In a prior post, I discussed the problem of college students having poor physical intuition both before and after taking university physics. In another post I will discuss David Hestenes' proposed solution to this problem, but first I wanted to provide some examples of what kinds of errors these students are making.

Hestenes et. al. developed a test called the Force Concept Inventory, but they've embargoed online versions of it (it's available to you if you can prove you are a physics teacher or professor.) The following questions are similar to questions on the FCI, but I've respected their embargo, and adapted them from similar questions. Their own questions are taken from other papers as well, as the literature is filled with examples of how physics students don't understand basic mechanics.

Here are two test questions, the first adapted from Students' preconceptions in introductory mechanics, J. Clement, Am. J. Phys. 50(1), Jan. 1982, and the second adapated from Rule-governed approaches to physics--Newton's third Law, D. P. Maloney, Phys. Educ., Vol 19, 1984. Note that my adaptations haven't been tested on thousands, so they may not be as crystal clear as I hope...

1. A ball is tossed from point A straight up into the air and caught at point E. It reaches its maximum height at point C, and points B and D are at the same height above the ground. IGNORE AIR RESISTANCE.

Try to imagine that "up" on the page is the z direction, and that the horizontal direction is x. No motion is occurring in x.

a. Draw with one or more arrows showing the direction of each force acting on the ball when it is at point B.

b. Is the speed of the ball at point B greater, lesser, or the same as at point A?

c. Is the speed of the ball at point D greater, lesser, or the same as at point B?

2. Consider the following diagrams of two blocks on a frictionless surface and answer the following questions. Ignore air resistance.

a. Assuming both blocks are at rest:

How does the force that A exerts on B compare to the force B exerts on A, if A and B are equal in mass?

How does the force that A exerts on B compare to the force B exerts on A, if A and B have different masses?

b. Assuming both blocks are moving to the right with velocity v:

How does the force that A exerts on B compare to the force B exerts on A, if A and B are equal in mass?

How does the force that A exerts on B compare to the force B exerts on A, if A and B have different masses?

c. Assuming both blocks are moving to the left with constant acceleration a:

How does the force that A exerts on B compare to the force B exerts on A, if A and B are equal in mass?

How does the force that A exerts on B compare to the force B exerts on A, if A and B have different masses?

---

While the actual test employed some randomization and various other elements (set values for the masses, e.g.) the results for this last question were that less than 10% of experienced students (those who had taken college physics) got the right answer using the right reasoning, and 0% of the novice students (those who had not yet taken college physics) got the right answer.

UPDATE: See, I told you I hadn't vetted the questions. Updates are above in BOLD. College Physics above means college students taking a standard first term mechanics course. In the test they did with question c, the students were junior or senior year chemistry students who were required to take a 1 year physics course as a prereq for their major. The author was at Creighton University, so presumably these students were at Creighton University as well. The author points out that at least half a dozen of these students who got these wrong had also taken the MCAT, and possibly had studied physics AGAIN as well.

SECOND UPDATE:

how about some answers?

Problem 1: a. There's one force on the ball. it's Gravity, pointed down. b. The speed of the ball at B is less than the speed at A. The speed drops continuously until we reach C, in fact. c. The speed of the ball at B is the same as at D. In fact, the speed of the ball at any height X above the initial A is the same whether going up or going down. The ball speed depends only on height above our origin.

Problem 2: the answer to all problems is the same: the force exerted by A on B is the same as the forced exerted by B on A.

Physics Education and Failures in Conceptual Understanding

Fixing Physics Education: Modeling Instruction

Physics Education Continued

More Modeling Instruction: Techniques

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## 43 comments:

How is "college physics" defined -- e.g. AP Physics?

But I realise you're just excerpting from the actual publication -- how many people got it right for the introductory section you quoted? (Rather than the whole test.) It should be much higher than 10%, right?

Are students allowed to use math to solve these problems, or are they just supposed to answer them based on what they think? Is concetual understanding something that always comes before math or does it derive from math? For many things I do, my understandings come from studying the math. I create equations that model a problem and then I analyze the equations to see whether they reflect reality.

