Wednesday, July 21, 2010

Grading Math Exams

This summer my son and daughter have been taking calculus, and are not entirely happy with the instructor. One of their complaints is that he grades them down on exams when they get the right answer but don't show the step by step procedure for getting it that they have been taught.

Thinking about the question, it seems to me that the instructor's policy is backwards. The ability to solve a problem by following a recipe, a step by step procedure that one has been taught for that kind of problem, is only weak evidence that the student understands what he is doing. Inventing a way of solving the problem that one has not been taught—and getting the right answer—is considerably better evidence.

I am now imagining that instructor as the schoolteacher who tried to keep a class of children—among them the young Gauss—quiet by having them add up the numbers from one to a hundred.

Opinions?

53 Comments:

At 11:06 AM, July 21, 2010, Blogger Micah said...

Yes and no. Math teachers want to see the students "show their work" because it's the only way to be sure that they're actually understanding the issue and not guessing or copying.

I once took a math test over the law of cosin without actually knowing the law of cosin. I tried to derive it during the test, but ended up with what was essentially the law of co-tangents. I solved all of the questions using this law, but got an F on the test because the teacher said my "work" was wrong. After I showed her what I did, though, she upgraded me to a B+.

If your kids have invented new ways of solving the problems, they should talk to the teacher and demonstrate their logic. If he's any good, he'll accept that. If they just can't be bothered to write down the steps, though, then they're likely to get little sympathy.

Not needing a recipe is one thing; not being able to follow one is another.

 
At 11:25 AM, July 21, 2010, Blogger Michael Duff said...

I think cheating is a much bigger concern than it used to be. Showing your work doesn't necessarily mean do it exactly like your teacher did.

I'm sure a student who showed a valid but alternate way to solve a problem would get credit.

 
At 11:35 AM, July 21, 2010, Blogger ___________________________ said...

Well, if the teacher is only stressing the taught method, but discredits the student showing work using any other method, then that is a problem. If the teacher just wants work shown, then I don't see the problem, as showing work allows for students to show the unique way to do it.

Most math classes I've had just emphasized showing the work, not using a particular method, unless the instructor was simply trying to teach that particular method at that point in time. Sometimes teachers teach multiple methods, but they want to make sure all are known because perhaps the different methods have different values or uses.

 
At 12:07 PM, July 21, 2010, Anonymous Anonymous said...

At least at a high level, math is all about the work, all about the proof of the answer, and not merely the correctness of the answer. How and why something is correct is an important topic in math and at a high level the student is often given the answer and told to prove it. Prove the Pythagorean theorem, for example.

 
At 12:08 PM, July 21, 2010, Anonymous David W said...

I'm with Micah on this one. Another point I discovered through my own experience is the value for myself in showing my work.

Eventually, no matter how smart you are, you'll run into a point where you can't hold the entire problem in your head while you work on it. If you don't, then you're not pushing yourself hard enough.

If you're in the habit of showing your work, you're also in the habit of using the paper to store some of the problem's details, which allows you to keep going once you hit that point.

For me, showing my work rapidly went from 'pointless, but I'll do it because the teacher says to' to essential - and that transition happened right when the math got complex enough that I couldn't hold it all in my head. It was right when I started making 'stupid' mistakes, like swapping the limits of integration and getting a negative result, or dropping a negative sign halfway though my calculation - stuff that didn't mean I failed to understand, yet still made me get the wrong answer.

A secondary benefit comes when doing things like large scale engineering in groups. Having your work down on paper allows others to check it and find your mistakes. Imagine if, say, the Mars Climate Orbiter software was written to require units be attached at all stages - they might have noticed the English/Metric conflict before the orbiter became a crater instead of after.

Regardless, I view being required to show your work on the same plane as being required to type with your fingers on home row instead of hunt and peck. It is annoying as you learn, temporarily slows you and feels pointless - yet it eventually allows you to perform at a higher level than if you didn't.

 
At 12:11 PM, July 21, 2010, Anonymous Anonymous said...

AFAIK, the story about Gauss includes his explaining how he got the answer so much faster than the other students. If he hadn't "shown his work" on this problem, the teacher would have been entirely right to assume that Gauss had copied the answer or perhaps known it from somewhere else.

