As I mentioned in a previous post, I’m terrible at remembering things—names, faces, dates, all become a blur for me very quickly. That is why, as a young child I was attracted to mathematics. There was very little memory work involved, and what little there was fit neatly into patterns, not as a jumble of unrelated factoids (which was my perception of all the history classes I ever had).

I recently read a critique of Lemov’s *Teach like a Champion* by The Number Warrior, that points out the similarity between Lemov’s Technique 13, Name the Steps, and Devlin’s satirical title In Math You Have to Remember, In Other Subjects You Can Think About It. I think that both Devlin and the Number Warrior are correct in pointing out a major flaw in much math teaching: emphasis on memorizing procedures rather than understanding them. Once you understand them most of the algorithms in elementary and secondary math are fairly trivial, but memorizing them without understanding is difficult and error-prone.

One of the great joys of math for me was that by learning only a few facts and some general thinking techniques, I could solve a huge variety of problems. I was the sort of kid who never bothered to memorize the quadratic formula, because I could rederive it when I needed it. (By college, I had used it enough that I did remember it, but I still derived most of the trig identities as I needed them, using only a few key facts, like that cos omega + i sin omega is the unit circle.)

In most of the courses I have taught, I am not interested in students learning many facts. What I want students to develop is skill at applying a few facts to a number of problems. The subjects that develop these skills best are math and computer programming. Computer programming is particularly good at developing skill at getting the details right (Lemov’s “Right is Right”), since the computer will do precisely what the students instruct it to, even if that is subtly or ludicrously wrong. The need to break big problems down into smaller pieces, work on the pieces separately, and still be able to put the pieces together again is particularly evident in programming.

I have little use for touchy-feely approaches to math education that result in neither memory nor understanding, or ones that insist that no child is capable of abstract thought so everything must be physically manipulable. I think that the best pedagogy starts with interesting questions (“the hook”), then helps the students learn the tools needed to answer those questions. What’s nice about math is that good craftsmen can get by with a fairly small toolbox, if they learn how to use the tools well.

Your comment about getting by with a fairly small toolbox reminded me of something Richard Rusczyk said at Math Prize for Girls last year. There’s a video of his speech here:

http://mathprize.atfoundation.org/archive/2009/rusczyk

It’s an hour long. The part I was thinking about begins around 27:50 and lasts only a minute. He’s talking about MOEMS and how they crafted the problems to be solvable many different ways. Someone asked him how he solved a problem and he explained. They took away one of his “tools” and he solved it again. They took away another “tool” and he found another way to solve it.

The whole talk is interesting (although I think it was more interesting to be there than to watch it online). His slides are available separately at http://mathprize.atfoundation.org/archive/2009/index, but they definitely don’t say everything that his talk does.

Comment by Jo In OKC — 2010 July 5 @ 10:46 |

I’m very fond of Rusczyk’s books, but I don’t know that I’d have the patience to sit through an hour-long video talk. I’ve not seen the MOEMS problems, but the AMC-8 and AMC-10 problems are certainly solvable in multiple ways.

I tried taking the AMC-8 and AMC-10 tests this year alongside my math team, to see how rusty my math skills have gotten. The AMC-8 was easy for me, but I did not finish the AMC-10. I scored 127.5 out of 150, which is respectable, but not stellar (I didn’t get any wrong, but I did not have time to do all the problems). I probably would have done better 40 years ago, when I still remembered some geometry. The AMC-10 is fast-paced enough that it does help to remember a bit more, as re-deriving things does take a little longer. Not having used any geometry for almost 40 years certainly slowed me down.

Comment by gasstationwithoutpumps — 2010 July 5 @ 11:26 |

I feel like you, if math didn’t involve creative thinking, I wouldn’t be doing it. I couldn’t teach in a program that revolved around students memorizing rules. I wonder about some of the other points Devlin makes though. He says,

“Of course, teaching math in the progressive way requires teachers with more mathematical knowledge than does the traditional approach (where a teacher with a weaker background can simply follow the textbook – which incidentally is why American math textbooks are so thick). It is also much more demanding to teach that way, which makes it a job that deserves a far higher status and better pay-scale than are presently the case. And it’s a lot harder to collect the data to measure the effectiveness of the education, since it means looking at the actual products of the process: real, live people.”

He’s making a whole bunch of claims here. He seems to be saying that the typical math teacher is simply not capable, ability-wise, to teach using an approach that requires independent thinking. This should be a controversial point, and is not necessarily the case.