I guess I don't understand the whole premise. What, exactly, does this test tell you? Do some people feel that the math will somehow be easier if there is a better understanding of the concepts? Is this about some sort of concepts first philosophy? What, exactly, is the problem?

I think concepts are good for problem-solving...

In engineering you're going to have situations where it's more than just solving a series of equations to find your desired specifications.

For example, if one didn't realise that (neglecting air resistance) the ball at points B and D has the same velocity, then one would be less likely to make the insight that intentionally speeding into a planet's gravitational field can help you get away from it.

(What equations would one plug in in order to obtain the idea?)

"What equations would one plug in in order to obtain the idea?"

The fundamental governing equations.

I don't mean to be a pain here, but I still don't understand the problem. In engineering, I don't just turn the crank with equations to come out with an answer. I can study them and try to apply them to different problems.

This seems to be all about mental or conceptual models that will help form a better framework for understanding. That's fine, but I don't see how that will eventually tie in with a mathematical understanding. There are too many times where it has been the math that has provided me with the understanding.

There's certainty no reason to do math to answer any of those questions assuming no air resistance, etc. And in fact with the blocks if you're planning on using some kind of force equations that indicates to me that you don't understand the problem.

I heard Hestenes give a talk at the University of Michigan back in the 90s -- just as I was finishing my Physics major and about to head off to study Theoretical Physics at Cambridge -- and he spoke about the Force Concept Inventory then. As far as I can recall his point then was that a very large proportion of students who successfully complete college Physics courses with good grades

nevertheless continue to fail at basic conceptual understandingas measured by the FCI.My interpretation of that phenomenon goes like this: in the context of doing a homework problem, or doing an exam question, students can correctly draw a force diagram, set up the equations, and solve the equations for the indicated variables. Thus, they seem to "know" Physics.

But out of that context, in a problem that lacks details (none of the examples Alison provided have sufficient information in then to set up a meaningful equation) students are unable to correctly make use of the general principles that

supposedlyare the basis of their knowledge.In other words, they have procedural understanding, but not conceptual understanding.

It's disheartening to know that students can learn to solve complicated Physics problems without actually understanding the general principles that supposedly are the basis of those solutions. But it's not really all that surprising.

What's complicated, and probably more controversial, is the question "What do you do about it?" Here there are two obvious knee-jerk responses:

Knee-jerk response (a): De-emphasize the mathematically complicated problems (since they demonstrably aren't helping students learn basic concepts) and focus on "conceptual" physics.

Knee-jerk response (b): Increase the emphasis on mathematically complicated problems (since only the certitude of math can overcome faulty intuition). If it hasn't worked so far, we must not be doing enough of it.

The problem with (a) is that there is an obvious risk of throwing out the baby with the bathwater. I think most people on this blog would view this as "dumbing down" the curriculum, with the predictable result that students end up learning neither procedures nor concepts.

The problem with (b) is that eventually you have to wonder if you're just throwing away good money after bad. If something has consistently failed to work (and Hestenes's findings are extremely robust, having been replicated at many institutions, with many instructors, and thousands of students), why on earth would you think the solution is to do more of it?

Both responses are naive and easy to criticize.

Of course, I don't have any solutions of my own. And I'm very curious to read Hestenes's proposal. And to see whether he has any data that it works.

By the way, just about everything said here about Physics applies equally well to Calculus.

I've seen hundreds of students who can take the equation of a function, differentiate it, set it equal to zero, find the critical points, and tell you where the local extrema are.

But show them a graph of a moving object's velocity, point to a part of the graph that is positive and decreasing, and ask them what the object is doing at that point -- and most of the time they'll tell you "It's going backwards". (Rather than the correct answer, "It's going forwards but slowing down.")

Decades and decades of "traditional" calculus instruction failed to change this. So 20 years ago Reform Calculus was born. And two decades later, students still do the same thing -- only now they can't find the local extrema either.

(a) is what we expect from K-12 education, and, apparently, with Reform Calculus in college. You have to watch out for K-12 creeping into K-16.

(b) won't work unless you really know what's going on. It sounds too much like guess and check.

