Advanced Math for Young Students

Phil Keller sent me a copy of his beautiful new book Advanced Math for Young Students.

(Here's his blog. And here's an early review!)

I told Phil, as soon as I laid eyes on the book, that it has the look and feel of an authored work, nothing like the many-authored splendor of a typical commercial textbook.

Plus (and I have to say this) the cover and pages are creamy and smooth, a throwback to the physical beauty of books pre-crash (although of late I've had the sense that decent paper may be making a comeback).

The book is so compelling I may have to buy a second copy to work through myself so the other one can stay on the coffee table.

From the Introduction:

For 26 years, I have been a high school physics teacher. I work in an excellent, well-regarded high school and I have been fortunate to have many talented students who soak up all the physics I can teach them, and more. But every year, I also teach students who struggle to master the topic, despite their great efforts and mine. And I know from discussions with colleagues, both within my school and arose the country, that we are not the only ones struggling. There is something getting in our way. Maybe this will seem obvious to anyone who has struggled in physics, but here's what I think: I think it's the math.

Physics applies math. It's all about finding relationships and solving the puzzles that the laws of physics present. For the most part, this work is done in the language of mathematics, and more specifically, the language of algebra. So to be comfortable learning physics, a student has to be fluent in that language. Algebra cannot just be a memorized set of procedures for finding 'x'. It has to be a symbolic way of representing ideas. But for many students, that level of fluency is not attained unjust one year of algebra--which is all that many students have had when they start studying physics. It's no wonder that some struggle.

It is not only physic students who struggle. For even more than 26 years, I have been teaching students how to prepare for the math portion of the SAT. What I have seen over the years is that most students are not fluent enough in algebra to successfully apply algebra on the SAT. One goal of my SAT course is to teach alternative, non-algebraic approaches to SAT problems. It is also a major theme of my math SAT book, The New Math SAT Game Plan. And I will tell you something you may find surprising (or even distressing): on the SAT, these non-algebraic methods work very nicely. They won't get you to an 800, but they will take you pretty far. And even my top scorers report that they like to mix in the non-algebraic methods along with the standard approaches (which, as top students, they also know how to use).

The non-algebraic methods, however, won't get you very far in physics. In fact, a student who does not really learn the language of algebra is going to struggle in all later math and science classes: physics, statistics, computer science and beyond. That STEM door is swinging closed because one year of algebra class did not lead to sufficient fluency. So why spend only one year? Why not start earlier?

I am not saying every 7th grader should be in a high-school version of Algebra I. But I am saying that every middle school student should, over the course of the middle school years, start learning about and thinking about the ideas of algebra (even some ideas that won't reappear until Algebra II or Pre-calculus). These are ideas that take some time to ponder.

stop the multiverse, I want to get off

So I was propounding my theory that Something Happened in 1985, a world-jarring event that catapulted us all into a parallel universe where Not Teaching is Teaching and Salman Khan is the man you summon to help you spice up your presentations.

Then a couple of minutes later I came across this: The Accidental Universe: Science's Crisis of Faith by Alan P. Lightman.

Why are we living in a world where Writing to the Point is out of print and Grammar to Enrich & Enhance Writing is in print?

I blame the multiverse.

Physics Education Continued

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

Fixing Physics Education: Modeling Instruction

In prior posts, I referred to work done by David Hestenes and his colleagues at Arizona State University addressing the dismal results of traditional university level physics instruction. Hestenes and others developed the Force Concept Inventory, FCI, to demonstrate that even after a year of traditional instruction, college students had failed to create proper models in their own minds for how mechanics actually works. Specifically,

"Before physics instruction, students hold naive beliefs about mechanics which are incompatible with Newtonian concepts in most respects.
• Such beliefs are a major determinant of student performance in introductory physics.
• Traditional (lecture-demonstration) physics instruction induces only a small change in the beliefs. This result is largely independent of the instructor’s knowledge, experience and teaching style. "

Hestenes has gone forward from there, and developed a new curriculum design for high school and college physics instruction, and has pushed this curriculum design out to high school and college physics teachers on his own, and more recently, through NSF backing. He calls this new type of instruction modeling instruction.

