Bob, this is a response to your question re Quotative (sometimes called "Measurement") and Partitive Division: >Speaking of unifying and simplifying, that is why >I teach "12 divided by 3" as meaning "12 has how >many 3's in it?" I don't understand the labored >distinction between the following two customary >interpretations of "12 divided by 3" >1. it could mean "having 12 widgets that are to >be separated into 3 groups, so how many widgets >will each group have?"; OR >2. it could mean "having 12 widgets and giving 3 >widgets to each group, so how many groups will >there be?" > >Am I missing something here? The distinction >between these two cases of division to me >seems trivial and worth far less than the time >it takes to even explain it. Rather than creating >confusion by insisting (to either kids or teachers >or future teachers) that they understand the >distinction, I prefer teaching it as I have come >to explain it: that "12 divided by 3" means "12 has >how many 3's in it?"
You are pointing to a classic distinction that is disacknowledged by formal abstract arithmetic (same operation) - but not by an arithmetic of quantities (different operations: Extensive quantity divided by an Intensive quantity vs Extensive Quantity divided by an Extensive Quantity). If you believe that childrens' mathematical knowledge arises from situations and their actions in them, then you need to attend to the difference between these two operations - because they are conceptualized differently and hence are elicited differently in different situations. They also yield very different treatments of "remainders," depending on differences in the situations.
To see evidence of the huge differences in the ways children interpret these operations, and large differences in performance, I recommend looking at a paper I did with Max West reflecting several years' work on this and related issues: (1994) Missing value proportional reasoning problems: Factors affecting informal reasoning patterns. In J. Confrey & G. Harel (Eds.) The development of multiplicative reasoning in the learning of mathematics. (Research in Mathematics Education Series) Albany, NY: State University of New York Press.
I recommend the book for teachers of teachers because multiplicative reasoning is a place where lots of children fall apart mathematically because our traditional approaches do not serve them very well. And the reason for this is that traditional approaches do not acknowledge, and hence run roughshod over, fundamental differences in the experiential/action roots of their mathematical thinking.
Jim Kaput Department of Mathematics University of Massachusetts-Dartmouth No. Dartmouth, MA 02747-2300 Tel: (508) 999-8321 or 999-8797 (SimCalc) or 999-8316 (Math Dept.) Fax: (508) 999-9215 (SimCalc) or 910-6917 (Math Dept.) SimCalc Website: http://www.simcalc.umassd.edu