That's my question. What, exactly, is the problem? Given that I had 7 1/2 years of full-time classes in engineering in two different fields, I have my own idea of the problem, but it could be quite different than what they are talking about here.

SteveH,

You ask,

"What, exactly, is the problem?"I can think of at least three possible interpretations of that question:"What is the phenomenon?"

"Why is the phenomenon undesirable?"

"What is the cause of the phenomenon, and what can be done to change it?"

Alison has done a pretty good job of answering the first question: the phenomenon is that a large number of students who can do the math are nevertheless incapable of correctly answering conceptual questions.

As to the second question -- maybe it's just a matter of taste and opinion, but I think most of us would agree that it's a sad state of affairs.

As for the third question -- that is, of course, the big mystery.

I have my own opinion, which I think I can sum up with the word "deconstructivism". That is: before you can start teaching students (or having them discover) what is

right, you have to create carefully-constructed situations in whichthey realizethat their prior misconceptions are wrong.Tellingstudents they are wrong won't do it -- we are all capable of cognitive dissonance. You have to create a situation in which their prior "knowledge" just simply breaks down. Like breaking up the earth before you can plant.Ineffective instruction, whether of the traditional lecture variety or of the group-learning constructivist variety, skips this step.

"... incapable of correctly answering conceptual questions."

This is too vague for me. This could be based on a lot of things. It also seems to imply that there is some sort of required conceptual understanding that is unrelated to the math.

For me, it was never that I had any misconceptions. I just had a lack of content and skills. Even after graduate school (twice), I felt that I had barely scratched the surface. I learned a lot of pieces, but little time was spent on putting them together; little time was spent on understanding how everything derived from fundamental governing equations.

I picked and poked at Bernoulli's principle applied to specific problems, but there was never any time to go back to see it for what it really is. I really wanted this, but there was no time. Could this be done beforehand and still cover the same material? I don't even like to call it conceptual. I would rather call it fundamental; the ability to derive and analyze everything from the governing equations.

Could the problem also be related to low-rigor that creeps up through all of the grades? My old advisor at Michigan says that he can see this dumbing down creep. I'll have to ask him about how he sees the last 35 years of PhD oral exams.

Assumption for problem 1: The graph is a graph of positions z and y, not z and t. This assumption arises from the fact that the ball is "thrown from postion A and caught at position E", which is most easily read as A and E being different spatial positions. This assumption would not be necessary if the scales were labeled.

So, 1-a is straightforward if you decompose the air resistance into a horizontal and vertical component. (Since the ball moves horizontally even though it was "tossed ... straight up into the air", there must be a wind from the left side of the drawing.)

1-b is moderately straightforward assuming a constant wind speed and assuming that the horizontal and vertical scales are the same. If the speed of the ball at B were higher or the same, the ball would move more horizontally before landing than is shown in the picture.

1-c is indeterminate, depending on the strength of the throw, the density of the ball, the speed of the wind, and the density of the air. With a high enough wind speed and a high enough ball density, the speed at D could be higher than it was at A.

In all of 2, the force exerted by A on B must always equal the force exerted by B on A. Assuming no air resistance, that force will always be 0, of course, which is perhaps confusing to some.

Newt's fourth law of successful physics: "When in a hole, don't throw dirt on your shoes (or something like that)".

Since the author didn't say anything about friction in either problem the clever student states up front that they are going to ignore the minimal force due to air friction. If you state it you can't be wrong. If you include it you run the risk of ruining the shine.

In other words, they have procedural understanding, but not conceptual understanding.This is like my biggest problem. I think it summarises the main issue of retention following graduation. I always strive for conceptual memory because concepts are less easy to forget than procedures.

It was something I was always frustrated at. My teachers (or profs) would throw out proofs for really intriguing or amazing statements, but sometimes I would be puzzled and ask, "but *why* is it that way?" But they would be puzzled in return, along the lines of, "because I've proven it that way" but I wouldn't be satisfied. Actually I'm looking for a good conceptual proof of e^(pi*i) = -1 but I can sort of already grasp a vague concept of a very strong link because of the Taylor series and simple harmonic motion.

(y'' = -ky is up there with e^(pi*i) for most mathematical zen-ness ever.)

