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June 13, 2007

Why Math Teachers Get Grumpy

Posted by John Baez

Did you ever have a math teacher who seemed grumpy? At least in America, schools are full of them. Why do they get that way?

Maybe it’s because they spent too long controlling rowdy students, or grading papers full of mistakes. Maybe they realized trying to force kids to think clearly was a losing proposition… but didn’t know what else to do. They may feel they’re on the front lines of a war against ignorance… a war the other side keeps steadily winning, year after year. Whatever causes this bitterness, it’s a terrible thing. Teachers like this make students hate the subject!

Teaching can be tough on the soul. For me, the hard part is grading midterms and final exams. I feel that’s when I earn my pay. Everything else about my job is lots of fun. Grading the same problem, over and over, 50 to 100 times… that’s really work! At the end my mind is reduced to jelly — I can barely total up the grades.

When things get tough, a sense of humor comes in handy. I just finished grading the finals for my undergraduate number theory course. It wears me down reading phrases like “suppose ii and jj are both distinct integers”, or “let pp be a unique prime.” But one proof by contradiction brought a smile to my face…

Here it is:

Prove that there are infinitely many prime numbers. Assume that there are a finite amount of prime numbers p 1,p 2,,p np_1, p_2, \dots, p_n for some number nn. That means at prime p np_n you can’t have any more prime numbers. What about p n+1p_{n+1} or p n+2p_{n+2} or p howevermanyittakestogettothenextprimenumberp_{however many it takes to get to the next prime number}? That is a contradiction. Because there is a next prime waiting. Somewhere.

I like that: “…there is a next prime waiting. Somewhere.”

If you teach math: do you have to work hard, sometimes, to keep from getting frustrated? What do you do?

(This is my second post on Mathematics: the Dark Side. As someone who loves mathematics, I think we should face up to its dark side and maybe see what we can do about it.)

Posted at June 13, 2007 12:01 AM UTC

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60 Comments & 1 Trackback

Re: Why Math Teachers Get Grumpy

In lower-level courses, where the students are still learning formal mathematical style (e.g. meaning of unique, distinct, …), I have found it useful to award 20% of the points for writing style. Then I try to make clear what this means. The two most important words to tell them are precise and concise. This seems to help them break the habit of writing down every single thing that occurs to them in the hope that somewhere in there the grader will find something worthwhile.

I also like questions that ask them to give either an example of X or a proof that no such example exists. I find them much less soul destroying to grade than the standard question that asks them to prove Y. For example, give a complex number that generates a degree 11 Galois extension of Q.

Maybe they’re more fun to grade because you often get unexpected but correct answers, so there are many ways for their mathematical personalities to shine through. Also you can often tell instantaneously if they got it right or not. I even think they are better measures of understanding. But I don’t know how you could make the proof of the infinitude of primes into something of this format, and it’s hard to get more important than that. But with that question, they can only either meet your preconceived ideal or come up short, so you’re almost setting yourself up for disappointment, in my opinion.

Posted by: James on June 13, 2007 8:18 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

At UC Riverside, this course on number theory is one of the first courses our math majors take that require lots of proofs.

They usually take a basic course on set theory first, but most faculty seem to agree that naive set theory is not a good place to start learning the art of proof. It’s not easy to formulate interesting conjectures in set theory that are easy to test but nontrivial to prove. Perhaps for this reason, the students coming to number theory are just beginning to learn what counts as a proof, and the special writing style that mathematicians use in proofs.

So, when I taught this course, it was roughly half about the joy of finding beautiful patterns in numbers, and half about learning to prove things — which in large part means: learning to write like a mathematician!

Given the current state of California’s public schools, this also means: learning to write!

I warned the students that if they thought math was the subject where you just did calculations and didn’t need to know how to write, they’d been fooled — and now they were learning otherwise.

I told them that at every stage of education, we fool our math students! We never just sit them down and tell them what math is all about. We make them learn it a tiny bit at a time, the hard way.

So, first they think math is about arithmetic with numbers. Tnhen they discover it’s about solving equations algebraically. Then they discover it’s about doing integrals and solving differential equations. Then they discover it’s about proving things in books. Then they discover it’s about coming up with your own things to prove. And finally, when they get a job, they discover it’s about teaching students to do arithmetic with numbers.

Since it’s not easy to write original proofs with a clock ticking, I downplayed tests, and put a lot of emphasis on homework. I spent each weekend grading 45 homeworks, each containing several proofs, writing comments on them. I distributed copies of some of the students’ proofs and critiqued them in class, showing them several different proofs of the same results, and trying to get them to discuss these proofs. I had them redo some proofs they’d already done.

It seemed to work reasonably well… some of the students improved a lot!

It did however have a bit of the ‘Proof Boot Camp’ feel to it — with me as drill sergeant. I tried to soften this with lots of jokes.

When it came time for the final exam, I told them a long list of definitions they needed to know, proofs they had be able to regurgitate, and so on. There were also calculation questions, some involving creativity.

So, proving that there were infinitely many primes was mainly a test to see if they could remember a mathematical argument and write it in a nice style.

But with that question, they can only either meet your preconceived ideal or come up short, so you’re almost setting yourself up for disappointment, in my opinion.

I see what you mean. A lot of them did it perfectly. A lot of them didn’t. But you’re right: nobody was able to wow me on this question. What pleased me more was how some people figured out how to compute

11 8mod17 11^{8} mod 17

without using Euler’s Criterion. I also enjoyed how some of them recycled some of my jokes, such as:

Next we need to show that pp is prime. This is obvious: if it weren’t prime, it wouldn’t be called ‘pp’.

… clearly understanding that this was a joke, not serious.

Grading the finals was still soul-sapping. I felt the need to vent my grumpiness yesterday, and I wanted to explore the question: why do math teachers so often sink into terminal grumpiness?

But, I’m already in a better mood today! Summer is here, and soon I’ll be travelling through Europe, giving talks, thinking about all sorts of fun questions.

Posted by: John Baez on June 13, 2007 6:29 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Terminal grumpiness is still better than Pi-like insanity. Come to think of it, the ratio of mentally disturbed mathematicians to relatively stable ones in modern movies is remarkably high. (Pi, Proof, A Beautiful Mind — all of ‘em I can think of except, I guess, Good Will Hunting.)

I suppose it’s hard to make people pay to see a movie about grumpiness and irascibility.

Posted by: Blake Stacey on June 14, 2007 3:52 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Blake wrote:

Terminal grumpiness is still better than Pi-like insanity.

Of course, what’s really best is ebullient eccentricity, as practiced by Conway, Erdos:

and The Wizard — although the Wiz gets too grumpy at times to be a good role model.

Posted by: John Baez on June 14, 2007 5:18 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

and The Wizard

I love these wizard stories. They were my very first encounter with you, back when.

I know that some part of myself, even though I kept ridiculing that part of mine, was disappointed when much later I saw your photograph and you didn’t have a long white beard.

(I mentioned that in Toronto, I think, it’s an interesting phenomenon: when reading a novel, even if the characters are only described very vaguely, one cannot help but create a mental image of all persons. That works subconciously, I think. I certainly cannot control it.

What makes this a curious phenomenon in the modern age is that the same mechanism is apparently at work when it comes to pure email acquaintances. At least for me.)

Posted by: urs on June 14, 2007 6:23 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Urs wrote:

I know that some part of myself, even though I kept ridiculing that part of mine, was disappointed when much later I saw your photograph and you didn’t have a long white beard.

What do you mean?! Here’s what I look like (without my hat):

Maybe you got fooled by Baez — sometimes that rascal pretends to be me.

Posted by: The Wizard on June 16, 2007 9:13 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Didn’t you fall into the Shadows at Moria?

Posted by: Thomas Larsson on June 16, 2007 3:16 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

That robe is neither grey nor white. I’d say it’s blue, which would make him one of the Ithryn Luin – either Alatar or Pallando. We don’t know much about them, as their mission was in the East, beyond the scope of the Red Book.

Posted by: John Armstrong on June 16, 2007 3:58 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I remember when I was a Part III student writing my essay on knot theory, learning about the fantastic Jones polynomial and having a strong image in my head of Vaughan Jones as a slightly reticent, rather tall Victorian gentleman with a top hat. I then went to a conference in Warwick, organised (I think) by the two Joneses, and for days I was looking round the audience trying to spot this chap. Someone eventually pointed out to me who he was…

Posted by: Simon Willerton on June 16, 2007 12:36 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

At least where I live, one problem is that too many students decide to become teachers not because they enjoy teaching, but because they fail in what used to be called “Diploma” studies here and what is now, following the Bologna process, called the “Master” you all know and love.

That leads to a vicious circle, disastrous for all those involved.

Because, let’s face it: math and physics are dry on first sight. That’s why there’s a national geographic TV channel but not a math TV channel.

The thing about math and physics is that these are esoteric subjects: only those initiated will be grasped by its fascination.