I was an instructor in math at university. Any student who didn't show their work was assumed to be cheating. Harsh, yes, but I had to grade quite a couple of students, and I believe that a slight bit of cooperation from them didn't hurt them. In fact, whenever I probed into a step that had been handwaved, I found that the student usually had *not* understood how he got from point A to point B. Showing their work was essential to prove that they had understood what they were doing.

After that, I worked in one of the bigger management consulting firms. Yes, the most important thing was the result - but if the client drilled into one of your recommendations during the final presentation and we weren't able to "show our work" and tell the client how we arrived at that recommendation, Bad Things happened.

Bottom line: showing your work is not unnecessary drudgery, and I am completely with your kids' instructor on this.

 
At 12:29 PM, July 21, 2010, Blogger Stephen R. Diamond said...

Expertise in math involves not only getting the correct answer but also demonstrating the answer's correctness.

 
At 12:37 PM, July 21, 2010, Blogger David Friedman said...

"If your kids have invented new ways of solving the problems, they should talk to the teacher and demonstrate their logic."

Tried. Didn't work.

 
At 12:38 PM, July 21, 2010, Blogger David Friedman said...

"I'm sure a student who showed a valid but alternate way to solve a problem would get credit."

You are mistaken. The experiment was tried.

 
At 12:39 PM, July 21, 2010, Blogger David Friedman said...

"At least at a high level, math is all about the work, all about the proof of the answer, and not merely the correctness of the answer."

The weren't being asked to prove theorems, merely to solve problems. For most math problems, the chances of getting the right answer by an incorrect method are low, so getting the right answer is good evidence that you did the problem in a correct way.

 
At 12:41 PM, July 21, 2010, Blogger David Friedman said...

David W makes a legitimate point about the usefulness of doing some of your thinking on paper--but that doesn't say how much. If you can keep track of the argument with three lines of equations, there is no reason to insist on your writing nine. Nor in your writing the same nine the instructor would have written.

 
At 12:43 PM, July 21, 2010, Blogger David Friedman said...

"Bottom line: showing your work is not unnecessary drudgery, and I am completely with your kids' instructor on this."

You mean you agree a student should be graded down if he does not show his work in full on the exam paper, is willing and able--like Gauss--to explain to the instructor how he solved the problem? That was the situation in the cases I am describing.

 
At 12:47 PM, July 21, 2010, Anonymous Anonymous said...

I had a chemistry teacher who gave poor grades if you got the answer but didn't "show work". I was a stubborn 10th grader so I accepted the poor grades rather than change my ways.

Didn't have the problem with any math class, but math is often about proving things rather than getting a numeric answer.

 
At 12:51 PM, July 21, 2010, Blogger DR said...

I think the problem is that most math instructors at least until you get to math departments at good schools, probably themselves don't have that strong a grasp of the material. This is especially true in the vast majority of US high schools.


So the teacher is sure that when the problem is solved in the usual method that it's correct. Otherwise there is uncertainty about whether the student's logic is sound or not.

By mandating that the students must do the problem the "correct" way, the teacher may commit some errors in incorrectly marking students. But the bargain is that it drastically cuts down on rent seeking by students who got the problem wrong, but say it was correct and alternative.

Given that the vast majority of students who don't get a problem, didn't get it because they simply didn't know it, rather than actually doing an alternative method, this rent seeking will spiral out of control.

 
At 12:52 PM, July 21, 2010, Blogger Darf Ferrara said...

I agree with virtually all the comments posted here. Based on your responses it sounds like they just have a lousy teacher.

 
At 12:53 PM, July 21, 2010, Anonymous BobW said...

Students don't take classes to solve problems, but to learn how to solve problems.

Mathematical algorithms are software that runs in the human brain. Proving that you've loaded the software correctly and can run it without error is worthwhile.

I'm sure your children are brilliant. However... they need to learn to write it down. If they didn't write it down, it didn't happen.

 
At 1:01 PM, July 21, 2010, Blogger DR said...

The other thing is that a teacher will have a much harder time giving partial credit if the work is done by an alternative route or just the answer is given.

I think the idea is that people have overconfidence bias that their answer is correct so they don't invest enough in going through the steps that will earn them partial credit. To make up for this teachers mandate that all students show work.

