kitchen table math, the sequel: Lockhart's Lament

Thursday, March 13, 2008

Lockhart's Lament

Lockhart's Lament via Devlin's Corner

The low down:

A PhD Mathematician teaching school age children suggests that since kids aren't really learning anything as it is in drill and kill math class, let's stop the pretense altogether and teach them "real" math.

And I'm all for "real" math, I really am.

It's not that he doesn't bring up some good points, but I'm not for playing chess and calling it math, like he suggests. And I'm also pretty sure that no matter how interesting of problems that he can come up with he's not going to get everyone excited. If kids thought that mankind's struggle to measure curves was really exciting you'd see more video games and movies about it.

And I also suspect that before one can appreciate all this poetic beauty, artistry, creativity, and finger snapping, that a student, even in pure math, might need to have mastered a few basic techniques and have memorized some definitions and axioms first. However, this too seems to be dismissed by Lockhardt as so much mindless formalism and dispensed with. Is it possible that instead of serving up a dose of "real" math, he is serving up math appreciation?

With all the references in that article to aesthetic appreciation Jacque Barzun's chapter on Occupational Disease: Verbal Inflation came to mind.

Then again, maybe I've just got a bad case of sour grapes. I spent an entire afternoon working on proving that the max of two different sets was the same by showing that the max of one set was less than or equal to the max of the other set and then showing that the max of the other set was less than or equal to the max of the first. There was nothing "charming" about it.

I can't do charming proofs. I can barely do ugly ones.

23 comments:

SteveH said...

The comparison of algebra to musical notation is ludicrous.


"Everyone knows that something is wrong. The politicians say, 'we need higher standards.'
The schools say, 'we need more money and equipment.' Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones
most often blamed and least often heard: the students. They say, 'math class is stupid and
boring,' and they are right."

Well, he can go start his own school and see how much ALL of the kids like his dreamy, discovery, arty math. The kids might not use the term boring, but I'll bet stupid will be right up there, and they will be right.

But you can be sure that my son knows about the circle of fifths and practices his scales and arpeggios. The author might like a top-down discovery approach to mastery, but it won't get the job done. Oh, that's right, the author doesn't believe that most kids need a rigorous mathematical education. Just like most kids will never use algebra when they grow up, and his idea of math education will guarantee that.

Brewhaha said...

His premise that just because students aren't learning we should stop teaching is a big part of the problem. If teachers don't expect their students to learn the foundational principles and if teachers don't believe that their students can learn these principles then the failure will continue.

Adrian said...

Well, I think the real point is that it just isn't true. It is true that a lot of students aren't learning. But, it certainly is true that plenty of students are learning just what we're teaching them. Where I would agree with him is that what we are teaching them isn't really math. But, that means a content change not a change in pedagogy.

So, fine. I'm all for covering things like the infinitude of primes. But, that means expanding the Euclidean Geometry we do not castigating it as the worst subject we teach and replacing it with mentioning that the primes are infinite over a nice game of chess.

Catherine Johnson said...

I can't do proofs at all.

That's next up on the agenda, proofs and logarithms.

That's assuming I don't get sidetracked by origami...

Karen A said...

Speaking of origami--

My older daughter (now a college freshman) despised origami (she had to do an origami project when she was in Honors Geometry), but my younger daughter (now a 9th grader) loves origami. In fact, the younger daughter (as a 4th grader) helped the older daughter with her origami project.

Younger daughter loves puzzles; older daughter, not so much. Older daughter is spatially challenged; spatial thinking comes much more naturally to younger daughter. Younger daughter loves art; older daughter, not so much.

I think there's a pattern here (although I realize that the Math Panel said to stop looking for patterns). hee hee

However, regardless of my kids' differences, I have been very clear about what must be the same, and that is that they each receive a solid education, grounded in the fundamentals of reading, writing and math, so that they can have the option to choose their respective paths in life.

SteveH said...

"The first thing to understand is that mathematics is an art."

This is a lousy analogy, and his premise that the best approach to art or math is a top-down approach is just wrong.