We had a funny story here about a student who went to a professor for advice and told him: “I am trying to become a teacher for physical education [since there are too many applicants, it’s hard to get into this program]. If that fails, I can still become a math teacher.”

Posted by: urs on June 13, 2007 10:33 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Is that meant to imply that you think National Geographic is not dry? If so, I suspect you need to get out more :)

Posted by: Eric on June 13, 2007 3:05 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Is that meant to imply that you think National Geographic is not dry?

Ah, you have been spoiled by our stacky Die Hard action movies.

;-)

But in any case, it seems I missed John Baez’s point:

I had experienced a couple of grumpy math teachers, and my impression was that the main reason for their grumpiness was that they had been neither interested in math nor in teaching students.

But I guess John’s point is that there is something about teaching math at a school or university that makes you grumpy even (or maybe especially) when you love both the subject and teaching the subject.

Posted by: urs on June 13, 2007 7:28 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Comparing teaching at the university since some time with years of private tutoring for (university) students, my impression is that:

- college/highschool pupils asking for private math tutoring are usually brighter than students in the math lecture.

- school trains students to learn by memory and that only fools try to learn by understanding. They are encouraged to play ‘learning’ .

- declined language skills (e.g. the average vocabulary fall by ca. 30% here the past 2 decades) causes problems in understanding math statements.

- the communication in private tutoring is much better than in the university.

Posted by: Thomas on June 13, 2007 3:45 PM | Permalink | Reply to this
Read the post Why Mathematics Is Boring
Weblog: The n-Category Café
Excerpt: Storytellers have developed many strategies for luring in readers and keeping them interested. Mathematicians systematically avoid these.
Tracked: June 13, 2007 5:36 PM

Anecdotally… Re: Why Math Teachers Get Grumpy

I taught 5 semesters of Elementary Algebra and Intermediate Algebra at Woodbury Universaity, in Burbank, California. My students were mostly majoring in Architecture, Business, Fashion Design, Interior Design, Graphic Arts, or Animation.

Hence I developed a curriculum, approved by the Math Coordinator, which played to the visual strength of the students.

I gave half credit on exam questions under the system: “If you don’t know the equation, draw me a good picture. If you don’t have the equation or a picture, write me a paragraph or so of narrative. I just want to know what is going on in your head, so that I can help you.”

I worked into the course slides of Leonardo da Vinci drawings, dozens of ping pong balls and whiffleballs and party hats cut and pasted to cones of differing angles and with various conic sections, and the breakdown of the Federal Budget as reported and graphed in the New York Times.

Best narrative answer on one of my exam questions:

Q: “How would you use a programmable calculator to solve the following problem…”

A: “I would go to someone who took this course before and say, ‘Dude, if you tell me the answer to Professor Post’s question, I’ll give you this programmable calculator.’”

I gave full credit for that. Obvious Management material.

This summer I am teaching, technically as a Substitute Summer School teacher, at a local High School, to predominantly poor Hispanic and African American students, many of whom have been unable to pass the Califonia High School Exit Exam.

In my interview, I asked the charismatic Principal (who’d been awarded “California Princcipal of the Year” in 2005): “If your car breaks down, do you go to someone who has read some books about cars, or to someone who fixes cars every day? I do Math every day because I love Mathematics.”

I like teaching the best students (my son just earned his double B.S. in Mathematicas and Computer Science, at age eighteen), and the worst students. Anyone can teach the ones in between.

I tell students, and demonstrate it every day: “Every number shows a picture. Every picture tells a story. So every number tells a story.”

Posted by: Jonathan Vos Post on June 13, 2007 6:03 PM | Permalink | Reply to this

Re: Anecdotally… Re: Why Math Teachers Get Grumpy

That narrative answer sounds suspiciously like one of the answers on the classic list of ways to find the height of a building with a barometer.

Posted by: John Armstrong on June 13, 2007 6:38 PM | Permalink | Reply to this

Barometric Spaces; Re: Anecdotally… Re: Why Math Teachers Get Grumpy

(1) Yes, but I swear it’s true. I keep my exam bluebooks from students archived. Many have very good drawings, often of Homer Simpson, a favorite there since one of my fellow faculty members had been an animator for The Simpsons and Futurama.

(2) I submitted a novel manuscript which I’d written in high school, 1967-1968, which had an extended version of the Barometer Story as a chapter. I was offered $5,000.000 for it by a New York book editor, Patrick LoBrutto, I believe then at Ace Books, said offer later rescinded.

(3) My wife remnds me that I need to finish my novel manuscripts “Fast Times at Stuyvesant High” and “Axiomatic Magic.” Each over 80% complete…

(4) The last of my short story manuscripts which my father approved (himself a famous editor) was called “Sex, Savagery, and Semiprimes.” It is not suitable for posting on the Web. But the sex and violence in integral to the plot, about a girl in a third-world children’s army who has modeled her life on Sophia Kovalevskya, but in less survivable circumstances.

Posted by: Jonathan Vos Post on June 13, 2007 8:08 PM | Permalink | Reply to this

Re: Anecdotally… Re: Why Math Teachers Get Grumpy

A friend of mine was asked to determine the height of a tall object using an unusual tool, although in his case (this was at a job interview after graduating MIT) he was told that he was sitting atop a flagpole on a cloudy day with a battery-powered oscilloscope. Using the oscilloscope, how tall is the flagpole?

The “good” answer the interviewer had in mind was to drop a shoe and time its fall using a timer function on the oscilloscope. My friend said, “I’d drop the oscilloscope.”

“How would you time its fall?”

“With my watch.”

And he unclasps the watch he wears hanging from a belt loop, which the interviewer had not noticed.

Posted by: Blake Stacey on June 14, 2007 3:42 PM | Permalink | Reply to this

Re: Anecdotally… Re: Why Math Teachers Get Grumpy

John Armstrong wrote:

That narrative answer sounds suspiciously like one of the answers on the classic list of ways to find the height of a building with a barometer.

Right. It also reminds me of a recent quiz where Associated Press asked a bunch of US presidential candidates what they’d most like to have if they were stranded on a desert island.

Only one of them gave the correct answer: “A boat”.

I guess this kind of joke goes back to Alexander cutting the Gordian knot.

Posted by: John Baez on June 14, 2007 6:09 PM | Permalink | Reply to this

Re: Anecdotally… Re: Why Math Teachers Get Grumpy

Calculator joke : funniest thing I’ve heard in ages! pic

How do people feel about Thurston’s article on Mathematical Education?

Posted by: Bruce Bartlett on June 13, 2007 8:01 PM | Permalink | Reply to this

Re: Anecdotally… Re: Why Math Teachers Get Grumpy

My brother emailed me this:

Architecture, Business, Fashion Design, Interior Design, Graphic Arts, Animation (my addition: Wood-working, and many others) all required math in any level.

My woodcraftsman friend volunteered teaching part-time (once a week) with deaf students from Deaf School in Fremont, California.

He said to students that it is important to learn math, then a student interrupted him and said, “Math? What for?”

Ron held his breath for a second and went on explaining to him for a better part of non-stop 30 minutes. That math is essential to survive in the world outside the school. He explained that if the student want to be a self employed (kind of rare in any professional), he would have to know MATH to do financial booking, inventory, bills, salary, etc.

Period.

The student shrank in his seat, all blushed while every students gasped and nodded.

Nicholas Post

Posted by: Jonathan Vos Post on June 14, 2007 8:28 PM | Permalink | Reply to this

Re: Anecdotally… Re: Why Math Teachers Get Grumpy

Of course, what’s really more important for the general population about mathematics than algebra (bane of WaPo columnists) is mathematical reasoning. Any given person may never need to find the roots of a quadratic polynomial, but what they will need is the rational thought processes involved in deriving the formula.

The problem is that I have no idea how to teach that sort of thing directly. I don’t know if anyone does. So we’re left with drilling these applications of rational thought and hoping that the rationality comes along for the ride.

Posted by: John Armstrong on June 14, 2007 10:50 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Here is why Department Chairs get grumpy:

Student number 1 who has not been advised for Fall semester walks in and does not understand that his advisor is on vacation during the summer term. Student has all but flunked out because during his first year in school he worked 3 jobs. He reports that he will take nothing but PE courses next semester to bring his GPA up. He wants to teach math at the high school level.

I explained to him that by taking English 101, elementary statistics course, and whichever math course he already flunked out of, and by studying, he could bring his GPA up. I guess the idea of studying while in school was too strange.

Student number 2 came in to complain about her instructor and wanted to change sections. Summer school here is well into the 3rd week. I was unwilling to do this for the student.

According to the student, the instructor told students that the material on the first test would not be taken directly from the homework and that there would be problems that they had not seen before. I explained as patiently as I possibly could that this meant that the student should look at the homework problems that HAD BEEN assigned, and then try to work the problems that HAD NOT BEEN assigned. I also explained that the instructor was trying to get students to think on their own, and to learn to synthesize the material.