 
At 2:52 PM, July 21, 2010, Blogger David Friedman said...

In response (in particular) to DR. This was a college course at a good, although not top of the line elite, college. My son, 17, was talking it as a "young scholar"--the college has a program for high school age kids who want to take college courses. My daughter, 20, was taking it as a part time student during the summer, to fill in a gap in her education before starting her third year of college.

On the other hand ... . I wouldn't be surprised, from my kids' description, if the math teacher didn't have really strong mathematical intuition, and so was more comfortable with the cookbook approach.

 
At 2:56 PM, July 21, 2010, Blogger David Friedman said...

DR writes:

"But the bargain is that it drastically cuts down on rent seeking by students who got the problem wrong, but say it was correct and alternative."

I don't think I follow that. If the student got the wrong answer, then his alternative way of doing it was wrong. If the student got the right answer, that's strong evidence that the way was correct. The answer is a simpler and less ambiguous test than the work.

Incidentally, I think this is a class with something like ten students in it, not a hundred, so there is no practical difficulty in the teacher listening to a student's explanation of how he did a problem. Nor in the professor making it difficult for students to cheat on tests--homework would be a different issue.

 
At 3:00 PM, July 21, 2010, Blogger David Friedman said...

"Students don't take classes to solve problems, but to learn how to solve problems. "

I disagree. They take math classes to learn mathematics. If all they are doing is memorizing a series of steps, without understanding why those steps work, they are wasting their time.

Every decade or two I do something that requires me to solve a constrained maximization problem. One standard approach is to use a Lagrange multiplier. Before I do it I sit down and spend half an hour or so convincing myself that that approach will indeed give the correct answer. If I didn't do that I would be quite likely to misuse the approach in one way or another.

As a referee, I've seen some pretty bizarre things done, in articles submitted for journal publication, by people who used mathematics without understanding it.

 
At 3:02 PM, July 21, 2010, Blogger David Friedman said...

DR writes: "The other thing is that a teacher will have a much harder time giving partial credit if the work is done by an alternative route or just the answer is given."

If the answer is wrong, the student who doesn't show his work should expect no credit, unless the form of the wrong answer makes it obvious that it was due to an arithmetic error or the like. But that's a risk the student should be free to take. If the answer is right, as in the cases I am discussing, the issue of partial credit shouldn't arise.

 
At 3:19 PM, July 21, 2010, Blogger Jonathan said...

I remember when I was a university student and was asked to write an essay about the relationship between two economic variables.

I had a mathematical background and thought that writing essays was cissy stuff. So I went to the library, dug up statistics, performed a regression, and on that basis declared that there was no relationship between those variables.

The lecturer said, "Great initiative. But now go away and write an essay about it."

At the time I was disgusted, and I suspect I never wrote the essay. I have more sympathy with the lecturer now, except that he never explained why he wanted me to write the essay: he just expected me to do as I was told. That's something I never liked about formal education.

 
At 4:45 PM, July 21, 2010, Blogger Jadagul said...

I'll preface this comment by saying that I am 1) a professional mathematician who 2) spent all of high school flatly refusing to show any work anywhere.

First, if what you're saying is strictly accurate, then the teacher is being quite silly. There's a difference between requiring your students to show work and requiring your students to show the same work that you would show. If the latter is what's happening then I don't really have an explanation (other than laziness--it really is easier to grade work if everyone does the problems the same way. I always hate grading the assignments that have like five or six equally valid ways to solve the problem--or even worse, five or six equally accurate ways to express the answer. Hyperbolic trig is an infamous culprit here. But even then, I feel like that's the work I've signed on for).

But second, I now have a much better understanding of why we need to show work. In the case where you're working out problems, and someone else knows the answer, you're right--the fact that I got the right answer usually demonstrates that I must have done other things right, because it won't happen by chance.

But the thing that makes math powerful is that not only can you get the right answers, you can assure both yourself and anyone else that the answers you got were right. If there's no mistake from line to line, and the inputs are right, we can be pretty confident the outputs are right. So you don't need to show work to do math; but you do need to show work for it to be really useful. So it's important to be able to do that.