What is meant by modeling instruction? He means that you learn physics by constructing appropriate models for the interactions in your system, and then you apply inference to your model to solve whatever problem you have. The emphasis is on getting the student to recognize the model at hand by getting them familiar with constructing models in the first place. What's a model? A model is a representation of your system and its properties. A model tells you everything you need to know. So a model tells you your system, the boundaries of your system, the state variables inside your system, the initial conditions of your system, the transition function for the state variables in that system, and whatever interactions you need to know. This sounds vague, but the point of the model is that you can explicitly say whether or not you've got everything you need to infer what happens if you write it all down.

In modeling instruction, the idea is that "the modeling method approaches the problem of restructuring students’ intuitions by engaging them in explicit construction and manipulation of externally structured representations. In the case of mechanics (6), we have found it advisable to engage students in explicit comparisons of the three major misconceptions in Box 1 with their Newtonian alternatives. When these three are adequately treated, many other misconceptions about mechanics fall away with them. "

To facilitate the teaching of mechanics by modeling instruction, the modeling group at ASU created course materials and curricula for a high school level mechanics class. Their modeling method for mechanics explicitly teaches 10 models, five of them models of motion (kinematical models): constant velocity, constant acceleration, the simple harmonic oscillator, uniform circular motion, and a collision model; and five models of force (causal models): the free particle, the constant force, the central force, the linear binding force, and the impulsive force. The idea is that by explicitly organizing the ideas of motion and force in this way, the student will see the common physics in each. This is as opposed to organizing ideas around "problems".

Here's an example. "oh, that's a projectile problem", " oh that's a block-siding problem" "oh, that's a orbit problem" is a typical way a student might think about the physics problem in front of them, but it doesn't help elucidate what were the relevant features of the problem at hand, whereas recognizing "oh, that's a constant velocity problem", "oh that's a free particle problem," "oh, that's a central force problem" leads you immediately to know or be able to infer the geometric structure, the interaction structure, the force structure, the changes over time, etc.

Enough talk! Let's jump in. Here are the course notes for unit on the free-particle model. The instruction goals are to use the free particle model to develop intuition for Newton's First Law (commonly stated as "an object in motion tends to stay in motion; an object at rest tends to stay at rest"), for Newton's Third Law , and to correctly be able to represent forces as vectors.


1. Newton’s 1st law (Galileo’s thought experiment)


Develop notion that a force is required to change velocity, not to produce motion
Constant velocity does not require an explanation.

2. Force concept

View force as an interaction between and agent and an object
Choose system to include objects, not agents
Express Newton’s 3rd law in terms of paired forces (agent-object notation)


3. Force diagrams

Correctly represent forces as vectors originating on object (point particle)
Use the superposition principle to show that the net force is the vector sum of the forces



4. Statics

•F = 0 produces same effect as no force acting on object decomposition of vectors into components

Continuing, the teacher's notes state "It is essential that you get students to see that the constant velocity condition does not require an explanation; that changes in velocity require an interaction between an agent and an object. We quantify this interaction by the concept of force. After the dry ice and normal force demos, one can use worksheet 1 as an opportunity to deploy the force concept in a qualitative way. It is important to carefully treat how to go about drawing force diagrams in which one represents the object as a point particle. Drawing the dotted lines around the object helps students distinguish between the object and the agent(s). "
And

"Newton’s Third Law ...Researchers have identified and categorized many such misconceptions, but two of them are particularly important, because they are persistent common sense alternatives to Newton’s Laws. Ignoring variations and nuances, these misconceptions can be formulated as intuitive principles.
I. The Impetus Principle: Force is an inherent or acquired property of objects that make them move.
II. The Dominance Principle: In an interaction between two objects, the larger or more active object exerts the greater force."

The notes then describe detailed demos and labs, with pre and post discussion points as well:

"It is an indirect goal of this activity to provide students an opportunity for arguing that a free particle, i.e. one subject to zero net force, will have a constant velocity. Also, students should conclude that any apparent change in velocity of an object indicates that a non-zero net force is acting upon it, provided that the observer is in an inertial frame of reference. ,,,
Make the point that when no force acts on the block in the horizontal direction, the block maintains constant velocity.
* Point out that an impulse applied perpendicular to the original trajectory does not result in the block making a right angle turn.
* Be sure to ask why they think the block continues to move once it leaves the hand. Some are likely to answer " due to the force of the hand."

