “1”. After all, mathematicians are interested in the question of “how little you have to assume to prove a particular thing” in addition to whether that thing is true.
The real interesting question would be “is there a useful mathematical framework in which 0.9 repeating is not exactly equal to 1”?
I think you need to do better than “it’s all over the internet” on that one, assuming you are suggesting such a thing.
Just because you can start with absurd frameworks where the two are not equal (it seems obvious that this can be done), so what? Is there the slightest utility in such a number system?
Brandon,
I think you are referring to the associative law in your example.
Carrick, I’ve definitely said there are useful frameworks arising from non-standard analysis, but I haven’t said the popular debate over whether or not .9 repeating equals 1 has anything to do with that. I had made a comment about there being a debate over that topic, and oliver said he was looking for evidence such a debate existed. I believe the fact there is lots of debate over it on the internet is compelling evidence there is lots of debate over the subject.
That has little to do with the technical aspects of the issue. The only really important thing that has to say on the technical aspect is whatever one may feel about the technical answer, people’s intuition will often be that infinitesimals do exist. So that’s something you’ll need to address.
I find it unhelpful to refer to things as “absurd” simply because you dislike or disagree with them. There is nothing inherently absurd with any number of frameworks which allow for the existence of non-zero infinitesimals. Or at least, nothing any more absurd than any number of commonly used mathematical frameworks. Then again, the first use of negative numbers in Europe had them quite literally labeled “absurd numbers,” and that was in the 15th century (3000 years after other civilizations had been using negative numbers)! So I guess there’s some form to calling things absurd.
Anyway, I actually referred to problems I’ve solved with non-standard analysis a bit upthread. I won’t say the problems couldn’t have been solved another way, but there are entire classes of problems where non-standard analysis is at the very least, more efficient than using standard analysis. There have certainly been many papers published using it, including new insights in a number of fields relying upon it.
It’s actually a really fascinating thing to watch develop if you’re a fan of math. One of the most fascinating things is just watching how mathematicians react to it since it is so “new.” It’s really only been around as a formal thing for ~50 years, and quite a few of mathematicians have reacted poorly to it, I suspect because of having viewed math a certain way for such a long time. The result is those hilarious overreactions that miss the point so badly even a person with little knowledge of the subject can understand why the guy is wrong.
(Which isn’t to say that’s a common reaction. I have no idea how common or uncommon it is. I just know what the guys who make big fusses say.)
Brandon Shollenberger (Comment #138623)
I didn’t realize what you meant was “there’s been a huge debate [on the Internet] for years.†I wouldn’t have bothered asking if I’d known.
The statement is absurd, and not simply because I disagree with it. From the discussions above, it seems that you personally believe that many professional mathematicians don’t understand math as well as you do. Even so, “the average person†is still a pretty low bar.
I was talking about proofs given by actual mathematicians, so I am not claiming anything about what random people arguing online do or do not say.
Also, there are “reasonable” assumptions about context and convention. If I give a seminar on analysis at a university, I am not going to preface every statement by reminding the audience that the axioms of arithmetic are still being assumed, or that when I write down 0.1666… I mean a real number. It will be assumed that these axioms will already have been encountered, in explicit form, by every member of the audience. If people have questions about assumptions, then they can ask.
Re: Carrick (Comment #138627)
Well, the answer is, of course, there such frameworks do exist. But I think it’s pretty far out in la-la land to go around arguing that the statement 0.999… = 1 is “false” (and anybody who claims the equality knows “nothing” about math) because frameworks can be constructed in which the equality does not hold.
The way this whole discussion is shaping up really reminds of the grand Internet arguments about the “order of operations problem,” which has everything to do with conventions and almost nothing to do with math.
To be fair, nonstandard analysis seems to be a creditable new idea which formalizes the Lebniz/Newton approach to calculus. It seems to have attracted interest from some big names in the field such as Terence Tao.
However, it seems that its uses, at least so far, have been largely to make certain proofs of theorems simpler in theoretical math, theoretical economics etc. The user of calculus already did not need to be familiar with the theoretical foundations of either approach.
For the layman, as the link in Comment #138622 states, the problems with intuitiveness seem to arise from representing real numbers by decimals instead of as a collection of sets with either one entry or two entries. And that the convention was chosen to be consistent with the algorithms of arithmetic for operations such as subtraction.
oliver:
Um, okay. I don’t know why you would assume somebody saying there’s a debate on a topic, linking to a post which talks about discussions going on in the public realm, would be limiting their remarks to arcane arguments held between mathematicians. I’d think somebody talking about what goes on in discussions involving people who know next to nothing nothing about math would have made it clear they weren’t referring to arguments between mathematicians, but… live and learn, I guess.
I don’t know why you think the statement is absurd. It’s true for many fields. The simple reality is you can get by in lot of fields with things like rote memorization without any genuine understanding of the underlying principles or concepts of the fields. Heck, a lot of math classes get taught that way. Why would it surprise you people can get by on it? (That’s not rhetorical.)
Somehow, I doubt you would be able to provide a single proof “given by actual mathematicians” on this topic. I mean, you completely misread what conversations my post was referring to, and now you just said:
Which shows you either didn’t understand the point of my post, are willfully ignoring it, or are just hand-waving it away without doing a single thing to show it is wrong. I can’t tell which of these it is because you’re not actually contributing anything to explain your derisive remarks.
But hey, sneering is fun!
RB:
I don’t try to keep up with all the developments in non-standard analysis as I wouldn’t be able to, but I have found work with non-standard analysis on probability theory to be rather interesting. Path analysis is another area I’ve found its application interesting, one where I’ve actually used it myself. A person I know insists harmonic analysis is the most important area for the use of nonstandard analysis right now, but I’ve always been terrible at that, so I have no idea what to make of their opinion.
Anyway, infinitesimals were commonly used back when calculus was first being developed. They were used before then too. It was only as calculus became more formalized that the idea (non-zero) infinitesimals could not exist became unto gospel. It took hundreds of years to find ways to formalize things so that infinitesimals didn’t have to be used.
Even if you genuinely believe frameworks which allow for their existence aren’t useful, I’d say infinitesimals themselves were pretty dang useful. We wouldn’t have had calculus without them. That means the worst you could possibly say of people who choose to use them is they’re using an outdated system.
Re: Brandon Shollenberger (Comment #138634)
I guess I just didn’t think someone would call something a “huge debate†if all it turned out to be was a big Internet argument involving mainly “people who know next to nothing†about the subject.
I also didn’t expect that a statement that “anyone who claims otherwise knows nothing about ___” was intended with the caveat that the statement is not intended to cover people who actually know something about ___.
I was clearly mistaken on both counts.
Well, first of all, I would be surprised if any professional mathematicians didn’t understand mathematics much better than the average person because upper-level undergraduate and graduate level math classes for math majors require a lot more than rote memorization — they emphasize construction of proofs, from explicitly stated axioms, no less.
Secondly, professional mathematicians are required to demonstrate a much higher level of math proficiency than the average person, which strongly implies that they have a better working knowledge of math than most, even if they haven’t spent much time lately thinking about the history and foundations of mathematics.
By this do you mean to that: (1) You don’t believe such a proof exists at all in the literature, (2) You don’t believe I could provide such a proof, or (3) Something else entirely?
Before I get any further into misreading, exactly which post are you now referring to?
I don’t think that the issues related to any non-standard analysis that yields $latex 0.999\ldots \ne 1$ have been fully addressed here, so here’s my two cents:
The principle issue I’m going to discuss is that the decimal system is nothing more than a representation of a real number in terms of (potentially) an infinite series. This generalizes to a “radix system” when you use a positive integer other than 10 to represent the real number.
If we restrict ourselves to the domain of real numbers on [0,1), then any real number in that range can be written as:
$latex {\cal R} = \sum_{n=1}^\infty a_n b^{-n}$
where $latex b$ is a positive integer known as the base of the representation and the $latex a_n$ are all non-negative integers less than $latex b$.
More compactly we can write in radix notation
$latex {\cal R } = 0.a_1a_2a_3\ldots$.
Now the problem here with arguing that $latex 0.999\ldots \ne 1$ is that the infinite sequence $latex 0.999\ldots$ or equivalently the infinite series
$latex \sum_{n=1}^\infty 9 \times 10^{-n}$
arises precisely because of the particular base (10) that is chosen.
You don’t necessarily get an infinite series in other bases when you perform radix arithmetic. More on that below.
First let’s consider one example for how to obtain $latex 0.999\ldots$ using base 10:
$latex (1/9) \times 9 = 0.111\ldots \times 9 = 0.999\ldots$.
Now we’d expect the two rational numbers to commute so
$latex (1/9) \times 9 = 9 \times (1/9) = 9/9 = 1$.
If you then claim that $latex 0.999\ldots \ne 1$ then you are left with the result that multiplication of rational numbers is no longer always commutative.
Now… what if I wrote down the same problem in base 9? Here we simply have:
$latex (1/10) \times 10 = 0.1 \times 10 = 1$.
So we are left with the absurd result that using different bases for our decimal notation yields numerically different results.
Toe me that seems like an absurd result.