During the course of the interview, I tried to help the student learn how to read the book, understand that chapter and section titles contained words that she was expected to understand, and I repeatedly asked her to explain concepts that had been introduced in the book. She did not do well at this task. The student repeatedly thought that the instructor’s job was to teach her — which it is — but she was not taking the personal initiative to learn the material. Apparently, reading the book and working a substantial number of non-assigned problems was thought to be unreasonable.

Since she was concerned about passing the test, I did everything that I could to tell her how to pass the test: Work the assigned problems, work the unassigned problems, read the book, and understand the concepts. I showed her where to find problems in the chapter review, and we went over some of the problems in the most recent section on which she had done homework.
While I was not sympathetic to her complaints about the instructor, I did tell her, as well as I could, what the instructor might be expecting. She thought that she had purchased instruction, and that her teacher was not teaching her. When she received instruction on how to study from me, she did not want to learn what I was teaching.

Her request was to change instructors; I learned after she left my office that the other instructor did not want her in his class. The other instructor had enough boneheads already in his class already.

Student number 2 was majoring in international business. I guess she won’t need much math there, will she?

A plethora of students who major in elementary education can currently get certified to teach with a D in precalculus algebra. “I won’t be teaching little kids THAT.” I keep wondering why not. When *I* was in college, Herstein’s remark in the Preface to Abstract Algebra seemed prescient, “The last few years have seen marked changes in the instruction given in mathematics at American Universities. This change is most notable at the upper undergraduate and beginning graduate levels. Topics that a few years ago were considered proper subject matter for semiadvanced graduate courses in algebra have filtered down to, and are being taught in, the very first course in abstract algebra. Convinced that this filtration will continue and will be intensified in the next few years, I have [written this book].”

Well gee, I don’t know about you all, but I am still waiting for group theory to make it into the high school curriculum. Why can’t it? Not because the students are particularly dumb, but their teachers think that students can’t understand abstraction.

Most of the children I talk to can handle abstraction pretty well. It is the adults who have trouble.

Every third grader that I have encountered can understand modular arithmetic. Most can learn to multiply using differences of squares, and with a little patience and a few floor tiles, they can understand that 1+3+5+…+ (2n-1) = n2.

College students after a semester of work still cannot compute the slope of a line, nor complete the square, and administrators are asking why more are not passing.

One local administrator, currently retired from the university and now an elected official on school board wanted me to articulate some set of skills that I thought every college student should have aquired. I thought but did not say, “Adding fractions would be nice.”

There is a culture against mathematical thinking because it is thought to be too difficult. Personally, I always found baseball to be much more difficult — a GOOD hitter gets on base less than 30% of the time. Coaches can yell at the team because it builds character, and it is for the good of the team. A teacher who tells students that they are not working is being condescending. What about the good of society? Bridges that don’t fall down, a careful analysis of planetary temperatures, and the ability to solve some problems?

That felt good!

Posted by: Scott Carter on June 13, 2007 10:03 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

There is a culture against mathematical thinking because it is thought to be too difficult .

It’s not just seen as hard, it’s seen as useless.

Posted by: John Armstrong on June 13, 2007 10:59 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Clearly Richard Cohen should be one of the first up against the wall when the revolution comes. Or maybe, we should just offer to compute his house payments for him.

Posted by: scott carter on June 13, 2007 11:16 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

For a fee of course. We can even give a quote in terms of percentages.

Posted by: John Armstrong on June 13, 2007 11:50 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I’ve found that the problem with students who are unable to do things like add fractions stems, mostly, from a lack of understanding of what fractions actually ARE, beyond some sort of symbol that can be manipulated using certain rules - which students can then forget.

I agree this is a problem with not teaching abstract ideas early enough. Just as children can acquire languages more easily than adults, I would guess that they can acquire abstract concepts more easily. I remember, at age 7 or so, looking in a first year college text on logic which my Dad had on his shelf. I had no particular trouble figuring out truth tables for compound propositions. Give me an equally simple abstraction today, and it takes quite a bit longer for me to internalize it.

I sometimes try to describe to non-mathematical people what kind of preparation students get for college math by analogy. It’s as if English majors arrived in their first year able to sound out words phonetically, and read and in some cases write simple sentences - but never having heard of a paragraph, let alone read a book, and unaware of the difference between a verb and an adjective. You could hardly blame the students if that were the high school curriculum. But you would have to take several years teaching them how to read and write beyond the level of single words.

Posted by: Jeffrey Morton on June 13, 2007 11:46 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

One of the things that strikes me about mathematics is that unlike, say, physics, chemistry and (to my limited knowledge) most other scientific subjects, there’s often a big difference between what gets covered in school (in UK at least) and what’s covered in a university course. Physicists, eg, learn a bit about mechanics, magnetism, electricity, the vague ideas behind special relativity, etc, at school and then get more intense, detailled studies at university. In mathematics, you aren’t given even a glimpse of most of the stuff you’ll be studying whilst at school. In the English analogy, it’d be like learning to how to read individual sentences and grammatical form without ever looking at longer pieces that embody dramatic/humorous/etc narrative, varieties of style, etc.

One thought that occasionally kicks at me is if there’s some way to combine computer graphics with some interesting and reasonably formalizable area so that school students get to see intricate, interesting things and are then in a position to do something semi-formal with them. (As a really bad example, come up with some way to automatically visually display a finite groups and subgroups and lead them into spotting Lagrange’s theorem. Of course, the “lead” bit is the difficulty.)

Posted by: dave tweed on June 14, 2007 10:20 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

My inclination would be instead to use computers for simulating physical (or biological, ecological, etc.) systems. For example, planets could be made to orbit a star around the screen; when the students go, “Aha! It looks like an ellipse,” the class could go on to look at proving how 1/r 21/r^2 gravity gives you conic-section orbits. A similar thing could be done for particles bouncing around inside a box, to motivate kinetic theory and statistical mechanics.

I read in Gleick’s Genius that Feynman proposed giving elementary-school students simple algebra problems and encouraging them to solve the problems by trial and error: “two times what plus four is ten?” Even if you don’t have the full apparatus to solve every conceivable algebra question, you can make progress by guessing, checking, estimating and so forth.

Isn’t this how numerical and analytical solutions interact? Before we know what the analytic solution is, we can try an order-of-magnitude calculation via dimensional analysis, and we can try building a simulation with a greater or lesser amount of detail — all aiming to see what the form of the solution will look like. It seems to me that this “do what you can with what you’ve got where you are” approach would be much more valuable than the ways computers are used in classrooms now: doing large numbers of rote arithmetic exercises, TEAL and so forth. It would also jump the students to the penultimate step of the progression Prof. Baez mentioned earlier:

So, first they think math is about arithmetic with numbers. Then they discover it’s about solving equations algebraically. Then they discover it’s about doing integrals and solving differential equations. Then they discover it’s about proving things in books. Then they discover it’s about coming up with your own things to prove. And finally, when they get a job, they discover it’s about teaching students to do arithmetic with numbers.

Posted by: Blake Stacey on June 14, 2007 7:34 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I wasn’t thinking that the students would particularly learn anything mathematical, but more that it would be a way to tackle a problem in (the UK’s at least) education: it’s easy to get the idea that maths is just putting stuff into equations and calculating, with the idea that mathematics is partly spotting hidden patterns put off until you’ve got the technical skills needed to do this “properly”. The idea would be to take kids of maybe 12–13 who don’t have much more algebra than they do now and show them “pattern spotting” and proving. The tactic would be to do something like a magicians “pick any card trick”: you’re not told what to do in a step-by-step way questions like lots of mathematical software, but have something cunningly designed so that features that appear to be for one purpose are actually designed to guide you into the position for spotting something. (In the group example, I was sort of thinking of a “tidy screen” option that just happens to put all the cycles of the subgroups in some way that highlights that they’re all divisors of the group size.) Of course, all the kids who weren’t paying attention at all wouldn’t get it, but at least some might get the idea there’s more to maths than cranking out formulas.

I know there were very occasional experiments carried out whilst I was at school looking at for instance, IIRC, by just telling us to look at what happened when you iterate linear rules. The problem was, doing all the evaluation by hand you had to (a) be very accurate (not me :-) ) to get results in which to see patterns and (b) it still gave you the idea that maths was “just doing algebra on numbers”.

The problem I suspect with simulating physical systems is that it’s all too complicated, so you get back to the teacher telling the class what to do step by step. (Conic sections crop up a lot in computer vision and even now I find analysing and dealing with them isn’t what I’d call trivial.)

Posted by: dave tweed on June 15, 2007 1:10 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Continuing the analogy, these students may have been taught for the first seven years of their schooling by people who didn’t know anything existed beyond the level of a word.

In the UK you can qualify as a primary school teacher with grade ‘C’ GCSE, our national exam for 16 year olds. All you have to do then is score around 55% on a paper at the Intermediate level, examples here.