The notes continue in this fashion, with specific notes on what demos/labs, how to guide them, what the appropriate leading questions are, when to ask for students' input, when to build to consensus.

This might seem like a fairly normal course, being taught with a normal lab. But the structure is different. More on that in the next post.


Physics Education and Failures in Conceptual Understanding
Fixing Physics Education: Modeling Instruction
Physics Education Continued
More Modeling Instruction: Techniques

Physics Education and Failures in Conceptual Understanding

For decades, physics professors in universities and colleges in the US have known there is something wrong with physics studies in their schools. Their own physics majors have a poor understanding of basic concepts in mechanics, electricity and magnetism, and quantum mechanics.

Journals like The American Journal of Physics (devoted to teaching and pedagogy at the university level) and Physics Teacher (the same for high school and lower) bring up these issues, with a variety of proposed changes and solutions both at the individual classroom level and at the higher theory-of-ed level. D. Hestenes at ASU and his colleagues have done work in this area, both in questioning the failures of pedagogy and developing some solutions. First, the problems.

D. Hestenes wrote in "What Do Graduate Oral Exams Tell Us?" (Am J. Phys. 63:1069 (1995)) of finding a quote from physicist W. F. G. Swann, in "The Teaching of Physics", (Am. J. Phys. 19, 182-187 (1950)):

"Much can be said about oral examinations for doctor’s degrees, and in my judgment not much can be said that is good. I have sat in innumerable examinations for Ph.D. at very many different universities, sometimes as a member of the permanent faculty and sometimes as a visitor. In almost every case the knowledge exhibited was such that if it represented the true state of mind of the student, he never should have passed. However, after the examination is concluded there is usually a discussion to the effect that: "Well, So-and-so got tied up pretty badly, but I happen to know that he is a very good man," etc., etc., and so finally he is passed."

Hestenes goes on to quote Swann as saying [A student] "passes his tests frequently [including graduate comprehensive exams], alas, with very little comprehension of what he has been doing."

Hestenes diagnoses the problem as this:

It seems not to have occurred to the faculty that dismal oral exams may be symptoms of a severe deficiency in the entire physics curriculum. I submit that there is good reason to believe that they are symptomatic of a general failure to develop student skills in qualitative modeling and analysis.

These general failures mean that even students who have the grades to appear to have excellent mastery of the material do not understand basic elements of the material they have "learned".

It also suggests that college students who fail to understand the material may end up there because their confusion prevents them from attaining the mastery the "good" students have.

Of course, the errors didn't just start in college. Generally speaking, proper physical intuition is lacking in students who took high school physics, even in those who did well. Hestenes writes in "Force Concept Inventory", (Physics Teacher, Vol. 30, March 1992, 141-158)

"it has been established that1 (1) commonsense beliefs about motion and force are incompatible with Newtonian concepts in most respects, (2) conventional physics instruction produces little change in these beliefs, and (3) this result is independent of the instructor and the mode of instruction. The implications could not be more serious. Since the students have evidently not learned the most basic Newtonian concepts, they must have failed to comprehend most of the material in the course. "

Hestenes et. al. wrote the Force Concept Inventory, a multiple choice test whose aim is to "to probe student beliefs on this matter and how these beliefs compare with the many dimensions of the Newtonian concept. " It poses questions that force a choice between the correct Newtonian answer for an explanation of a given system, and other commonsense explanations that are actually misconceptions. After the test, interviews are done to determine students' reasoning.

Here's an example of a misconception that the FCI aims to tease out of a student:

[The misconception of "impetus":]
The term "impetus" dates back to pre-Galilean times before the concept was discredited scientifically. Of course, students never use the word "impetus"; they might use any of a number of terms, but "force" is perhaps the most common. Impetus is conceived to be an inanimate "motive power" or "intrinsic force" that keeps things moving. This, of course, contradicts Newton’s First Law, which is why Impetus in Table II is assigned the same number as the First Law in Table I. Evidence that a student believes in some kind of impetus is therefore evidence that the First Law is not understood.