This doesn't discount my first point; your children's teacher may still be quite bad. I have no idea. I don't know who it is.

 
At 5:47 PM, July 21, 2010, Blogger Lex Spoon said...

I have a different view, after having taught and been taught in a number of circumstances. The point of teaching is to guide someone down a path you have already been down, and to bring them along much more quickly than it took you. Tests, then, check whether the student has come down that path after all.

As an example, if I was teaching integration by parts, I'd want to see the student separate out the formula according that particular technique. A student that can guess and check an answer without doing integration by parts is many wonderful things, and they might even be a fine mathematician. Such a student did not, however, learn what I was teaching them.

 
At 6:24 PM, July 21, 2010, Anonymous RKN said...

Might it be the case that the particular example of the problem that appeared on the exam was solvable by a shortcut or novel approach, but the novel approach may not generalize to all problems of that class of problems, and that this was the basis of the instructor's objection?

I think this actually happened to me once, but if it did it was so long ago I don't recall the details.

 
At 6:45 PM, July 21, 2010, Anonymous Anonymous said...

Did question explicitly say "Solve ... Show your work."

Then not showing work is, strictly, not answering the question. Without that preface, the teacher is on weaker ground.

This policy is basically a social insurance against bad grades arising from arithmetic mistakes.

A few brilliant, yet stubborn, students fall through the cracks but the average student benefits from partial credit. Sometimes question requires several techniques and if you know only 2 out of 3, you won't solve it but can still pass that section.

Also, there's a fundamental difference between multiple choice and short-answer math questions!

 
At 7:09 PM, July 21, 2010, Anonymous Anonymous said...

My degree is in mathematical logic, where getting "the right answer" at the end is often worth nothing unless all the steps along the way are valid. For example, consider the following "proof":

David is on my Friends list.

Some of the people on my Friends list are professors.

Therefore David is a professor.

The hypotheses are true, and so is the conclusion, but it doesn't follow logically from the hypotheses. No credit.

Yes, I know the class in question wasn't about proof, but my point is that often the steps are more important (pedagogically) than the answer.

Since this was a class in calculus, consider that integration is largely a collection of techniques, of which any one, or several, or none, may apply to any given problem. I would consider it an excellent assignment to choose a problem that could be solved in several different ways and specifically tell students to solve it in each of those ways. There is no way to grade that assignment without seeing the steps. Of course, if all I wanted to do was test the students' mastery of each technique, I could give five different problems each amenable to only one technique, but that wouldn't make the same pedagogical point.

For yet another example, I frequently teach beginning computer programming, using a very strict, step-by-step "design recipe". Students who produce working programs without following all the steps of the recipe get only partial credit. Some of the sharper students inevitably skip the design recipe (or do it after the fact to meet my stupid, close-minded requirements rather than to help them solve the problem). Halfway through the course they hit a problem that they can't solve through pure brilliance (usually "given a list of objects, produce a list of all its permutations"), and they haven't learned the recipe well enough to apply it to these problems.

(I did have one student a few years ago with some learning disabilities: he could frequently reason out how to solve simple problems, but he could not follow the recipe; I could see his face closing up when I insisted on it. When he got to more difficult problems, he had no tools whatsoever with which to solve them.)

I sometimes justify this insistence on following the recipe by analogy with the movie "The Karate Kid": you do seemingly-pointless exercises in order to internalize the motions, so that by the time you need those motions for a real task, they're fluent and easy.

-- hudebnik (whose identity OpenID is inexplicably refusing to recognize)

 
At 7:26 PM, July 21, 2010, Anonymous Anonymous said...

I think this is a class with something like ten students in it, not a hundred, so there is no practical difficulty in the teacher listening to a student's explanation of how he did a problem. Nor in the professor making it difficult for students to cheat on tests--homework would be a different issue.

So your suggestion would be that for small class sizes, the professor should simply keep the students far apart while taking the test and be willing to listen to explanations after the fact, whereas for larger classes, that won't work so the professor should insist on seeing the work on the exam paper to detect cheating and confirm understanding?

