kitchen table math, the sequel

Back from the Open, so summer is officially over, as opposed to emotionally over, which it was the instant Chris set foot inside his dorm.

Hate the empty nest! hate! hate! hate!

Anyway, getting back to my interrupted post on precision teaching, when you learn something to fluency, you "get the MESAG":

Maintenance	"You never forget how to ride a bicycle." When you learn content or skills to fluency, you remember them.
Endurance	This one surprised me. Fluency in content and skills means you can perform the content or skill for as long as you need to perform them. You have stamina.
Stability	Another surprise, rife with implications for our ADHD epidemic. Fluent knowledge and skills  are impervious to distraction. A noisy classroom has no effect on skills a child knows so well he can do them in his sleep. If he can do long division in his sleep, he can do long division inside Penn Station.
Application	Transferring semi-old knowledge to new contexts is hard. My favorite story re: transfer of knowledge is the little autistic boy whose parents and teachers spent months painstakingly teaching him to butter his bread. Finally he learned! Everyone was happy until, a few weeks later, they discovered that the little boy had no idea how to spread peanut butter on bread. Spreading butter on bread and spreading peanut butter on bread were two different things, and they had to start all over again. Fluency allows you to apply your bread-buttering skills to peanut-buttering bread. Autistic children, by the way, are rarely taught anything to fluency. "Discrete trial" teaching puts a ceiling on the number of repetitions a child can do in a minute. 80% correct does not equal fluency.
Generativity	Fluent knowledge and skill "repertoires" readily recombine to produce new skills that don't have to be directly taught.

Is fluency the magic that makes inflexible knowledge flexible?

The contribution of handwriting to learning to write was examined in an experimental training study involving beginning writers with and without an identified disability. First-grade children experiencing handwriting and writing difficulties participated in 27 fifteen-min sessions designed to improve the accuracy and fluency of their handwriting. In comparison to their peers in a contact control condition receiving instruction in phonological awareness, students in the handwriting condition made greater gains in handwriting as well as compositional fluency immediately following instruction and 6 months later, The effects of instruction were similar for students with and without an identified disability. These findings indicate that handwriting is causally related to writing and that explicit and supplemental handwriting instruction is an important element in preventing writing difficulties in the primary grades.

Is handwriting causally related to learning to write? Treatment of handwriting problems in beginning writers.
Graham, Steve; Harris, Karen R.; Fink, Barbara
Journal of Educational Psychology, Vol 92(4), Dec 2000, 620-633

Another dispatch from my two weeks at Morningside's Summer School Institute:

"Get the MESAG" is Morningside's acronym for the results of fluency: get the MESAG.

Maintenance
Endurance
Stability
Application
Generativity

More tomorrow ---

More from the Education Week article on Common Core assessments:

He pointed to one illustrative example in PARCC’s materials that tries to gauge students’ fluency in division and multiplication. It offers five equations, such as 54÷9=24÷6, and asks 3rd graders to specify whether each is true or false.

“I like that it does multiple assessments in one item,” he said. “It asks kids to work each of those problems easily and be comfortable with it, which is what fluency is.”
Published in Print: August 22, 2012
Consortia Provide Preview of Common Assessments
By Catherine Gewertz

Offhand, I don't see how a set of five equations like these can test fluency. Fluency isn't simply a matter of accuracy, ease, and comfort. Fluency includes speed, and to assess fluency you need fluency aims, or standardized rates of performance.

How fast should a third grade student be able to answer these 5 questions? That's what you would need to know to use these equations to assess fluency, and one set of 5 simple equations probably isn't enough to measure speed.

Beyond that, I'm skeptical this is a fluency test at all. Seems to me it's more an application-of-knowledge test than a test of fluency per se.

Here's the list of fluency aims Rick Kubina culled from the precision teaching literature.

Maybe I should send a copy to PARCC.

I heard a lot about Terry while I was at Morningside Academy's Summer School Institute.

Eric [Haugton, one of the creators of precision teaching] helped his wife Elizabeth plan for a kindergarten student named Terry Harris. Terry had cerebral palsy, and walked with crutches. Elizabeth was teaching him to write his name. It had taken from September to Christmas vacation to teach Terry how to write his first name. Elizabeth wondered if there wasn’t a better way to teach him to write his last name. Even though there were only four new letters to teach, it still seemed like a daunting task. Eric asked her if Terry could write 250 to 200 vertical stokes in a minute. Elizabeth mentioned that Terry was quadriplegic — Eric replied, “I didn’t ask what he looks like, Elizabeth — can he do 250 to 200 vertical strokes per minute or not?” Elizabeth admitted that she didn’t think so. “Can he do 140 to 120 zero’s in a minute?” Again Elizabeth said he probably could not. “Those are the elements that make up the compounds for every letter or number we write. If they are not fluent, then learning to write numbers and letters will fail.”

Returning to school Elizabeth and Terry spent the next three weeks working on strokes and 00s. Terry went from about 50 vertical strokes to over 175, and from 25 zero’s to over 90. “But Terry and I were getting tired of this drill, and we were ready to try going back to writing his name.” So they did; how long did it take for Terry to learn to write Harris?