I'll give you an example. This past weekend, my son was in a piano competition at our university. Pianists (up through high school) from three states came to compete in the concert hall. They had to play 10-15 minutes of music from memory. They were judged and awards were given out. Many of the pianists were amazing, not just in their technical abilities, but in their musicality. They did not get that way using a top-down constructive approach. These kids were far from robots. They practice hours a day on scales, arpeggios, and etudes.

During breaks, we wandered around the building and looked at some of the work made by the university's art students. The contrast in "art" was astounding. Right behind the concert hall were student art objects made out of black and red swizzle sticks. (Last year it was popsicle sticks. I'm not kidding.) One of the objects was a very bad swizzle stick representation of a grand piano. At least it looked like it had 88 keys. Compare this with the art required to build the real thing. This is art we get from college students who have had 12+ years of intense, constructive, thematic, top-down training in art. And these are kids who are majoring in art!

The music was so wonderful that it made you cry, and the art was so bad that it made you cry in a different way.

By the way, one of the organizers said that it was easy to find my son because he was the only one there with blond hair. You don't get any educational handwringing from the Asian or Russian piano teachers or parents, and they don't look to ed schools for guidance.

It would be one thing if a top-down discovery approach to teaching and learning was rigorous and required a lot of hard work, but it's never that way. People somehow think that less can be more.

Ben Calvin said...

I'm a top-down type person. It's my "natural" learning style. Over reliance on my ability to "get it" is why I've never learned much math past algebra and logic.

(Thank god I completed those by 10th grade as I do computer programming for a living!)

My son is learning piano. He has a good deal of natural talent. But his progress only comes by practice and repetition.

Math has a conceptual, big picture dimension. But it can only be accessed by learning a variety of skills that have to be practiced, internalized and built on. Those skills have utility whether you ever get to the philosophical aspect of math or not.

I noted another thread talked about a school bringing in animation and film-making software into the classroom. I would prefer the schools concentrate on the math skills these kids require to write animation and film-making software.

Catherine Johnson said...

I noted another thread talked about a school bringing in animation and film-making software into the classroom. I would prefer the schools concentrate on the math skills these kids require to write animation and film-making software.


You said it.

Lsquared said...

I completely disagree with Devlin on this, and I do think mathematics is an art for (most mathematicians believe this, I think). But--it's an art form you need to know an awful lot to do or even appreciate. Steveh's comment on how much practice in the basics musicians need is a really good analogy. You really don't have much hope of doing math like a mathematician does, unless you have the basics. You can't discover and prove things about prime numbers until factoring is something you can almost do in your sleep. You can't invent new calculus until you know calculus thoroughly. Even the boring Geometry theorems have their place because they teach you to be careful with your axioms (I've taught enough of that recently to decide that 2 column proofs are the easiest for most people starting out). The rotating the triangle in a semi-circle proof is really cool, but if a student were doing it I would ask, now, have you already proved that a parallelogram with congruent diagonals has to be a rectangle? You do have to learn these things you know (you have to be able to pin down the details as well as coming up with a good idea).

Math is an art form, and letting students conjecture, and explain, and explore is good, but they're never going to make it unless they practice those scales (or multiplication tables, as the case may be).

Lsquared said...

Oops, I disagree with Lockhart, not Devlin... Maybe I wouldn't even disagree with him in person, but I do think that you need to know all of that "boring" stuff if you're ever going to be able to do the cool stuff.

Anonymous said...

I'm an English major who hated math through high school and college. I loved Lockhart's example proofs, and I caught a glimpse of what makes people enjoy math, perhaps for the first time.

I think your doubt that such problems can enthrall kids is cynical and destructive.

SteveH said...

"...and I caught a glimpse of what makes people enjoy math, perhaps for the first time."

You miss the point.

Many of us at KTM know first hand the thrill of math on a daily basis in learning, teaching, and application. Many of the teachers here also try very hard to offer that view and experience to their students. Developing that thrill in students is not tied to one curriculum or another.

But what is the rest of the curriculum?