Also, the statement that the rational numbers don’t always commute also would depend on the base that was chosen. Given a different choice of base, and suddenly the exact same two rational numbers once again commute.
oliver:
(2). It was me expressing my view that your opinion of what proofs given by mathematicians would and would not contain isn’t founded on any real knowledge of what proofs given by mathematicians would actually contain, but doing so in a way which wasn’t aggressive and provided you an easy opportunity to indicate I was wrong.
There’s only one post I’ve referred to. Other people may not make the distinction, but when I say post, I refer to head posts, not comments submitted underneath them. And as for what I said in my blog post on the issue of whether or not .9 repeating equals 1, any decent mathematician would agree with it.
When a person expresses doubt .9 repeating equals 1, they show their intuition is (non-zero) infinitesimals exist. Responding by providing them proofs which are built upon the unstated assumption infinitesimals do not exist does nothing to address their concerns. That means the response is wrong. Either the person giving it is being dishonest, or as I say in the post, they don’t understand math as they apparently think that there is somehow only one “right” framework you can use.
And yes, the phrase “know nothing” is an exaggeration. Just like my choice of the word “false” was not literally correct. As reading the post makes clear, those word choices were made for rhetorical effect, with the full meaning being made clear further in the post.
Carrick:
But you’ve ultimately just done the same thing I point out people do in my post. What you posted looks fine and all, but it’s purely an optical illusion built upon a framework whose assumptions people intuitively suspect. In fact, your argument hinges upon one of the classic formulations of the “proof” .9 repeating equals 1, that you can do:
1/9 = .111…
9 * 1/9 = 9 * .111…
1 = .999…
But the response to that proof has always been, “How do you know 9 multiplied by .1 repeating equals .9 repeating?” When you look at the actual math underlying the answer to that, it rests upon the assumption infinitesimals do not exist. That assumption is what leads to the conclusion you can have an infinite chain of multiplications that just “end” because we know what the result is. In over-simplified terms, your calculation relies on taking a limit.
Remove the assumption an infinite chain of multiplication can effectively be terminated with the leftover part disregarded, and you don’t get that result. What you get is:
9 * .111… = .999… + x
Where x is an infinitesimal component representing the “leftover” part of the chain of multiplications. That gives the correct equation for what you thought was absurd as:
1 = .999… + x
Which is completely unremarkable because all that says is there is some infinitesimal difference between 1 and .9 repeating, exactly what people arguing about this issue intuitively think.
TL;DR: The only reason you get an “absurd” result is you’re applying assumptions from one framework to a different framework which doesn’t use them.
Hummm… seems all a bit like asking how many angels fit on the head of an infinitely small pin.
SteveF,
Yes. If the angels are infinitesimals, perhaps many.
Brandon,
Any rational which leads to a conclusion that (1/x) • x is not equal to 1 is indeed absurd. Changing the numerical representation of 1/x, and the limitations of any chosen representation, can’t change the logical requirement that any number divided by itself must equal 1.
SteveF, I think you misunderstood what I said. Remember, there were two sides of the formulation. This is how it works in the real number system:
1/9 = .111…
9 * 1/9 = 9 * .111…
1 = .999…
Under a different framework, it just changes to:
1/9 = .111…
9 * 1/9 = 9 * .111…
1 = .999… + x
Where x is an infinitesimal. As you see, 9 * 1/9 is still equal to 1. It’s just not equal to .9 repeating anymore.
Brandon,
Too small a pin, and too many angels. Give it a rest.
SteveF, you made a comment which seemed to indicate you had misunderstood that which was being discussed, as what you described didn’t fit anything anyone had described. I don’t think the appropriate response to someone correcting your misunderstanding is, “Give it a rest.”
Brandon,
I have no misunderstanding. Nitpicking nonsense about infinitesimals and numerical representations is the issue, but based on what you have written, I rather suspect you can’t see that. I really do suggest you give it a rest… and that is intended to be helpful advice, not critique.
Brandon Shollenberger (Comment #138639)
Hmm. Are you claiming that there is some infinitesimally small difference
$latex \displaystyle \delta = (\mathrm{Passive})- (\mathrm{Aggressive})$?
(Sorry, I couldn’t resist. 😀 )
But okay. Since you are providing (in the infinitesimal moment that we are freezing the system to examine it) an opportunity to indicate that you are (in a continuing sense) wrong: You were wrong. I have undergrad and graduate degrees in math, so I’ve seen some proofs before. Also, I’m pretty sure I could search on Google Scholar or Wikipedia if pressed.
Please indulge me for the moment and pretend that I am decent mathematician. I earlier raised what I think is a fundamental problem with your blog post, which is that you speak about 0.999 = or != 1 in terms of “belief†and lack of awareness of alternative viewpoints. By contrast, mathematicians are aware of the axiomatic underpinnings of such statements. In general, mathematical proofs follow upon the structure of: “Based on the premises ___, what necessarily follows is ___.â€
SteveF:
You had said:
Nobody has said anything which would remotely imply the existence of any framework where “(1/x) • x is not equal to 1.” Now, if you want to insist you “have no misunderstanding,” I can’t rule out the possibility you’re telling the truth. If that’s the case though, I have to ask, why did you write a comment to call something absolutely nobody had even hinted at absurd?
Personally, I still think you misunderstood something, and that led you to believe a remark about the possibility where “(1/x) • x is not equal to 1” was somehow relevant. If not though, I have no idea why you thought your comment was relevant, and I’m entirely baffled as to why you would make a completely irrelevant comment then tell me I’m nitpicking nonsense and wasting time. That would seem all sorts of silly.
The problem of how to treat infinitesimals (standard vs non-standard) remains regardless of the radix system used. A similar situation would occur for 0.(8) recurring in base 9, 0.(7) recurring in base 8 etc.
The way I see it, you run into the various issues of how to treat the familiar algorithmic rules of arithmetic.
For instance, the algorithmic way of performing the subtraction 1.00000…. – 0.9999…. = 0.000000….
(or, recursively the first number would become 0.9999…. for performing the subtraction)
With non-standard analysis, one would say that you need to invent new structures called hyper-reals and you cannot perform such operations as above.
The math of non-standard analysis seems to be of use to those who are deeply involved in certain disciplines of theoretical math who desire greater rigor in manipulating infinitesimals.
oliver:
Just so you know, saying you could search on Wikipedia goes a long way in undermining your claim that I am wrong.
You ignore the fact almost all proofs are built upon unstated unassumptions. This doesn’t make them wrong as assumptions can be implied via context, but once you rely upon implied assumptions, you remove the need for one to be aware of them. Many mathematicians cannot list the fundamental axioms the real number system is based upon. What percentage of practicing mathematicians do you think could derive the real number system from axioms on the spot? In my experience, it’s a very small percentage.
That’s not a slight against mathematicians. Most of them simply have no need to be able to do that. They never do anything where those axioms are questioned. They know how to construct a proof, but they never do anything where they have to work from first derivatives. When you combine that with a lack of awareness of things like non-standard analysis, which isn’t even widely taught at the moment, there is no clear line between what most mathematicians do and “belief.”
Regardless, even if one agreed with your complaint, it would be a trivial one. The reality is mathematicians are responsible for disseminating knowledge about math. They’ve taken it upon themselves to spread knowledge about the subject. Even if all mathematicians know and understand the nature of how their choice of what axioms they use affect answers to questions like, “Does .9 repeating equal 1,” it’s clear that knowledge has not made it to the public to any meaningful extent.
So if you want to praise mathematicians for supposedly being aware of all this, go ahead. My personal experience suggests you give them far too much credit. Regardless, if they are as aware as you say, they need to do a far better job spreading that awareness so people interested in math can understand it. Right now, a student interested in math can go through high school and college trying to figure out that problem and never get a genuine answer. That’s inexcusable.
Brandon, You asked how we know that 0.111… x 9 = 0.999…
That’s basically a cop-out.
We have rules for how to do radix multiplication, which if you follow you get 0.999… from 0.111… x 9. Within the framework of radix representations, we find this 0.999… has the same meaning as “1”, from which we discover that particular radix representations are not always unique.
If you are saying the 0.999… you are talking about is a different entity than the 0.999… that comes from standard radix math operations, then you need to provide a prescription for how to compute it, and a proof that it’s not the same 0.999… series.
Fundamentally the case I gave you has two rational numbers:
1/9 and 9.
Multiplying them together must yield 1, regardless of the radix you’ve chosen, otherwise there’s something wrong with mathematics at such a fundamental level that basically everything ends up broken.
I think this discussion of what if $latex 0.999\ldots \ne 1$ is more interesting as an example of the ripple effect from “what if this is wrong”.
And since I just don’t see how you prevent the entire artifice of modern mathematical analysis from being destroyed here, I’d argue the ripple effect is very large in this case.
RB, as I understand it, non-standard analysis seeks to remove the concept of infinitesimals and replace it with a more rigorous mathematical framework involving limits, rather than vice versus.
Saying that $latex 0.111\ldots \times 9 = 0.999\ldots$ really involves the limit as $latex N\rightarrow \infty$ of the well defined series multiplication:
$latex \left[\sum_{n=1}^N 10^{-n}\right] \times 9 = \sum_{n=1}^N 9 \times 10^{-n}$.