My teachers' responses to my conceptual puzzledness would usually be along the lines to practice more and do more problems. They always seemed to assume that conceptual understanding would magically spring from enough procedural mastery. The funny thing is I did eventually gain insight from homework, but often not from the homework for the class I was puzzled about!

It's funny because when you do linear algebra, a lot of amazing ("because I said so") facts from calculus suddenly all make sense.

y'' = -ky is magic. Magic magic. The kind of magic where at first the general solution is stated as an unproven fact for you (doing a proof from ordered differential equations is a no-no for the AP Physics level), but yet the kind where everything magically falls into place, without having to solve, simplify or do anything computationally intensive.

I remember in HS during the simple harmonic motion unit, I was flipping to the back of the book to check my answer when I noticed the pre-computed trigonometric tables. Nowadays we generally don't use these because of calculators -- the tables supposedly become obsolete in the classroom, but it was the tables that pointed out what otherwise should have been an obvious fact from doing all those calculus trig identities:

sin(0.0100) ~= 0.0100

sin(0.0200) ~= 0.0200

sin(0.0300) ~= 0.0300

etc.

and I was like, "whoa! How did I not notice this before?"

And I had not noticed this before because students tend to do their work in either radians or degrees, with radians in terms of pi divided by some integer. Never in decimals. (See? The problem of proceduralism.)

But that discovery really had me thinking. Like the fact that at least in that portion of the unit circle, sin(x) seemed to be increasing at a fairly constant rate. Which should have been obvious (it gets more obvious with Taylor series approximations, since the first term of sin(x)'s Taylor equation is x), but at that time, classwork emphasised the non-linear and periodic nature of the sin(x) of course. I mean look at that sexy, curvaceous graph y=sin(x). With all the curved troughs and valleys, a lot of beginner eyes might not initially notice you can make a good linear approximation around the origin.

And here too, I was amazed -- "why should it be linear? The derivative of y=sin x is y=cos x, which too is curvaceous and nonlinear ..." and then it struck me that the cos(x) was indeed 1 around the origin, and then I noticed that the derivative of cos x, that is, (-sin x) is roughly zero around the origin. Duhh... but not so duh. And then I looked at the other part of the unit circle at 90 deg, noting that cos x started from zero but changed linearly, then noting that conveniently that cos''(x) was also roughly zero at that area. Interesting, what a coincidence. On one hand, I was marvelling at how it was so convenient that sin x and cos x somehow managed to have such an amazing correspondence to support such a feat of complementing each other's linear zones, but on the other hand, I was like, "duh, this is obvious, we learnt the proof for the derivative of y = sin x a year ago..."

A gap between conceptual understanding and procedural understanding. See, I had memorised sin'(x) = cos x as a cute and convenient formula, along with then cos'(x) = -sin(x) and the whole cycle, and I had gone ahead and solved my homework problems, but I had not *really* thought about what those equations meant.

Cuz when you go into simple harmonic motion with restoring forces, etc. and where the point of fastest velocity is also the point of zero acceleration, and the point of zero velocity is the point of maximum acceleration, suddenly everything clicked. Like really clicked. I had initially been confused because, "how did circular functions get involved in the linear, one-dimensional motion of a weighted spring?!!" This train of thought led me to consider why a simple thing like a circle (points equidistant from a centre) gave rise to such complex yet elegant relationships, as well as infinite differentiation. (And also led me to especially consider how the circular functions are defined "circularly" to one another...)

A lot of relationships suddenly made sense, in a way that wouldn't have made sense by doing just the calculus homework assigned to us.

So how much more can be understood in the early stages of learning and what can only be known after other pieces of the puzzle have fallen into place? I had many pieces fit together after I had to teach a course in linear algebra many years after I graduated. These pieces weren't strictly conceptual. They were tied directly with the math.

I've always been fascinated with line integrals, but I use them only for specific purposes. How do you go about teaching them in the most general way?

I'm sure that there are many of ways to improve the teaching of early conceptual frameworks, but the devil is in the details. It seems to me that, like the Physics First issues in high school, you have to separate better from more. You can always do better if you require more courses.

Sorry for confusions, will add updates now.