Posted by: David Corfield on June 14, 2007 10:31 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Speaking as an undergraduate student, I think the most beneficial course structure I have encountered was from Dr. Gierz in both, Math 145B, and in Math 172 this quarter. Although it may seem to some that he is making it a bit easy on us by taking all of the quiz and exam questions directly from the homework, with no alterations to the problems, it was actually very helpful. The reason I say this was beneficial is because he graded primarily on the writing of our proofs. We still had to learn the concepts and theorems, but it was our proof writing ability that he was focused on, and I feel as though I have improved ten-fold in that area.

Posted by: Jason Payne on June 14, 2007 2:10 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I’ve just graduated with a BS in Math while tutoring calculus and differential equations students during my final semester, so my experiences with these same issues are very fresh.

I saw instructors making two key mistakes. The first was to assume that assigning homework makes the students get practice solving the problems. The second was in not using a variety of methods to explain what is really going on when teaching crucial concepts and expecting the algebraic proof to also be a sufficient explanation.

–As a tutor, I saw how networked the students are. You can make a simple model in which each student has a probability P of being able to solve each problem on the first attempt. A student with a P close to 1 has little incentive to join a study network, but may choose to join one for social reasons. A study network has little incentive to accept a member with a P close to 0, but may choose to accept such a member for social reasons. What you end up with is a network with a lot of unconnected A and D students who know either all or none of the material, and clusters of B and C students who collectively know the material but do not know it individually.

The remedy to this “efficient” homework computation seems to be to give short weekly quizzes in class so that students are forced to grapple with the problems individually within your sight and so they feel immediate pressure to know that week’s material. Students then change focus from telling each other homework solutions to telling each other how to solve each type of problem. The few classes I took with weekly quizzes seemed to produce a few more drop slips, but also a lot more As at the end of the semester. Sure, grading quizzes on the weekend is annoying, but hopefully the pleasure of grading the better midterm and final exams makes up for the misery of grading the quizzes. (there’s also the altruistic payoff of helping students to learn math, if you’re into that)

–I had about a dozen different tutees last semester. A large proportion of them said in our first meeting that they were totally lost in lecture. Their notebooks would typically be full of notes from the first quarter of half of each lecture and then doodles after that point. (a clue that they are visual learners?) Some of them had found lecture so uninformative that their attendance had become sporadic. They had different lecturers but similar complaints about a lecturer who did little but fill the board with symbols without ever interacting with the class. Some students came in with no idea how to even start working on the problems. After I explained that week’s concepts using pictures or English and answered a question or two, all of my students were able to solve the problems related to those concepts. The lecturers thought that their job was to prove theorems to the students, while I thought my job was to explain what the theorems mean and answer their questions.

I know that the Math department here is trying to get professors to accept that students have different learning styles. They are supposed to try to present the concepts to visual learners, experimental learners, auditory learners, et cetera. I don’t think most professors really believe that there exists more than one valid way to learn mathematics: the way that they learned it. I don’t want to sound like I’m blaming the professors, but I also don’t think it’s useful for professors to blame the students or become bitter. I had a very old and grumpy professor in Linear Algebra wait until the meeting after students had filled out the professor evaluation bubble sheets to tell everyone that math education students were ruining his classes and forcing him to dumb them down so much that teaching wasn’t any fun.

Sadly, there seems to be nothing that professors can do directly about the problem of ill-prepared students. Universities, under financial pressure, are increasingly using TAs and lecturers in place of full professors for lower level courses. These replacements may not really understand the theorems themselves! They would then be understandably reluctant to expose themselves to questions. Indeed, the problem has roots deep within the public school system. From what I understand, the teacher’s union has repeatedly rejected calls for differential pay based on area of expertise. This has led to a shortage of teachers with lucrative math and science degrees such that fully 50% of public school teachers of these subjects do not have the relevant degree! Why shouldn’t students of these teachers give their professors headaches when they finally arrive at their junior-level classes with little more than low level symbol manipulation skills and lots of unintegrated bits of knowledge? I have no choice but to agree with my bitter Linear Algebra professor. The students who don’t have a deep understanding of the subject are usually the ones who go into math education. But to give up on these students is to perpetuate the problem and make future professors become even more grumpy! As an ancient Roman poet could have said: who teaches the teachers?

Posted by: David Lyon on June 14, 2007 11:59 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I actually think they get grumpy as a defense tactic. It makes perfect sense and it is a very strong strategic move. Imagine there are two sections in an undergraduate modern linear algebra class: a class that is generally required for math majors but would be considered by boneheads to be on the higher end of difficult undergraduate courses. The two professors teaching the two sections have very different reputations. “You’re taking linear algebra next year? AVOID PROFESSOR XYZ! If you value your gpa sign up for the other section. Professor ABC is so much nicer and more fair of a grader (i.e. he is more likely to give you a decent grade despite how poorly you perform because he is jaded and/or simply doesn’t care as much). Professor XYZ is an embittered and unfair tyrant in the classroom (i.e. he doesnt take bullsh*t from slackers and has a higher standard for learning). Then what happens is that all the slackers sign up for Professor ABC’s class and get C’s. The minority of people who have an actual interest in mathematics will sign up for Professor XYZ’s class along with a few slackers who were too lazy to register for classes in a timely manner, and the overall experience will be more pleasurable for Professor XYZ.

Posted by: Michael on June 14, 2007 1:52 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Have to say there seems to be a lot of truth to this. One other thing I’ve witnessed is lecturers who go out of their way to scare off as many (of the presumably academically poorer) students as they can so that they can actually teach the material that they want to. I’m not sure if it’s a good or a bad thing, given that such courses that I’ve attended I’ve been able to get a lot out of.

Posted by: stephen on June 14, 2007 3:02 PM | Permalink | Reply to this

Off topic: comment entry box

I suspect this is probably not something easily fixable in the blogging software but: is there a reason why the comment box appears to be a fixed size in pixels? It’s a bit annoying because I’ve got a largish screen and use larger than default-size fonts but the comment box resolutely sticks to its size, which means I get about 36 characters on a line in the comment box. This makes seeing larger stretches of the text for editing annoying.

Posted by: dave tweed on June 15, 2007 1:21 PM | Permalink | Reply to this

Re: Off topic: comment entry box

It’s a problem with the textarea element: you have to specify its width in columns rather than as a percentage of the page. I suppose that they could dynamically set the number of columns on a page resize event using javascript, but there’s no simple attribute that you can set.

My latest problem is that on Linux I get a Greek font for stuff that shouldn’t be Greek.

Posted by: Mike Stay on June 15, 2007 6:50 PM | Permalink | Reply to this

High Schoolers Hate Math; Re: Why Math Teachers Get Grumpy

I asked 9th, 10th, 11th, and 12th grade summer school students at a Pasadena high school to write a paragraph on (their choice) “Why I Love Math” or “Why I Hate Math.”

Their responses (names removed for privacy, no spelling or grammar corrections):

=====================

I wouldn’t necessarily say that I hate math, I just dislike it a great deal. Algebra is currently not my best subject and has never been. The reason as to why
I do not like math is because it is confusing and I get headaches from trying to solve all of these equation and word problems. I am very good at basic math, but when it comes to Algebra, I am terrible. If teachers found a way to make math more fun and interesting for their students they would be more interested in learning it. And not all the students
have the same learning style so you would have to put that into mind too. But as of now, it is hard, boring, confusing and at our age we just don’t see the need for more of it in our future. But then again,
that is my personal opinion. I’m sure that there are many people who love math and find it very easy and fun to learn. We’re all different. There will be
things that you like in life and things that you don’t like. There will be things that you find extremely easy and things that you find extremely hard. That is
just the way life is. Think of it as an obstacle that you have to overcome to get where you want to be.

=====================

Math is my least favorite subject. I really do not like math one bit. Although I think math is unique because it is everywhere you look and is the same in
every country, I can not seem to understand it. All my life I have not been able to understand the math. I honestly don’t think I should have passed
Pre-Algerbra but, I am very grateful that I did.

I have never been good at math… why? It could be because I’m just useless when it comes to math, or I just don’t try hard enough.

=====================

I didn’t always hate math, in a matter of fact I used to like it a lot. It wasn’t till I started 8th grade that I began to lose my love for the subject. Up till
now, math has become very troublesome now that I’m having such a hard time with it almost all the time. What I think I need to do is to slowly go thought all of it, to improve my overall understanding

=====================

Well I don’t hate math I just don’t understand. How to do it Sometimes but sometimes I do hate math maybe because of the teacher, Classmates, or because I
didn’t understand it and the teacher call on me then I don’t answer question quickly but yes I don’t hate math I just don’t understand it but I’m willing to learn How.

=====================

The reason why I hate math because it really doesn’t interest me anymore. Math used to be my favorite subject when I was younger but not it’s whatever to me. Math can be very easy or it can be very hard to
you. But personally math isn’t my subject anymore. The reason I hate math, because it’s no spark to it. I just hate math period.