Most K-6 schools now offer math curricula based on low expectation of mastery, child-centered, group discovery learning. It doesn't get the job done. Lockhart wants to change the job, but he doesn't go into many details. It's easy to describe a few dreamy examples, but quite another to define and implement a full curiculum.


"I think your doubt that such problems can enthrall kids is cynical and destructive."

Nobody here said or even thinks that.

A thrilling (all the time?) approach to math will not motivate students to do all of the hard mastery work. Even Lockhart says that math is hard work, but he doesn't explain how to make sure that kids know the times table or can manipulate fractions. It's easy to get excited when math gurus get up on their soapboxes, but what happens the next night when the student is trying to do the required homework set? Oh yes, Lockhart doesn't believe in that.

Excitement is a nice goal, but it doesn't get the job done, even if you try to redefine the job.

Barry Garelick said...

Beware of what Keith Devlin chooses to talk about. He writes on a level of math appreciation, and Lockhart's Lament fits right into that. There are many interesting things about math, for sure and kids should be exposed to them. But they also need to be taught the tools so that eventually they can work with and analyze these media-ready cool things Devlin likes to write about.

Working with such skills and content-based mastery, students can progress from problems like "The sum of three consecutive numbers is 15, find the numbers" to "Prove that the sum of an even and an odd number is always odd" and with even more work to problems such as "Prove that ratios in the form 1+3/5+7, or 1+3+5/7+9+11, and so on, always equal 1/3." Or "Prove the the product of four consecutive numbers is always one less than a perfect square."

These last two are not easy but can be solved with mastery of algebraic techniques and some insight gained from working with problems like the first few that many deem to be too boring to waste students' time on.

Math is like anything else. Start with the easy stuff and build upwards, and practice, practice, practice.

Steve said...

I just came upon Lockhart's Lament today for the first time and I think it is right on target. My own experience is in physics, but the same issues of students doing exercises that are boring and mindless rather than something that makes them think more deeply is thoroughly entrenched.

For the past few months I have been helping some high school students with their math homework and I find it very repetitive and unmotivating. So do they. No wonder they hate doing it.

Last year I hooked up with a teacher who is coaching a bunch of math students for an area MathLeague competition. Those students were really getting into the challenge of trying to solve some tough problems under the pressure of a time limit.

I was very impressed. I asked one of them why they found it fun, and she said part of it was the sense of pride she got from being able to figure it out.

I don't see how a student could ever feel that way after doing most of the tedious homework problems I helped with. It was just the feeling of relief that the torture was over for the moment.

What is needed is problems to work on that make the students feel they figured something out by reasoning rather than just parroting an example.

Steve in Duluth MN

Catherine Johnson said...

Hi Steve!

You're going to have to come see the Westchester Math Lab teachers.

The class I observed used Primary Mathematics as the core text, along with one page of Math League problems and several word problems/brain teasers translated from Russian math textbooks.

Apparently there are zillions of fantastic Russian math textbooks posted on the web.

Problem is: they're in Russian.

SusanJ said...

I just got too busy to follow this wonderful blog regularly but I've missed you guys! Now I need some quick help. I just ran across a positive review of "Lockhart's Lament" (which I'd never heard of before) by a blogger mother of a 4-year old who seems innocent of the Math Wars as do her commenters, most of whom are also parents. I'd love to be able to make a short helpful comment. My question is whether Lockhart's article is intentionally supportive of the "discovery math" side of the Math Wars (in contrast to most of you) or is it simply an isolated essay with some good points and some bad points?

Catherine Johnson said...

Hi Susan!

Great to see you!

I miss you, too!

I'm gonna let everyone else answer this ---

SteveH said...

"My question is whether Lockhart's article is intentionally supportive of the 'discovery math' side of the Math Wars (in contrast to most of you) or is it simply an isolated essay with some good points and some bad points."

