Commentary: Operations Reflection Plus Number Patterns

 

            This is a two-part homework assignment: a reflection and a write-up.  The first part was intended to get feedback on students’ understanding of place value, multiplication, and division.  We had been doing similar activities in class  in fact, the day I first gave this assignment, students had worked in groups trying to find models to show why multiplying by ten looks like “adding a zero.”  All groups came up with visual ways, with or without manipulatives, to show why multiplying by ten “shifts” place values to the left.  Nonetheless, when given this assignment, a significant proportion of the class resorted answered the first question with statements such as, “Multiplying by ten is like adding a zero because the zero is just a place-holder.”  My sense is that these students don’t really understand what it means to explain WHY something works… so that while it makes sense to them that ten ones make ten, ten tens makes one hundred, etc, this explanation is not any more compelling than a phrase like “zero is a place-holder,” which doesn’t even mention anything about multiplying or about ten.  They don’t expect to understand deeply, so they don’t, even when they have all the pieces.  Another thought is that they are more used to thinking in class than when doing homework, so they go into some kind of rote, spit-things-out mode when doing homework that I don’t allow in class. In any case, working on such deeper understanding is one of my top priorities for my Concepts and Processes (Math for preK-6 Teachers) classes this year.

            The write-up questions are all nice number tricks.  We had done some of this kind of algebraic thinking earlier, and were getting ready for a unit on number theory.  Students generally enjoy the tricks.  Those who are comfortable with variables can explain them, perhaps with some help getting started.  Others are more likely to try a few examples, decide that the trick works, and then (hopefully) say they don’t know why (rather than proclaim the examples sufficient to show that the trick works). 7/05.

 

To Activity                   Doc File                       PDF File                      Home