Of course, some students in larger classes will complain of the "unfairness" that they (and not their roommate who took the class over the summer) have to show their work and can't justify it after the fact, so the professor will have to explain that. And justify exactly what class size qualifies as "large". And rewrite the lecture notes in a "large class" and a "small class" version. From the professor's point of view, that's a significant cost, for little or no benefit; one can certainly sympathize with the prof's impulse to apply a single consistent standard in all classes instead.

-- hudebnik again

 
At 7:42 PM, July 21, 2010, Anonymous Anonymous said...

Sounds like a poor teacher to me, if showing a valid (but different) path to the answer didn't work.

When I was in grade school I was forever getting in trouble for not showing all my work. It wasn't that I wasn't showing any; it's that I was combining steps that were "obvious" (like inverting sign and moving an operand across an equal sign; I didn't show two steps there). I trained myself to produce answers that would net full credit by asking myself how I would teach this solution to a lesser student and wrote that down. Darn frustrating, though.

I did get credit for alternate correct solutions, but usually not on the first try.

 
At 8:04 PM, July 21, 2010, Blogger David Friedman said...

One minor point--none of these were multiple choice questions. For those it would be legitimate to ask the student to show work--but in practice people aren't asked to show their work on multiple choice tests, because part of the point of giving such tests is that they can be graded mechanically.

 
At 8:14 PM, July 21, 2010, Anonymous Anonymous said...

If the answer is wrong, the student who doesn't show his work should expect no credit, unless the form of the wrong answer makes it obvious that it was due to an arithmetic error or the like. But that's a risk the student should be free to take.

A few years ago I taught a sophomore-level programming class. I decided to treat the students as adults. I told them they were to demonstrate as many as possible of a list of techniques, and they could do it by turning in as many or as few programs as they wished, whenever they wished. No programs came in until 2/3 of the way through the semester. Then a trickle, then a flood during exam week. As a result, (a) I didn't have time to grade all the programs at all; (b) the students didn't get feedback in time to be useful to them in subsequent programs; and (c) most of them demonstrated very few of the list of techniques.

The next year I added the rule that "I will accept at most one program per student per week." This solved problem (a), but had little or no effect on problems (b) and (c).

This past year I told students "You must turn in at least one program in Category A by this date, another by that date, at least on in Category B by this third date," and so on. Almost all the students met the deadlines, they got timelier feedback, they demonstrated more mastery, and they had a higher opinion of my "fairness" as a teacher.

Dan Ariely, in Predictably Irrational, describes a similar experiment, with similar results... except that he also surveyed both groups of students afterward, and they overwhelmingly preferred to be given deadlines spaced throughout the semester.

What's happening, of course, is that when a student (or anyone else) has a bunch of tasks to do in limited time, the "urgent" ones get done and the "flexible" ones don't. Oddly enough, if more of them are "urgent", more of them get done, with surprisingly little loss of quality; time is somewhat fungible in this sense.

This is a situation in which real people (not theoretical rational choosers) actually derive more utility when they have fewer options.

-- hudebnik

 
At 8:35 PM, July 21, 2010, Anonymous Anonymous said...

I suggest scanning the exam in and letting the choir judge. There's a little too much s/he said here, and seeing the exam would go a long way towards determining who is slacking here, the students or the teacher.

 
At 11:32 PM, July 21, 2010, Anonymous Miko said...

As another professional mathematician, I'll second everything that Jadagul said.

Also, while I will accept any method the vast majority of the time, there are special circumstances in which I will require that a certain method be used (and clearly note this in the problem). For example: early in Calc, students learn how to find derivatives using limits. Later on, they learn rules that make computing derivatives for elementary functions much easier. However, it's likely that a fair number of them will later get to a point where they'll be dealing with more complicated functions, or with functions generated from data, or with abstract functions, at which point they'll need to work with limits directly. As such, it's important that they know how to find derivatives using limits, even if the particular problem given to them is one that they know how to solve using special rules.

 
At 2:26 AM, July 22, 2010, Anonymous Kid said...

A lot of teachers seem to defend the idea of requiring a specific method to solve the problem, because you want to teach the students that method. In particular, I see a lot of "this problem might be possible to solve in an alternative way, but for more complicated problems the specific method I am requiring really is necessary".

The danger is that the students don't see why they are learning it, which doesn't help motivation.