Since I don’t think modern non-standard analysis attempts to do away with limits, I don’t expect they would ever disagree with the limit of that expression being:
$latex 0.111\ldots \times 9 = 0.999\ldots$,
or disagree that
$latex 0.999\ldots = 1$
when we stipulate that $latex 0.999\ldots$ was obtained in the limit of a finite series expression in the manner that I obtained it above.
Similarly, the issue of infinitesimals in other radix representations never need show up, as long as we restrict ourselves to the infinite series being the limit of a finite series.
Probably we should view infinitesimals as a heuristic method (shorthand) for performing (more rigorous) limit operations. Like any heuristic, you’d expect it to yield inconsistencies when you push it too far (and I’m pretty sure it does, such as the problems that Brandon notices).
But the cure seems to me to be to rely on the theory of limits and stop worrying about whether you can formulate a rigorous self-consistent mathematical framework using the theory of infinitesimals while eschewing the theory of limits.
The idea we retain infinitesimals and throw away limits seems a bit bonkers to me. Or cart then horse.
Carrick,
The argument seems to be that when you do radix math operations on say 9*0.111… you are making the assumption that the decimal numbers (e.g., 0.1111….) represent real numbers (thus allowing you to perform those operations in the first place) and therefore the manipulated result is also a real number. Instead, non-standard analysis makes use of structures such as hyper-real numbers. For reasons I won’t understand but Terry Tao describes on his blog, non-standard analysis has some advantages that make proofs simpler for things such as epsilon management, etc.
Standard analysis instead defines certain decimal representations to be real numbers by means of Stevin’s construction, Cauchy sequences etc , thus we define 0.999… to be the same as 1.
Carrick,
Yup. All agree 1/9 (base 10) is equal to 0.11111…..
All ( I think!) agree 9*0.11111…. is equal to 0.99999….
All agree that (1/9) • 9 = 1, therefore, 0.9999….. must logically be equal to 1. The rest of the arm waving strikes me as silly nitpicking…. and a waste of time to boot. Why would anybody get their panties twisted into a knot because we smart primates happen to have 10 fingers and so most often calculate things in base 10? I have no answer to that question, and I am honestly puzzled…. but if they were smart enough to do math, I think chimps would likely prefer base 10. Soon after, some chimps would worry about the meaning of 0.11111…. No need for them to worry, it all comes from our fingers.
Re: Brandon Shollenberger (Comment #138655)
Do you suppose they’d rescind my education if I were ever found reading Wikipedia?
Re: Brandon Shollenberger (Comment #138655)
This is probably a reason why a big part of the curriculum for mathematicians focuses on constructing proofs from explicitly stated axioms (as I said earlier).
Come to think of it, I remember mathematical axioms and proof from axioms being taught in high school geometry.
You’re right. It sounds genuinely traumatic.
This is seriously hurting my head guys. The points aren’t that difficult. Carrick, you say:
Everything you just said is predicated upon the standard analysis framework, the one I’ve been repeatedly pointing out doesn’t have to be used. You say there are rules which if you follow lead to certain conclusions. Of course that’s true. If you follow the rules used in standard analysis, you get the results we’ve always gotten in standard analysis. Nobody has questioned that!
The entire point of non-standard analysis is uses a different set of axioms than standard analysis does. That means it doesn’t require you follow the rules standard analysis would have you follow. That’s a very simple concept. It only gets complicated when you look at what rules there are in non-standard analysis. And in that vein:
I have no idea where you got this idea. The reality is non-standard analysis is a class of frameworks which covers many different frameworks. As such, I cannot rule out the possibility there might be some framework which fits your description. I sincerely doubt it though as I can’t even conceptualize what such a framework would be like. I don’t even know what change you have in mind since you seem to be describing standard analysis, not non-standard analysis.
In any event, the more popular classes of non-standard analysis, and the ones we’ve been focusing on in this topic, definitely do not fit the description you give. They do not seek “to remove the concept of infinitesimals.” They make extensive use of infinitesimals. You seem to have things completely backwards in regard to what non-standard analysis is. So when you say:
I don’t know what to make of that. There is no reason one would need to worry like you describe as there is no question about whether or not we can formulate such a framework. It’s already been done. It was done some 50 years ago. You may find it difficult to believe, but mathematicians have been using the very thing you scoff at for decades.
I had a text book not ten feet from me explaining how one derives much of one of these frameworks just yesterday. It’s buried in my barn now though. I replaced the dresser I was using in my room, and in the process, I moved a bunch of junk out of my room. Oddly enough, I found my copy of Godel, Escher, Bach: an Eternal Golden Braid in the process. I hadn’t seen that in almost four years.
RB:
For what it’s worth, radix math operations will often work the same in many non-standard frameworks as they do with real numbers. The main time you’ll see differences is with numbers which are infinite series. That’s because when using real numbers, we effectively take limits of the series. Limits are based on the assumption infinitesimals do not exist. If you remove that assumption, you can’t take limits of the series,* and thus, arithmetic operations on numbers which are infinite series becomes different.
In other words, as long as you’re not working with something involving infinity (it hasn’t been discussed, but non-standard analysis can often diverge in how it handles infinitely large numbers as well), the frameworks are usually going to be the same. It’s just how infinity is treated that’s different.
.
*Because non-standard analysis is a class of frameworks basically defined as anything other than standard analysis, there are as many different types of it as people choose to create. That means statements like this can be technically false even if they’re true for the most popular forms of non-standard analysis. For instance, I do know of one form of non-standard analysis which has both infinitesimals and limits at the same time. It’s really wonky. I’m trying not to over-complicate things by worrying about nuances like this, but I don’t want to mislead people either. So try to forgive me if I’m not always completely precise, or if I’m more precise than you might care for.
See, this is what bugs me. SteveF says:
Even though I explicitly told him whether that statement is true or false depends on which mathematical framework you use, because he had misunderstood something I had said before. He responded by basically telling me my comment was worthless, then later insisting there had been no misunderstanding. Yet here he is, repeating the very same misunderstanding. And yet, he has the audacity to say:
We wouldn’t have so much “waste of time” if people would just try to read what other people say.
Brandon,
From any practical POV it is a complete waste of time. Really, this is a crazy subject to argue over. If you ever wonder why people become frustrated with you, you need only read over this thread.
I took a look at Brandon’s post on his blog. I believe this statement is not true
The non-standard analysis framework implies that there is an infinitesimal, but it does not say that it ends with a 1.
Actually the thing with angels and pins was about whether an angel had a finite volume. There was also discussion about if an angel went from point A to point B, did it travel through the intermediate distance or not. It seemed important at the time.
SteveF:
If people become frustrated with me because they refuse to try to read simple sentences, then make numerous comments pointing out how much of a waste of time it is that there are comments arising from the fact they refused to try to read simple sentences, well… that’s just sad. In both the “depressing” and “pathetic” meanings of the word.
RB:
It depends on the framework since, as I’ve pointed out, non-standard analysis is a class of frameworks, not a single framework, but in at least some representations, yes, that difference does end with a 1. Not necessarily because it has to be a 1, but because that’s how it people choose to represent it.
But even if it weren’t the case that some frameworks already do it that way, it would be reasonable for a person to roll their own framework which does for demonstration purposes. As long as the rules you choose for your framework are coherent and understood by your reader, it’s fine. And I’m pretty sure readers would have no problem understanding the idea the difference between 1 and .9 repeating is an infinitesimal I chose to represent as 0.000…1.
Let me be say it differently, I doubt there is any representation such as 0.0000…1 – a very standard notation. Some examples of the hyperreal representation are in the section on infinitesimals here.
https://en.wikipedia.org/wiki/0.999…
Well, I think you have a standard notation intuition of what constitutes the difference while arguing against your readers’ intuition from a standard perspective where using traditional arithmetic algorithms and recursion, 1.0000.. – 0.999… = 0
I also doubt that your example upstream of 0.55… was a practical application of what really constitutes the non-standard understanding of 0.999…
I’ve always thought of the 0.9999…. = 1 debate as usually involving a clash between a Platonic view, in which numbers are *really* things out there in the universe & math is supposed to describe their real properties; and a formalist view, in which numbers are abstract things which exist only within & as defined by a theory.
In the second kind of view, setting 0.999…. = 1 is fundamentally no more & no less valid than setting 0.9999…. = 1-a for some infinitesimal a. The important thing is that “numbers” are different things depending on which approach you take – using common symbols obscures that. Of course there will be mappings from one into the other, preserving many properties, but fundamentally they don’t say different things about the same objects – because the objects, existing only within different theories, are necessarily different.
If you want to, you can say that one approach is a better model for the real world than the other, but a non-Platonist like me will tend to find that puzzling.
“God created the integers …”
I’ve said this before and will likely say it again. We would have been a lot better off if the Athenians had slaughtered Socrates’ entire school rather than just forcing him to kill himself.
RB thanks for your response above–that was very helpful and corrected my misunderstandings/delusions about “non-standard analysis”.