=====================

Why I hate math its because all the fractions and the ather things and all the Decimals and metric system, percent thats why Hate Math.

==========================

Mathematics is the most challenging subject for me. I don’t like math because I get lost so quickly. When I don’t understand it after the teacher just explains it I get frustrated. When I ask for the teacher to repeat it all the kids get upset that makes me feel embarrased. I sometime like math when I understand
it and get good grades. But when it comes to fractions or memorizing formulas I completely hate math.

==========================

I hate math because there is to much thinking and that makes my head hurt. It also gets me tired because it is boring. I never like math because when I was a
little kid I wouldn’t like to work with numbers. I also hated the way they teached it. Math to me is very boring.

==========================

The reason I don’t like math is because you have to memorise alot of things. You have to memorize alot of formulas and alot of ways of doing the things. I don’t like math also because some problems are very
hard to understand, or to do Math just makes you think and work to much.

==========================

Everyone has a Subject that they dislike. I hate math Just because it’s a group of numbers put into one problem. And Just the chances at getting a probelm Correct is low chances. And it affect people
emotionally they start thinking and well it can not be over with.

==========================

I hate math because math is really hard to do. Math has so many rules to follow to be able to solve a problem and if u don’t follow them the problem will be completely wrong. The problems are very complicated
and sometimes can be confusing. I don’t think that I would need to use this kind of math besides just the basics in our lives. All these fractions and decimals
may be common but math is my worst subject.

============================

I hate math because it confuses me. Similar to girls, it confuses me more when I try to understand it (them). I have never been at ease with the subject. I feel intimidated by it, and as a result, I tend to second guess myself when solving problems.

============================

I hate math because it make’s you think too much and it hurts my brain and I don’t like thatabt.

============================

I don’t like math because it is boring I really dont like math It makes my head hurt and math doesn’t really make me think. Math makes me sleepy. I use to
like math when I was in 5th grade but when I hit middle school I started hating math I don’t seem to get math know to me math is hard.

============================

Hi my name is ******* ********. and I really don’t care for math. I have a hard time understanding certain parts of the math. When I get stuck on a problem I lose focus and then give up. But I would
like to get help so that being able to understand math will make it more fun and easy.

============================

First. I don’t like math because never passt this class and I don’t understen Nothing I want passt math in summer school because I don’t want repid more math. Second. The math little bet is my faborite class but want understen math this class is Hart. Thirth. matematic I don’t like but make my major idea for understen and passt the math class.

=================================

Math is the worst subject Invented. Its so hard and intricate. URCTH. It involves way too much thinking and brainwork. I rarely do the work w/out a calculator. And I cheat a lot every chance I get now
don’t get me wrong. I do good at everything (SUBJECT) else I do. So maybe I could learn to discipline to myself to focus and excel at math, cuz I’ve BEEN
failing since about 2nd grade. So do what you need.

=================================

I honestly dislike math because of of one good reason, and that is all because I have taken Alegbra 1 for about 3 years now. It is the same stuff over and over
again. Having to go through the same course so much has gotten on my nerves. I used to like math, until my 8th grade teacher did not want to transfer me to a
higher level of math because I had gotten a C average. That upset me in such an enormous way because I PASSED!

=================================

Math is like whatever to me I like it when I get what their teaching me. And when I don’t get it, I hate math. Overall math is OK, I don’t necessarily hate it
or like it.

=================================

What I think about Math. There are some things I like about math and Somethings I don’t like. I like learning new math consepts. I also like the feeling
of getting my answers right. I do not like taking math test. I do not like the tests because of the memorizations.
=================================

Right now I hate math because it’s complicated for me, back then it wasn’t because it was all basic for me and easy, but not when I entered Pre-Algebra that’s
when I said to myself “damn this isn’t basic’s any more”. After that it just started to get complicated in Algebra or I just started to forget the equations or lesson the teacher told me. Another reason I hate math is because I always get low grades on tests no matter what I just get a D- or F. Another reason I hate math especially when it is mathematical word
problems, man they are so confusing, I don’t even read the problem or equation.

==============================

Although I’m in summer school to take math again, I like math. I like doing math sometimes. Sometimes I don’t like it. I used to be good at math, really good. Then I just froze, and since then I haven’t really been good at it any more.

==============================

I don’t hate math or like math I just find it unfavorable.

For me it is always been harder to understand than most subjects.

There is just something about numbers that is confusing especially fractions.

I know that I need Algebra 1 i just wish it was easier to understand.

==============================

The things I hate about math is this doing homework. I mean whats the point of doing it at home. I mean where not in the classroom. Another thing I hate doing is working out long problems, I really don’t
feel like doing it I mean who does.

Now lets talk about things I like about math is the learning experience. I mean even though I’m very smart, and intelligent individual. And when I feel
like working I and really focus.

So that’s the things I like and hate about math.

==============================

I love math because when you get job you have to no math to count the money you get. Another reason why I love math is because it is more easy then something
else and you will need it if you want a good paying job.

==============================

I love math because without math I can’t count my money.

And also the more math you learn the more money you make. You use numbers everyday, without knowing math, Algebra, how are you gonna know if your getting the right change back from the money you spend. So that’s why I take math.

But I still don’t like to sit in a math class.

==============================

Copyright (c) 2007 by Computer Futures, Inc.

Posted by: Jonathan Vos Post on July 1, 2007 7:13 AM | Permalink | Reply to this

Re: High Schoolers Hate Math; Re: Why Math Teachers Get Grumpy

So the kids who like math like it for counting their prospective money.

Ironically, it’d be lying to tell them to get into math for the money involved.

Posted by: John Armstrong on July 1, 2007 8:31 AM | Permalink | Reply to this

Money; Re: High Schoolers Hate Math; Re: Why Math Teachers Get Grumpy

John Armstrong, bless your heart. I tell my kids over and over how much of a pay cut I took to have the honor of being in their classroom. I wear a coat and tie in a hot summer, out of respect for them. I tell them how I earned $800,000 per year during the dotcom boom by being on the board of a software start-up acquired by a NASDAQ firm. I tell them how I was in a rock band, and how Mick Jagger got $1,000,000 in advance for an unrecorded album. One of my students wants to own a restaurant, two to work in Animation, one to be a District Attorney. It’s not about the money. But almost all of my students come from impovershed families. Math might help, if only for discipline, graduation, thinking hard without headaches… And there will be many business/money/coin problems in their homework.

Posted by: Jonathan Vos Post on July 1, 2007 5:37 PM | Permalink | Reply to this

Midterms and Magic; Re: Money; Re: High Schoolers Hate Math; Re: Why Math Teachers Get Grumpy

My high school students (I’ve just finished teaching my 2nd full week there, and today will be grading the dreaded Midterm Exam, which is EXACTLY on-topic here) were puzzled by my love for Harry Potter books and films. They were awed that J. K. Rowling’s literary agent has become one of the 20 richest people in Great Britain. “Writers have agents?”

I spend a LOT of time telling them the truth about Money, and how to get rich, which neither their parents or the other teachers here could do. Los Angeles County just became the first in the USA to have assessed real estate value of $1 Trillion. Really! Now, students, write that using exponents of 10…

They were equally dubious that I’d spoken for an hour with Johnny Depp, think him a great actor, and dig the Pirates films. “Is he hot?” the girls asked. When I agreed, they suddenly questioned my gender-preferences. “And how do you feel about Orlando Bloom?” they asked.

I said that I liked him better in LOTR than Pirates, and reminded them that I was a happily married man, and not gay. “Not that there’s anything wrong with that,” as they say on Seinfeld.

Speaking about Wizards (that was another thread here, with Merlin, Gandalf, Witten) I’m serious about those open questions in LOTR [Lord of the Rings], which emerged from discussions with Dr. George Hockney at JPL.

J. R. R. Tolkien was a great writer. His oeuvre constitutes the greatest Fantasy ever written. I grew up on it, with my mother reading to me from the newly minted hardcover British first edition. The Peter Jackson trilogy is the greatest Fantasy film ever shot. BUT…

J. K. Rowling, however derivative [not Lie derivative], however sometimes tone-deaf and in need of a good editor, has a better
sense of humor than Tolkien.

J. K. Rowling is better at setting up and solving mysteries.

J. K. Rowling has more plausible female characters.

J. K. Rowling shows us more about education.

Question about the Harry Potter world: is the Muggle reality actually OUR reality? With no USSR? No King/Queen of England? Was there a Newton – Alchemist? A Kepler – Astrologer? An Einstein?

Final questions: is there any religion as such in LOTR (not, of course, in the deeply theological “The Silmarillion”)?

What Magic is there in LOTR, which is not some Sufficiently Advanced Technology (the Palantir-web 2.0, the Ring-net, Mithril metal, etcetera).

We’ll know in a very few days, via Book 7 of Potter:

(1) Which world IS that, in the multiverse?

(2) Whose side is Snape really on – is he a triple agent or a quadruple agent?