You would win a lot of goodwill with me as a student, and probably with other students like me, by showing me one of those more complicated problems first. If you are right, and I am unable to solve it with my alternative method, I will much more easily accept solving more simple problems in tedious ways in order to learn new (and now for me relevant), techniques.

Of course, the teacher has to understand those more complicated problems himself, before claiming that the techniques he is teaching are necessary for them. Most of you will, but some teachers are hiding their own incompetence behind that argument.

 
At 9:07 AM, July 22, 2010, Blogger Albert Ling said...

In some deep philosophical sense, all math equations are equally "obvious" and the mathematical "steps" we use are just because we are so limited and stupid. Every "solution" can thus be equated to an axiom.

So I think in the future, when we enhance our brains, all these "steps" kids learn at school will seem as trivial as "proving" that 1+1=2

 
At 9:10 AM, July 22, 2010, Blogger Curmudgeon said...

Two thoughts:
a) Many who teach Calculus are used to the AP grading methods and purposefully teach accordingly. This means that a 9-pt problem has points for each part of the problem: 1pt for the correct limits of integration; 2pts for Primative; 1pts for F(7); and so on. If the limits of integration are wrong, then the rest is wrong but AP requires that certain bits of information be specifically stated in order to receive this point or that one.

B) Simplistically: reduce 26/65. If you cross out the sixes, you get 2/5, which is the correct answer. Sometimes, you get the right answer in an incorrect fashion. I want to see how you obtained that. If you crossed off the sixes, it's wrong. ( 49/98 and two other fractions work in this fashion, if you're interested)

I do agree, though, that a class of 10 or so summer calculus HS advanced kids is the PERFECT GROUP OF STUDENTS and the teacher should listen to the explanations and grade according to understanding instead of on scribblings on paper. Pissing off this kind of a group is self-defeating.

 
At 9:49 AM, July 22, 2010, Anonymous Bruce said...

The appropriateness or inappropriateness of the teacher's approach might depend on the actual problem. Could you post that?

The issue reminds me of a similar (but happier and funnier) event from high school. We were doing integration by trigonometric substitution, and the teacher had each of us come up to the board (in sets of 3 or 4) and write out our answer to particular problems. Most of the problems were fairly complex, and solving them took many lines.

One of the problems was the integral of 3 / (1 + x^2). This actually is quite an easy problem, and one that takes only one step. I believe that the authors of our book included this problem to remind us that not all problems that look complicated are complicated and not to get stuck in a rut.

The student who was assigned that problem said in a deadpan voice, "You remove the 3, so you get 3 x the integral of 1 / (1 + x^2). You then look at it and think 'arctan'" He then wrote 3 arctan x + c and sat down. The class cracked up, but he was exactly right. That saying -- "you look at it and think arctan" -- became a funny phrase that we used for the rest of the year (and for years afterwards) for those "aha" moments.

The fact that I remember this after 30 years shows the cleverness of the problem.

 
At 10:11 AM, July 22, 2010, Blogger David Friedman said...

To Curmudgeon:

Note that this wasn't a high school class. It was a college class at a pretty good university which has a program that lets high school age kids, usually in the summer, take a college class. So far as I know, only one of the students in the class (my son) was high school age.

 
At 2:39 PM, July 22, 2010, Blogger Albert Ling said...

David, escaping a bit from the subject, what are your thought about heretability of IQ (what % is genetics and what % is enviornmental)?

Also, what are your thoughts on jewish ancestry correlating so highly with above average IQ?

 
At 7:11 PM, July 22, 2010, Anonymous StrangeLoop said...

As a teacher myself, the teacher

(1) can demand work to be shown; this is part of the stated assignment and can be graded as such

(2) should accept alterantive problem-solving procedures unless these techniques are explicitly excluded.

"They take math classes to learn mathematics. If all they are doing is memorizing a series of steps, without understanding why those steps work, they are wasting their time."

I consider mathematical truth to be purely syntactical in nature; that is, we follow rules. "Understanding why those steps works" should be important in the instruction (and, I believe, produces the primary joy in learning maths), but, let's face it: conforming to a prefabricated process is what most of mathematics is. That's not even easy to many people, as mathematical illiteracy is widely admitted.