I’m going to back out of this discussion due to time issues, but I did find it interesting, though I remain perplexed on the utility of non-standard analysis.
RB:
Leaving aside, for the moment, whether or not any formalized mathematical frameworks exist with this property, let’s remember I was writing a blog post for the casual reader on the internet. These people wouldn’t likely care whether or not there is a formal framework which uses a representation or not if that representation makes sense as an explanation of things. With that in mind, I’ll quote the article you just linked to:
If people really want to discuss whether or not that description fits any formalized mathematical frameworks, we can. I don’t know why it would really matter to my post, but hey, I like math, so I’m cool with it. But before we do, can we all at least agree my description would make perfect intuitive sense to the target audience of the post? I’d like to feel like we’ve at least made some progress here, and that seems a point we could all agree on since you’re even linking to sources which say the same thing I did.
This may just be a dumb moment on my part, but I can’t make heads or tails out of this remark. I think you’re disagreeing with the idea my readers would have understood what I wrote, but I have no idea why you would. It seems very intuitive that 1 – .999… would equal .000…1.
Hey, would you look at that. My comment went through. There’s no telling when I’ll have an IP that’s banned and when I won’t. I really wish I had a different ISP >.<
“This may just be a dumb moment on my part, but I can’t make heads or tails out of this remark. I think you’re disagreeing with the idea my readers would have understood what I wrote, but I have no idea why you would. It seems very intuitive that 1 – .999… would equal .000…1.”
there is no final 9
from BS:
“.999=x
10x=9.999
10x – x = 9x
9x=9
1x=1.
.999 = 1.
The hand-waving is obvious. How does one multiply an infinite series of 9s by 10? What happens to the zero you’d get when multiplying by ten? Are we to believe it just disappears?”
As with the subtraction example, where it is imagined that there is a final 9, in the above example, a similar faulty imagination is used, as if there is a final 9 to multiply by zero.
However, Brandon is “correct” he just has a horrible time explaining himself.
.9 repeating is defined as 1. That’s the simplest way to explain it.
Attempts to “prove” the definition will always rely on or import the concept of infintesimals.
you can choose to define .9 repeating as some number such that
1-x = .9999.. what’s 1/2 of x? nevermind.
To illustrate what I mean by horrible time explaining himself
start with his title. alway bold you know.
“0.999… DOES NOT EQUAL 1”
Sounds pretty emphatic .
and he closes with
“If you feel 0.999… does not equal 1, you’re right. If you feel it does equal one, you’re right too. Which answer is “right†just depends on which type of math you feel most comfortable with. It’s purely a matter of personal choice.”
So he start with the emphatic X is wrong, beats up some folks along the way, and then closes with.. nevermind its a personal choice.
you basically have two choices when it comes to structuring this kind of piece.
Choice 1. The brandon approach. basically it allows you to criticize the ‘standard” approach. you get to call professional mathematicians no better than average people. you get to offer up your own silly “proof” of the opposite:
“Of course not. The proof is invalid. It’s just an optical illusion relying on tricking the reader by hoping they don’t notice the hand-waving. The reality is no “proof†can address the issue better than simply looking at the two values. If the two are equal to one another, subtracting one from the other must give an answer of 0.”
The proof here is “Looking” so much for math rigor. And if you know in the end that its convention, why play around with your own proofs?
What is the other choice?
Choice number 2. You Start with your Actual position
.9999 = 1, is a mathematical convention. There is an alternative convention in which 1-.9999 equals a non zero entity.
ya, start with what you think the truth is. novel approach.
The other thing that is funny is that brandon thinks the choice is personal
“Which answer is “right†just depends on which type of math you feel most comfortable with. It’s purely a matter of personal choice.”
Now of course one has to ask, what if I feel equally comfortable with both.. haha what if my comfort level with one definition is ‘1’ and my comfort level with the other is .99999 repeating.?
But seriously, the choice probably has less to do with how people feel (.9999 = 1, never felt right to me at all!! ) BUT, if I accepted that definition there was a bunch of stuff I could DO. Accepting that convention wasnt a personal choice ( ok I made it ) but personally I’d rather not make that choice. And again, there are problems that can be solved if you accept the convention. So, I can imagine going into the boss and saying — while its against my personal preference I will accept that .9 repeating = 1 and you will have your answer tommorrow.
The notion that picking a convention is personal seems to be somewhat suspect.
So again, you basically have two choices when writing this kind of piece — start out with the attack on the standard– and call everybody stupid— but end with a ‘nevermind its all personal choice” OR start out with what your really believe “Its a convention and here is why I choose option X” of course oppositionally defiant personality types get to make a choice of which approach they choose… or do they?
as for what professional mathematicians say.. there is this.. read up to page 60
https://books.google.com/books?id=DBxSM7TIq48C&lpg=PA66&pg=PA56#v=onepage&q&f=false
Steve Mosher,
So if it is all ‘personal choice’ can we agree this is an incredibly stooooopid subject to discuss on a (painfully) long blog thread?
(not a rhetorical question)
lucia (Comment #138626)
“The main thing about the pause is it may finally be ending.”
By definition a pause is something that ends.
Either by breaking up or down
“The announcements surrounding the paper suggests ‘the pause’ has become very important to “themâ€. But to me, the important issue is discrepancy between models and projections”
Of less importance is which data set or combination of data sets one uses.
The satellites show no sign of the pause ending.
The recently modified upward surface/sea series do.
Should we discuss the data in terms of original adjustments, current adjustments or future adjustments?
SteveF,
While the topic has gone pretty far astray, it does connect back to questions about physics testing, and testing in general. In particular: How much is it reasonable to assume? How much “acceptance of premises” should be required vs. how much leeway should one have for “interpretation”? (Note: these are meant as actual, not rhetorical, questions.)
Also, although I don’t think Brandon has gone about it in the best way, I don’t think these are ridiculous types of questions for the physics or math student to discuss. Students should be shown good reasons why 0.999… = 1, or why our framework of physics incorporates instantaneous accelerations. (Also: Why both the tortoise and the hare do or don’t reach the finish line).
Brandon:
OK, and it probably made sense to many of your readers.
I’ll leave it there.
Carrick,
I learned quite a bit from these discussions too, particularly, I learned quite a few new things about the real number system.
Lulz. Steven Mosher writes a lengthy comment here to say I’m terrible at explaining myself when just earlier today, he came to my blog and responded to a post criticizing his slanderous responses to criticisms of BEST to say nothing more than:
When the post had explicitly addressed that very point. Apparently I’m so terrible at explaining myself so Mosher will write lengthy comments to explain how terrible I am. But when I explicitly discuss something, Mosher will just somehow fail to see it so he can ask me about it and ignore any substantive points I may have raised. Somehow I don’t think the problem is my ability to explain myself. I’m pretty sure nothing I say could stop reactions like that.
There’s a reason I don’t put much stock in criticisms of my writing style or my behavior. Sort of like how SteveF just asked:
I’m fine with not discussing a subject if people don’t want to, but constantly commenting in a non-constructive manner to complain that there are too many comments is…
On a completely different topic, I’m happy to report the gaming project I mentioned above has taken a surprising turn. I was talking to a couple people about it when one of them suggested turning it into a game of its own. A number of people I asked think it could actually work fairly well as one. I’m pleasantly surprised by that. I like the idea of a game which is just a collection of mathematical equations the player has to try to optimize.
Oliver,
I agree that students have to be exposed to the basic rational for concluding 0.9999…. =1 (base 10), and why that depends on the base used. Students should also understand the concepts of differentials, acceleration at a point in time, and why the tortoise and hare both cross the finish line. But a first year calculus course seems to me the place.
.
WRT how much leeway there should be for interpretation in tests: poorly stated questions seem to me the real problem; there is no need for leeway in interpretation if the question is properly written. A test question should evaluate if a student adequately understands the concepts involved and can apply those concepts. A question which fails to do that is just a dumb question, and not something I find very interesting….. certainly less interesting than the rate of solar energy flux as a function of depth in the ocean. 🙂
Oliver,
Off topic: I think I remember that you work in oceanography in the NE of the USA. I have noted this summer (digging hard-shell clams and fishing) that the water in Nantucket Sound is warmer than usual for this time of year. Do you know if water along the East Coast is in fact warmer than typical for this date?
Let me sign off on this with this Q&A which I found very illuminating on many issues discussed here. Among other things, it discusses these:
(1) On why 3*1/3 = 3*0.333.. = 0.999… = 1 is not a proof (Questions 8.1 and 8.2)
(2) The limit in standard analysis is replaced by a standard part function in non-standard analysis, which is the core contribution of Abraham Robinson.
(3) The decimal representation for a hyper-real number is based on the Lightstone extended decimal system, invented by Robinson’s student. In this system, 0.999…..; …..9999…. is also exactly equal to 1 while 0.9999….; ….999 is not. Thus the residual from 1 for the latter would be the hyperreal number 0.0000…..;……01. There aren’t “alternative frameworks” for representing hyper-reals using decimals.
When I attempt to comprehend the phrase “an infinite terminating string of 9s” (from RB’s link) it hurts my head.