(3) Dumbledore – a Schrodinger cat?

(4) Historically, how did the House Elves get enslaved, and what role do they play in Book 7? The little silly Hobbits saved Middle Earth, after all…

Posted by: Jonathan Vos Post on July 13, 2007 5:09 PM | Permalink | Reply to this

Harvard UVa study on Math/Science; Re: Why Math Teachers Get Grumpy

A study about the be published in Science, and my subjective reaction.

Contact: Steve Bradt
617.496.8070

College Science Success Linked To Math And Same-Subject Preparation

Cambridge, Mass. - July 26, 2007 - Researchers at Harvard University and the University of Virginia have found that high school coursework in one of the sciences generally does not predict better college performance in other scientific disciplines. But there’s one notable exception: Students with the most rigorous high school preparation in mathematics perform significantly better in college courses in biology, chemistry, and physics.

The work will be published this week in the journal Science.

Authors Philip M. Sadler of Harvard and Robert H. Tai of Virginia say the findings run counter to the claims of an educational movement called “Physics First,” which argues that physics underlies biology and chemistry, and therefore the traditional order of high school science education – biology, chemistry, physics – should be reversed.

“Our findings knock out one of the primary claims of ‘Physics First’ advocates,” says Sadler, F.W. Wright Senior Lecturer in Astronomy in Harvard’s Faculty of Arts and Sciences and director of the Science Education Department at the Harvard-Smithsonian Center for Astrophysics. “Taking more physics does not appear to improve students’ subsequent performance in either chemistry or biology courses.”

“Many arguments have been made for chemistry and physics preparation to benefit the learning of biology,” says Tai, an assistant professor in Virginia’s Curry School of Education. “On the scale of single cells, many processes are physical, such as neurons ‘firing’ electrically. Also, the complex molecules at the root of life obey chemical laws that are manifested in macroscopic processes. Yet our analysis provides no support for the argument that physics and chemistry principles are inherently beneficial to the study of biology at the introductory level.”

Sadler and Tai surveyed 8,474 students enrolled in introductory science courses at 63 randomly selected four-year colleges and universities across the U.S. The students reported on their high school coursework (0, 1, or 2 years) in biology, chemistry, physics, and mathematics; this data was then correlated with their ultimate performance in their introductory college science courses. Sadler and Tai subjected this raw data to robust modeling to correct for socioeconomic factors that may advantage some students, including race, parental education level, and mean educational level of students’ home communities, as defined by ZIP code.

Not surprisingly, the controlled data indicated that high school preparation in any of the scientific disciplines – biology, chemistry, or physics – boosted college performance in the same subject. Also, students with the most coursework in high school mathematics performed strikingly better in their introductory biology and chemistry courses in college; introductory college-level physics performance also benefited. Conversely, little correlation was seen between the amount of high school coursework in biology, chemistry, or physics and college performance in any of the other disciplines in this trio.

“The link between math and biology is not exactly an intuitive one, but biology has become an increasingly quantitative discipline,” Sadler says. “Many high school students are now performing statistical analysis of genetic outcomes in addition to dissecting frogs and studying cells under a microscope.”

The current order of high school science education was established in the 1890s, in an attempt to standardize what was then a system of wildly disparate science education in high schools across the U.S. Biology was given primacy in that ordering in part because the late 19th century experienced a flowering of interest in the natural world, and also because it was perceived to be less daunting intellectually than either chemistry or physics.

Sadler and Tai’s work was funded by the National Science Foundation’s Interagency Educational Research Initiative.

===============

This is good news indeed.

What my wife and I found, teaching Science courses in colleges and universities, was the problem of students being under-prepared for Science, particularly because of weak Math.

The Harvard University/ University of Virginia study is, to me, another justification for my nearly completed time in the trenches of an inner city high school, teaching Algebra. A few of my students have a chance, now, if they go on to college.

In early September, I’ll start teaching Chemistry, Earth Sciences, and Biotechnology at a school-witin-a-school (Health Careers Academy) at that high school. This will allow me to asses, for this school, the level of hands-on lab technique, and qualitative abstract reasoning, with the dreadful level of math that I observed.

Social Promotion seems to be one culprit. Some of my summerschool students explicitly, in writing, wondred why they had been allowed to pass “pre-algebra.” As a result, I spent more time reviewing “pre-algebra” – i.e. how to add, subtract, multiply, and divide fractions and decimals, than I did hammering home what equations were, how to manipulate them, and how they related to the physical and social world.

Yesterday being the anniversary of Syncom 2 (26 July 1963), I took my class back to the Age of the Beatles, told them stories about how Arthur C. Clarke lost a billion dollars in his spare time, and guided the kids through finding diameter, and circumference, of cricular orbits of various radii, and the speed a stallite needed to go to be synchronous, and how fast the Earth was moving around the sun.
In the process, I had to confiscate a skateboard and 3 iPods being used in violation of posted classroom rules. had I moved faster, I’d have had 2 skateboards in use (inside!) and 6 iPods. These get locked in the Principal’s office safe, and need a parent to retrieve.

Those students who cooperate get to (quietly) play Chess and cards during class.

Carrot and stick.

But Math seems to be at the root, beyond behavioral problems, for these poor kids. Some live in Group Homes. Some are technically homeless. Many do not see a father.

If I cannot save my students, with all my passion and expertise, I don’t know who can. If I can make a difference, then I can have hope for Western Civilization.

Public education is badly dysfunctional in America. Better to light a candle than curse the darkness.

===============

Posted by: Jonathan Vos Post on July 27, 2007 3:45 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

– – – – –
definition → theorem → proof
the words “obvious” and “trivial”
general statements
jargon
facing the board

++++++
pictures
examples
curiosity
questions
handouts
personification
applications
answering the question “Who cares?”


Students don’t need someone to copy a proof from a book to the blackboard in front of them, or to be told that an eigenvector is “defined as a vector that contains all the eigenvalues of a linear transformation”.

They need someone to get them excited about why eigenvectors are important, what you do with them, what they’re LIKE, and then tell them what they are.

Start with questions, or examples. Then build to the general.

Posted by: Chris on March 14, 2010 7:23 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I partially agree with you. Certainly, the point shouldn’t be to just blindly copy down a textbook of your own from the lecturer’s board to your notebook.

That said, the courses I have hated the most have always been the ones with handouts. If I write along, I learn - I’m kinetic that way. Also, if the lecturer (and this goes for me a LOT nowadays) is forced to write himself for most of his lecture, chances are he’ll slow down enough to not outrun everyone.

But the narrative _is_ important; John Baez goes on and on about it here, and on the This Week’s Find - and the focus on the narrative of mathematics is one of the most valuable things reading these have brought me.

In my own teaching, I strive to provide narrative. Also, I tend to use blackboard exclusively, unless I have graphics or computer programs to show.

Posted by: Mikael Vejdemo Johansson on March 14, 2010 11:49 PM | Permalink | PGP Sig | Reply to this

Re: Why Math Teachers Get Grumpy

The conventional style of writing proofs up on the board is a waste of students’ time. Every theorem taught in undergrad has been looked over by 10^n mathematicians who are more likely to see mistakes than someone just hearing it.

Students need explanations, not proofs. Pictures, stories, characters.

Chemistry teachers talk about bonds and chemicals having personalities. In physics you might say a ball “wants” to roll into a potential well. Science teachers also lie to their students. Economists say there is “an” interest rate, and chemists say orbitals are circular. Electricity “flows” and, you know what, it’s all a metaphor anyhow. No one needs to waste their attention

Math teachers need to perform more, lie more, draw more, and face the board less.

Always start with either a motivating question, a curious puzzle, or an applied example, and move to general proofs later.

And, ALWAYS give students print-outs rather than making them copy down funny symbols.

Lastly, the following question should never be dismissed:

“So? Who cares? Why should I spend my time on this?”

Posted by: Chris on March 14, 2010 7:31 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy


And, ALWAYS give students print-outs rather than making them copy down funny symbols.

I don’t agree.

As I stated above, handouts/printouts handed out is a common criterion in all the classes I learned the least from. I think that taking notes
1) is a good way to keep you alert
2) is a good way to train you to take notes
3) helps at least the kinetic learners get ahead. The visual and the auditory learners are already halfway provided for by the existence of the lecture in the first place.

In general, a mediocre lecturer that still gets his point across can generate okay notes from the note-takers; a mediocre lecturer that ends up discouraging note-taking, and hands out mediocre notes, will in general produce results that are less helpful for studying.

