kitchen table math, the sequel: listmania

Tuesday, June 29, 2010

listmania

Have I mentioned that I collect Listmanias?

Well, I do.

This week's find: best listmania on SAT prep I've seen thus far.

Thanks to Tertullian, I've spent the past two days reading The Talent Code and Arvin Vohra. Arvin Vohra's book is a revelation, especially the part where he advises parents to use Everyday Math after explaining the importance of a coherent, hierarchically organized curriculum and extensive memorization of math facts and formulas from an early age.

Say what?

The Asian Method

16 comments:

Catherine Johnson said...

I'll post excerpts from the first 50 pages tomorrow --

I've never read anything like it. Strikes me as very close to Wu.

Catherine Johnson said...

His recommendation of Everyday Math is predicated on the teacher being extremely good at math himself. Vohra majored in math at Brown. He's also talking about one-to-one teaching.

Still and all, I was nonplussed by the EM recommendation.

Allison said...

Being very good at math is a hindrance to being a good teacher of math for most people, just as being very good at anything is usually a hindrance to being a good teacher of that subject.

A good teacher must be able to vocalize explicitly the steps a novice needs to know. This means that a good teacher must already recognize how different a novice is from a non-novice.

Most people who excel at something didn't spend much time in the novice phase. They didn't hit a wall during noviceland where they were forced to question their basic and wrong understandings. The rest of novices generally do hit that wall. Whether they move past being novices depends on how they are able to analyze what's wrong with their understanding/implementation/technique. A good teacher or coach does exactly that, and breaks it down into manageable steps for someone to master.

But most people who excel in math never spend any time teasing out when math became a coherent whole for them. They didn't know *why* something was true in math, because they'd become so good at using it, that they just used it without worrying about why. By the time they needed to think about why, they'd already known other things predicated on it, or seen other things that implied it, and it all fit together. It was synthesized.

So EM can appeal to that kind of a mind, because to an expert, it is saying "look, here's a new way to approach thinking about this". And experts, who already have mastered the concepts, love to hear how other experts think about what they already know. To them, this provides insight.

To the novice, it is gobbledygook. If forced to look at only what the book gives you, and assuming no outside contextual knowledge to back you up, it is an incoherent mess.

Bostonian said...

Allison wrote, "Being very good at math is a hindrance to being a good teacher of math for most people, just as being very good at anything is usually a hindrance to being a good teacher of that subject."

I doubt this. There has been research finding that student achievement in math is positively correlated with the teacher's math knowledge. This must be true overall -- you can't teach what you don't know. For Allison's statement to be true, there must be some threshold of math knowledge/ability above which teaching ability declines. That is possible theoretically, but I will consider it unlikely until I see evidence for such a threshold.

Lsquared said...

In reaction to Bostonian's reaction to Allison's remark, I believe that studies have shown that math knowledge positively correlates with teaching ability for secondary teachers, but not necessarily for elementary teachers.

On the other hand, as someone who teaches math to future elementary teachers, there are some who I'm convinced will make better teachers of elementary math than others, and often (though not always) they are good at math in general. Of the ones that I privately think to myself--I would be delighted for my child to be in his/her class, maybe 1/3 of them are math minors, as opposed to maybe 1/8 of a typical class being math minors...And there are a few mathematicians (not at the university where I teach now) who I'm not sure should be teaching math to college students, much less elementary students, because they have no talent or interest in figuring out how learners think.

momof4 said...

I think I know what Allison meant by "being very good at math is a hindrance for being a good teacher of math". I think she was referring to those people for whom math was pretty instinctive and who could easily make leaps of understanding, combine or skip steps and move through new material at a rapid pace, with little explanation needed. Those people are likely to be very valuable and effective working with kids at the top of the cognitive spectrum; people like themselves. They would be likely be wonderful assets in honors and AP levels of physics, chemistry and math.

However, I think what Allison is saying is that they might not be as effective with the middle and lower levels as someone who knows the material well but had to work harder to have learned it; to have needed explicit instruction and repetition at each step and who knows where likely pitfalls and misunderstandings are. There are probably more of the latter than the former in k-12.

All that being said, I think too little knowledge of math is a much larger problem, especially at the elementary and middle school levels. Too many people really don't know enough to be able to teach effectively.

Crimson Wife said...

"So EM can appeal to that kind of a mind, because to an expert, it is saying 'look, here's a new way to approach thinking about this'. And experts, who already have mastered the concepts, love to hear how other experts think about what they already know. To them, this provides insight."

This is almost exactly what my DH said when I showed him the handout given at my local district's "math forum for parents".

IIRC, his exact words were: "that seems like something a math professor would come up with when he bored. And then other math professors saw it and thought, 'hey, that looks like fun.' But it's a terrible way to teach kids who do not get math intuitively."

Bostonian said...