Can we talk about the levels of infinity next? No, let’s not.
RB, one thing I would caution people about that Q&A is it begins by referring to “Abraham Robison’s non-standard analysis” then just refers to to “non-standard analysis,” making it appear there is only one form of non-standard analysis. Hyperreals are a perfectly valid system, but they’re not the only “alternative” to reals like a person reading that might think.
Other than that though, it’s a really good reference. I highly recommend anyone who’s followed/participated in this discussion take a look at it.
DeWitt,
How about infinity raised to the infinity power, or infinity times a differential? See, there is really no ‘limit’ to how useless and silly such discussions can become. Number systems are human abstractions that are used to facilitate quantification; they have nothing to do with physical reality…. nature has no number systems, no rational or irrational numbers, no imaginary numbers and no differentials; it’s all us.
Perhaps like the tortoise and the hare it is a time problem
consider a number of 0.99999 repeating having to exist at a certain point in time
Like to do a maths problem.
While you are going to do it, it has an infinitesimal, whatever that is, but when you add it to 1 and get 2, in the moment that you have added it, it turns into a 1.
In other words while you are considering it it is always less than 1 but whenever you use it it becomes 1.
problem solved.
TM angech 2015
@angech,
Needs more cats.
Steve Mosher,
So if it is all ‘personal choice’ can we agree this is an incredibly stooooopid subject to discuss on a (painfully) long blog thread?
(not a rhetorical question)
×**********
Yes. But it was fun to read.
over my head Earle but thanks anyway.
OK, I actually did the experiment in my University Physics 1 class yesterday: I asked them Lucia’s question 7 as an in-class quiz. I had just introduced the concept of acceleration the previous day.
Among the non-honors students (sample size: 235), 27% got it wrong, whereas among the honors students (sample size: 83), only 16% got it wrong.
Also, virtually nobody chose answer d. By far the most popular wrong answer was b.
That would appear to disconfirm my theory. Well… there is always “false positive”!
@angech
Sorry for being too cryptic. I think your thought experiment needs a cat to make it more understandable.
https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat
SteveF says: “How about infinity raised to the infinity power, or infinity times a differential?”
And here I always thought infinity + 1 sufficed.
Kan,
Sure, it suffices if you want to actually point out the failure of logic and ‘mathurbation’ involved; but on a thread like this, you need to aim higher… into the really silly stuff.
julio (Comment #138709),
Data talks, speculation walks…..you are da man!
Earle
Got it, thanks.
I was thinking of the musical by the F1 guy,not the quantum stuff by the Phy guy.
Same answer for both if we consider it.
Cat is alive and dead but the moment later you open the door you know the answer.
SteveF, your view is spot on.
Reminds me of the story about Gauss. His professor wanted to keep the class busy for a while as he left the room, so asked them to sum all the integers from 1 to 100. While the others sweated over the large number of additions, Gauss wrote down the answer and left the room.
How did he do it?
He broke it down into pairs of numbers that added to a hundred
(1 + 99< etc) and found there were 49 pairs, plus 100 and 50. QED, 5050.
I think.
JimB, that’s one of those stories which has been told about tons of different people, and likely isn’t true for any of them. Or if it is, it’s only by chance that we might associate it with them. I know I figured out the same shortcut when I was in grade school because I had some problem involving a triangular pattern like with bowling pins and noticed:
4 * 5 = 20 = 10 * 2
5 * 6 = 30 = 15 * 2
It only took me a minute to realize that wasn’t just coincidence. I bet there are plenty of other people who made the same discovery at early ages as well. But discovering it after actually having that specific task assigned to them? I doubt it.
Brandon,
I think lots of kids figure it out too. It’s hard to believe a professor even in Gauss’s time would not have already known the trick or that most kids wouldn’t figure it out.
That said: it’s sometimes difficult to compare what one ‘figures out’ now a days relative to before because to some extent good teaching means material is presented in a way one might be primed to figure something out.
I remember in 5th grade I’d already figured out how to never lose at tick-tack toe (and win if my opponent didn’t know the right way to play). I won a few games then my friend asked me how I could do it. So I told her my strategy. After that, tick-tack toe was boring for both of us and we went out and played. I’m sure other kids figure this out too.
“JimB, that’s one of those stories which has been told about tons of different people, and likely isn’t true for any of them. Or if it is, it’s only by chance that we might associate it with them.”
The story was first told about Gauss in the 1856 biography by waltershausen, as I recall. its been a while.
It is described as a story Gauss told about himself. waltershausen was his close friend.
An english translation by Gauss’ great grand daughter of the biography resides in the stanford library if you cant read german
In The story as told by gauss he did not leave the room. He finished the problem took his slate and place it on the table
muttered ‘there it lies’ and then returned to his seat. the oother students finished and put their slates on top of his. At the end of the hour the teacher turned the whole stack over and started by reading gauss’ answer first
I don’t know what reminded JimB of Gauss, but it might be worth mentioning this other anecdote:
I’ll always be amused by what Steven Mosher deigns to respond to and what he can’t be bothered to address. More amusing, however, is his implication this story has credibility due to how it was written about. All that shows is the story was attributed to Gauss some time back, and hasn’t been widely questioned since. That’s hardly a compelling point.
Also of note is the story told in the original source referred to by Mosher doesn’t even match the story told nowadays. It didn’t contain any mention of a number of details, including rather important ones regarding mathematical details. Namely, it doesn’t say what numbers Gauss supposedly managed to sum (there’s no mention of 1-100), nor does it say anything about how Gauss supposedly managed to solve the problem.
Sure, Gauss may have told a story to people while he was living similiar to this one. And sure, one of those people may have written the story down as he remembered it in a biography. But that’s all there is to suggeset this ever happened. Gauss supposedly told people a story about his childhood, and one of them wrote it down. That’s not evidence.
And it ignores the fact the stories about this have contradicted one another many times throughout time. As well as the fact the original story is highly implausible.
So… yeah. Stories morph over time, and they may start with some kernel of truth, but this is one of those ones where if there was any truth to it, it’s so buried in a sea of misinformation nobody could possibly hope to find that truth.
Brandon– Could have happened.
Mosher– How old were the students supposed to be? If they were college students (which professor suggests) the rest were a bunch of dim bulbs. If they were 4th graders, it’s explicable.
The Gauss anecdote as written by Sartorius (Baron von Waltershausen), has Gauss being only 7 years old.
JimB:
You can also do it by adding (1 + 100) + (2 + 99) + (3 + 98) + … (50 + 51)
There are 50 such pairs of 101 so 50 * 101 = 5050
Seems pretty plausible that a gifted student could have worked this out.
Carrick,
I would suspect the purpose of the assignment was for the students to be encouraged to start thinking about doing math at a somewhat higher level. It’s possibly a bit early for 2nd graders, but I think this sort of stuff was discussed in 4th grade and pretty much everyone was expected to grasp it.
( I don’t have firm recollections of what was covered in 2nd grade, but I do remember 4th because we had a really mean, mean teacher. One who made one of the kids who was bad at flashcards stand in the garbage can as a penalty for losing to much. She did cover stuff like this, and stuff like predicting the next number in a series and so on.)
Piomas up a tick so should be an encouraging re freeze in the Arctic.
One of the earliest re freezes of the last 33 years.
Go ice.
Carrick –
Yes, or perhaps some would use the geometric equivalent. That is, take a stair-step shape, with each step being 1 unit high, the top step is 1 unit wide, increasing until the bottom step is 100 units wide. Its area is 1+2+…+100. Pair it with an equal shape rotated 180 degrees to form a rectangle 100 units high and 101 wide. From that, it’s easy to see that the original sum is 100*101/2.
Sometimes it’s easier for me to “get” something when presented as a visual demonstration.
lucia, HaroldW, I found a piece discussing why the story is, at best, an unreliable one. It’s pretty good. I recommend it to anyone interested in the issue.
There are a number of reasons I called the story implausible. Age has nothing to do with it. It has more to do with details like the story claiming there were ~100 students, each of whom had to write their answer on a slate which they then stacked one on top of another. That was an important detail as Gauss’s slate was on bottom and sat there long before others turned theirs in, so when the teacher turned over the pile of slates, Gauss’s was on top and was the first he read.
It’s difficult to imagine people stacking 100 slates one on top of another than a teacher turning them over. There are a number of other problems with the story like that which make it seem implausible. For instance, depending on the version of the story, some say the teacher had no idea you could solve the problem that way. If so, how would he have known the right answer? It’s difficult to imagine a teacher sat there and added the numbers 1 through 100 up just so he could ask his students to do the same.
Plus, if you did try to add all 100 numbers together (on a single slate, no less), you’d almost have to see some patterns which would let you shortcut parts of the problem. And what would the teacher have been trying to accomplish? The original version of the story didn’t have him leaving the room or taking a nap. It had him in the classroom the entire time, casting venomous stares at Gauss. I can’t imagine what the purpose of such an assignment might have been.
As far as I can tell, there’s no real reason to believe this ever happened in the way the story is commonly told. Maybe there is some nugget of truth buried deep within the story, but ultimately, it’s just a story. It can be useful and entertaining as a story, but it is never going to be anything more than just a story.