The FCI has been given to thousands of college and high school students. The above paper details the results on the FCI, given as a pre and post test to both high school and undergraduate physics courses, with tremendous detail on similarities and differences across classrooms in the country. More, it provides strong evidence that traditional college physics pedagogy isn't doing anything to teach physics to the students who take it:

"The pretest/post test Inventory scores of 52/63 for [The Regular Physics Mechanics course at Arizona State University] are nearly identical to the 51/64 scores obtained with the Diagnostic for the same course...we have post test averages of 60 and 63 for two other professors teaching the same course. Thus, we have the incredible result of nearly identical post test scores for seven different professors (with more than a thousand students). It is hard to imagine stronger statistical evidence for the original conclusion that Diagnostic posttest scores for conventional instruction are independent of the instructor. One might infer from this that the modest 11% gain for Arizona State Reg. in Table III is achieved by the students on their own. "

Which brings us back to the state of physics majors going to graduate school:

One of us (Hestenes) interviewed 16 first-year graduate students beginning graduate mechanics at Arizona State University. The interviews were in depth on the questions they had missed on the Inventory (more than half an hour for most students). Half the students were American and half were foreign nationals (mostly Chinese). Only two of the students (both Chinese) exhibited a perfect understanding of all physical concepts on the Inventory, though one of them missed several questions because of a severe English deficiency. These two also turned out to be far and away the best students in the mechanics class, with near perfect scores on every test and problem assignment. Every one of the other students exhibited a deficient understanding of buoyancy, as mentioned earlier. The most severe misconceptions were found in three Americans who clearly did not understand Newton’s Third Law (detected by missing question 13) and exhibited reading deficiencies to boot. Two of these still retained the Impetus concept, while the other had misconceptions about friction. Not surprisingly, the student with the most severe misconceptions failed graduate mechanics miserably, while the other two managed to squeak through the first year of graduate school on probation.

Is it just that the Chinese students who manage to get into US physics grad schools are such creme de la creme that they are perfect? Or is Chinese instruction vastly superior?

(And for those who wonder about American instruction in other subjects, read this and weep:)

One disturbing observation from the interviews was that five of the eight Americans, as well as five of the others, exhibited moderate to severe difficulty understanding English text. In most cases the difficulty could be traced to overlooking the critical role of "little words" such as prepositions in determining meaning. As a consequence, we discarded two interesting problems from our original version of the Inventory because they were misread more often than not.

And yet, those who make it through physics graduate school to professordom mostly correct these errors, at least in mechanics. (Though not necessarily. In quantum mechanics, new professors are notorious for teaching elements of the material incorrectly. In special relativity, David Mermin, prof at Cornell, believes many professors teach the entire subject wrong. (He discusses this in a paper called something like "how to teach Special Relativity.") Hestenes suggests this is due to the realities of post quals grad school: the day in, day out teaching and researching refine one's intuition over and over again.

I think this implies something else as well. Error correction in intuition can only occur and stick if the mastery of the manipulation of the equations is so strong that you (correctly) believe what they tell you. If you can be forced to do the math on the board, and forced to read and think about what it says, then you can learn the truth counter to what your intuition tells you, but only if you are utterly sure you did the math on the board correctly.

If instead, you doubt yourself, doubt your manipulation of equations, doubt your application of the laws as you understand them, then you will get confused, doubt your answer, default to your intuition, and scrap learning the correct way to think.

That means you need a tremendous amount of mastery. How in the world to achieve that?

Hestenes' answer --changing how physics is taught in high school and in college--will be explored in a week or so.


Physics Education and Failures in Conceptual Understanding
Fixing Physics Education: Modeling Instruction
Physics Education Continued
More Modeling Instruction: Techniques

Mastery and Conceptual Understanding in Physics

For decades, physics professors in universities and colleges in the US have known there is something wrong with physics studies in their schools. Their own physics majors have a poor understanding of basic concepts in mechanics, electricity and magnetism, and quantum mechanics.

Journals like The American Journal of Physics (devoted to teaching and pedagogy at the university level) and Physics Teacher (the same for high school and lower) bring up these issues, with a variety of proposed changes and solutions both at the individual classroom level and at the higher theory-of-ed level. D. Hestenes at ASU and his colleagues have done work in this area, both in questioning the failures of pedagogy and developing some solutions. First, the problems.

