Arithmetic vs. Mathematics

I am teaching a content course for elementary education majors this semester, and I had them do presentations yesterday. Their task: “explain why an unusual multiplication or division algorithm (e.g. “Lattice Multiplication”) gives the correct answer to a multiplication/division problem.” The idea, of course, is to show how the algorithm is really just doing a definition of multiplication (usually “repeated addition”) or division (either partitive or measurement).

Several presentations were good, and one was dazzling. However, most of the presentations missed the question entirely, and simply demonstrated how to do the algorithm (and assuming that it is an algorithm worth doing). This has been an on-going challenge this semester (and every other semester I have taught this course). I have had some successes, but they are usually short-lived for most of the class.

I am also giving the students feedback on drafts of papers that explain why some algorithm gives the correct answer (according to the definition). We will see how much this helps.

Does anyone have any suggestions on how I could communicate the question (or that a question even exists)? The students are very focused on the arithmetic, but ignore the mathematics.



7 Responses to “Arithmetic vs. Mathematics”

  1. Andy Rundquist Says:

    With an 8-yr-old at home I find myself often thinking about the math behind some trick he’s learned. Take the finger trick for 9’s multiplication. It’s an easy trick to learn but I’m not sure he knows why it works. Of course if it helps him finish his homework I don’t know if he cares how.

    Another great example is the 3’s and 9’s trick for determining whether something is a multiple of 3 or 9. Again it’s easy to do but harder to prove. I am really impressed by how you focus the class on this, Bret, but I feel for you when not everyone embraces it. I really like how a math colleague of mine deflects all calls for him to do some quick multiplication or something in a meeting by saying “that’s arithmetic, not math and I’m bad at arithmetic.” I’ve tried out the line a few times too 🙂

  2. bretbenesh Says:

    Hi Andy,

    I loooove thinking about the finger trick and divisibility results, too. I just love it.

    In fact, those are two projects that I assigned my students for my school’s Scholarship and Creativity Day. My students are going to have a poster session where they explain these ideas to others.

    To be fair—I think that my students are trying to embrace this idea (most of them, anyway). I think that most of them want to see it, but I think that it is just a very difficult thing to see.

  3. Siouxgeonz Says:

    Perhaps if you talked about repeated addition… had them generate some examples of it… and then ask them to *find* the repeated addition in lattice multiplication?
    I am afraid that, in all probability, your students do not have the faintest, foggiest idea how to embrace the question “why does that get the same answer?” When I’m doing tutoring here, an awful lot of people who are totally familiar with the arithmetic procedures are unaware that the math we do when you’re adding exactly the same thing again and again is multiplication… that it evenhas the word “times” in it…
    That’s one reason why, when I read blogs of people saying “oh, heavens!! Someone is teaching that multiplication is repeated addition! The horror!” I want to hurl and I fervently hope they climb up their little ivory tower and stay there.

    • bretbenesh Says:


      Thanks for the suggestion! The exact problem is how to get them to understand that there is something to “find” in the lattice algorithm. We have been using the phrase “repeated addition” (although it was only one of many contexts in which multiplication can apply, so no one needs to write “The horror!”),

      My students are very slowly getting the idea, so I am pleased with that. However, I am hoping that they get the idea before the end of the semester, which is rapidly approaching.

      I am off to try again! Bret

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