Oh, sorry. I didn’t follow your link HarodlW so I didn’t see you had linked to the same one I had linked to. Great minds think alike, I suppose.
Thanks to JimB, this list is even more incomplete now !
Versions of the Gauss Schoolroom Anecdote
Brandon,
I agree it is difficult to imagine a 100 7 year olds stacking slates one on top of each other. Perhaps Gauss’s memory about the incident was embroidered.
It is hard to believe a teacher wouldn’t know the trick.
Brandon
” More amusing, however, is his implication this story has credibility due to how it was written about. All that shows is the story was attributed to Gauss some time back, and hasn’t been widely questioned since. That’s hardly a compelling point.”
HUH? implication of credibility? Where?
I gave folks a pointer to the original source.
I make no judgement on the veracity. One cant.
As I said, the story appears in a biography written by a FRIEND.
And the friend recounts the story Gauss is reputed to have told
in his old age.
So
1. you dont know if his friend recounted the story right
2. you dont know if gauss recounted the story right.
The main point was that in the Ur text, gauss did not leave the room as was asserted above, but he stayed, according to the Ur text.
There is a version on the internet of the English translation
https://ia800306.us.archive.org/18/items/gauss00waltgoog/gauss00waltgoog.pdf
However, I would suggest that folks should read the original german if they want the real Ur text.
next up, his teacher’s diary?
I would like to discuss a different educational issue — schools’ regulating the snacks that children can eat. Last year when my daughter was in the 3rd grade, the school wouldn’t let the third graders eat sweets. My daughter (and she said her classmates) was always hungry at school and came home hungry. She refused to eat fruits and the other foods that the school permitted.
This year she is in the fourth grade in a different (and better district), and the school has the same policy. My 14 year-old son has always eaten junk foods and about 2 years ago he started to lift weights and regularly do aerobic exercises. I do both myself, but he started the lifting and aerobic exercises on his own with no direct encouragement by me other than fruitlessly encouraging exercise when he was about 8. Right now there is not an ounce of fat on his body. I rarely eat junk foods, but since my cooking is so bad I don’t mind my children eating junk foods. My daughter is not the least bit heavy and is healthy.
When I asked the teacher if my daughter could, for instance, bring chocolate chip cookies she reacted in horror. She said that might hurt kids with diabetes. (As far as I know there are no kids with diabetes in the class, but even then a rule could be followed that under no circumstances are the children to share their snacks.) Also, she said that my daughter would feel out of place if one rule was applied to her but not the other children.
…..
I have a number of problems with this policy. 1. Who is the teacher to tell me what my child can eat? 2. The policy smacks of intolerance and small minded prohibition. 3. As far as I know, Michele Obama’s attempt to regulate what children eat has been a failure. 4. I understand my children better than the teacher and my policies should govern as opposed to that of the teacher. 5. The teacher and my daughter’s previous teacher are not in particularly good shape — they have no business telling others how to eat.
…..
Am wondering what others think.
JD
JD Ohio,
I think this is an outrage. Surely you, of all people, could devise a letter to the board of education which would put an end to this nonsense.
It seems likely that you are compensating at home for the damage being done to your children at school, but maybe you need to send them to a better school.
If your kids realize that their teachers are a bit screwy will their education suffer?
I can’t remember thinking a teacher was a bit off until I got to sociology in high school. I hung on “Sociology is a science equal in stature to chemistry and physics.” I couldn’t understand why it mattered so much to her.
I’m sure most who read here would agree that the realization that some of what you are told in all sincerity will be nonsense probably shows up in high school. And further, that you would not be a complete person without having realized this. It might also be that having your cookies denied you at too early an age could be counter-productive. When is the best time for doubt to arise? Maybe when you realize that Santa Claus is a “silly story” as my five year old daughter, now the professor, told me.
JD, I think you face a very serious problem here, and wish you well with it.
As if the climate catastrophe wasn’t enough. Now they want our cookies.
Lucia,
it might be good to increase the editing opportunity a bit. As it is, it’s not long enough to figure out how to fix really bad sentences like the one with ‘realization’ above.
john
JD Ohio,
You obviously didn’t get Hillary’s “it takes a village” message…. the “village” should control most everything, even what people can eat. The left focuses always on public control of private activities, and limiting parental choices for their kid’s foods is just the logical extension of that philosophy. The left is, and has always been, focused on instituting long term “social progress”, and that means brainwashing children via education. Heck, when I was in 6th grade, my leftist teacher used to read books to the class which consistently had the same leftist messages… equality of outcomes, not equality of opportunity…. “fairness” in all things…. “suppression” of the poor by the rich, etc. The bizarre regulation of speech on college campuses is but one result; virtual “litmus tests” of leftist purity in the hiring of professors and administrators is another.
.
Seems to me you have three options: 1) explain to your kids that you disagree with the policy, but can do nothing about it, 2) pull your kids out of public school and send them to private schools, 3) fight the policy via lawsuit. If I were you, I would also make sure your kids understand that the schools’ efforts to control what they eat is the result of a certain POV about the proper role and scope of government, and the extent of control that POV logically leads to is without limit.
SteveF,
Would you do choice 1?
jferguson,
The schools are no doubt expecting people to take option 1, but no, I would probably explain to the principal that the food policy had to change, or option 3 was likely. If I were JD, I might also mention in passing that I was a lawyer.
JD’s problem might be that he would have to get another attorney to handle it.
I’ve just remembered that in the late ’40s, in primary school in Minneapolis, we children were given milk in little bottles at lunch each day. We provided the lunch, but someone subsidized the milk. Maybe that isn’t all that different from someone denying the cookies, but IIRC we weren’t forced to drink it.
I’d add that it’s hard to believe that JD’s school authority wouldn’t respond appropriately to a visit from him or even better, JD and some other parents.
jferguson,
In the 1950’s I also received a small bottle of milk for lunch… but we had to pay for it. I don’t know if the milk was subsidized, but receiving it was optional. I don’t think subsidized milk is at all similar to prohibitions on the kind of foods parents can send their kids to school with. Lots of kids had thermos bottles in their lunch boxes, and brought their own drinks. Of course, grade schoolers tended to break thermos bottles with some regularity… so buying the milk was probably a reasonable option.
Maybe subsidized arugula?
JD Ohio,
Oh heavens. My sister has celiac. Would the teacher have banned all gluten lest she feel bad watching other kids eat sandwiches?
I have a friend who is allergic to whey– which would knock milk out.
This one makes me snort. The obvious answer to that is her theory about how your daughter would feel about being allowed to eat chocolate chip cookies is inaccurate. Your daughter would feel just fine. (Perhaps other kids would be jealous but that wouldn’t necessarily make your daughter feel bad. The real problem is other parents would ask for their children to get to eat cookies too.)
I think schools setting up arbitrary rules about ‘junk’ foods is ridiculous. It should not be their place.
I understand the ‘no sharing’ for classes with kids less than 12 years old because of food allergies. Some kids are so allergic to peanuts they will go into shock, and very young kids might not be responsible enough to monitor everything. So that sort of protection is not arbitrary.
I’m a bit dubious of rules that don’t let kids eat peanut butter sandwiches let a few peanut molecules get into the air. My thought is if a kid is that allergic, perhaps they should be homeschooled. But I could be dissuaded from that final view.
We also got bottles of milk. We paid for it, but I think there was a potential for a subsidy. Our government still has a subsidized lunch program for kids who qualify.
jferguson,
I increased edit time to 15 minutes.
JD Ohio,
As you are a lawyer, you might want to look into the legality of what they are doing. Quick blogging leads to a page with states creating laws– but the issue is what does the law really permit or require schools to do?
This is obviously a blurb. But presumably whatever this statute is, it says something. I wouldn’t be surprised if it tells schools what to do about foods they provide, but they might be going overboard.
In anycase, it seems ridiculous to me that someone can prevent a kids from bringing chocolate chip cookies, m&m’s, twinkies (if they still exist) and ho-hos to school.
JD,
The feds evidently have a food law:
http://www.ptotoday.com/pto-today-articles/article/311-school-snacks-under-attack
[…]
Whatever the mandated “wellness” policy required of the school might be, it doesn’t seem to require the school to prohibit kids bringing their own cookies. So you may need to get some other parents on board with you to get rid of this utterly stoooooopid policy. (I’m sure it can be done.)
I checked with Mom, who at 97 can remember that I was sent to school with a quarter on Mondays, so I did buy the milk. And likely it wasn’t subsidized because the little bottles at the Art Institute also cost $0.05 which I wouldn’t think had been subsidized.
The prevalence of all of these allergies seems much greater than when I was a kid.
Thanks for the 15 minutes.
Re: Comment #138714
Thanks, Steve! I was just lucky to be in a position to run the test, and it was useful information for me too. The hardest part was keeping exactly the original wording; I would have liked to make it more “grammatical,” or at least more natural, but I restrained myself.
That said, we’ll never know how my students were different from the other group, but I guess that’s the thing with “false positives,” as Lucia would put it: you never know why they happen. About all we know now is that the original result is not generalizable.