D. Hestenes wrote in "What Do Graduate Oral Exams Tell Us?" (Am J. Phys. 63:1069 (1995)) of finding a quote from physicist W. F. G. Swann, in "The Teaching of Physics", (Am. J. Phys. 19, 182-187 (1950)):

"Much can be said about oral examinations for doctor’s degrees, and in my judgment not much can be said that is good. I have sat in innumerable examinations for Ph.D. at very many different universities, sometimes as a member of the permanent faculty and sometimes as a visitor. In almost every case the knowledge exhibited was such that if it represented the true state of mind of the student, he never should have passed. However, after the examination is concluded there is usually a discussion to the effect that: "Well, So-and-so got tied up pretty badly, but I happen to know that he is a very good man," etc., etc., and so finally he is passed."

Hestenes goes on to quote Swann as saying [A student] "passes his tests frequently [including graduate comprehensive exams], alas, with very little comprehension of what he has been doing."

Hestenes diagnoses the problem as this:

It seems not to have occurred to the faculty that dismal oral exams may be symptoms of a severe deficiency in the entire physics curriculum. I submit that there is good reason to believe that they are symptomatic of a general failure to develop student skills in qualitative modeling and analysis.

Of course, the errors didn't just start in college. Generally speaking, proper physical intuition is lacking in students who took high school physics, even in those who did well. Hestenes writes in "Force Concept Inventory", (Physics Teacher, Vol. 30, March 1992, 141-158)

"it has been established that1 (1) commonsense beliefs about motion and force are incompatible with Newtonian concepts in most respects, (2) conventional physics instruction produces little change in these beliefs, and (3) this result is independent of the instructor and the mode of instruction. The implications could not be more serious. Since the students have evidently not learned the most basic Newtonian concepts, they must have failed to comprehend most of the material in the course. "

Hestenes et. al. wrote the Force Concept Inventory, a multiple choice test whose aim is to "to probe student beliefs on this matter and how these beliefs compare with the many dimensions of the Newtonian concept. " It poses questions that force a choice between the correct Newtonian answer for an explanation of a given system, and other commonsense explanations that are actually misconceptions. After the test, interviews are done to determine students' reasoning.

Here's an example of a misconception that the FCI aims to tease out of a student:

[The misconception of "impetus":]
The term "impetus" dates back to pre-Galilean times before the concept was discredited scientifically. Of course, students never use the word "impetus"; they might use any of a number of terms, but "force" is perhaps the most common. Impetus is conceived to be an inanimate "motive power" or "intrinsic force" that keeps things moving. This, of course, contradicts Newton’s First Law, which is why Impetus in Table II is assigned the same number as the First Law in Table I. Evidence that a student believes in some kind of impetus is therefore evidence that the First Law is not understood.

The FCI has been given to thousands of college and high school students. The above paper details the results on the FCI, given as a pre and post test to both high school and undergraduate physics courses, with tremendous detail on similarities and differences across classrooms in the country. More, it provides strong evidence that traditional college physics pedagogy isn't doing anything to teach physics to the students who take it:

"The pretest/post test Inventory scores of 52/63 for [The Regular Physics Mechanics course at Arizona State University] are nearly identical to the 51/64 scores obtained with the Diagnostic for the same course...we have post test averages of 60 and 63 for two other professors teaching the same course. Thus, we have the incredible result of nearly identical post test scores for seven different professors (with more than a thousand students). It is hard to imagine stronger statistical evidence for the original conclusion that Diagnostic posttest scores for conventional instruction are independent of the instructor. One might infer from this that the modest 11% gain for Arizona State Reg. in Table III is achieved by the students on their own. "

Which brings us back to the state of physics majors going to graduate school:

One of us (Hestenes) interviewed 16 first-year graduate students beginning graduate mechanics at Arizona State University. The interviews were in depth on the questions they had missed on the Inventory (more than half an hour for most students). Half the students were American and half were foreign nationals (mostly Chinese). Only two of the students (both Chinese) exhibited a perfect understanding of all physical concepts on the Inventory, though one of them missed several questions because of a severe English deficiency. These two also turned out to be far and away the best students in the mechanics class, with near perfect scores on every test and problem assignment. Every one of the other students exhibited a deficient understanding of buoyancy, as mentioned earlier. The most severe misconceptions were found in three Americans who clearly did not understand Newton’s Third Law (detected by missing question 13) and exhibited reading deficiencies to boot. Two of these still retained the Impetus concept, while the other had misconceptions about friction. Not surprisingly, the student with the most severe misconceptions failed graduate mechanics miserably, while the other two managed to squeak through the first year of graduate school on probation.

