Posted by: groupact | July 6, 2013

What Is A Number? (The Lesson)

My “What Is A Number?” prompt came in a lesson from the first day of a unit on complex numbers that I did in the final three weeks of a semester introducing complex numbers.  I’ll write more about the unit and its goals later, but I want to start by focusing on the goals of the task because I think it could be helpful in other settings as well.   My goal was for students to understand numbers as both algebraic and geometric objects.Image

I had students start by discussing the question of what is a number at their tables, and after about 5 minutes I asked for a response from each table which I recorded on the whiteboard. (I now wish I had kept a more permanent record) I recall that in each class I had several responses with the key words of either “how much,” “amount,” or “quantity.” I consider these responses as dealing with a geometric aspects of number since geometry focuses on measurement. I had at least one response in each class along the lines of “things you can add.” I consider this response as dealing with an algebraic aspect since the focus is less contextual and more on the formal structure. So far so good, but the last key idea I needed from students was an agreement that there was also a geometric aspect to NEGATIVE numbers. I did get from one table in one class, “It is an amount or lack of an amount.” I followed up on this and applauded the consideration of negative numbers, but I wasn’t convinced that lacking 5 watches was the same as having -5 watches. Christopher Danielson has a great post on thinking about integers in which (among other things) he passes on some work from Chisty Pettis and Aran Glancy from the University of Minnesota on characteristics of good integer contexts. The key aspect I teased out of the students after we considered a couple of examples was that negative numbers should imply in some contextual sense an opposite direction which is exactly what we see in the more abstract setting of the number line. Addition has a geometric meaning on the number line in terms of translation which can be in the positive or negative direction. My recollection is that I followed up on the more difficult concept of the geometric meaning of multiplication the following day, which is what I will do in this post as well. To be continued…

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