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The Mis-Education of Mathematics Teachers


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Wow, thanks. It's uncanny how well that article dovetails with material I've been reading lately, like Ma's "Knowing and Teaching Elementary Mathematics", Lockhart's Lament, and Devlin's "Introduction to Mathematical Thinking". A great piece of the puzzle, and now I want the author's since-published book...ugh, the price though. http://www.amazon.com/Understanding-Numbers-Elementary-Mathematics-Monograph/dp/0821852604

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all right, I really did not need to read that! I'm having my semi-annual teaching-math-and-physics panic attack!! :scared: Feeling inadequate.  :unsure:

 

I thought that way too, but on the other hand you look at the Chinese teachers who provide the best teaching, and they've only had two years of post-secondary education. Their big strength is that they study their teaching materials in order to understand them as well as possible before teaching; they also talk to other teachers about interesting situations that come up in the teaching.

 

So, before buying the $75 text on the precision and coherence of the mathematics underlying k-12 instruction, I think I'm just going to get the HIG and textbook for the next level of Singpore that I'm about to teach, and start studyin'. :D

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So, before buying the $75 text on the precision and coherence of the mathematics underlying k-12 instruction, I think I'm just going to get the HIG and textbook for the next level of Singpore that I'm about to teach, and start studyin'. :D

Only I am staring at a large pizza sliced up to explain fractions in SM 3B :), so I guess the same faulty methods.

Beast taught fractions on a number line, but even they didn't escape the tyranny of the pizza. :)

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all right, I really did not need to read that! I'm having my semi-annual teaching-math-and-physics panic attack!! :scared: Feeling inadequate.  :unsure:

 

I don't see why you would feel that. The author is arguing that the math that is taught to future teachers is too hard. Also it is very abstract and there is no obvious relationship with what the teachers are supposed to teach in elementary and middle school. It would be much better, at least in theory, to teach the future teachers a more thorough exposition of the material they are going to teach.  Now if that would work better in practice or not, I am not sure. If it were for me I would make the teachers college only one year of theory (some basic principles of psychology and pedagogy) and three years teaching under heavy supervision. But of course it is much cheaper to pay a professor to lecture in front of a class of 100 than to have a qualified instructor supervise a candidate teacher one on one.

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Wow, thanks. It's uncanny how well that article dovetails with material I've been reading lately, like Ma's "Knowing and Teaching Elementary Mathematics", Lockhart's Lament, and Devlin's "Introduction to Mathematical Thinking". A great piece of the puzzle, and now I want the author's since-published book...ugh, the price though. http://www.amazon.com/Understanding-Numbers-Elementary-Mathematics-Monograph/dp/0821852604

 

You can get some parts of this book for free from author's website http://math.berkeley.edu/~wu/.

 

For example:

http://math.berkeley.edu/~wu/EMI1c.pdf

http://math.berkeley.edu/~wu/EMI2a.pdf

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Only I am staring at a large pizza sliced up to explain fractions in SM 3B :), so I guess the same faulty methods.

Beast taught fractions on a number line, but even they didn't escape the tyranny of the pizza. :)

 

The pizza isn't faulty in and of itself, it's a natural representation. I think the trick is moving on to expand the concept to things that are not round foods. ;) Hopefully SM does this before long? If the materials know that there is a reason that a piece of pizza relates to the number of boys:girls in the classroom which relates to the result of dividing a larger number into a smaller one; and if the teacher is willing to do the work to study the materials and grasp the underlying coherence in order to systematically answer the question "WHY" when it arises, then I think there's hope for reg'lar teachers without a background in academic mathematics (not even an academic treatment of school mathematics as proposed in the article.)

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The pizza isn't faulty in and of itself, it's a natural representation. I think the trick is moving on to expand the concept to things that are not round foods. ;) Hopefully SM does this before long? If the materials know that there is a reason that a piece of pizza relates to the number of boys:girls in the classroom which relates to the result of dividing a larger number into a smaller one; and if the teacher is willing to do the work to study the materials and grasp the underlying coherence in order to systematically answer the question "WHY" when it arises, then I think there's hope for reg'lar teachers without a background in academic mathematics (not even an academic treatment of school mathematics as proposed in the article.)

Beast introduces fractions as numbers on the number line and I think it is the least confusing method. At this point I trust anything that comes out of Aops to be of top quality, so yes, I am not worried. SM is our main program and I feel the same about it. :)

 

I think the reason for me to try to put my kids eventually through Aops has a lot to do with their books acting as teachers, correcting major deficiencies with "live" teacher my kids are stuck with at home. :)

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Beast introduces fractions as numbers on the number line and I think it is the least confusing method. At this point I trust anything that comes out of Aops to be of top quality, so yes, I am not worried. SM is our main program and I feel the same about it. :)

 

I think the reason for me to try to put my kids eventually through Aops has a lot to do with their books acting as teachers, correcting major deficiencies with "live" teacher my kids are stuck with at home. :)

 

Dr. Wu agrees that starting with the definition on the number line is not only the easiest, but also the most defensible from the perspective of having a definition we can rely upon when it comes to doing the same operations we've done on whole numbers, but now on fractions. (In fact, comparing the Beast Academy sample pages with the Fractions chapter from Dr. Wu's website, it's almost verbatim!)

 

I love math but I hate that my grasp of actual mathematics is weak. I love that the AoPS books are written by actual mathematicians who get actual mathematics much better than I do. :)

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The discussion of how to represent fractions reminds me of one of the scenarios Liping Ma used when interviewing teachers for her research. Here is the scenario (from p. 55 in my edition of Knowing and Teaching Elementary Mathematics):

 

"People seem to have different approaches to solving problems involving division with fractions. How do you solve a problem like this one? 1 3/4 : 1/2 [I'm using the colon to represent the traditional division symbol]

 

Imagine that you are teaching division with fractions. To make this meaningful for the kids, something that many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content. What would you say would be a good story or model for 1 3/4 : 1/2 ?"