Is it just that the Chinese students who manage to get into US physics grad schools are such creme de la creme that they are perfect? Or is Chinese instruction vastly superior?

(And for those who wonder about American instruction in other subjects, read this and weep:)

One disturbing observation from the interviews was that five of the eight Americans, as well as five of the others, exhibited moderate to severe difficulty understanding English text. In most cases the difficulty could be traced to overlooking the critical role of "little words" such as prepositions in determining meaning. As a consequence, we discarded two interesting problems from our original version of the Inventory because they were misread more often than not.

And yet, those who make it through physics graduate school to professordom mostly correct these errors, at least in mechanics. (Though not necessarily. In quantum mechanics, new professors are notorious for teaching elements of the material incorrectly. In special relativity, David Mermin, prof at Cornell, believes many professors teach the entire subject wrong. (He discusses this in a paper called something like "how to teach Special Relativity.") Hestenes suggests this is due to the realities of post quals grad school: the day in, day out teaching and researching refine one's intuition over and over again.

I think this implies something else he doesn't say. Error correction in intuition can only occur and stick if the mastery of the manipulation of the equations is so strong that you believe what they tell you. If you can be forced to do the math on the board, and forced to read and think about what it says, then you can learn the truth counter to what your intuition tells you, but only if you are utterly sure you did the math on the board correctly.

If instead, you doubt yourself, doubt your manipulation of equations, doubt your application of the laws as you understand them, then you will get confused, doubt your answer, default to your intuition, and scrap learning the correct way to think.

That means you need a tremendous amount of mastery. How in the world to achieve that?

Hestenes' answer --changing how physics is taught in high school and in college--will be explored in a week or so.


Physics Education and Failures in Conceptual Understanding
Fixing Physics Education: Modeling Instruction
Physics Education Continued
More Modeling Instruction: Techniques

Barry G on learning physics in high school

We had a bit of physics in 8th and 9th grades. My first real course was as a senior in high school. The teacher was very good and I was totally amazed by the subject. Although it was not calculus-based, it did use trig which we were studying at the time, and math and science merged very nicely. I still remember the lecture on the derivation of the formula for distance of a uniformly accelerating body, relating it to the "area under the curve" which in this case was a straight line, and thus was the area of a triangle. I wondered why he called a straight line a "curve", and the next year in freshman calculus, it again all made sense, when we learned integration.

I can't wait to learn calculus.

Or physics.

Speaking of subjects I intend to learn, The Teaching Company has high school courses. I ordered the chemistry course. ^* I'm thinking of asking C. to work through as much of it as he can next summer, before he takes chemistry at Hogwarts. (He's taking AP biology next year, when he'll be a sophomore.)

I have Nature of the Earth, too, and Building Great Sentences, which is fantastic. Building Great Sentences is so fantastic that the other night, when Ed and an attorney friend were discussing the critical importance of writing short sentences, I rolled my eyes.

"You don't try to write short sentences?" Ed asked.

"No," I said curtly, leaving the men to wonder why.

While it is true that I personally have not read Strunk & White, I have listened to two lectures from Building Great Sentences, and thus feel confirmed in my view that short sentences are neither here nor there.


^*"This course was an absolute joy."

The constructivist MIT: doing away with the big lecture

Fascinating article on MIT's shift away from teaching physics in the "standard" way, with a professor doing a main lecture 3 hours a week.

The physics department has replaced the traditional large introductory lecture with smaller classes that emphasize hands-on, interactive, collaborative learning. Last fall, after years of experimentation and debate and resistance from students, who initially petitioned against it, the department made the change permanent. Already, attendance is up and the failure rate has dropped by more than 50 percent.

Some bits you might find interesting: clickers! Homework due several times a week! And the biggest of all: attendance counts!

