Math as Art rather than Science

[Ryan A]

A friend of mine passed along this article (http://www.maa.org/devlin/LockhartsLament.pdf) that laments the current state of math education. As I know we have a few mathematicians and teachers around here, I figured I'd post it. I'm curious what everyone thinks.

[Robert Sharp]

I'm not a mathematician, but I basically agree that the rote memorization of formulae route breaks the spirits of many potential math artists (so to speak). I've heard lots of people talk about how much they loved math at first only to have some "bad teacher" kill that love around 7th or 8th grade (sometimes HS). In some sense, I'm one of them. I was on a math team in 5th and 6th grade. We even won our region (yeah, SE US, but still...). I had fun with that sort of thing. Somewhere along the way, I started to hate math, or at least math classes. I think that's part of why I couldn't go through with becoming an engineer, my original major. I couldn't drag myself to another math class, or even to open the book anymore.

[Ryan A]

Here's my favorite quote from the article:
Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught by someone who doesn’t even play an instrument, and has never listened to a piece of music in their lives? Would you accept as an art teacher someone who has never picked up a pencil or stepped foot in a museum? Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? What kind of a teacher is that? How can someone teach something that they themselves don’t do?

Of course the danger seems to be in swinging the pendulum too far to one extreme or the other, as it could be argued that the "Integrated Math" curriculum that has been adopted by most public secondary schools in WA state has resulted in lots of kids without the ability to think or speak in simple algebraic terms.

[Mordrak]

I certainly agree that math isn't often enough treated as a process of discovery and imagination. But discovery requires the time to explore and courses (I can't speak to HS, since I was barely there) often feel like they never have enough time. They are trying to shove as much information as they can in that 10-15 weeks as possible and they still fall short of their goals.

One of my physics professors said he was a late bloomer in physics and it wasn't until his masters work that it all began to click. I don't doubt that was in part he was actually given room to explore rather than just absorb information.

Here's my favorite quote from the article:


Of course the danger seems to be in swinging the pendulum too far to one extreme or the other, as it could be argued that the "Integrated Math" curriculum that has been adopted by most public secondary schools in WA state has resulted in lots of kids without the ability to think or speak in simple algebraic terms.

Requiring every teacher of math to have authored an original piece of mathematics seems a bit extreme. We're already low on math teachers, but I agree that we should want teachers that appreciate it, as well as enjoy it.

[Hawkeye Fierce]

I agree with his general thesis. I often felt that I would have appreciated math much more had there been more effort to explain not just what to do, but why to do it that way.

However, while this isn't a defense of the current curriculum, he seems to have no understanding of the actual dynamics of a classroom. He seems to want all teachers to be masters of extemporaneous problem solving and virtuosos of the Socratic method and, more importantly, he expects that all the students will respond positively to that. It's almost a cliche at this point, but it's easy to teach a child that wants to learn. Most of a teacher's effort is focused on those that are resistant, or are having difficulty. Where do those kids fit in the idyllic world of beautiful problem solving he envisions?

Ryan, as for the quote - do we expect our history teachers to have published historical works? Our English teachers to have written novels? Mastery of a subject for the purposes of teaching is very different from the ability to produce professional work in said subject. Professional practitioners of various academic subjects often make terrible teachers. He also plays a little rhetorical trick there, by asking if we'd accept other teachers that know nothing at all about their subjects, and then moving the goalposts when it comes to math teachers. Demanding that our teachers have a solid understanding of the concepts they teach is a no brainer, but going farther and lamenting that they themselves aren't producing original work is silly.

Anyway, I too, think that math (and science) education could be vastly improved. But I think he's unfairly dismissive (or ignorant) of many of the obstacles to doing so.

[Jason McCullough]

To be fair, everything is taught in the same terrible way he describes in K-12 schools.

[Hawkeye Fierce]

Doing a little googling, I see he is a teacher of mathematics at Saint Ann's School in Brooklyn, which is "a school centered on gifted students who were encouraged to learn for the sake of the learning itself rather than the reinforcement of grades." Which is admirable, but I suspect his manifesto might be ignoring the fact that many of the nation's students are not gifted students who enjoy learning for its own sake.