Posted by: Mikael Vejdemo Johansson on March 14, 2010 11:55 PM | Permalink | PGP Sig | Reply to this

Re: Why Math Teachers Get Grumpy

There’s one issue with this: for a lecture that’s going at typical speed you spend most of your concentration on making sure you don’t miss writing anything down rather than thinking about what’s being explained. (I can remember several courses, particularly those where there wasn’t a precisely specified syllabus so that you could look things up later, where I was primarily sitting there trying to write all the proofs and examples down and didn’t spend virtually any time thinking about the material.) On the other hand, as you say having to be active helps prevent the audience drifting away. One way to handle this is to give out basic printed notes at the beginning which have maybe one key item (equation, etc) every “couple of minutes” that’s blanked out so that the audience has to keep alert to write it in(you don’t mention in your presentation when you discuss the blanked out item) without them having to copy down so much stuff that they (a) aren’t thinking about what you’re saying and (b) don’t have time for writing personal perspective notes.

Posted by: bane on March 16, 2010 2:18 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I love the concept of a kinetic learner! I think it is an excellent description of the kind of learning that goes on when taking notes in a lecture: full of energy and motion, but sadly lacking in any potential.

I suspect that the word you were looking for was kina?esthetic. But the connection between taking notes and kinaesthetic learning is about as tenuous as that between watching a lecture and visual learning, and listening to a lecture and auditory learning.

I find that if I’m in a note-taking situation then no learning actually takes place during the lecture. And if no learning is taking place then the lecture, frankly, is a complete waste of everyone’s time.

To paraphrase a great Briton, “attending a lecture is the worst way of learning a subject”. But unfortunately, all the better ways of learning require far more time and resources than universities are prepared to spend (well, I should say than governments are prepared to spend) and there’s a good argument for why they shouldn’t radically alter the system: at the end of a university education, a student should be able to learn stuff without needing any props. So we gradually teach them with fewer and fewer extras until they can attend a standard seminar and come away having learned something.

Reading these comments, I can’t help thinking that the correct interpretation is “A dire lecturer is less dire if they use the blackboard. Most lecturers are awful, so all lecturers should use the blackboard.”. Also, I pick up on a related theme: “I was taught by taking notes, I have learnt a lot of mathematics, therefore I learnt it by taking notes.”.

Chris makes an excellent point (amongst many) which explains my motivation for switching from blackboard to computer:

And, ALWAYS give students print-outs rather than making them copy down funny symbols.

I teach in English, but most of my students are Norwegian. So for them, words like “arbitrary” are “funny symbols”. The waste of time that bane talks about at the end of his comment is magnified ten-fold for me since the students are listening, reading, and writing in a non-native language.

This statement is just fantastic:

As I stated above, handouts/printouts handed out is a common criterion in all the classes I learned the least from.

The subject I learnt best at university was the one where the lecturer was the absolute worst. By a long way. The lectures were so confused and confusing that I had to learn the material from other sources (mainly from an absolutely fantastic TA). But I was motivated to do so because I was already hooked on mathematics. I can think of only two universities in the world where one can make that assumption (okay, okay, hyperbole; you could probably add Cambridge to make it a third). Everywhere else, there will be a tiny minority who are sufficiently motivated to go and learn the stuff themselves, but the rest will say “So what?”.

Finally, of course it is crucial to keep the students actively involved during the lecture. But actively involved with the lecture, not with taking their notes. And there are so many ways to do this, but I feel that in mathematics we’re a bit behind the times and think that it “dirties” our subject by having to motivate it or “find the story” - the maths should talk for itself; except that so few know how to listen that we have to be the interpreters and translate what the maths is saying to us into some form that others can understand.

Posted by: Andrew Stacey on March 19, 2010 12:37 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

A lot of good stuff there, especially that last sentence.

As to

The lectures were so confused and confusing that I had to learn the material from other sources (mainly from an absolutely fantastic TA).

The same can be true of the written version, cf. the story of how Serre and Borel came to master spectral sequences.

But I was motivated to do so because I was already hooked on mathematics. I can think of only two universities in the world where one can make that assumption (okay, okay, hyperbole; you could probably add Cambridge to make it a third).

Cam implies Ox but what’s the third?

What assumption? That ALL the students are already hooked on math?

Posted by: jim stasheff on March 19, 2010 1:44 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Hull!

(I know that Google can’t be relied upon to give the same results to the same search to different people, but when I just tried it then the top four hits for “Oxford Cambridge Hull” all gave the correct reference, and one of those is the Wikipedia page for the University of Hull, so if anyone is bemused as to why these three, it shouldn’t be too hard to find out. And it may explain a lot.)

Posted by: Andrew Stacey on March 21, 2010 1:19 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I am not arguing against a narrative approach to mathematics, quite the opposite - finding the story, motivating the material, engaging the students - with whatever means connects with them - to the material at hand I view as absolutely crucial.

My argument was with the categorical condemnation of blackboards in favour of notes, and my argument was based on my own personal experiences as a student. I’m sure my view will refine and change when I start accumulating serious experience as a lecturer, but right now my main foundation is my time spent as an undergrad and grad student, and the courses I attended in my own studies.

And what I’m saying above is that I feel that I learn better if I’m taking notes and that I feel that I learn worse if I’m equipped with preprinted notes. Especially if there are preprinted notes to go with a Beamer presentation. One of the reasons I feel this way is that I almost always am able to take notes swiftly enough that I can think about the material too as long as the lecturer is working on a board; but with a preprinted Beamer presentation, the tendency is to run on faster than I can keep up; and I don’t end up with the same kind of personal connection, and the same kind of burnt in memory, when dealing with notes I didn’t take myself.

Thus, when I sit down and review - with notes I wrote myself, I almost always have felt that I know almost everything during review, while with preprinted notes, I feel disconnected and need to go back and relearn in a completely different manner.

These may well be too personal experiences to extrapolate from reliably. In that is so, I’m sure my viewpoint will change drastically over the next few years.

Posted by: Mikael Vejdemo Johansson on March 19, 2010 4:00 PM | Permalink | PGP Sig | Reply to this

Re: Why Math Teachers Get Grumpy

I tended to and still do take notes NOT by copying everything on the board but by mentally ‘editing’ as I go along and even jotting down my own thoughts, questions etc.

Are beamer presentations any different in speed than the old transparencies which were some times flashed at the speed of light? ;-)

Posted by: jim stasheff on March 20, 2010 2:11 PM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

There is, of course, a temptation with beamer (and other) presentations to go too fast and to try to cram too much onto a slide and into a talk. Simply giving a beamer presentation is no guarantee of a decent lecture, and - for essentially the reasons that Mikael says - it’s easier to be a lot worse with beamer than with a blackboard.

But I think that it’s easier to be a lot better with beamer than with a blackboard. The problem is that it’s a new thing, particularly in mathematics, and we don’t have much experience of how to make it work so we’re having to try things out. Some work, some don’t.

For example, in my lectures I’ve started colouring all the mathematical things according to what they are. Sets are one colour, functions another, elements a third, and so on. The idea being that it’s much easier to look at a mathematical expression and understand what each piece is doing. I couldn’t do that on a blackboard.

I do find myself in two minds about this and do have some sympathy with what Mikael says. After all, it’s how I learnt my mathematics. Or at least, it’s how I think that I learnt my mathematics. I suspect that it had much more to do with other factors than the lectures since, as I’ve said, the quality of the lecture and lecturer does not correlate well with my understanding of the subject!

But the way I learnt mathematics is fairly irrelevant because I’m not teaching myself. The vast majority of the students that I teach are nothing like I was. So I need to adapt my teaching style to match their learning style. Otherwise, what’s the point of the lectures?

To illustrate my point, a student recently complained to me that I was putting in too much “story”. He just wanted the details, not the extra padding. But he was retaking that particular course and was a very bright student, so of course he just wanted the details. But he was completely atypical of the students in the room so it would have been completely wrong to base my teaching on what he wanted.

Posted by: Andrew Stacey on March 21, 2010 1:32 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Andrew, do you have your beamer lectures posted on your web page? (I couldn’t find them after a quick, cursory look.) I’m curious to see how much detail you include, how long they end up being for one lecture, etc., not to mention seeing this coloring scheme. (By the way, I assume you use macros for that, and I’m also curious about how exactly you chose to implement them.)

On a different subject, I think this point:

But the way I learnt mathematics is fairly irrelevant because I’m not teaching myself. The vast majority of the students that I teach are nothing like I was. So I need to adapt my teaching style to match their learning style. Otherwise, what’s the point of the lectures?

is one that many lecturers should keep in mind more.

Posted by: Mark Meckes on March 21, 2010 2:24 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

For example, in my lectures I’ve started colouring all the mathematical things according to what they are. Sets are one colour, functions another, elements a third, and so on. The idea being that it’s much easier to look at a mathematical expression and understand what each piece is doing. I couldn’t do that on a blackboard.

I don’t quite go so far as to color each single symbol according to its type - most of the time - but lecture, seminar talk or recitation I prefer to use a lot of color on the blackboard.

But the way I learnt mathematics is fairly irrelevant because I’m not teaching myself. The vast majority of the students that I teach are nothing like I was. So I need to adapt my teaching style to match their learning style. Otherwise, what’s the point of the lectures?