The new approach at M.I.T. is known by its acronym, TEAL, for Technology Enhanced Active Learning....A $10 million donation from the late Alex d’Arbeloff, an M.I.T. alumnus, co-founder of the high-tech company Teradyne, and former M.I.T. corporation chairman, made the switch to TEAL possible. The two state-of-the-art TEAL classrooms alone cost $2.5 million, Professor Belcher said.

The article says the failure rate is down, and performance is up. But then again, with required attendance, the failure rate could easily drop.

Is this constructivism? Are the teachers teaching any more? Or guides on the side? Well, guess what? The students hate it.

from the MIT paper, The Tech, in 2006:

"Most students do not bother to hide their dislike for TEAL. Their list of grievances is long and oft-repeated: the physical set-up of small tables makes it difficult to see the lecturer, the numerous homework assignments are tedious, the in-class problems are gone over too quickly, the students strong in physics end up doing all the work, and so on."


it continues:

Though student complaints are numerous, a number of changes have in fact been integrated into TEAL since its inception. Professor Eric Hudson, course administrator for 8.02T, has worked on modifications including more undergraduate teaching assistants in the classroom, fewer experiments (a drop from 18 to 10), and an emphasis on faculty training. Still being tested is the new AIM screenname iheart802, which will allow students to instant message a TA during class.

But even with the changes, the irrefutable fact remains: students are uninspired by the course. Dourmashkin admits that “students don’t like to go to class,” while Professor John Joannopoulos, who teaches a section of 8.02T this semester, said that there is a “tendency for students to be lax and lose concentration.”

Freshman Sarah Levin ’09, currently a TEAL student, said that “all of TEAL is so unmotivating because it’s so tedious that I don’t put any effort into the class and because of it I’m losing a good percentage of my grade just by lack of attendance.”

Shaw sees this problem as well. “Students come out of TEAL with a dislike for physics, and they seem less inclined to major in physics. TEAL has never done a good job in instilling a sense of why [learning] this is important.”

but I think this is the giveaway:


There are “lots of ways to do active learning,” Belcher said, citing a study conducted at Harvard that exhibited stronger learner gains than TEAL in a lecture environment with regular student involvement. “The important thing is to have students interact with students,” he said.

ah, yes, that's why I was such a poor student at MIT. Because I interacted so seldomly with STUDENTS! uh....no....not exactly.


another student's comments on TEAL:

I strongly suspect the NOOLT ("No One Likes Teal") phenomenon occured because TEAL, as I overheard someone whose name I can't remember say, "is the perfect example of when too much technology can be a bad thing."

We sit in tables of nine in groups of three. Each group has a computer to enable the learning process. Most of the time, though, it's used to watch the power point that's already projected in four (or more) different places around the room. (Sometimes these computers are used for Facebook. We're going to ignore that data.) In the beginning of the year, we took a diagnostic test and we were assigned to tables in a fashion that would keep an even distribution of physics background at the tables (meaning that all the people who took AP Physics in high school wouldn't sit in the same place).

This is all geared towards collaborative learning, which is nice in theory, but what happened in my experience is that the people at the table who knew what they're doing would work through the problem, and I would be left in the dark in terms of where this equation came from and what that one means. The idea was to learn from eachother, except that I feel that we do plenty of this while working on p(roblem)-sets. Personally, I'd like classtime to be geared more towards learning from the teacher.

And finally, this one culled from the comments on the NYT article:

This article is wildly misleading about the success of TEAL. As a member of the class of 2009, I was one of the first students required to participate in TEAL of I chose to take 8.01 (Mechanics). I then took TEAL again for Electricity and Magnetism (8.02).

If you notice in the pitcure, the TEAL classroom is a windowless, dark room that causes drowsiness better than any cold medicine. Each class is 2 hours long and you work with two other people that you have not chosen yourself. On fridays, you are to complete a small quiz with these people and all three of you recieve a grade for it. What ends up happening is the one person in the group does the problem and has no real motivation to explain it to you other than common courtesy.

The grades may have gotten better, but that is only because you get a grade for sitting there as well as about a thousand other assignments that are due at a thousand different times.

