A Mathematicians Lament- have you read it?

[Just Another Jen]

http://www.maa.org/devlin/LockhartsLament.pdf

 

This is long but worth the read. What do you think?

[Jane in NC]

Thank you ever so much for posting this link. I have mentioned it on the high school board where we often have some interesting discussions on mathematics and mathematics education. Perhaps some comments will surface over there.

 

Jane

[Myrtle]

I've read through a few pages and I really will have to read through the entire thing carefully to make an attempt at decent response, but my first thoughts are that so far he has spent a lot of time saying that math is about the "whys."

 

And I think that is wonderfu, but what "why" means to a mathematician is completely different than what the rest of us do. So far, he has not clarified this vaguery or acknowledge that we have different notions of what qualifies as "why" in math.

 

When I first started teaching Singapore I was thrilled to think that I had finally discovered "why" arithmetic worked and how numbers worked only to be told that this wasn't why at all that the explanations were heuristics and not mathematical explanations. What was that supposed to mean? It's all very frustrating when you start down this road because everyone tells you what math is not, but no one will tell you what math is. And then when you can get a straight answer about what math is, it's meaningless. It only began to start to make sense after a long time.

 

An anecdote, one day I had discovered the "casting out nines" gimmick for double checking calculations and I asked my father in law if he could write up a proof that it did really work. And he deadpans to me, "You go figure out why it works and tell me, and then I'll write the proof." I was clueless why he'd make such a response to me.

 

Secondly, I can see where those promoting constructivist pedagogy might think that Lockart is giving them ammunition for their cause and I'd like to see him disambiguate what he's doing from this other notion of math.

 

...

 

Okay, I'm reading more...

 

"A proof should be like a splash of cool water, a beacon of light--it should refresh the spirit and illuminate the mind. And it should be charming"

 

Hmm, I was always told proofs were about being correct.

 

It's entertaining to have a charming proof but when I'm in the midst of showing that the max {a,b} + max {c,d} ≧ max {a,b,c,d} all I care about is getting the proof at all...in any form...if I wanted charm I'd stick a candle and bouquet of roses on the table while I work out the details.

 

Why is it that I concern myself with mindless boring formalism and "technical details" for weeks on end? Because if I delay gratification I can get the pay off in the end by doing what I really want to do which is knowing why the Minkowski inequality works, or knowing why the Holder inequality works. And it turns out that the mindless formalism and technical details really are there for a reason after all.

 

He says that rigorous formal proof isn't important. You soon discover how important it is when someone instantly pokes a hole in your reasoning because you were not precise enough, didn't take certain things into consideration, etc. All rigor is (and he means matheamtical rigor, not academic rogor there is a difference) , is making sure you account for everything, your assumptions, etc. Rigor means delivering an airtight argument, it's the ultimate in prosecution, and no defense attorney anywhere can raise any doubt about anything you've said; you've accounted for every last detail and it's because you've accounted for every assumption, every premise, every definition, and every last inference that you have absolute knowledge and thus Truth (with a capital T) , about the validity of your conclusions.

 

And yes, that takes a LONG time to write out, to think about, to edit, to recheck, and it's not fun. But much like cleaning the house, no one really much likes the tedium of the process, but we all love the results for the hard work.

[Jane in NC]

For me the article was more amusing than informative until page 14. Under the heading of "The Mathematics Curriculum", Lockhart bemoans the fact that little mathematics is actually being taught in today's mathematics curriculum. This had me sit up and take notice.

 

I cannot tell you how many students I have had in college math classes comment that math was like Latin: a dead subject. From how mathematics is taught, one would think that nothing new has happened in math since Newton and Leibniz, that they marked the end of the subject with the creation of this thing called Calculus. There are times when I think that those of us who have studied mathematics are to blame for the state of things. We have created a fairly exclusive club. In order to understand 20th century mathematics, one first earns a BS in math and then goes to grad school. We are not asked to explain what we do to non-mathematicians.

 

The self discovery aspect that is suggested in the article is what makes me a bit nervous. Self discovery is a wonderful thing for a young child. But it seems that some formal indoctrination into arithmetic is not a bad thing.