Steve, I will post a bunch of comments to yours later tonight. Let's hold off on the PhysFirst comments til after I get the Hestenes' modeling instruction post up.

To answer your questions about how students can solve these problems,

they can do whatever they like.

Just do part a of the first one: draw the free body diagram. Do it, and if you've done it correctly, you should be able to reason out the answer to the 2nd part of the question: is the speed at point B higher, lower, or the same?

As to the B/D points speed part, my intuition was unclear. But conservation of energy should solve it for you in less than 30 seconds.

But if you DON'T draw the free body diagram correctly, then it's hopeless. You've got the wrong intuition and it led you to the wrong physics and math system: if you draw it wrong, all of your equations will be wrong.

What can be known only after other pieces fall in place? It's hard to do that kind of well controlled study, so it's hard to know. But showing that a different kind of instruction improved FCI scores when traditional instruction doesn't could be evidence that you don't need those other pieces to fall in place if you get the instruction right.

lrg,

I don't have individual scores for the individual questions for the two-mass problem, because they didn't structure the test that way. (They randomized a selection of those two mass problems, 8 in all 1) equal masses at rest, 2) different masses at rest, 3) different masses at rest, 4) different masses, moving with constant v, where the greater mass is the "Cause", 5) where the greater mass is "opposed" to the movement, 6) same masses but with constant acceleration instead of velocity, 7) different masses with const a, but where greater mass is the "cause", 8) same where greater mass is "opposed". They cycled through these questions, and were asked to give reasoning.

So the paper actually gives the percentages for the top "Reasons" for their answers. The main 6 reasons were categorized, and the test results were based on the reasoning. e.g. "mass is the only determination for all states of motion; greater motion exerts greater force" was the rule designation 1a, and 11% of the students used that rule. Other top rules were "at rest forces are equal, but for moving systems, greater mass exerts greater force", etc.

Does the FCI test go beyond kinematics? I mean, cuz this excerpt is basically one chapter of mechanics. No crazy resistor-capacitor-inductor circuits that could account for students tripping up the rest of the test?

The "tossed up into the air" portion was particularly tricky. I would be uncomfortable assuming no air resistance when a problem explicitly mentions the word air...

"But showing that a different kind of instruction improved FCI scores when traditional instruction doesn't could be evidence that you don't need those other pieces to fall in place if you get the instruction right."

This reminds me of one of my very first thoughts about math education when my son was in pre-school. I remember thinking about all of the things I didn't like about my traditional math education. Then I learned that our school used MathLand and claimed that it improved conceptual understanding and problem solving. Since then, I have had to deal with the the idea that all we parents want is what we had when we were growing up and that we are afraid of change. So, concepts and understanding have a lot of baggage and different interpretations. When it comes to K-12 education, the implementation makes all of the difference.

As for conceptual (and usable) models, one could claim that pictures of pie for fractions or bar models work better than what was used before, but is that all there is that's going on? Spending more time on conservation of energy or "follow the energy" is probably a great idea, but not if a high school course spends most of it's class time in groups covering half the material they could and avoiding math because it's not necessary at this stage. Implementation can ruin a great idea. K-12 education is very good at co-opting ideas and terms an turning them into an image of themselves.

Then later on, do these models really make a good connection to a mathematical understanding. Do bar models solve the whole problem? Do they make the transition to an algebraic understanding easier?

I can't see any downside to a better approach to concepts like conservation of energy, but I don't know what that really means in terms of time and coverage of other knowledge. It could mean that it works best only for those who don't have any problems in math. Is there some claim (like in K-8) that a conceptual approach will drive a better learning of the math? At best, it seems that the connection is just on motivation. How many kids get A's in their Project Lead the Way classes, but can't get into a school of engineering because they don't have the math SAT scores?

How far can we trust intuition, even with good models? Students might do better on some tests, but is that all there is to the problem? There is the time, experience, and hard work issue. I've applied fourier transforms to many problems, but I don't fully understand how they fit into some larger picture. I don't think it's just a lack of a proper conceptual model. I just didn't have enough time.

I've been watching this post like a peep show. I didn't want to contribute (didn't want to make a fool of myself I guess) yet I can't take my eyes off of it.