[Demon G Sides]

7th Grade Math killed it for me.

[tiohn]

I was extremely fortunate to have a math teacher in high school that was constantly engaged in recreational mathematics. I attribute most of my love of mathematics to him. Until I had this teacher, I took math classes because they were easy A's for me, but I didn't really start to enjoy mathematics until I took his classes. He took the time to teach us WHY particular formulas and methods work, rather than just insisting that they do. Thanks to him, I didn't memorize much of anything, as he equipped us with the knowledge and confidence to derive anything we would need on a test.

Now that I am approaching the end of my B.S. in Mathematics (FINALLY), it's only been in the last year that I've really started to love it again. Calculus I-III and the required course in differential equations were all excruciating for me, as those classes are designed to churn out engineers and as such placed little emphasis on more than memorization.

What really worries me are the math education majors that I have classes with. Only a small percentage of them seem to have any real passion for mathematics and many of them seem to actively hate their actual math courses. I'm not sure how any of that makes sense, really. On the other hand, I have a small interest in math education, but can't bring myself to take any courses in education since, at this point, that represents time I could spend in another math class, which I would much rather take.

[Ryan A]

While I think the author goes too far in his wildly arrogant claims that "teaching can't be taught" I can understand where he's coming from. Far too many (the vast majority of?) teacher preparation programs churn out ineffective teachers and far too many ineffective teachers practice slavish devotion to their canned lesson plans because they lack the ability or willingness to adjust their instructional behaviors in response to the needs of their students.

As for Hawkeye's question, I can't imagine I would ever want to see an English teacher hired who did not themselves have a deep love for reading and writing. The very worst English teachers any of us have ever had most likely hated literature (or loved only hateful literature and hated children).

[Hawkeye Fierce]

As for Hawkeye's question, I can't imagine I would ever want to see an English teacher hired who did not themselves have a deep love for reading and writing. Of course not, but that's not actually the standard he was holding math teachers up to. As I said, he moves the goalposts. He claims our math teachers "have never produced an original piece of mathematics..." which is an absurd criteria. It is the equivalent, as I said, of expecting our English teachers to be published authors.

He also claims that they "know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students" which is a wild generalization without any supporting evidence.

I think he is, as you say, wildly arrogant regarding teaching. That does not necessarily invalidate his larger point that today's math curriculum is sub par and actively discourages study in mathematics, but I think his disdain for other teachers is misplaced.

I also have a minor issue with his high-minded flighty view of mathematics as a beautiful art, but hey, I'm not a mathematician. I tend to lean towards the sciences and thus view mathematics as a practical tool, a view for which he seems to have some contempt.

[Gav]

I also have a minor issue with his high-minded flighty view of mathematics as a beautiful art, but hey, I'm not a mathematician. I tend to lean towards the sciences and thus view mathematics as a practical tool, a view for which he seems to have some contempt.
I'm not a mathematician either, but I know a few, and they're not really all of one mind on this subject. There are some who definitely are completely uninterested in any practical applications for what they do, but there are some who are happy to consult to companies (like one guy who did a lot of work using his graph theory expertise).

I also have a feeling that he wouldn't actually be too happy if we started teaching math the way we teach art -- my kids do math 5 days a week, for a fair amount of the day. They do art and music once a week each for 40 minutes. In most school districts, art classes will be dropped before math classes, if there's a budget crunch.

[Ryan A]

They do art and music once a week each for 40 minutes. In most school districts, art classes will be dropped before math classes, if there's a budget crunch.

Behold, the Tacoma School of the Arts (http://www.tacoma.k12.wa.us/schools/hs/sota/).

[MikeSofaer]

Of course not, but that's not actually the standard he was holding math teachers up to. As I said, he moves the goalposts. He claims our math teachers "have never produced an original piece of mathematics..." which is an absurd criteria. It is the equivalent, as I said, of expecting our English teachers to be published authors.
I think that in the context of the article it's expecting them to be writers, published or not. I think he's using a pretty loose definition of "original" here, meaning coming up with it yourself, not being the first to come up with it.

