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As a mathematics instructor, what is the most valuable (to you) thing that classroom experience taught you about mathematics education?

To be specific, I'm looking for answers not about mathematics itself, or student discipline unrelated to mathematics, but about the learning, teaching, or reasoning of mathematics.

[To split some hairs, there is knowledge of mathematics that falls into this category; the difference between quotative and partitive division is a good example of pedagogical content knowledge. It's something most math users don't need to know about, but it helps teachers understand student conceptions of division. If you learned something like that through classroom experience, it would be a great answer and probably a great story as well.]

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    $\begingroup$ Care to explain quotative/partitive? We here are domain experts (or play same in front of the class), but most of us undergraduate teachers haven't ever had even a far encounter with education training. In my case, I had a bit of training, but in Spanish, and I doubt this was ever discussed there. $\endgroup$
    – vonbrand
    Jun 16 '14 at 3:25
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    $\begingroup$ Partitive division can be thought of as "sharing" division. One and a half pizzas are shared among five people - how big is a serving? Quotative division is sometimes called "measurement division." You have 64 mini chocolate bars and would like to make gift bags with 6 bars each in them. How many gift bags can you make? $\endgroup$
    – JPBurke
    Jun 16 '14 at 3:42
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    $\begingroup$ The distinction is often important when thinking about fraction division. Quotative division evokes counting down, or repeated subtraction. You have 3 2/5 pans of brownies, and one serving of brownie is 1/5 of a pan. How many servings do you have? In this case it is easy to see how you can have a division problem that results in a number larger than both the dividend and divisor (if you use 1/5 as a measurement of a serving, and count the servings). The idea of division producing smaller numbers can be a tenacious conception picked up from early arithmetic. $\endgroup$
    – JPBurke
    Jun 16 '14 at 3:47
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    $\begingroup$ Thinking only of partitive division only can be confusing in modeling a division problem with fractions. 30 divided by 1/3 -- You have 30 sandwiches you would like to distribute among 1/3 people... Hmmm. $\endgroup$
    – JPBurke
    Jun 16 '14 at 3:52
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I have loved the zen of teaching. Accept where a student is, and figure out how to help them move forward. I remember years ago being a bit bored teaching algebra for the nth time. And then it got less boring, because I was able to focus more on the students and less on the content. Or rather, my focus was on how the students were seeing the content. Don't know if this is detailed enough to be useful...

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    $\begingroup$ It sounds like a significant qualitative change in your experience as a mathematics teacher. $\endgroup$
    – JPBurke
    Jun 20 '14 at 7:56
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    $\begingroup$ +1 for "focus more on the students and less on the content" $\endgroup$ Jun 20 '14 at 16:34
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It was quite a shock to me when I found out I had the expert's curse (can't understand the problems newbies have because at their root is something so familiar that you can't imagine somebody doesn't know it). Now I know I'm cursed, dunno if it has done my victims much good...

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One thing that hit me this past summer while teaching Algebra 2 was that exponential functions and geometric series were basically the same thing, exponential functions being over the reals and geometric series being over the integers. I had spent years studying mathematics and math education, and never made the connections between those two topics.

For example: 1,2,4,8,16 Is a geometric series, with a ratio of 2.

But you could also write it as an exponential function, with each term being 2^(n-1).

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