Here is a study finding that teachers' knowledge of math, as measured by the "Content Knowledge for Teaching Mathematics" test, *does* matter. The paper has two sample questions from the test.

http://www.sii.soe.umich.edu/documents/Hill_Rowan_Ball_030105.pdf

Effects of Teachers' Mathematical Knowledge for Teaching on Student
Achievement
Heather C. Hill
Brian Rowan
Deborah Loewenberg Ball
Abstract
This study explored whether and how teachers’ mathematical knowledge for teaching
contributes to gains in students’ mathematics achievement. We used linear mixed model
methodology in which first (n=1190) and third (n=1773) graders’ mathematical
achievement gains over a year were nested within teachers (n=334 and n=365), who in
turn were nested within schools (n=115). We found teachers’ mathematical knowledge
was significantly related to student achievement gains in both first and third grades,
controlling for key student and teacher-level covariates. While this result is consonant
with findings from the educational production function literature, our result was obtained
using a measure of the specialized mathematical knowledge and skills used in teaching
mathematics. This result provides support for policy initiatives designed to improve
students’ mathematics achievement by improving teachers’ mathematical knowledge.
KEYWORDS: educational policy; mathematics; student achievement; teacher knowledge

Allison said...

Bostonian,

i'm sorry if I am not expressing myself clearly. Of course I think teachers should have more math knowledge--I just started a math institute to provide just that.

But that's a far cry from people who are naturally good at mathematics being good teachers.

Yes, more math knowledge, means that odds are, you've corrected more misconceptions so you don't have them. It's obviously better to teach a subject where you don't have misconceptions than one where you do.

But just because you've don't have misconceptions now doesn't mean you consciously corrected them, or recognize them in others. A good teacher is someone who recognizes others' misconceptions, finds their source, and creates an environment where making that misconception is as improbable as possible.

Now, we can teach mathematical knowledge, and we should. We should do so explicitly, because it helps both the weak and the strong student, because it helps the teacher. Those who are weak need it taught explicitly to help math make sense in a clear, coherent, precise way. Without that clarity, it falls apart early for them.

Those who find themselves good at math are often procedurally good at it, and still haven't received good grounding. They deserve good grounding, because they too are often missing the precision and clarity, but they have gotten a lot farther along the path before that's a problem.

It's one of the reasons kids hit the wall in college, because they think they know things they don't know, and what worked up until that point doesn't work anymore. The best students are still good procedurally, and manage to use this to carry themselves so far that eventually, they assimilate truth and overcome their errors in grounding--often right before their prelims in grad school. The students who aren't the best collapse somewhere under the weight of their misconceptions, unless they've had teachers who forced them to correct them.

But this experience is really not different qualitatively from what happens to a 4th grader--both have the same problem, where they can no longer go forward procedurally because of their fundamental misconceptions confusing their understanding, and preventing them from reaching mastery.

So those who are naturally good at math may have little experience helping those who get stuck, because they never consciously corrected or were aware of their own misconceptions. They don't know where they went away, and they don't remember not understanding. They don't know how to linearize the mathematical whole they have in their heads as well as someone who has been taught to do so.

And again, this was with respect to a curriculum, like EM: those who already get math may think EM is insightful, because it is TO THEM. It has little to do with how to reach someone who doesn't already know what's true.

Allison said...

The idea that above a certain threshold, knowledge makes you a bad teacher is born out by lots of people's experience with the geniuses, of course.

Newton was a terrible teacher. No student to learn from him, no proteges or apprentices followed in his footsteps. No one could understand his work, because he was so far ahead.

Feynman's freshman lectures were legendary in their terribleness to the students. Within week, the only people attending his lectures were THE OTHER PROFESSORS at Caltech. His courses produced more freshman physics failures than anything other professor who taught the sequence--basically no students could learn anything from him. He had no grad students, because he solved every problem he gave to them within hours or days, and all of his undergrad and grad friends and colleagues quit physics as a field! because they saw him and thought that was what it took to be a physicist.

Einstein was similar. I can find more examples if you need, but nearly everyone who had a math or science professor in college had one that was brilliant and truly dreadful, no?

Crimson Wife said...

I took organic chemistry from the guy who synthesized Taxol. I can remember thinking it was a good thing that he was an awesome research chemist because he was a TERRIBLE instructor.

Catherine Johnson said...

Same deal with the top high-risk obstetrician in LA.

Catherine Johnson said...

Just to take a little free-associational leap into the blue there...

SteveH said...

I always thought it was a bad omen to have a class taught by the person who wrote the textbook. Then again, my advisor was no slouch, but he was also a great teacher. I had a course in vibration taught by a professor who was a wee bit outside his specialty. It was not successful. I also had a world-renowned researcher for a professor who liked to smoke cigars in class. It was awful ... the class too. However, many of the poor professors I had couldn't fall back on any sort of genius label. With the geniuses, you get some sort of historical payoff. In music, you can talk about being a "descendant" (student-wise) of Liszt or Paderewski. At least it sounds good.

Anonymous said...

Allison wrote: "Feynman's freshman lectures were legendary in their terribleness to the students. Within week, the only people attending his lectures were THE OTHER PROFESSORS at Caltech. His courses produced more freshman physics failures than anything other professor who taught the sequence--basically no students could learn anything from him."

Yes--my father took freshman physics from Feynman and every time the man's name comes up he goes on about how terrible a teacher Feynman was. It's been over 60 years and he's still annoyed about it.

I also agree with Allison about it being advantageous to a teacher to have struggled with the material being taught. I know I am a far better math teacher to my children (I homeschool) than I am a grammar teacher because I have really had to think about why math is the way it is and actually study it. I understand about the misconceptions because I've had them. I can empathize with my kids when they are confused because I've been confused. Grammar, on the other hand, is so totally obvious to me that when my kids ask me questions about it my first inclination is to say, "Well, duh!"

Allison said...

Typo earlier should read "within WEEKS, the only people attending Feynman's lectures..."