[Dave Perkins]

I'm a math prof at Houghton College.

This morning, I figured out how the Indian mathematician Madhava of Sangamagrama (http://en.wikipedia.org/wiki/Madhava_of_Sangamagrama) came up with this formula for pi in the 1300's:

http://upload.wikimedia.org/math/0/0/0/000ef86aee7800e9ae19c8f3bd7b496a.png

He probably used the polynomial approximation for arctan(x) which can be found using trig, derivatives, and the generalized binomial theorem. I was pretty excited that I retraced his steps on my own. This wasn't an original piece of mathematics on my part, but it still felt good, and I bet the author of the article would appreciate the effort.

Whenever my class comes up with a conjecture that we investigate during classtime, often finding it to be false but repairable, and then provable, I say, "Now that was math. We did some actual math there." Of course, we'd have no shot at having that experience without having learned some techniques, but there's no substitute for keeping people curious by engaging in real math in the classroom. It keeps me interested, anyhow.

[fire]

Agreed. Mistakes are what makes us learn.

In class, we've been teaching that Moore's Law states that the number of "active components" (transistors) on a chip doubles every 18 months. The other day, one of the students protested that he answered on the homework that Moore's Law was 24 months (which he found on Wikipedia, apparently the answer to everything). In class, we looked it up in a textbook and the law actually says 18-24 months.

Now, I will always remember Moore's Law.

[hong]

Agreed. Mistakes are what makes us learn.

In class, we've been teaching that Moore's Law states that the number of "active components" (transistors) on a chip doubles every 18 months. The other day, one of the students protested that he answered on the homework that Moore's Law was 24 months (which he found on Wikipedia, apparently the answer to everything). In class, we looked it up in a textbook and the law actually says 18-24 months.

Now, I will always remember Moore's Law.
Clearly, Moore was a statistician!

[Sol Invictus]

I don't understand how anyone can expect a math teacher to come up with an original piece of math. It's like asking someone to publish a theory in a peer reviewed magazine like Nature or Science before he can teach high school science.

Absurd.

[Rimbo]

Requiring every teacher of math to have authored an original piece of mathematics seems a bit extreme. We're already low on math teachers, but I agree that we should want teachers that appreciate it, as well as enjoy it.

Mordrak, the point he's making is that the whole reason we're low on Math teachers is because of the way we teach Mathematics..

His description of Geometry class is so apt it's painful. It brought back some painful memories of my own. I was always good at following instructions, so I did well in Math classes, but I hated it as a subject, and I finally understand why after reading this. It all rings true for me.

There is a kind of applied Mathematics that was taught well in school, at least for me: Music. Music is intensely (if not purely) Mathematical, and yet the way it's taught in school is utterly different from how Mathematics is taught. And how we teach Music seems to fit the description of how he'd like to see Mathematics taught.

[Rimbo]

I don't understand how anyone can expect a math teacher to come up with an original piece of math. It's like asking someone to publish a theory in a peer reviewed magazine like Nature or Science before he can teach high school science.

Absurd.

The fact that you're only considering something a graduate student would do proves the man's point; it only sounds extreme and absurd because, as I pointed out to Mordrak, we've been teaching it the wrong way for so damned long that people have been doing it wrong all that time, and it's become part of our culture to do it this obscene and bizarre way. If it's taught the right way to begin with, having people who know how to come up with theories would be as commonplace as those who have written their own music.

[Nengjanggo]

The fact that you're only considering something a graduate student would do proves the man's point; it only sounds extreme and absurd because, as I pointed out to Mordrak, we've been teaching it the wrong way for so damned long that people have been doing it wrong all that time, and it's become part of our culture to do it this obscene and bizarre way. If it's taught the right way to begin with, having people who know how to come up with theories would be as commonplace as those who have written their own music.