All of which reminds me of the story about the social scientist, the natural scientist, and the mathematician, who are traveling abroad when they see a sheep on a field, and the social scientist says–
“Hey, how about that? Sheep are black in this country!”
To which the natural scientist replies–
“Oh come on, you can’t say that! It’s the first sheep we’ve seen! All we know is that they have some black sheep over here.”
And the mathematician retorts:
“You do not even know that! All you know is that in this country there is at least one sheep, at least one of whose sides is black!”
jferguson,
The allergies may have increased. People tendency to self-diagnose when they don’t likely has also increased.
My sister had celiac before ‘gluten intolerance’ became fashionable. She was diagnosed in infancy by means of an inspection of tissue (colon? Large intestine? Something.)
She is thrilled “gluten intolerance” is fashionable, but also suspicious that many of these people are just part of the latest food fad. Bit from her view, who cares? It means there are ton of new gluten free foods she can pick from whereas before quite a bit of investigation involving turning over labels and reading every ingredient in details was required to find things. (Modified food starch? It’s in tons of stuff. Could be wheat. Or might be potato. Or something else. One can’t know. But if it says “gluten free” it won’t be wheat.)
julio,
Yes. I know the Regional Office of Education exam continues to be given in some manner. It would be interesting to be able to get the data about answers. (Heck, it might be FOIA-able). I doubt one could get the questions though. FOIA exempts tests themselves– but not other stuff. Which is an interesting feature.
But that means that unless someone with access write another MS thesis with the Q’s in the appendix and the results, and the next test includes this exact question, I can’t learn whether it repeats itself with a similar batch of high-school students.
It was great you repeated with yours.
(FWIW: I do hope the teachers write fresh questions. If nothing else: these are in the appendix of a MS thesis. And even if it weren’t, over multiple years, I’m sure the temptation to ‘teach to this specific test would be too much for at least some fraction of teachers. Some would assign these specific questions or use them in class or so on. If the do write fresh questions, it would be great if they published them the way the NY Regents does. But… doubt they will.)
Lucia: “As you are a lawyer, you might want to look into the legality of what they are doing.”
Thanks for your quick research. It is interesting. However, I suspect that it mostly deals with what the school sells to the students — not what snacks parents give to their children. I could be wrong however.
……
What really frosts me is how totally oblivious the teachers are to my rights as a parent. In fact, the school prides itself on its commitment to diversity (in fairness, there are a lot of Indians & a fair amount of Chinese), but the teachers have no clue that a one-size fits all policy conflicts with their stated desire for diversity. I suspect that a substantial part of the hysteria associated with sweet snacks is the idea that sugar causes hyperactivity in children. However, the scientific case for that is weak. See http://www.webmd.com/parenting/features/busting-sugar-hyperactivity-myth I also suspect that the teachers think that sugar will make controlling the class harder and they then develop bs excuses to justify their policy.
…..
To show you the extent of the support for a nanny state, an urban planning professor (whose wife teaches elementary school) who I used to respect commented to me that allowing parents to give sweet snacks to their children was like allowing children to smoke — According to him one child eating snacks contaminated the other children in a manner similar to second hand smoke.
……
One of the reasons I posted it here was it supplies background as to how incompetent people, such as Mann & Holdren, survive in academia — the standards are low and the professors live in a simplistic bubble, much like my daughter’s teachers.
JD
JFerguson “JD Ohio, I think this is an outrage. Surely you, of all people, could devise a letter to the board of education which would put an end to this nonsense.”
Generally, the school district is very good and my son in the first month is doing exceptionally well with the increased exposure to other smart students. I simply have to absorb the idiocy and be happy for the 85% of the school that is very good. Particularly, since I have only been in the district for 3 weeks, I don’t want to start a ruckus. The teachers think they are sophisticated, knowledgeable thinkers, and if I attack that self-perception (which would be inevitable between their lack of respect for parental rights and their lack of scientific and legal knowledge), it would only make them defensive and would make matters more difficult for my daughter.
…..
To show you how stupid things have gotten, the kids can’t even have cake or food on their birthday to class. They are supposed to bring games to share with the other kids.
…..
I would add that when my son was attending the 6th grade, his teacher, prior to the school year, stated that “it takes a village to raise a child,” and I almost vomited. It turned out later that she was extremely complimentary to my son (“I LOVE having M* in my class”), and she helped him a lot notwithstanding her goofy comment. So, I would rather see how matters play out before I would directly challenge the policy at this point in time. We obviously don’t live in a perfect world.
JD
JD Ohio,
I suspect it applies only to what schools sell too. But I also suspect someone at the school will make some sort of claim based on one of these laws. After all: that’s typically what bureaucracies do– even when it’s their “interpretation” of what they are permitted or mandated with that “interpretation” being little more than “what some prefer anyway”.
It seems to me the Ohio specific and Federal laws are ones you’ll want to be familiar with.
Well… yes. And we see oddities like parents suing school districts who force kids into the “English as a Second Language” track despite the fact the kid’s only language is English:
http://blogs.edweek.org/edweek/learning-the-language/2007/10/parents_sue_over_placement_in.html
If that’s their argument then they should make it. Once made it can be discussed.
In fact one of the reasons to push back– possibly with other parents– is to force these silly arguments to be aired in public. Then you can present the evidence showing the idea sugar is causing misbehavior is just a superstition.
If it turns out that most parents are on the side of the school board, you might end up losing (or at least not being able to win in a time frame that make any difference to your kids). But I suspect it will turn out that most parents think these rules are silly. Possibly a few parents did push for them in an earlier period and the others weren’t paying attention. But now the silly rule is in force, and it will require push back to get rid of it.
How? I know that snacks are delicious– which is precisely why people like them. And kid2 seeing kid1 eat something delicious will likely make kid2 want to eat the delicious thing too. And when kid2 tastes it, he will discover it’s delicious. But that hardly makes it like “smoke”.
But you know, we were allowed cookies in school. Our lunches typically contained some cookies– and sandwiches and carrots etc. Parent’s didn’t fill the bags up with 2 dozen oreos and then just stop there.
JD Ohio,
I think your approach to this is the right one, not rocking a boat that otherwise is providing a good ride.
I was complemented by spouse after I kept quiet through a long exegesis about how spot-on Naomi Klein was in her latest book at a dinner hosted by very dear friends. The scars on my tongue can still be seen, not that you’d want to.
Despite this, there might be some subtle non-threatening way to inoculate the powers-that-be with the notion that their dietary restrictions might possibly be a bit extreme.
Good luck with all of this.
JDOhio
Sadly, I think this is often the only wise course with public schools. The are likely wrong, but you might need to pick your battles.
Sadly even though many parents may agree with you — and you are all likely right– there is probably no pre-existing mechanism to push without risking fall out to your own kids. And the latter is too dear a price even for those who would be willing to risk fall out on themselves.
Often true.
To some extent, the situation you describe is precisely the reason I like the notion of introducing choice — through vouchers or charters– to public schools. Do private schools sometimes have odd rules? Sure. But when some degree of choice is present, parents can ‘vote’ against some stupid rules by moving their kids, and that does grant some power against schools making zillions of stupid arbitrary rules.
On this
As with all platitudes, there is some truth to this phrase. It is wonderful to live in a neighborhood where lots of parents know each other and interact with each others kids (in a positive way.) It is wonderful and necessary for kids to play and interact with other kids. It is wonderful that an adult with a unique skill set (soccer? cooking?) share that with many children and another adult shares their skill (sewing? Algebra?)
So yes, “It takes a village”.
The problem is that it is often used to justify why parents choices should be over ruled and some sort of “community” philosophy be imposed.
Lucia: “As with all platitudes, there is some truth to this phrase. It is wonderful to live in a neighborhood where lots of parents know each other and interact with each others kids…..”
…..
You make as good a defense of that phrase as is possible. However, in the situations that you describe, I would say that it is good for a child to interact with other children and adults. I wouldn’t say that the village is “raising” the child. I am responsible for my children, and it is my duty to raise the child in a way that he gets along with other children and adults. To me, the village has little to do with raising my children; it is merely a good, enriching environment for children.
JD
I think part of the concern arises from national concerns about childhood obesity rates and pre-diabetes in adolescents.
Pediatricians can be found advising moms worried about their kids being underweight that it is actually the opposite issue that is of concern.
So, on the CDC website you find this:
RB “I think part of the concern arises from national concerns about childhood obesity rates and pre-diabetes in adolescents.”
……
I agree.
CDC “Schools play a particularly critical role by establishing a safe and supportive environment with policies and practices that support healthy behaviors.”
……
I question whether this is true (this is the job of the parents), but even by its own terms, the CDC is talking about a supportive environment, not mandatory rules overriding parental choices.
JD
JD Ohio
I don’t disagree with you.
