Pascal’s Triangle and Pascal’s Identity: Contextualizing and Decontextualizing

PurposesEffective teaching; Professional development activity
DescriptionAccording to the Common Core State Standards, the ability to contextualize and the ability to decontextualize are important mathematical skills for students to develop. Learners should have "the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents--and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved" (CCSS, 2012).

This analytic focuses on a problem-solving session involving four high school students named Ankur, Jeff, Michael, and Romina, participants in a long-term study of students’ mathematical learning (Maher, 2002). (A fifth student, Brian, joins the group briefly, but does not participate in the discussions highlighted in this analytic.) The students make sense of Pascal’s Triangle and Pascal’s Identity (the addition rule for Pascal’s Triangle) by referring to combinatorics problems they know well, including the pizza problem. * The analysis of this session, called the Night Session,** includes seven events. In the first two, students demonstrate the ability to contextualize the numbers in Pascal’s Triangle by referring to the pizza problem and to binary numbers. In the third and fourth events, they demonstrate the ability to decontextualize the pizza problem in order to generate Pascal’s Identity in general form using standard mathematical notation. The last three events provide further examples of decontextualization: working with symbolic notation only, the students convert Pascal’s Identity to factorial notation, correct an arithmetic error, and simplify the equation.

*The generalized pizza problem is: How many pizzas is it possible to make when there are N toppings to choose from? The answer is: there are 2^N possible pizzas because there are two choices for each of the N toppings: on or off the pizza.

**The Night Session took place on the evening of May 12, 1999, during the students’ junior year in high school. The five students had participated in the longitudinal study since elementary school. At the time of the Night Session, they had worked together on combinatorics problems involving Pascal’s Triangle on seven occasions between December 1997 and January 1999 (Uptegrove, 2005).

References

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved February 11, 2014, from www.corestandards.org/standards/mathematics

Maher, C. (2002). How students structure their own investigations and educate us: What we have learned from a fourteen year study. In A. Cockburn & E. Nardi (Eds.), Proceedings of the Twenty-sixth Annual Meeting of the International Group for the Psychology of Mathematics Education (PME26), Vol I, pp. 31-46. Norwich, England: School of Education and Professional Development, University of East Anglia.

Uptegrove, E. (2005). To symbols from meaning: Students’ investigations in counting. (Unpublished doctoral dissertation) Rutgers University, NJ.
Created on2014-02-11T15:10:55-0400
Published on2015-06-16T17:02:26-0400
Persistent URLhttp://dx.doi.org/doi:10.7282/T31N82WG