Terry learned it in five minutes.

LESSONS LEARNED: Eric Haughton and the importance of fluency
Wicked Local Hingham | January 22, 2012

Autistic children often spend years learning the same things over and over and over again in school.

What would happen if all of these students were moved from "discrete trial"/80% mastery criteria to precision teaching/fluency training?

C. and I took another math section of the SAT today, and it looks like I am now definitively finishing math sections. In fact, I'm starting to finish with a minute or two to spare. C. is finishing, too, although he's still not able to do all the problems. He skipped 3 today and probably got everything else right. I skipped none and probably missed 2. (We can't check our answers 'til we enter them online.)

I'm still making dumb mistakes. Typically, I'll miss one of the very first questions on the test, which C. never does but enjoys watching me do. We're starting to have a ritual: I miss question 2 or 3, and C. says, "You always miss the easy one and get the hard ones right."

Today I misread the number 6 on a graph as the number 5. Arrrgh.

The good news: I seem to have stopped making bubbling errors. Thank God. Losing points to bubbling errors when you can't even finish the damn test is uniquely demoralizing.

These days I have enough time to check each page for bubbling errors, but I don't find any when I check. More and more, I think speed is the answer -- whatever speed means, exactly, which is more than simply finishing early.

I'm probably talking about fluency.

As I become more fluent in SAT math, I'm becoming more fluent in bubbling, too.

Next challenge: no more dumb mistakes.

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

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

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

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

According to a recent study, somewhere between 10% to 30% of children have difficulty learning to produce rapid, legible hand-written work(1). Handwriting difficulty is often linked with other problems such as attention deficit disorder. Poor quality of handwriting of children with handwriting problems seems particularly related to a deficiency in visual-motor integration. (2)

Children who do not acquire fluent, legible handwriting in the early years often experience far-reaching negative effects on both academic success and self-esteem.(1)

“Handwriting is one of the basic building blocks of good writing and plays a critical role in learning,” Graham, Currey Ingram Professor of Special Education at Vanderbilt University’s Peabody College, said. “Young children who have difficulty mastering this skill often avoid writing and their writing development may be arrested. They also may have trouble taking notes and following along in class, which will further impede their development.”

There are three possible sources of children developing handwriting difficulties: a problem with the child, a problem with the teacher, or a problem with the curricula (and related materials).

In " How do primary grade teachers teach handwriting? A national survey",(3) the authors found that

Nine out of every ten teachers indicated that they taught handwriting, averaging 70 minutes of instruction per week. Only 12% of teachers, however, indicated that the education courses taken in college adequately prepared them to teach handwriting. Despite this lack of formal preparation, the majority of teachers used a variety of recommended instructional practices for teaching handwriting. The application of such practices, though, was applied unevenly, raising concerns about the quality of handwriting instruction for all children.

In a less-formal presentation of the national study data, Steve Graham is intereviewed:

Graham suggests that a return to consistent handwriting instruction, with an understanding of the challenges different children face, would not only result in more legible papers but also support overall learning across subjects.

“Teachers need to continue to teach their students how to properly form and join letters. We found that this sort of instruction takes place for 10 minutes or less a day in most schools, down from two hours a week in the 1950s,” Graham said. “At home, there are many things that parents can do to help their young children improve their penmanship. Activities such as identifying and tracing letters, forming letters from memory, copying words and playing timed games to see how quickly they can accurately produce written letters and words all go toward building this skill.”

There are two common handwriting approaches or curricula used in U.S. schools--one: traditional, based on the Palmer method, and two: "italicized" -- more flowing. The most popular of the former is Zaner-Bloser and the most popular of the latter is d'Nealian (developed by Donald Thurber).

There is very little research on the relative effectiveness and efficiency of each approach.

There is, however, a third way: Handwriting Without Tears.

I spent Friday and Saturday at a Handwriting Without Tears seminar. Handwriting Without Tears (HWT) was created in the 1970s by an occupational therapist as a remedial program, and over the decades, has grown into (1) a pre-K through 5th grade classroom curriculum and (2) a remedial program..

It works as a classroom curriculum because the letters are taught in logical order, the letter formation skills are taught to mastery, and the curriculum uses multisensory methods. The teacher watches for, and immediately corrects, errors in letter formation, and the curriculum includes frequent "Review and Mastery" opportunities.

It works as a remedial program because HWR's authors have structured remediation in small, precise steps.

I highly recommend Handwriting Without Tears.


1. Feder KP, Majnemer A. Dev Med Child Neurol. 2007 Apr;49(4):312-7.
2. Volman MJ, van Schendel BM, Jongmans MJ. Am J Occup Ther. 2006 Jul-Aug;60(4):451-60.
3. Graham S, Harris KR, Mason L, Fink-Chorzempa B, Moran S, Saddler B Reading and Writing 2008 21(1-2):49-69.

Elsewhere:

Handwriting Key to Learming, Newsweek, November 12 2007

LD Podcast: Dr. Steve Graham on writing development.

Interview with Steve Graham

Previous KTM Posts referencing Handwriting
Somewhere in a Well to Do District

Learning in a Castle of Fear

Speed Test