The trouble with platitudes is they are glib and one needs to decide “what it means”. If all “it takes a village means” is that you would be unwise to lock your kid up in a tower (like Rapunzel) and keep her away from all and sundry, it’s ‘correct’. If it means ‘some self appointed group in the village (e.g. school teachers) claim the right to “raise” your kids and trump your decisions”, the platitude is wrong. When intoned, it often means the latter with the person intoning the platitude means ‘You, mere parent, must given into my demands because, to coin a phrase, “Le Village, c’est moi!” “
JFerguson: “I was complemented by spouse after I kept quiet through a long exegesis about how spot-on Naomi Klein was in her latest book at a dinner hosted by very dear friends.”
You have my sympathy. Klein is particularly pompous, ignorant and judgmental. I would have simply stated I disagree with her for a couple of reasons and that I doubted that there was any way to resolve the issue — and, then avoided talking about it.
JD
JD Ohio,
I agree nutrition is a mostly the parents job. I do support schools not having soft-drinks in vending machines, and making their own decisions about what the school cafeteria will make available. Schools have always been able to do that– and my grade school did not have soft-drinks in vending machines.
My grade school did not serve food in the cafeteria– but they did have a private contractor selling things like apples, sandwiches and such.
No matter what the school provides or what concern they might have about obesity or diabetes, kids should be able to bring their own lunches and eat those. If those lunches contain oreo, so be it.
For what it’s worth: We had home ec. Balanced diet was discussed in home ec. We also baked cookies. That’s a good early cooking project as it’s easier than cake. Whole wheat yeast bread is impossible– there is not enough time for it to rise.
Lucia ” I do support schools not having soft-drinks in vending machines, and making their own decisions about what the school cafeteria will make available.”
I don’t have a problem with soft drinks in the schools, but it is up to the schools to decide what they will sell. I generally agree with schools greatly reducing the amount of junk food that they sell so long as they have food the children will eat. However, it is not the job of the schools to tell parents what snacks their children can bring to school.
JD
JD–
I think our positions are either similar or identical. Schools get to decide what to sell, and if they don’t stock soft drinks, I have no problem with that.
When I was a kid, most grade and middle schools did not stock food because most kids lived in walking distance and many walked home for lunch. The ones who did not walk home for lunch brought sack lunches. We brought whatever we liked.
Many of us would have loved the school to ban one of the little boy’s choice of sardines owning to the smell. But he wanted sardines; he ate sardines.
Lucia,
“Many of us would have loved the school to ban one of the little boy’s choice of sardines owning to the smell. But he wanted sardines; he ate sardines.”
.
I never encountered a grade schooler who like sardines, but I remember lots of kids who were quite fond of egg salad sandwiches. Between the time that mom or dad made the sanwich and Junior got to eat it at lunch, the paper bag was beginning to resemble low tide at the seashore….. lots of emitted hydrogen sulfide. People would say “Oooo what is that smell?”, but that never seem to inhibit the kids who liked egg salad.
.
JD Ohio,
You’re a lot more tolerant than I am… some rant about Naomi Kline would be impossible for me to resist commenting on rather harshly (words like fool, psychosis, and id!ot almost certainly would be involved).
SteveF,
I was asked later in the evening after the offending guests had left, and my word for her was crackpot. I suppose that might beg the question as to whether only men can be crackpots. I don’t know.
JD Ohio,
Would your school allow orange juice or apple juice? There’s lots of sugar in those too. 8 ounces of apple juice averages 113 calories, mostly from 24g sugars. OJ averages 112 calories, mostly from 20.9g sugars. One medium size, 16g, chocolate chip cookie is rated at 78 calories, but only 9g total carbohydrates.
Possibly worse, there’s lots of fructose in apple juice and more fructose than glucose in orange juice. Fructose can only be metabolized in the liver, much like ethanol. There is some research that suggests the increased incidence of pediatric nonalcoholic fatty liver disease may not just be due to childhood obesity but be caused by drinking too much fruit juice and drinks sweetened by high fructose corn syrup, especially when the ratio of fructose to glucose exceeds one. Apple juice has an F/G ratio of 2.7 and OJ is 1.2.
Which brings up the subject of the recent ‘epidemic’ of anti-hunger drives. If no child goes to bed hungry, will we have even more obese children? Dunno, but it seems likely to me.
jferguson,
I believe women can aspire to be crackpots and achieve that status on the same basis as men.
DeWitte/JD,
I wonder if “fruit roll ups” are allowed:
https://en.wikipedia.org/wiki/Fruit_Roll-Ups
These are on par with ‘Starbursts’ and those old timey fruit flavored slices:
http://www.hoffmans.com/Barton-s-Fruit-Flavored-Slices-p/11056.htm
You know… with respect to JD’s issue it might be useful if he obtained a written policy of what is allowed.
Are granola bars allowed? If yes, homemade granola bars? If yes, can you add melted marshmellows to the recipe? Can you add rice krispies to the oat/nut/honey mix? Is that any different from home made oatmeal cookies? (Is everything ok as long as it’s a ‘bar’ and not a “drop cookie”? )
Fruit leathers? If yes, homemade ones?
Dried fruit? If yes, candied dried fruit?
Hot cross buns and/ or panetonne? With icing? If yes can we go for coffee cake? If yes, how about pound cake? Dare we move to sponge cake?
Where, precisely, is the line? Do they have a list of items?
Clearly, we need a list. And a print out of the formal policy. I say FOIA it to spare JD the risk his child will be exposed!! Heh.
Lucia: “You know… with respect to JD’s issue it might be useful if he obtained a written policy of what is allowed.”
…..
Interesting issue. Both of my children like dried and salted seaweed, which is a Chinese snack similar to potato chips. May try to get it in, although my daughter has sort of bought in to the propaganda and doesn’t want to take traditional snacks in now. However, she still, pretty much, doesn’t want to take fruit to school. Unfortunately, she prefers to be hungry and eat when she gets home.
JD
Meant to add that the seaweed would be a nice challenge to their supposed commitment to diversity.
JD
Some of you may remember the bumper sticker often seen on pickup trucks in the ’90s: “It takes a village to raise an idiot”
JD,
I’m still sort of curious to find out whether they have a formal written policy.
Lucia: “I’m still sort of curious to find out whether they have a formal written policy”
None that I could find. There is a district wellness committee and it states: “The committee’s goal is to encourage teachers, students and staff to lead healthy lives while providing tools and support to the XXX Community Schools system. The hope is for XXX to be a healthy school environment in which the students are able to engage in educational experiences that will equip XXX students with critical skills that promote the intellectual, social, emotional, and physical growth needed to be highly successful in life.” Nothing more specific.
In fairness to the district, some parents are on the committee.
JD
JD,
Hmmm… although it may be counter productive to say this to the teacher, it seems to me teachers dictating food choices violates the stated policy.
If they wish to “XXX students with critical skills that promote the intellectual, social, emotional, and physical growth needed to be highly successful in life.” teachers need to permit students to collect information and present argued reasons for why their own choices are acceptable. Moreover, the teachers need to set their own preferences aside and often accept the argument the student advances. For example: if a kid presents this argumet
(a) I am super active in ‘xxx’ sport.
(b) I have consulted with my physician and he says it’s ok for me to eat jelly beans during the day.
(c) I eat a mostly balanced diet during the day so
(d) it’s fine if I eat jelly beans (even if the teacher prefers I eat fruit.) and moreover,
(e) those kids who are diabetes prone need to develop the ability to self regulate their diet and exercise rather than insist I be unnecessarily regulated to for their benefit and finally
(f) regulating my food consumption for the hypothetical benefit of these other kids is likely futile anyway.
I should think that shows they have developed “critical thinking skills”. The teacher refusing to accept it would suppress the development of said skills, and thus be violating the code!
But I suspect the teachers would consider this “smart alecky”. Because, rather often, teachers actually don’t like critical thinking skills as that will often suppress teacher authority. The “critical thinking skill” many teachers prefer is:
(a) The teacher has a rule s/he likes.
(b) If I present the teacher a good argument why their preferred rule is counter productive, the teacher may retaliate against me.
(c) In addition, they won’t life the silly rule anyway.
(d) To succeed in life, I should indulge the teachers irrational behavior even if it causes me some inconvenience.
This is, of course, also a ‘critical thinking skill’, and it is a lesson many people learn (and need to learn.) That said, it’s thought pattern teachers prefer to admit they wish to instill in students.
To get back more to the subject of the original post, here are my experiences with testing with respect to my now 14-year-old son, M.
…..
When he was in the fourth grade he only scored in the top 23% in Ohio for reading and writing. This was not nearly good enough for him. I had been hands off because he enjoyed school and I didn’t want it to become work.
…..
After I got the scores, I implemented a bonus system for both grades and scores. In the next year, he got all “A”s and scored in the top 5.1% on his Ohio testing scores. Other than the bonus system, I did virtually nothing special during the year — the money really motivated him, and in fact, he about broke the bank and earned $300. If he had scored .1% higher, he would have earned another $300.
……
My basic point being that although the test scores don’t come close to measuring everything of importance, they do provide a useful marker. In this instance, I was alerted to the fact that M’s performance was substantially below his abilities (he had scored one point below gifted in the First Grade.) and was able to substantially improve his performance. I am glad that he was tested.
…..
I would also add that in my experience, virtually everyone who tests well is academically smart. However, not everyone who tests poorly is academically deficient.
JD