He also makes another point I agree with: there’s a mistaken reliance (in America, at least) on quick and easy quantitative tests as assessment tools.

Comment by esivel — 2010 August 12 @ 17:17 |

Mathematics came about because man (mankind) wanted a better and more efficient way of doing things. The Inca Empire used the Quipu (knotted ropes using a positional decimal system). This was use for crops, taxes, population and many other data. The Egyptians used the Pythagorean Theorem for construction, architecture and measurement. These methods weren’t about remembering but about critical-thinking and action. Somewhere along the life’s line we lost the true meaning of Mathematics and it became rote. Math became an intellectual sport that didn’t require reasoning, just memorization and application. We must begin figure out how to get the gas from the gas station that has no pumps.

Devlin was right when he said that ‘Mathematics is a way of thinking about problems and issues in the world. Get the thinking right and the skills come largely for free.’ I also agree that we must start with an interesting “hook”.

Comment by gk1as — 2010 August 20 @ 11:47 |

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I agree that the best pedagogy begins with a good “hook” as well–it is important to create student buy-in before we ask them to digest a bunch of facts. I disagree, however, that teaching students to memorize the processes to arriving at an answer is a bad idea. I haven’t read Lemov’s book although it is on my list. I am familiar with his work because the instructional model used at the school where I teach is partly based on his reccomendations. This is the first time that I have taught using direct instruction, and I have found it very interesting to hear the many arguments both against and for this style of teaching. I think a lot of it has to do with the setting in which a teacher finds him/herself. Many schools that advocate for such models are found in urban areas where many students arrive in 7th grade, for example, performing on a 3rd grade level. How else can we get these students to progress if we don’t focus on skills, skills, skills? Is there a better way?

Comment by semhouston1 — 2010 August 20 @ 14:43 |

I agree that students need to develop skills, but rote memorization of algorithms is a hard way to get there. If the algorithms make sense to the kids, the algorithms are a lot easier to remember (or to recreate if not remembered). Too many math teachers, particularly in the elementary levels, are relying purely on rote memorization of techniques that are, as far as the kids are concerned, totally arbitrary sequences of steps. That sort of skills-learning does not benefit the kids at all.

Comment by gasstationwithoutpumps — 2010 August 20 @ 15:07 |

We had several recent conversations at Math 2.0 interest group http://mathfuture.wikispaces.com/events and Natural Math email group http://groups.google.com/group/naturalmath/browse_thread/thread/5ad316e0deff66c8 on “expert modes of thinking.” Here is what seems to be the best of both worlds:

– Explore the topic, pose and solve problems, discover, mess with it

– DEVELOP (actively using the past human endeavors and the math expert(s) present in your group, such as the teacher) reliable, debugged heuristics and algorithms

– Write those down in a well-organized manner (diagrams, mindmaps, and other data visualization techniques help)

– Use these reference materials till your algorithms are internalized

Students need to firmly believe that given enough time and support, they could have derived any algorithm they use. For that to happen, they need to derive “enough” algorithms they use.

Comment by Maria Droujkova — 2010 August 21 @ 05:34 |

I don’t know that it is necessary for students to

derivethe algorithms they use, but it is necessary that theyunderstandthem. Some algorithms are easily re-derived once understood, others remain tricky. Most of the elementary-school algorithms are easy to re-derive.Note that being able to re-derive an algorithm once it is understood is very different from being able to discover it. Few students would be able to discover reliable algorithms for arithmetic.

Comment by gasstationwithoutpumps — 2010 August 21 @ 10:10 |

Let’s distinguish several very different activities.

1 – Derive an algorithm at the boundary of the current collective knowledge, where very few specialists work and the content is otherwise sparse.

2 – Re-discover or create a version of an algorithm at the “student” (kid) level, in the middle of plentiful content in a variety of representations, with easily available help from peers and mentors

3 – Follow other people’s explanations of how they derived an algorithm.

4 – Use an algorithm.

I don’t know if many students can do Activity #1. That is research math. I maintain 2, 3, and 4 are all student activities. Even re-discovering or modifying algorithms is a slow activity. I want to help students do enough of it to believe that they COULD (if so they wished) do it for every other algorithm they are studying (3, 4) as well.

Comment by Maria Droujkova — 2010 August 21 @ 13:53 |

[…] teaching, video camera I have previously posted on Doug Lemov’s Teach Like a Champion (and a critique of his “Name the Steps” as applied to math). But the book seems practical enough to be […]

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