I'm not so sure about that. The standards for what counts as "coming up with a theory" in math are different, and more rigorous, than what counts as writing a piece of music. I'd posit that the number of novel, interesting and plausible theories in math which are humanly createable in a reasonable amount of time (say, less than 10 years) is far fewer than the number of interesting and plausibly-decent-to-some-audience pieces of music. That is based on talking to the people I know who are PhDs in math, and on my own experience doing philosophy (which has at times something in common with math). I do think you could get a greater number of people who have a deeper appreciation of what goes into the sort of math he's talking about, but I don't think you could get as many amateur mathematicians as amateur musicians.

[Nengjanggo]

My big question is, why try so hard to teach an appreciation of pure math? Why not teach an appreciation of applied math? I love algebra because I know how useful it is for things that I am really interested in, like gambling/decision theory, and statistics. When it finally dawned on my that calculus was useful in physics (which none of my teachers in either class told me) I finally developed an appreciation of calculus. When I did graphics programming, I developed an appreciation of trig. An appreciation of applied math would be both easier to teach (because it wouldn't require super scarce resources, like pure mathematicians, to teach it, and because it is closer to the surface in more people) and actually practical.

I think there are many practical applications of complex math that are widely interesting to a wide variety of people, or would be if they were taught decently. And teaching them decently doesn't require (no offense) pie in the sky hopes that we can produce a nation of pure mathematicians. It is pretty doable in my experience as a teacher teaching exactly this stuff, to generate some interest in decision theory, probability, and applied statistics in middle school kids. If math were integrated into other aspects of curricula -- with economics which was integrated with history, so that students could think about how population pressure, for example, drove expansion, or with physics and engineering and history, so that people could see how all these drove each other, or with applied economics and decision theory, so students could evaluate their own lives -- math could be very interesting and very practical. I frankly think that would be more valuable than teaching the love of pure math (although I do think that has some place in a curricula).

[Rimbo]

I'm not so sure about that. The standards for what counts as "coming up with a theory" in math are different, and more rigorous, than what counts as writing a piece of music. I'd posit that the number of novel, interesting and plausible theories in math which are humanly createable in a reasonable amount of time (say, less than 10 years) is far fewer than the number of interesting and plausibly-decent-to-some-audience pieces of music. That is based on talking to the people I know who are PhDs in math, and on my own experience doing philosophy (which has at times something in common with math). I do think you could get a greater number of people who have a deeper appreciation of what goes into the sort of math he's talking about, but I don't think you could get as many amateur mathematicians as amateur musicians.

I think you're grossly overstating the originality of different works of composition (http://www.youtube.com/watch?v=JdxkVQy7QLM), or underestimating how much they borrow from each other. A work doesn't have to be unique to be original. Take the example given in TFA of what the 7th grader came up with; it's far from an original idea, but it's in his own words, which is no different than what Paravonian's demonstrating in the video.

And again, the idea that we would have fewer amateur mathematicians than amateur musicians comes from this same problem that music is taught in schools correctly and mathematics is not. You're one of several people who've pointed to this as a problem with his theory, when it's in actuality a symptom of the disease he describes.

[Sol Invictus]

The fact that you're only considering something a graduate student would do proves the man's point; it only sounds extreme and absurd because, as I pointed out to Mordrak, we've been teaching it the wrong way for so damned long that people have been doing it wrong all that time, and it's become part of our culture to do it this obscene and bizarre way. If it's taught the right way to begin with, having people who know how to come up with theories would be as commonplace as those who have written their own music.

I'm rethinking what I said, based on your response and I've come up with this:

Math is a very strict language. It's limited in the amount of expressions you can do with it unless you start playing around with transcendental numbers (but that's another issue). Math has to be practical in its uses as a language.

With that in mind, I suppose that working with mathematics is a lot like working with a programming language. If we can use programming languages to create programs and software, why not mathematics to create formulas? Or as you mentioned, it'd be like creating music.

I can't see people coming up with 'new math', though. You can't reinvent division and multiplication any more than you can invent new notes and octaves. I think the problem with society is that we tend to view math as being little more than notes and octaves rather than full compositions.