I'm not a natural puzzle solver and I think Allison has cleverly positioned this series of posts as such so I've been in lurker mode. I have been thinking though, and thinking a lot does help you learn. Finally, I think I get it.

It's not about the math at all. In fact strength in math with open ended questions like this can be an impediment. Here's why. It's a trick my wife taught me. She's a buyer and one of her intuitions is (during a negotiation) to keep asking why. She claims it never takes more than 5 whys to uncover the truth of a matter. Usually by the second or third why, sales people are spilling their guts to her.

I'm a bit like Steve in that I initially jumped on this with math tools. Then I started asking why and the math failed me. In the first problem I knew the equations that governed the flight of the ball. I knew the height vs time curve (without friction)was a parabola and I knew parabolas were symmetrical. Why is the flight of a ball symmetrical? The math doesn't tell me why. It just is. To answer why you've got to go deeper into the physics and not with math, with reason and logic only. And the reason and logic doesn't happen unless you really understand what's happening in terms of Newton's laws which are after all just words, not equations.

So hats off to you Allison. You forced me out of my comfort zone. You made me think and now I think I can see where you're going.

One of the things that really made me think more deeply was looking back at the misconceptions that existed before Newton. If you do that, it gives strength to Newton's insights and helps you see why students go down the same road.

The best way to learn anything (math or physics) is to experience and analyze mistakes. Perhaps learning math, or physics, or really anything is to experience the journey that humankind took to get to where we are. Every one of our breakthroughs was, in fact, the shattering of misconceptions and knowing them is essential to avoiding them.

lrg,

The FCI is only the three laws. Kinematics and Dynamics, nothing else. NOTHING ELSE.

It's basically a 6 scenario test with multiple questions for all the scenarios, but they are like the tossed ball question.

If you can get to a library, you can read it in the AM J Phys citation:

I. A. Halloun and D. Hestenes, "The initial knowledge state of college

physics students," Am. J. Phys. 53, 1043 (1985).

Steve,

Anecdotally, I was one of the kids who was exceptional at math, and my physics intuition was worse than Galilean.

Truly, I couldn't have gotten ANY ANY ANY of these questions right by intuition after freshman mechanics, upper division mechanics, and grad mechanics.

And the math failed me, too, because without recognizing the forces at play, I couldn't be sure Id' done the right math on the right system.

It's like the difference between solving the problem right, and solving the right problem. Knowing the math doesn't help you solve the right problem.

Paul,

Kind of you to call my posts clever. It should more properly be chalked up to the need to attend to my 3 yr old, my about-to-be-1 yr old, and my parents and in laws in town for a birthday party... :)

but how about some solutions?

OK

1a. There is but one force on the ball at point B, gravity is the cause and it is a vector pointing down.

1b. The speed (|V|) is lesser at B than A because in the absence of air, energy must be preserved. During the trip up, the ball is converting kinetic energy ( function of v) to potential (function of height). The sum (of the two energies) must be constant as is mass, and (practically) gravity, so v must be decreasing during the entire upward flight. Said another, more intuitive way, the ball is giving up velocity in order to convert kinetic energy to potential energy.

1c. B and D are the same height so they have the same potential energy and the same total energy (must be preserved remember) so the kinetic energies are equal which means the velocities are equal and opposite. Speeds are equal.

With the same logic, air is easily introduced. With air the ball is always giving up energy in the form of heat due to friction. Considering B and D then you still have equal potential energies but unequal kinetic energies as the ball has less total energy at D than it had at B. This means the kinetic energy at D must be less than at B and therefore speed is less at D.

2. For all of #2 you can safely ignore vertical forces as they would only contribute to friction which has been eliminated.

2a. With A and B at rest, Newton's first law says with bodies at rest or in constant motion there is no external net force being applied. If there is no external net force there can be no internal net force. It doesn't matter what the masses are. A and B are not pushing on each other.

2b. The vector seems to be drawn wrong but it really doesn't matter. The first law again; bodies in motion with constant velocity have no external forces applied (exactly the same situation as 2a). Therefore no forces between A and B regardless of their masses.

