Ok, so I have no idea how other people do it, but I feel like I am on the verge of drowning. Getting ready for classes, grading constantly, now we'll start math circles for teachers, need to write grants. Yikes.

So, we've been going for almost three weeks now. Apart from feeling like there should be 36 hours in a day, I'm enjoying both my classes a lot. Since I have some grading to do and I have to get ready for tomorrow's class I'll only write two observations.

El. math teachers course: From what I can observe (and this class isn't exception, I don't think) my students believe that in order to learn mathematics they only need to solve problems. But how can you find a complement of a set if you don't know what a complement of a set is. It is hard work to make them learn definitions. Or even to see the value of definitions.

Secondary: There was a homework problem that I assigned that asked "How would you want your students to respond to question: Explain why the sum of the digits of any multiple of 3 is itself divisible by 3." The question came form Principles and Standards, so one might imagine that this is material that these students will be expected to teach once they become teachers. I wonder what response you think I got. I'll tell you about three that I found most surprising: they said that they wouldn't expect their students to know the answer since they themselves don't know it. What was surprising, or rather what made me sad, was the fact that not only did they not know, but they didn't express any desire to find out. They didn't say "We'll look it up" nor "Can you tell us?", and they definitely didn't say "I'll work it out." I guess it's only a beginning of the semester :) Maybe we'll see some improvements as the class goes on.

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## 4 comments:

I'm not so surprised that a theoretical problem fell flat.

Who were your students? Who are their students?

I don't think I would expect very many secondary school students to encounter strong explanations of this divisibility rule. (note, the explanations of rules involving multiples of 2 and 5 are more accessible)

Did you see my train trip post?

Hi Jonathan,

Sorry it took a while. I have fallen completely behind both with reading and writing.

My students are pre-service secondary teachers, which answers both questions. None of them student teach yet, so they have no students. As for accessibility, I agree that 2 and 5 rule are more accessible, but the 3 is pretty neat, and I don't think too hard. By the way, have you ever heard of the term "casting out nines"? One of my student mentioned it and I have found only one other person who has ever heard of it (not me).

And, yes, I did see your train post. There are always problems in our neck of woods. If somebody thought of getting a patent out, we would have been very rich (but still not too smart). But, to be fair, most of us are at least fun to hang out with, when we're not on a war path. Anyway, next year you should come through Sarajevo. We'll have a coffee or two.

El Math course first: Elementary students do not learn definitions the way that mathematicians do, and neither do elementary math teachers. If you read the section in NCTM Standards--I think it's K-2 geometry, but I can't find my copy right now--about examples and non-examples in teaching triangles, you'll see what I mean. A mathematician takes a definition and reads it closely, and picks it apart until they understand it. Students at a lower level work from the ground up: they learn definitions first from examples and non examples of the term. Elementary students, and elementary teachers all start at Van Heile level 0 or 1. I think the best approach to appreciating definitions is to have students write some. You'll find some good example/non-example pairs in Michael Serra's HS geometry book (first chapter?). With a bit of work, you should be able to get them to function at Van Heile level 2 (finding properties of mathematical objects, and being able to compare things like rhombi and parallelograms). Most of them are pretty far from level 3, though, so there's not much hope for them to be competent with definitions, though they may start to appreciate them. I think you'll find after a few times through that you don't have time to sweat the small stuff in covering sets (too much else to get to). If you can get them to understand that the intersection of blue things and red things is empty, and not the set of blue and red things you should count it a success.

For the Secondary students, I would hope that they would expect their students to look at a bunch of numbers that are divisible by 3 and at least verify that the rule works. The best explanations of why that I've found are in Math for El Ed Teachers books, by the way (I'm fond of Bassarear). Of course, if you started by asking them to learn all of the elementary math stuff you'd hope they know, you could use up the whole semester and never touch the secondary curriculum at all (sigh).

Good luck with the classes. They are tough to teach the first time around. If you have time to look through elementary and secondary books for good examples, that helps a lot with getting the students involved

I don't know that I will do the train ride again, given the bad experience. But the company indeed was pleasant. People really did want to communicate, and my Russian allowed at least a possibility. Sarajevo? (Bosna Sarai!) Is it a good place to visit again? Molim tursku kavu. Hvala Vam. That's my entire vocabulary, but if it's a visit for coffee, then maybe I am all set.

I would hope that your students could understand your explanation of why the divisibility rules work.

Just yesterday I showed my freshmen why multiples of 3 form a closed set under addition. Little bits of theory... In tiny doses, they make an impact. Did they understand? Almost all of them had a clue. Could they reproduce the argument? A couple. But that wasn't the point. I wanted them to know the argument, the rationale, the proof actually exists. Math is not religion to be believed.

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