And this is probably why we’re even arguing about this. ;-)
I will freely admit to having less than extensive experience teaching, and while I know on an intellectual level that I’m not teaching clones of myself, it’ll be until I’ve actually done it myself before it sinks in on a visceral level.

All which said, I still will prefer the blackboard whenever I talk about mathematics. At least most of the time. For now.

Posted by: Mikael Vejdemo Johansson on March 21, 2010 2:30 AM | Permalink | PGP Sig | Reply to this

Re: Why Math Teachers Get Grumpy

Andrew wrote, “Simply giving a beamer presentation is no guarantee of a decent lecture…”

To me, this is like saying that drinking poison is not guaranteed to make you healthy! I thought that pretty much everyone agreed that the vast majority of beamer and slide talks are much worse than blackboard talks. (Not all, by any means. And maybe things will get better as the decades go by.) Maybe I’m just living in a weird parallel universe…

Posted by: James on March 21, 2010 4:11 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I find that the vast majority of people who rant about the quality of other people’s talks do say that

the vast majority of beamer and slide talks are much worse than blackboard talks

but my personal experience, at least, is rather different. Of the truly excellent talks I’ve attended recently, I’d say more were beamer than blackboard (and few if any used non-computer slides), but not by a wide margin. Of the truly awful talks, almost none used beamer.

However, that’s just for research talks. I’ve never considered using beamer for teaching, or known of someone else who did, which is why I’m trying to get more specifics from Andrew up above.

Posted by: Mark Meckes on March 21, 2010 5:00 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

I didn’t mean to suggest anything about the present company! I agree that it’s a good idea to think and talk about how to give the best beamer talk you can.

Posted by: James on March 21, 2010 7:38 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

(This is in reply to all of the above, but a bit out of order)

I think that there are several issues in this which are in danger of being convoluted but which really are orthogonal. In order of importance (first being most), they are:

  1. What should the audience be doing during a lecture?

    Within this, there’s two obvious separate cases: a lecture to students and a lecture to peers. In the former, one hopes that the students are learning something (though this simplistic answer doesn’t address the question above, it’s merely to distinguish the two cases). In the latter, it can be much more complicated. One might simply wish to interest people in a paper, or persuade them to give you a job, or lots of things.

  2. Which is better: blackboard or beamer?

    This is, almost certainly, unanswerable. At first sight, the main problem is that it’s too subjective. However, a deeper problem is that without an answer to the first question, there’s no point answering this question. The point is, one has to be able to say “better for what”. And if someone has thought through the first question, then I suspect that the answer to this question will be “whatever I choose” because whatever method of delivery that person has picked will be used to the best of their abilities to achieve the effect desired in answer to the first question.

So for me, the fact that I use beamer is because I’ve thought a bit about what I want my students to be doing in a lecture, and I’ve decided that I don’t want them taking notes, I want them to be listening to what I say for the whole time (well, sometimes I set them mini-tasks to do). Initially, part of the reason was the language question, but I think that even in an English university, I’d do the same, now.

I should say that I try to make sure that the students can get a hold of a copy of the notes before the lecture so that they can follow along; then if there is something that they want to make a note of, they can do so at the relevant point in the notes.

Of course, the resulting lecture notes are absolutely useless for the students when they review the course, but that’s because they were designed for the “right here, right now”. So in one course, where there wasn’t a decent book, we supplemented those notes with a wiki which could contain more detailed stuff.

When answering the first question, above, you do have to first answer the corresponding question for the whole course, otherwise it’s impossible to answer it for the actual lecture. And I think it’s useful also to ask: “What do I want the students to be able to remember a year from now?” It can be fairly sobering to think about what they are likely to remember! And if the answer is “not a lot”, then rather than despairing at that, try to ensure that the “not a lot” that they remember is the really important bit! (This can be a bit depressing to think about, except that if you ask someone how much they remember of a book that they read a year ago, then it’s quite likely that the answer will also be “not a lot”. I just got halfway through a book without realising that I’d read it before! And not that long ago, either.)

So, Mikael, as you teach more then you will learn more. The important thing is to remember that communication is 90% of our job and so it’s worth doing well. When you wrote your first comments, it felt as though you were closed to adapting your style. From what you’ve written since, I’m less worried about you! (Well, less worried about your students, I should say.)

To answer Mark’s questions about my beamer presentations, you can find everything off my webpage but it may be a little tricky to track it down since I sign-post it by course rather than by technology! The slides and so forth for my current course can be found at:

http://wiki.math.ntnu.no/tma4115/2010v/stacey/forelesninger

When looking at those (particularly with regard to length), it’s important to remember a few things:

  1. The standard lecture here in Norway is 45+45 minutes (i.e. two 45 slots with a short break in between).
  2. I usually prepare a little extra but fully expect to stop early.

I also use a pen on the screen to annotate the slides as and when appropriate. This way I can interact with the presentation a little more. (I didn’t switch to beamer for teaching until I was able to do this.)

I started this method last semester so there’s also slides available from that:

http://www.math.ntnu.no/~stacey/Teaching/TMA4145h2009/index.html

(this was also the course with the wiki). You’ll notice that I got the idea for colouring stuff about halfway through that course!

On that, I’m not completely happy with how I implement the colouring, but I don’t know enough about how characters get encoded to do it at a lower level. The way it works is that I declare a symbol to have a “type”, something like:

\type{variable}{x}   
\type{function}{\sin}

Then in the document, I have a command that invokes the typed version of a symbol. So I could type \ty\sin(\tyx) to get a nicely coloured version of sin(x). I can also redeclare types, locally declare types, and simply temporarily override. The TeX to do this is not overly complicated, but it does use a little hackery to make it as easy to use as possible.

One unlooked for benefit is that this keeps an eye on reusing symbols: if it came out the wrong colour then I must have used it as something else earlier and the students might get confused if I change the type!

On a final note, I do think that it is interesting that in this conversation we’ve discussed the difference between blackboards and beamer. In most discussions about this with non-mathematicians, they tend to use the word “powerpoint” and I have to keep correcting them! I have no such compunction with the word “beamer”! I would hazard a guess that the difference between the good beamer talks and the downright awful ones is that those who’ve read the beamer user guide tend to give good talks, whilst those who think that beamer is just a way to get their paper onto the screen tend to give awful talks! Till Tantau’s documentation is about as good as the packages that he’s developed.

Posted by: Andrew Stacey on March 22, 2010 12:14 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

Thanks for all that! I have no idea whether I’ll ever experiment with using beamer for teaching (although I usually use it now for conference talks, I’ve been a pretty firm blackboard (really whiteboard at my university) traditionalist for teaching), but it should provide some good food for thought at least.

By the way, I’ve seen some math talks that were literally done in powerpoint. These are mostly applied math talks, I’m guessing in communities that started giving computer talks before beamer existed. The results are usually not good, at least compared with beamer.

As for the beamer user guide, I have to admit I find a few things to disagree with about how to give talks, and the insistent preachiness gets old. But if you look past that it’s a great guide to best practices for both style and technical issues. On a related note, I wish more people would read the AMS’s Short Math Guide for LaTeX, which I believe would result in more people writing better-looking papers.

Posted by: Mark Meckes on March 22, 2010 12:48 AM | Permalink | Reply to this

Re: Why Math Teachers Get Grumpy

On your two questions: HEAR! HEAR! A perfect summary of what the discussion is about, and why personal tastes and miscommunications makes otherwise sane and sensible people says thing you’d read as insane and preposterous.

Of course, the resulting lecture notes are absolutely useless for the students when they review the course, but that’s because they were designed for the “right here, right now”. So in one course, where there wasn’t a decent book, we supplemented those notes with a wiki which could contain more detailed stuff.

The first lecture course I did myself, I wrote up with extensive lecture notes on the Haskell wiki (it was on relating category theory to the Haskell programming language); and gave my talks at the blackboard, wielding the notes as reference.

Certainly taking notes in the classroom shouldn’t be the only way to get hold of the contents of the lecture - if I gave that impression above, I’d need to apologize.

So, Mikael, as you teach more then you will learn more. The important thing is to remember that communication is 90% of our job and so it’s worth doing well. When you wrote your first comments, it felt as though you were closed to adapting your style. From what you’ve written since, I’m less worried about you! (Well, less worried about your students, I should say.)

I’m sorry for giving that reaction. My first few posts were in direct response to the comments from Chris that included a very categorical condemnation of blackboards and a categorical call to hand out notes in advance. The strength and lack of flexibility I perceived in Chris’ comments was what called forth the intensity of my own comments.

I also use a pen on the screen to annotate the slides as and when appropriate. This way I can interact with the presentation a little more. (I didn’t switch to beamer for teaching until I was able to do this.)

Yeah, if I had access to a tablet, I could probably be convinced to consider beamer for teaching purposes more often. As it is now, a beamer talk still feels to inflexible for me to be comfortable with it in a lecture-to-students.

Posted by: Mikael Vejdemo Johansson on March 22, 2010 2:06 AM | Permalink | PGP Sig | Reply to this

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