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Another wonderful contribution to a great thread.

Countless messages on this forum have described a child's precocity in math and asked what to do about it. Well, there are as many people in the bottom 10% of math talent as in the top 10%, and we should not pretend otherwise. For a dose of realism one can read the following article from a college math instructor.

http://www.brianrude.com/fractionsquiz2.htm
Fractions My Algebra Students Can't Do.
Brian D. Rude
(Revised) 2008

That is sobering. Do you think any of the people in that borderline-remedial class may have been the victims of poor instruction, or do you think that they're all in the bottom 10% for native ability?

They are 1st-year college students, most of whom graduated from high school, so I think they are in the bottom 50%. Many in the bottom 10% cannot get a high school diploma. Rude addresses the question you raise:

"Surely we have to ask why there should be such a lack of understanding of fractions. Two possibilities come to mind. Possibility one is that we are doing something wrong in the teaching of fractions. Possibility two is that our teaching is okay, but the human species is just not that mathematically inclined. Perhaps under the best of circumstances only a minority of people can actually gain a useful understanding of fractions.

Obviously I don't like the pessimism of this second possibility. I prefer to think that probably we are doing something wrong. Of course I have been saying that for a long time. I think the perspective of math offered by the NCTM in the 1989 and 2000 standards are not well thought out. I think this NCTM perspective has to be doing substantial damage."

I take the pessimistic view, but if better math curricula proves me wrong, good.

I suspect that lots of problems drive the situation described in that article.

I spent this past school year watching a risibly incompetent teacher water-down an already simplistic geometry textbook. The book eschews rigorous proofs, opting instead to list the steps and ask students to fill in the blanks. The book's problems are simplistic and mostly ask students to memorize and regurgitate. There are no A-, B-, and C-level problems: most of them are all the relatively simplistic A-type questions. Solving them requires using the exact same technique repeatedly. The books makes the problems "different" by rotating triangles or making the circle bigger. This book used all over the state of California.

Yet DS's teacher abandoned the simplistic proofs after Christmas. She also abandoned constructions around the same time and, as I reported in another thread, she skipped a lot of stuff and jumped back and forth through chapters.

My understanding is that the algebra class she taught last year had the same problems. She didn't finish that course, either.

Yet the school defends her.

I expect that many of her students will go to college and wonder why they get placed in remedial math when they got As and Bs in algebra and geometry.

Our society says that everyone should go to college, so we must have college students who really shouldn't be in college. So I'm sure there's an ability issue, too.

And I'm sure there are a plethora of other problems too.

But it's probably hard to tease out what's what by the time kids are 18 or 19. Individual teachers can probably figure some stuff in some of their students, but not everything.

My hunch would be that it's more interesting than either of those. I once set an extremely simple percentage problem in an exam (accidentally - I was examining something else and this was part of the set up; it didn't occur to me that this part might be a problem to anyone!). Think "what is 40% of 10" - it was that level, requiring nothing more than an understanding of what "percent" means. Around half (I forget, now, whether it was a bit more or a bit less) of the students got it wrong. Now, admittedly, this was part of a stressful exam and it will have been obvious to the students that this sum wasn't the point of the question - but I was shocked that it was even possible for a substantial proportion of them to have got it wrong! The course these students were on requires them to have an A grade at A level mathematics, or equivalent; that puts them easily into the top 10% of the population for mathematics achievement.

The spiral methods criticised by the author of that article are not widely used in this country, so we can't blame them here. I think there's a large group of people who can easily learn to apply a mathematical method while they're in the process of being taught it and tested on it, but who do not consider it worth giving brain space to afterwards. For me, and probably for you (dear Reader ;-) fractions and percentages form part of the basic mathematical language that it's hard to conceive of forgetting, but I think what's in that language and what's out is more variable than we sometimes consider. I noticed the other day that I have forgotten some of the trigonometric identities that DS is about to learn, which would have shocked me once, but does so no more!

I think one place where this discussion gets bogged down is in the distinction between repetition and practice. Experts spend vast amounts of time in what I recall the Cambridge Handbook refering to as "mindful practice".

There is a difference between doing AoPS problems or doing sheets of cookie cutter work. If the work is not in the child's zone of proximate development there is unlikely to be any benefit. So when looking at school work the question is whether it is vital practice or mindless repetition. In this view repetition is unlikely to be beneficial and is always suboptimal.
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The other issue is that the problems parents are usually objecting to are highly algorithmic. I think there is a valid concern that excessive overuse of drill can reduce the extent to which the material is generalized and available for creative use. I've read studies looking new math(CSMP) vs traditional. CSMP taught more varied approaches to elementary math so with similar class room time had less repetition of the traditional approaches. In follow-ups the CSMP kids had similar computation scores to the traditional kids but had significantly higher outcomes on novel problem solving tasks. I don't want to get in to a debate about new math :); I've seen similar studies in the sciences when richer curricula are used.

The issue is not that repetition is harmful per se. Its that repetition crowds out other much more beneficial work.

Absolutely - sorry, I thought I'd been clear about that! I had no interest in testing my students' abilities to do percentages; it was just an accidental part of testing them on something else. Yes, I'm sure this was an important aspect of what happened - but I was still shocked!

Thanks for the recent comments. The next FAQ I am revising relates to the appallingly bad methods for teaching fraction arithmetic in United States schools (other countries do MUCH better at teaching fraction arithmetic) and what parents in the United States can do about that to help their children.

Added: a few years ago, I had a student who had finished a biology degree from a highly rated college. He was taking courses where I worked to strengthen his applications to PhD programs. He was a very bright guy and learned very quickly. But he had trouble with fractions in a formula we were using. He had found a book that listed the same formula, but instead of multiplying by, say, 3/4, as I had been teaching, the new source said you should divide by 4/3. I had to teach him that dividing by a fraction was the same as multiplying by its inverse. Turned out the whole class benefited from that discussion and I ended up giving a quick lecture on dividing with fractions.

I'm a biologist. I know that lack of mathematical skill among undergraduate biology students (including majors) has been a serious concern for a while now. I attended a conference on biology education a couple years ago and was in a mathematics session. Most of the people in my group taught mathematics, and they said that part of the problem stems from the biology teachers not knowing a lot of math and/or not incorporating it into their classes.

ETA: Another one of my students who had finished her degree was a math tutor at a local highly rated high school. She took the job expecting to tutor trig and calculus and suchlike, and was very frustrated when she ended up teaching a lot of her students how to do long division and other elementary-level skills.