 

Of the 23 US teachers interviewed, 21 attempted the calculation but only nine reached the correct answer. Only 1 of the American teachers was able to come up with a conceptually correct story problem to illustrate division by fractions. 6 of the teachers could not come up with a story problem at all, and the other 16 came up with incorrect representations (mostly stories that actually called for division by 2 or multiplication by 1/2). 

 

Of the 72 Chinese teachers interviewed, 65 were able to come up with appropriate story problems.

 

That was the most depressing segment of the book for me, it truly was a condemnation of American mathematics and teacher instruction.

 

Just as an aside, I was trying to think of a story representation myself and came up with this:

I have 1 3/4 gallons of paint in a bucket and I want to paint a fence, I know that I need 1/2 gallon of paint to cover a segment of fence between two fence posts. How many fence segments can I paint with the paint in my bucket? 

 

What do you think, does that problem work?

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Yup, I think that is definitely a correct story problem, Alef. I have to admit that because it was a school fraction problem, my mind went immediately to pizza: if you have 1 3/4 pizzas and want to serve each person half of a pizza, how many servings do you have?

 

That is an extremely depressing statistic about American math teachers. As a former fifth grade teacher, I can't say I'm surprised, though.

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Yup, I think that is definitely a correct story problem, Alef. I have to admit that because it was a school fraction problem, my mind went immediately to pizza: if you have 1 3/4 pizzas and want to serve each person half of a pizza, how many servings do you have?

 

That is an extremely depressing statistic about American math teachers. As a former fifth grade teacher, I can't say I'm surprised, though.

 

 

Hey, at least you could think of a problem! The sample of American teachers in the book wasn't particularly large, but it is troubling that all but one of them apparently didn't understand what dividing by a fraction really means. Since these were elementary teachers, I imagine some of the difference is that Chinese math teachers are subject specialists, while almost all elementary teachers in the US are generalists (and often don't like math or have confidence in their own abilities and understanding). I know a young woman who is teaching in a bilingual immersion school--her only qualification for teaching is a degree in the immersion language, but since math classes are taught in that language she is the math teacher. I find it interesting that the school recognizes the need for an expert for language instruction, but not for math; I suppose the assumption is that a college graduate should know enough math to teach elementary school. From what I understand even teachers who do have elementary education degrees have only minimal instruction in teaching math--maybe one class in the course of their degree studies.

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I am very interested in doing some reading myself on math understanding now.  I just ordered some of the suggested math books from the library.

 

For you with older students, would you think saxon in the upper grades is still sufficient or is AoPS better at correct concept math study?

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The discussion of how to represent fractions reminds me of one of the scenarios Liping Ma used when interviewing teachers for her research.

 

Yep, that's what I was alluding to when I mentioned "round food" -- few of the American teachers even strayed from that representation, a couple used rectangular food or proportions in cooking (but still misrepresented the problem). It made me giggle when Ma kept referring to the pizza/pie obsession as "round food". ;)

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From what I understand even teachers who do have elementary education degrees have only minimal instruction in teaching math--maybe one class in the course of their degree studies.

 

This depends on the state. At my undergrad, there were two four-credit courses taken sequentially and then a methods of teaching course at the upper-division. At my grad school, they took a 'math for teachers' course and then a two-semester sequence in 'how to teach math' that included a significant amount of math as well. Where I work now, they take a two-semester sequence in elementary school mathematics and then a one-semester course in how to teach it.

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For you with older students, would you think saxon in the upper grades is still sufficient or is AoPS better at correct concept math study?

Saxon and AOPS are just about at opposite ends of the spectrum and there are a lot of programs in between. I really don't think AOPS is suitable for the majority of students, but there are a lot of other good, conceptual math programs out there.

 

Some students do very well with Saxon and are able to conceptually understand the material as well as apply it to novel situations. There are a lot of success stories.

 

Some students are able to compute well, but only able to solve problems if they are similar to Saxon's problems. Saxon's incremental teaching works so well at drilling procedures into students heads that they may become very adept at computation without actually understanding what or why they're doing.

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This depends on the state. At my undergrad, there were two four-credit courses taken sequentially and then a methods of teaching course at the upper-division. At my grad school, they took a 'math for teachers' course and then a two-semester sequence in 'how to teach math' that included a significant amount of math as well. Where I work now, they take a two-semester sequence in elementary school mathematics and then a one-semester course in how to teach it.

That's wonderful that your state requires such thorough preparation. My provisional elementary certificate preparation in Massachusetts ten years ago only included about three two-hour classes (and by classes, I mean individual class sessions, not semester-long courses!) I was not majoring in education, but this met the state requirements for normal certification.

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Just as an aside, I was trying to think of a story representation myself and came up with this:

I have 1 3/4 gallons of paint in a bucket and I want to paint a fence, I know that I need 1/2 gallon of paint to cover a segment of fence between two fence posts. How many fence segments can I paint with the paint in my bucket?

 

What do you think, does that problem work?

I was mulling over this problem and was thinking about the connection to whole number division. Here's another angle to it. We both came up with problems where an amount was divided by a part of that amount. (I.e., gallons of paint divided by gallons and pizzas divided by pizzas). It's analogous to having a group of 10 kids and wanting to divide into groups of 5 kids. So, 10 kids divided by 5 kids equals 2 groups.

 

But in whole number division, we often divide by the number of groups we want to create. So, in the above problem, we could also do 10 kids divided by 2 groups equals 5 kids per group. I'm having trouble figuring out an analogous and plausible fraction situation...especially one that would make sense to a kid. Anyone want to take a stab at it? :)

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