My view (from just that article) is that it is NOT intentionally supportive of the discovery math "side". It seems more to me like a viewpoint of a mathematician who has no clue about what acutally goes on in real schools. It reminds me of a new parent to my son's school (a few years ago) who had a masters degree in applied math and thought that Everyday Math seemed pretty good. He liked their high level talk of concepts, understanding, and alternative methods. He even offered to teach an extra afterschool class on problem solving. By the end of the year, he had completely changed his mind about Everyday Math.

The problem with discovery and constructivism is that they are not wrong per se. I have been involved with many confusing blog and math forum discussions over the years about this, and everything came down to the details. Often, discovery is implemented as mixed-ability group learning in EVERY class, with the teacher taking a minor role. This is very inefficient and guarantees that less material will be covered, unless, as in Everyday Math, you just barge ahead and claim that it will all work out in the end.

I see Lockhart's Lament as arguing the dream, but ignoring the reality of the details. As for the discovery "side", their dream is not really discovery, it's mixed-ability, child-centered, hands-on group work in every class. There is a big difference. I have had many light bulb discovery moments during direct instruction and during homework. However, discovery is neither necessary or sufficient. There is a goal and not just a process.

SusanJ said...

Hi Steve. Thanks for the very useful and well-presented information. I'd forgotten the important information about how discovery math plays out in mixed ability groups. And I love your phrase about "arguing the dream, but ignoring the reality of the details."

I just reading a new art instruction book where the artist author writes in the Introduction, "Life is too short to learn painting only by trial and error." And I would add math and many other items as well as "painting".

Thanks again!

Kurt Lovelace said...

I have read and re-read Lockhart's paper, all 25 pages of it -- 3 times now, with breaks of a week in-between readings, just to make certain that I had time to recover after tripping on one or another piece of emotive bait he lays out for the thin-skinned or unweary reader.

I believe Lockhart is correct although somewhat too vehement in his approach to stating his ideas.

But perhaps this is Lockhart's intended effect: getting teachers and textbook executives and school administrators of math programs extremely irate with his essay -- as these posts here provide ample evidence. How can it be otherwise? Lockhart’s aim is directly at the aforementioned audience.

His battle is with a system so self-reflective and shielded from outsiders that it has congealed into obsidian. All of Lockhart's shouting will not nick obsidian. But it is fun to watch the sparks fly!

Most of the posts therefore appear fatuous. Why?

Because Lockhart is teaching children in the classroom himself. Therefore, any post here saying that his ideas are not realistic in that context have chosen to ignore that portion of his essay. Lockhard is for rote learning. Lockhart is for memorization. But he is also for enjoyment and exploration.

Reading Lockhart, I kept thinking that he was too one-sided. But with each re-reading, I found he did address my misgivings, even if he did not go into detail.

Now, Lockhart has been under discussion between my self and a few other friends. I am Oxford trained in pure maths and software engineering, yet do a great deal of creative writing. A friend of mine is and MIT maths and engineer trained thinker. Both of us are heavily into pure research mathematics and applications to real-world solutions via computer programming. We both have discussed Lockhart at length. We both have children in elementary school. So, I wish to share a few observations we arrived at after many discussions during the past few years in the area of math education:


[1] The Chicago Math and its ilk of curriculum are very nonsensical approaches to teaching mathematics. Our children’s exposure to this destructive collectivist muddle of ideas and techniques has taken much work at home to counter. Even the teachers admit openly that the techniques are pointless and conceded that our children may use traditional long-hand methods and efficient short-cuts to do their “work” in class.

[2] Teacher’s we know and talk to are indeed trapped -- whether aware of it or not -- within a system that lays out each day’s “teaching” on their internal website. And precisely lays it out, down to which sections are to be covered that day, and what exercises are to be done that night. Why? Because they are preparing their students for the “TAKS” exam or the “Stanford” exam. But it boils down to bloody politics. With a massive school union and both elected and appointed “administrators” looking over their shoulders, there is nothing else they may do. Then there are their yearly bonuses, dependent on how many children pass the standardized exams. This is the obsidian, in its purest form, that I spoke of earlier. Read the play “Huis Clos” by Sartre -- a near perfect, satirical, black comedy analogy of our current school systems in America.