Cheating today is easier than ever: caculators can store data, wireless gadgets can be easily hidden, even cheat sheets can still be easily concealed. Proving one's knowledge without much additional cost (the effort and time of writing one's calculations qua the steps the instructor demands) seems an effective measure by the teacher to insure that learning is actually happening. That's really the root of this practice: answering the question--did the student learn?

 
At 7:20 PM, July 22, 2010, Anonymous StrangeLoop said...

Albert Ling, are you espousing the anti-anti-Semitic view that Ashkenazi Jews are smarter than the rest of us?

I resent that. Despite the evidence.

P.S. A nice popular survey of twin studies and other data that indicate a high heritability of intelligence is Pinker's The Blank Slate: The Modern Denial of Human Nature (2002).

Although, to confirm any elitism you may have, The Bell Curve (1994) is a fascinating read.

I still think I'm pretty smart, even though I'm a lousy gentile.

P.S.S. When you ask the "nature or nurture" question nowadays, most informed non-geneticists will respond, "Well, everyone knows it's both--we must avoid the extremes of blank slaters and genetic determinists!"

 
At 5:53 PM, July 23, 2010, Blogger robert said...

Fermat didn't show his work, and it took his classmates over 350 years to figure out whether he got the right answer. He did. Ramanujan often didn't show his work either. The problems were very hard, at least to anyone other than him. He was usually right, but occasionally wrong. His classmate Hardy, who was very good at the 'show your work' part for hard problems, helped him with this part of the assignment. These being distinguishable, though highly correlated, aspects of mathematical capabilities, the two friends made a good team.

 
At 9:17 PM, July 23, 2010, Anonymous Anonymous said...

We lucked out with Josh's math teacher. The guy was a pro.. He demonstrated to both Josh and us that even if it was a tough slogg through the process that he was teaching -- that he did have a reason for approaching the lessons the way that he was doing it. He explained that what he was doing was slowly building up a foundation that they would eventually be able to build in any direction if they studied any kind of math in the future.

This guy was great with Josh - was available every day after school to meet with any kid that wanted extra help -- and Josh ended up spending almost every afternoon afterschool doing not only his math homework with the teacher - but also his other homework too. He just really liked this teacher.

We are going to keep Josh in his present school even though we are moving this month because he wants to keep the same teacher...

The teacher told us Josh has a very natural aptitude for Math -- and he was patient and more than happy to help Josh try and figure out his own way of doing things in solving problems.

He explained to Josh that doing that kind of thinking and trying to really understand what he was doing by trying to solve it on his own was part of what would stand him in good stead in the serious pursuit of math later...

He did demand that Josh do the routine "show your work" -- but his explanation as for why that was the case satisfied Josh - particularly since he was willing to explore other ways of doing it with Josh -- in the after-class time.

I expect that his reasons for doing so, and the reason for Bill and Becca's tutor being a hard-ass have to do with trying to cut down on work... They have to teach a certain number of things and they have figured out (in their view) what the easiest way to do it is... And they don't want to waste time with duplicating effort or having to go over someting again and again when it's not needed.

The lesson plan that Josh's teacher had was to teach a whole group of kids a certain series of math principles and make them reasonably proficient in all in the time given... I suspect that this is where the rigidity comes from... If kids are not showing their work --they might find it harder to assess where problems in their understanding is arising from.

Josh's teacher was willing to deviate from his methods -- in this regard... but only for the kids like Josh who showed that *they* were willing to make just as much effort as he (the teacher) would be making in working with them outside of the methods/curricula he had devised for the class.

Meaghan

 
At 3:10 PM, July 26, 2010, Anonymous Paul Birch said...

Bruce:
Maybe I'm being thick, but I can't see how to integrate 3/(1+x**2) in a single step (other than by knowing the answer).
The best I can come up with is
Let x=tanu, then dx=(1/cos**2u)du=(1+x**2)du.
So Integral(dx/(1+x**2))=Integral(du)=u=arctanx.
And even that leaves out quite a few "trivial" steps where one can easily go wrong.

 
At 8:56 PM, July 27, 2010, Blogger Karl said...

Given what you've said didn't work, I think it's time to consider looking for a different teacher.