2c. Once again the vector appears to be wrong but no matter, this would just mean the acceleration is a deceleration. Newton's third law says two bodies in contact with one acting on the other experience equal and opposite forces at their boundary. It doesn't matter what their masses are. The forces are always equal and opposite. This is a bit counterintuitive but if you think of it as a system with an exteranlly applied force, that force is distributed across the two bodies in proportion to their masses. If A is receiving external force Fs then it experiences a net force of Fs - Fb, where Fb is the force of B pushing on A. B experiences an external force of Fa which is equal to Fb.

And I still believe this is a clever proposition. Until I actually did the hard thinking I was ready to dismiss your premise. Instead of the costly debate that would have entailed, you made me argue with myself for two days while you took care of business. Very clever indeed.

BTW am I correct?

For 1, if the ball is thrown straight up, the ball will describe a line in space, not a parabola. That said, if you graph in z and t, the ball will describe a parabola in time (which is a bit conceptually strange, but gives the right result). Your ultimate answers are correct.

I would say that by the terms of 2c, there is no indication that A is forcing the acceleration of B or the reverse. The problem states: "Assuming both blocks are moving to the left with constant acceleration a"; it is at least as reasonable to assume that both blocks are being accelerated by some outside agent, which means no force of A on B or B on A. Conveniently, 0 = 0, of course.

Another problem that exposes conceptual gaps in some people:

You have an iron plate 1 cm thick. This hole is tapped for a bolt with 10 threads per cm. You apply even heat that causes the plate to expand. (Assume that every part of the plate is heated to the same temperature.)

What happens to the size of the hole and the spacing of the threads?

If I drew an arrow for 2c, saying that some external force is doing the accelerating of the system, but that you don't need to worry about that, would it help?

or assume there's a (massless) rope pulling both blocks in a given direction, tied to the "back" block, and the rope pulls with constant acceleration.

so doug, you're close but not right. the external agent doesn't act independently on each block.

--"deconstructivism". That is: before you can start teaching students (or having them discover) what is right, you have to create carefully-constructed situations in which they realize that their prior misconceptions are wrong. Telling students they are wrong won't do it -- we are all capable of cognitive dissonance. You have to create a situation in which their prior "knowledge" just simply breaks down. Like breaking up the earth before you can plant.

so succinct and correct! and it is exactly where Hestenes goes with his mechanics notes, which I promise I'll post about tonight.

Paul, let me elaborate on the above comment, as my above comment was for you. The arrows are correct: assume something ELSE is causing the system to move.

So in 2b, let's just say something else pushed on B in the direction of -x for a while and let go.

In 2c, let's just say that something ELSE is accelerating A by pushing on it in the +x direction.

And then I ask you to think about newton's third law. which none of you understand apparently :)

As a COMPLETE sidenote, I am ASTONISHED at how much better I am at these problems NOW than I was the first, second, or THIRD time I took mechanics. It's been over 15 years since I took a mechanics class, and I am better at them now than I ever was while taking them as a physics major. We really do continue to build mastery during our sleep!

Michael,

Your comment about the problems in calculus are spot-on. One of the references that Hestenes et. al. site is a paper on motion IN 1 dimension, and it's basically a perfect example of how college students can't understand a graph of position vs time for the trajectories of two balls.

Imagine a trajectory for the first ball is the line y = t, and the trajectory for the second is y = 4 + (1/2)t. Consistently, when asked what it means on the graph where the lines intersect, they say "that the balls have the same speed when they meet". This stays true even if you make the slope of one of the trajectories negative.

"Knowing the math doesn't help you solve the right problem."

I guess I don't know what you mean by "the math". Is the problem that they pick the wrong governing equations or that the problem can't be solved using math? Some problems have multiple approaches, one of which is real simple if you can "see" what's going on. Conservation of energy is one of these approaches. However, simplified conceptual approaches don't exist for everything.

I guess I don't see the dichotomy between math and concepts. I see it more as a lack of integration. For the early years in college, my math was far ahead of what I was doing in my engineering classes, and it was taught out of context. Even when the applications caught up to (and surpassed) the math, there was still no cohesiveness. My learning was in lots of pieces. Perhaps better models or concepts would have helped earlier on, but I think that's just part of the problem. If you don't figure out how to tie the models and concepts in with the math, then you're stuck with only part of the solution.