If we think of math as building blocks like notes in a song, we could very well train it as an art form. After all, even though music and programming are sciences at their very core, they're expressed creatively as art.

[Rimbo]

With that in mind, I suppose that working with mathematics is a lot like working with a programming language. If we can use programming languages to create programs and software, why not mathematics to create formulas? Or as you mentioned, it'd be like creating music.

Yeah, yeah. I'd go as far as to say that programming languages and music themselves are Mathematics. It's not a simile, or a metaphor... it's the same thing.

[Pishtaco]

Pure maths research is certainly an art, but there's also a lot of the scientific method in it. You have to come up with creative ideas, but more of your time is spent checking, in boring detail, that they are true. I don't really see the connection with music. But a mathematical proof is, a lot of the time, the same thing as a computer program, although one written in an ambiguous, very high level language (English with some notation) and which you have to debug without being able to compile it.

I'm not sure what this has to do with the applied mathematics and calculation that people should learn in school. I suppose it would be good if students had some idea how new mathematics is made, but I can't see it being a large part of the curriculum.

[MikeSofaer]

I don't see where people are getting the idea that he expects teachers to have made contributions to the field. He just wants them to be able to approach math problems a priori, and thinks that many grade school teachers can't.

[Gav]

Yeah, yeah. I'd go as far as to say that programming languages and music themselves are Mathematics. It's not a simile, or a metaphor... it's the same thing.

No, they're not. I can determine if a proof is correct. How do I determine the correctness of a Haydn symphony (or, say, something by Cage or Varese)?

Programming seems related, but doing the two doesn't feel similar to me at all. (I program for a living, and did some reasonably heavy-duty abstract math in college, which I still like to keep up on). I think it's because, as a programmer, I have to worry about all the stupid things that a math person can assume away :-). (Assume a user smart enough to read directions.)

[Gav]

On a totally different note, I think he idealizes abstraction too much. He says that students don't get into made-up problems (Mary is half Sue's age plus 6), but the right problems can be very useful, and great for motivating kids. My oldest son (4th grade) recently did a project where each kid designed a dream house, where the rooms would be all different shapes and sizes (within limits for the shapes), and then had to figure out the area of the house. He was pretty excited about the whole thing, and it gave them a practical explanation of some of the area formulas -- they saw how the formulas they use work out with actual graph paper squares.

I'd argue that that sort of playing around is closer to how we teach music -- playing around with it to see how it works. Teaching abstract math would be like trying to teach first-graders about sonata-allegro form.

My wife was a teacher, and I used to occasionally look through papers she had to read when she was studying for her certificate, and this essay really resembles them -- some gifted and enthusiastic teacher (let's assume that this guy is actually a very good teacher) tries out an approach to teaching, and it works. But not necessarily because the method is so great, it's that he himself is a gifted enthusiastic teacher -- Lockhart clearly knows his math, and I'll bet he could put together a great applied math curriculum, and then maybe we'd be discussing whether an applied-math curriculum is the best.

My wife's specialty happened to be English, rather than math, but you'd see these essays by a teacher who'd explain why a whole-word approach is the only way to teach reading, because she'd tried it, and her students were so enthusiastic and motivated compared to when they entered he classroom. Then there'd be another essay by some other teacher who liked a phonics-based approach, because the students who came into her classroom had been really messed up by the whole-word approach, but by the time she was done with them, they really loved reading. And so on.

[Hawkeye Fierce]

I don't see where people are getting the idea that he expects teachers to have made contributions to the field. He just wants them to be able to approach math problems a priori, and thinks that many grade school teachers can't.

We were speaking directly to his quote that we accept math teachers that have "never produced an original piece of mathematics." Seems pretty clear to me.

Honestly, it's not all that major a quibble with the article. I just think he's unfairly blaming math teachers for a curriculum which is largely out of their control. There are a lot of factors which ensure, in a public school at least, that math teachers don't have the freedom to color outside the lines, so to speak.