In college, I experienced two calculus professors. One was very particular, to the point where if you used curly braces where he used brackets, he marked you down. The other one was much more flexible, and was much more concerned that students could show they got their answer by a valid method. The difference was worth one grade point for me.

 
At 8:35 AM, July 28, 2010, Anonymous bob@wakefield.net said...

I disagree. They take math classes to learn mathematics. If all they are doing is memorizing a series of steps, without understanding why those steps work, they are wasting their time.

Before you decide whether or not to use a particular method to solve a problem, you need to have mastered that method.

There are three components to mastery: talent, understanding, and skill. You need the basic equipment to do the work, the knowlege of the subject, and enough practice to do it without errors reliably.

Smart people tend to undervalue skill. Practice is boring, and in class they tend to pick things up faster than the students around them. All too often those slower students excel because they have practiced with the tools and can use them efficiently and accurately.

"Genius is 1% inspiration and 99% perspiration."

 
At 7:57 AM, August 08, 2010, Blogger Ilíon said...

"I am now imagining that instructor as the schoolteacher who tried to keep a class of children—among them the young Gauss—quiet by having them add up the numbers from one to a hundred."

Ah, so it was Gauss (whose name I did not catch, because it meant nothing to me at the time).

When I was in 8th grade, our math teacher instructed the class to sum the number from 1 to 100 ... which I did in just a couple of minutes (most of which I spent verifying that I'd really hit upon such a simple and time-saving way to get the correct answer), using a modification of a method (*) I'd come up with years before. This prompted him to ask whether I'd already been taught about this before transfering to that school (I hadn't). Embarassingly to me, he referred to Gauss as "a genius" ... you know, the whole social structure of children-in-groups thingie: one is allowed to be smart, but not too smart.

(*) that is, when presented with a column of numbers to sum, I'd mentally rearrange them so that I could count/add by 10s and 5s. My sister also does this, but we don't know whether I taught it her (I'm 3.5 years older) or she independently hit upon it.

 
At 8:03 AM, August 08, 2010, Blogger Ilíon said...

This comment has been removed by the author.

 
At 8:05 AM, August 08, 2010, Blogger Ilíon said...

From the link I found: "The formula, if you will, is to add 1 +100, 2+99, 3+98, ...48+53, 49+52, 50+51. So, we have the number 101 fifty times or 5050."

In my case, because it was by habit to mentally rearrange a series to be summed into multiples of 10s, the "formula" I came up with was (0+100) + (1+99) + (2+98) + (3+97) ... + (49+51) + 50 = 50 * 100 + 50 = 5050

 
At 9:00 AM, August 08, 2010, Blogger Ilíon said...

It seems to me that several commenters are missing that Mr Friedman said:

"One of their complaints is that he grades them down on exams when they get the right answer but don't show the step by step procedure for getting it that they have been taught."

That is, it appears that the issue is *not* that they are not "showing their work," but that they are not solving the problem in the One True Way.

WV: misses ... yeah, it seems that the instructor misses the point of education.

 
At 12:51 AM, August 19, 2010, Blogger Mark Horning said...

If you can keep track of the argument with three lines of equations, there is no reason to insist on your writing nine.

I agree entirely. And for some folks, it's actually difficult to find smaller steps that are not really necessary.

Showing work serves two purposes. One, it allows one to explain to another how to go about solving the problem. Two, it allows the professor to give partial credit for a wrong answer so long as they can follow the steps and determine where the mistake was made. (otherwise known as if you don't know the answer put in lots of work)

I assume this is not a case of showing Zero Work, but merely that the students think in larger steps than the professor.

Although I do recall once in grad school (Physics) where I was asked to do a problem which looked very complicated. I looked at it and wrote down the answer. Pi/2.

You could have done it in two pages of work or you could simply invoke symmetry and know what the answer was using the properties of trigonometric functions.

 
At 7:49 PM, September 19, 2010, Anonymous John said...

I always felt very cheated when the same professor who was demanding that we show our work would respond to questions about his examples with "that's intuitively obvious" or simply "Oh, it's obvious". Another variant was for him to write the question then say, "clearly then..." and write the answer.

 
At 11:49 AM, September 23, 2010, Blogger Ilíon said...

Thats' a good point, John. *grin*

 

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