Here is a rundown of what you have to do for a TEAL class:

Weekly problem sets (4-10 hours), class time (5 hours), 1 quiz (1 hour), twice weekly "mastering physics" assignments online (each can take as little as 5 min and as much as two days to complete), Office hours, almost always necessary (3 hours)

The system does not foster an interest in Physics, but further enhances your distaste for it. My memories of the classes have nothing to do with the material, but with trudging through the snow to get to sunday office hours because despite all of this technology, the problems were STILL too difficult to do without help, with sitting in my room with 5 other friends trying to finish the online mastering physics assignment before the midnight deadline looking for the midnight deadline, and waking up at 8AM for a 9AM TEAL class knowing I'd be asleep by 9:15.

Do not be fooled by MIT's spokespeople. TEAL is very unpopular among students. Especially me.

Of course, MIT is an odd place, where the number one pastime is hating MIT. The unofficial student motto is "IHTFP", which stands for "I hate this place." So maybe this is just par for the course.

The best of hands-on, real-world, interdisciplinary, multi-media learning

From In Memoriam: William R. Bennett Jr., Laser Inventor and Collector of Data, in this past week's Yale Bulletin:

Over his 38 years at Yale, Bennett carried out research in diverse fields ranging from atomic physics to computer science and acoustics...

Many of the approaches Bennett used to collect data for his projects provided much amusement to his students and colleagues. For one project, he rented a truck and filled it with equipment and a mattress and, together with his wife and dog, set out to measure the "Fifth Force" at a site where a large body of water changed height rapidly. The site he chose was the locks on the Snake River in Washington, which gave him special dispensation to camp there with his truck for the summer.

He was also frequently seen at various sites around the Yale campus collecting data for his popular course on "The Computer as a Research Tool." For this course he was named one of the 10 best professors at Yale for many years in a row. His lectures in that course were multi-media events and included demonstrations of firestorms, removal of warts by laser, calculations of how long it would take monkeys sitting at the typewriter to produce phrases recognized from great works of literature, and comparisons of the sound waveforms of the French horn and the garden hose.

One time the professor was spotted dressed in scuba gear and pushing scales and other gadgets at the bottom of the Yale swimming pool, measuring drag coefficients.

...He used his expertise in physics and sound to make calculations on how to decrease the noise levels in the Yale dining halls and used those successfully to improve the ability to converse and to enjoy chamber music concerts there. He also measured magnetic fields around campus and around New Haven. With the magnetic field data, he showed that it was improbable that those fields could cause cancer.

(Cross-posted at Out in Left Field).

my field trip to Barnes and Noble

Lots of goodies!

Physics for Entertainment by Yakov Perelman

Earth Science SparkChart!^* (Picked up an SAT Math SparkChart, too, just because.)

Algebra 101 (terrific layout with no seductive details)


looks good, remains to be investigated further:

Theory of Almost Everything by Robert Oertner


^* Speaking of earth science, I read an earth science article in the TIMES the other day, and I understood (practically) every word. If this keeps up I'll do great on my Regents Earth Science exam!

MIT physics lectures on the web

Walter H. G. Lewin, 71, a physics professor, has long had a cult following at M.I.T. And he has now emerged as an international Internet guru, thanks to the global classroom the institute created to spread knowledge through cyberspace.

Professor Lewin’s videotaped physics lectures, free online on the OpenCourseWare of the Massachusetts Institute of Technology, have won him devotees across the country and beyond who stuff his e-mail in-box with praise.

“Through your inspiring video lectures i have managed to see just how BEAUTIFUL Physics is, both astounding and simple,” a 17-year-old from India e-mailed recently.

Steve Boigon, 62, a florist from San Diego, wrote, “I walk with a new spring in my step and I look at life through physics-colored eyes.”

Professor Lewin delivers his lectures with the panache of Julia Child bringing French cooking to amateurs and the zany theatricality of YouTube’s greatest hits. He is part of a new generation of academic stars who hold forth in cyberspace on their college Web sites and even, without charge, on iTunes U, which went up in May on Apple’s iTunes Store.

At 71, Physics Professor is a Web Star

Or, if physics doesn't interest you, you can go audit the fancy-shmancy Yale course on death.


Chronicle of Higher Education on Yale online courses
Yale Offers Free Online Courses
Open Learning Initiative at Carnegie Mellon
MIT Open Courseware