"so doug, you're close but not right. the external agent doesn't act independently on each block."

Sorry, I'm just answering the question you asked, which seems not to be the question you intended to ask. 8-)

I don't think an arrow would be sufficient to indicate the intent. You would need something like: "A and B are accelerated to the right with constant acceleration

aby a force acting on A." (Working with the picture as drawn rather than the description.)BTW, I would choose different designations for the blocks to avoid confusion of A and a. Such confusion seems to be outside what the question is intended to test.

AT LAST THE LIE HAS BEEN EXPOSED !!!!

Yes, for all the praise for direct instruction, for drill and practice, Allison has finally let us in on the true secret:

"We really do continue to build mastery during our sleep!"

The answers for 2a, 2b, and 2c are all the same: the forces exerted by the two blocks on each other are equal and opposite. That's Newton's third law, and it has nothing to do with whether the blocks are moving, stationary, or accelerating.

Nor is there any reason to assume that the forces are zero. The forces in question are the normal forces. One way to get an intuition is to imagine a bathroom scale wedged between the two blocks. Facing to the right, it measures the force exerted by B on A. Facing to the left, it measures the force exerted by A on B.

The problem is conceptually the same as the man-standing-in-a-moving-elevator problem. The force exerted by the man on the elevator is the same as the force exerted by the elevator on the man (Newton's 3rd law). There is thus zero net internal force. If the system as a whole is accelerating, it is because of external forces.

I made the same mistakes when I saw Hestenes speak in 1994, and boy was I embarrassed.

Absent acceleration, the forces must be zero, since the surface is frictionless. With acceleration, the forces might be zero and there is no reason from the description to expect anything else. If the forces are non-zero, they must be of equal magnitude, of course (and if they are zero, they are definitionally of equal magnitude).

I think the whole equal-and-opposite thing shouts "omg obvious!" but you kinda had to look at the problem carefully to make sure you weren't being tripped up by a trick question.

Conceptually you don't really need a whole lot of math to understand why throwing a ball into frictionless air is symmetrical.

First gravity has to reduce the velocity acting against its direction to zero. This takes time. It takes the same amount of time to speed up the ball in the opposite direction to a velocity of the same magnitude (but different sign). No parabola-math required.

It's kind of as intuitive as in a perfectly elastic collision why the ball bounces to the same height as before. In fact, you could flip the problem the other way round -- a ball being bounced against the floor rather than a ball being tossed up.

Because of conservation of momentum (also derived from Newton's 3rd law), the graph the function of y against t will also be a parabola. (No real math required.)

Well, if the ball bounces repeatedly, you'd have a series of parabolae, and the point of each bounce would be indifferentiable.

--With acceleration, the forces might be zero and there is no reason from the description to expect anything else.

no, the block A is pushing on B in one case and being pushed on by B in the other. and normal forces--B on A and A on B respectively push back. The forces between A and B are not zero.

The secret to intuitively understanding Newton's 3rd law is to imagine why flying balls (normally) don't break through walls.

I actually realise how good my HS physics teacher was at teaching a lot of basic concepts because he brought up the case of a big bully attacking a defenceless kid on the playground.

"Wait," he says, after hearing a student's explanation of a problem. "You mean if a big, mean, nasty bully hit me, even if I didn't throw any punches I would exert an equal and opposite force back?"

I think the other classic example is of a guy pushing on a cart. Guy pushes on cart, cart pushes equally and oppositely back, and nothing ever gets done ... until it gets pointed out that the forces are exerted on different objects.

When I realised a reaction force was needed to "consume" the guy's momentum and prevent the cart (or the guy) from accelerating indefinitely (even in the absence of friction), it was such an "oh!" moment.

Doug,

you're right about the confusion of my independent-vs-non-independent comment. sorry. it's difficult to be precise enough to not mis-teach.

would it help clarity if I told you that the value of acceleration a was g? and you pretended that A was a book and B was a table? or if I suggested we were just mislabeling x and z?

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