 

My son is now working on trigonometry so Lockhart's comments on the subject led us both to chuckle. I was asked to be a long term sub in a precalculus course a few years ago. The students had been given two pages (!) of trig formulae to memorize. I told them that that they needed to know a few definitions concerning relationships of sides of triangle and everything else is derived for free. There is no need to waste students time on this sort of nonsense, particularly when this group intended to take calculus and would have retained none of the memorized garbage. Talk about silly! And talk about turning people off to math!

 

While Lockhart has valid comments on the problems of today's standard mathematics curriculum, I cannot say that I would abandon it all for fuzzy self discovery. Teacher led discovery could be interesting--some sort of framework is needed.

 

Jane

[Guest Lorna]

I think he has several good points. Sometimes mathematicians and text books try too hard to be story-like and 'word problems' can turn into English comprehension questions. By trying to make a complex equation sound like it can be applied to every day life the beauty and art of finding the solution can be lost.

I have painful memories of spending years working out hire-purchase agreements and gas bills in maths.

I don't agree that the foundations of arithmetic and the grammar of maths can be done away with but he has a point that much more maths education time should be spent exploring patterns and experimenting. This takes time and would be very tricky to do in a public school environment.

Thank you so much for pointing out this article. It has helped me become aware of what is lacking in our day-to-day maths.

[Myrtle]

He's dissing the rules of math for being stifling and what does he want to put in their place? The rules of chess, the rules of Go. That's not ridding ourselves of rules, that's just replacing one set of rules for another set of rules.

 

While it may be the case that logic can be used while playing chess, it doesn't follow that it will be used. Kids will turn the crank and not think at what they are doing when playing chess and care as little for it as they do for math itself. What's that Bertrand Russell quote? "Many people would sooner die than think; in fact, they do so." And it isn't that they aren't problem solving because they don't have the right kind of problem to solve, anymore than I'm disinterested in a same sex relationship because I haven't met the "right" person.

 

It is necessary to have good and interesting problems along the way for the minority that might be interested, but it is not sufficient. When teaching my kids I always try to come up with some sort of "discovery" activity in which to explore and introduce a concept, but the reality is that sometimes they'd rather gaze at their navel and flip the pencil eraser across the room. There is no cure-all for this. I can't force my kid to be interested in a particular problem and to want to think about it, but I can train him how to balance his checkbook. I can't force an aesthetic appreciation for math anymore than I can manipulate someone into liking country music or hard liquor.

 

While it is true that "Math Methods in the physical sciences K14" is not math, neither is "Math Appreciation."

 

But for that matter, until the basics of technique and proof have been mastered how can aesthetic appreciation of proof occur? What's "charm" to someone who's never experienced ugly?

 

We have millions of adults wandering around with "negative b plus or minus the square root of b squared minus 4ac all over 2a" in their heads, and absolutely no idea whatsoever what it means. And the reason is that they were never given the chance to discover or invent such things for themselves.

 

No one ever prevented a curious person from getting a pencil and paper and figuring out how to derive the the quadratic formula from a quadratic equation. The problem with people not knowing the why isn't that they are prematurely told, as if this were the plot to a movie they were in the process of watching when someone shouted out, "The butler did it!" and it killed the suspense, The problem is that people don't know why they should care about whodunit to begin with.

 

Lockhart is offering a King Xerxes promise, a royal road: Go his way and you will have endless joys, playing chess (appropriately enough invented by the Persians) Those cruel math meanies, the Greeks, gave you Euclid and required that you stand, all Lockhardt requires is that you kneel.

[Myrtle]

By trying to make a complex equation sound like it can be applied to every day life the beauty and art of finding the solution can be lost.

I have painful memories of spending years working out hire-purchase agreements and gas bills in maths. QUOTE]

 

 

Me too.

 

These kinds of problems are based on the pretense of "real life." I always hated pretending that I was a check out clerk or "researcher" like these problems required. But a problem about a four dimensional cube has a way of capturing the imagination. The teacher doesn't need elaborate pretend "real" life scenarios, he only needs to say the one magic word, "Suppose..."

 

At the same time, the conclusions arrived at by accepting the assumptions are only as valid as the technique used to arrive at them. Perhaps those with natural talent and a teacher with a PhD in math who personally worked with Erdos will quickly pick up on this technique without any direct instruction, the rest of us chickens have to study books and work at it and wait a long time for the pay off.