I think that a larger objection is that his view of how math should be taught is unworkable in a large number of classroom settings. Gav hits on why I think that:

My wife was a teacher, and I used to occasionally look through papers she had to read when she was studying for her certificate, and this essay really resembles them -- some gifted and enthusiastic teacher (let's assume that this guy is actually a very good teacher) tries out an approach to teaching, and it works. But not necessarily because the method is so great, it's that he himself is a gifted enthusiastic teacher -- Lockhart clearly knows his math, and I'll bet he could put together a great applied math curriculum, and then maybe we'd be discussing whether an applied-math curriculum is the best.

I suspect not only is he a gifted enthusiastic teacher, but he has gifted enthusiastic students, given his school's stated purpose. Not every classroom is like that, and I think it's a mistake for him to assume that his methods would be effective elsewhere without difficulty.

[Rimbo]

No, they're not. I can determine if a proof is correct. How do I determine the correctness of a Haydn symphony (or, say, something by Cage or Varese)?

You take an arbitrary set of an assumptions and an arbitrary set of transformations and see if the symphony follows those in the expression of its themes. I mean this was the definition of Classical music for a good long while; you'd come up with a theme, but in order to be "proper" you'd have to follow certain transformations. I think this is also the appeal of Bach, especially to geeks like us.

[shift6]

A friend of mine passed along this article (http://www.maa.org/devlin/LockhartsLament.pdf) that laments the current state of math education. As I know we have a few mathematicians and teachers around here, I figured I'd post it. I'm curious what everyone thinks.
I'm a mathematician by education and a corporate number cruncher by trade. Early on, my degree was to be in so-called "pure math", and I focused on topics like geometry, combinatorics, and set theory. When I transferred to UC Davis, which does not have a "pure math" program, I changed it to "general math" and finished out my sentence doing advanced calculus and such.

I found that, in general, I learned better and more thoroughly when I learned how something was done, as opposed to memorizing which numbers to plug in to which variables.

My best example for this was one of my final courses: partial differential equations. During the first part of the semester, I struggled because while others were essentially passing info around in "study groups", I was trying to learn the underlying substance of the thing. My goal was to fully understand the propositions of a given problem so that I wouldn't need to refer to a notecard or other rote tool as a crutch. At the end of the semester, this happened. I hate to be totally cliche, but something clicked and all of a sudden, I grokked diffey-cues to a point where I could explain them, in my own terms, and talk about what the variables meant. I could do them without needing to look up notecards and so on, because the underlying rules made intuitive sense to me. I did very well on the final, and finished much quicker than most.

I can see the value in attempting to teach kids (K-college) how to grok mathematics just as much as critical thinking in any other basic soft skill (reading, writing, etc). There is little value in memorizing anything from multiplication tables to Taylor Series expansions except to pass the next test. On the other hand, in a culture where everything is ladder-based, as the author explains, it would be difficult for one teacher at one grade to change; any kid leaving the class might be a richer person, but the next year's teacher will expect they've done p. 1-200 in the text. In a private or home-school setting, there might be something to it.

[Nengjanggo]

There is little value in memorizing anything from multiplication tables to Taylor Series expansions except to pass the next test.
I disagree with this in part; I don't know what a Taylor Series expansion is so it might be totally valueless to memorize, but certainly a lot of basic math is hugely useful to memorize. The ability to easily manipulate numbers one encounters in everyday life is useful. That ability comes in part from a fluency with basic operations, such as addition, multiplication, etc. Further, the ability to deal with numbers comfortably has a strong connection to the ability to do lots of pure math (and certainly to do much complicated applied math). This comfort and fluency absolutely requires memorization and drill. If you have to constantly think about stuff like multiplication, or consult a calculator to do it, you are going to have a hard time doing anything with numbers. You probably can't really get numbers or how they are related to each other.

Somewhat anecdotal evidence: My mother's experience teaching middle school students (in an environment where many of these students are way behind what we'd expect middle school students to known) is that students who do not have the basics down cold -- and multiplication tables are something she explicitly talks about as being hugely important -- feel totally uncomfortable with numbers and with math in general.