[3] Lockhart says little about how we might “get to” his ideal classroom. The Devil is indeed in the details. So, people will have to roll-up their sleeves and work on producing a “path-based” set of examples that a teacher would need to get children exploring mathematics in the fashion he suggests. We believe this is quite a challenge, given the inertia that currently exists within the systems as we encounter it everyday with our children. We teach them more math at home then they are getting in school, with more shortcuts, more thinking, and more rapid progression -- all of which causes its own problems when they return to their classrooms where no “staggered” path forward exists.

[part 1 of 2]

Kurt Lovelace said...

[part 2/2]

[4] For one glimpse of how to at least cut-though the detritus of the current textbooks, take a look at “Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry” by George F. Simmons. He is a teacher who one day realized that the entire 3 1/2 years of High School mathematics might be condensed into 119 succinct pages and taught as a one semester course to motivated students, pretty much as one contiguous subject. It is an utterly astonishing jewel of a text. A teacher would choose the exercises and students would have to think through much of this, guided by their teacher, yet still covering a very traditional curriculum but getting to the guts of the matter. This text is not following Lockhart’s thesis, yet it is one insider’s approach to cutting through the cruft that heavily impedes students when presented with 500 pages of pre-algebra, 500 pages of algebra, 500 pages of analytic geometry and trigonometry, and 500 pages of pre-calculus and four years of working exercises.

[5] There are no easy answers. However, closing ranks and not openly admitting the degree of “quantum inseparability” inherent within the current system is doing our children and ourselves a grave injustice. It is indeed a “self perpetuating” mockery and illusion of what real mathematics is about. Real maths are problems which guide a person into thinking up more problems as they attempt to answer the initial one. It is a self-expending journey of discovery and exploration.

[6] I have found numerous papers written and published on the NCTM’s own website going into these issues at great length. So, there is some internal dialog occurring. But just like Simmon’s text, is any of it being used to move more students to think math is anything other than boring?

This is the key question Lockhart is wanting everyone reading his essay to address: all 5 grade students I ask think math is pointless, painful, boring, and uninteresting. Most non-technically fielded adults I ask say the same things and explain in embarrassed tones why they hated algebra. No non-technical folks I ask can say they have ever used algebra to solve an everyday problem. Lockhart is onto something huge here, if only more people within the system put down their guards and stepped-back to look at it this way. Something is rotten in Denmark! Lockhart is not the answer, only a guide attempting to show us some too few attempted paths out of our current insanity that passes for mathematical education.

SteveH said...

"Lockhart is onto something huge here, if only more people within the system put down their guards and stepped-back to look at it this way."

My impression is that you are somewhat new to this problem. If it were only a matter of stepping back and having a nice discussion. There is no process for this. Schools decide and then hold Math Nights to inform parents about best practices and understanding in math. Parents who are mathematicians and engineers sit in little chairs and dare not question fundamental assumptions that have already been made.

The problem is not constructivism or understanding, but low expectations. It's also about how K-8 educators are redefining math in their own image. This is not about two alternate approaches to the same goal, but two different goals.


It's easy for everyone to nod their heads and say that there is a problem (not the same one), but quite another to get down to the dirty details of curriculum, especially when many educators view the process as the goal, not content and mastery of skills. There is no process for addressing these issues. Schools work really hard to keep it that way.


"...all 5 grade students I ask think math is pointless, painful, boring, and uninteresting. Most non-technically fielded adults I ask say the same things and explain in embarrassed tones why they hated algebra. No non-technical folks I ask can say they have ever used algebra to solve an everyday problem."

Is this the real problem? I disagree. This is just a symptom.

Anonymous said...

" Most non-technically fielded adults I ask say the same things and explain in embarrassed tones why they hated algebra. No non-technical folks I ask can say they have ever used algebra to solve an everyday problem. "

How do you define non technical folks? My husband is a farmer and he and I are constantly using algebra. How else do you figure out rates of chemical, fertilizer or seed application, irrigation scheduling, or net present values. How does anyone run a business without a fairly high degree of competence in algebra?


Jane