One might claim that this is because they are doing waste of time middle school math, and if they were doing pure math they wouldn't need to be comfortable with numbers. I'm dubious of this; it seems to me that you can't really grok certain relationships between numbers unless you are fluent with the basic operations (this comes from my personal experience working through seeing how the operations derive from the Peano axioms, which is kind of pure math, and also thinking about stuff like Goldbach's conjecture and patterns in prime numbers).

Further, even if you can totally get pure math without fluency in the basics, you certainly would be helpless around numbers. Anyone who's been in a restaurant or store when the cash registers are down probably thinks its ridiculous that people can't function without these things; this is how you'd be without knowing your multiplication tables. You wouldn't be able to do things that are useful like a) converting metric to English in your head, b) figuring out how to scale recipes while in the kitchen, c) calculate a tip in the absence of a calculator, d) allocate resources amongst groups quickly, etc etc etc.

Sorry about that long reply; maybe you meant something a bit weaker than I read you as saying, maybe I'm nitpicking.

[Funkula]

Well, I thought the article expressed some really good ideas, but Jesus Christ that guy is full of himself. So full he's coming out his own ears.

But yeah, I personally find the ability to perform arithmetic operations quickly in my head to be invaluable. I do think drilling multiplication has value because it honestly is something you can use every day. I also derived a lot of good from algebra classes, specifically the ability to move elements around in equations and the ability to think in a cartesian plane.

[Gourmand]

Nengjanggo,

I think Shift6 is trying to say that just saying "ok memorize this" will allow the kids to memorize it, but they'll forget it the next day.

Saying "Ok, this is really cool because ..." and showing examples, while scaffolding the information in a palatable manner will allow that child to retain the knowledge much much longer.

The problem is, memorization is step 1 on Bloom's Taxonomy, and you can do all of fuck all with it. Teaching a kid to really understand the material will naturally allow them to memorize the material. Ideally, you want to move higher into thought processes like analysis and synthesis (where student can take a part and then put back together a problem).

Unfortunately, the system isn't really setup to reward that kind of teaching.

[shift6]

Sorry about that long reply; maybe you meant something a bit weaker than I read you as saying, maybe I'm nitpicking.
Not at all, I just gave a bad example. Certain memorizing multiplication tables makes sense for 7 year olds to the extent that it helps begin to form their head for numbers, although I still agree with the author that a more qualitative approach would help; he gave a good example of thinking of multiplication along the lines of "5x7 = five sets of seven or seven sets of five" instead of just memorizing 5x7=35. FWIW, there are many interesting and fun mnemonics that people who intuit math discover in their own heads at this age, such as the multiplciation tables for 9s and 11s that totally bypass memorization.

A better example from me may have been something higher up the food chain such as memorizing sine/cosine/tangent rules in trig. For the life of me, even now, I can't tell you what a secant actually is, but I can rattle off the useless formulaic sec = 1/cos. I can, however, describe in prose the sine, cosine, and tangent functions, and tell someone all about them.

My best friend in college had alot of trouble passing calculus 1 because he was trying to do his sin/cos work by rote instead of grokking the notion of the unit triangle. If you think of trig this way, cosine is just the length of the X-axis side of the triangle, sine is just the length of the Y-axis side. That's it: just sides of a triangle. Once I got through to him about this, other things just started falling into place for him (like tangent: oh, it's just a fraction of the Y-side divided by the X-side).

[Nengjanggo]

I agree: you want students to understand the math facts, not just memorize them. Presenting them qualitatively is a good way to help accomplish that. There is a place for rote memorization and drills as well, because certain ideas need to be not only understood, but usable without thinking. When I have taught math or logic in the past I try to get my students to understand what it is they are doing and why it works, and then I make them memorize it and try to make using it habitual through repetition (and I make sure my students understand why that is important). I have no idea to what extent this is relevant to higher math because I stopped at AP Calculus myself.