LeVae Heiner What's Right About Looking at What's Wrong?

LeVae Heiner Literature Review

Title:  What’s Right About Looking at What’s Wrong?

Author:  Deborah Schifter

Source: http://www.ascd.org/publications/educationalleadership/nov07/vol65/num03/What%27s-Right-About-Looking-at-What%27s-Wrong%C2%A2.aspx

Summary:

Deborah Schifter shows how both students and teachers gain new mathematical understanding by examining the reasoning behind a student’s incorrect answers.

1. A 5^th grade teacher asked her students to find at least two ways to determine the products of several multi-digit multiplication problems.  They then met as a group at the end of the period to discuss their work.  She asked Thomas, (a student) to write his strategy for solving one of the problems (36 x 17) on the board.  Then, for a homework assignment, she asked the class to use what he was thinking in the 1^st steps, then revise it and come up with the right answer.

36 + 4 = 40     17 + 3 = 20      40 X 20 = 800 -4 = 796 – 3= 793

2. James envisioned the problem in a context, where the calculation could be used.  He thought of 35 x 17 as 36 bowls, each holding 17 cotton balls.  He explained that first he arranged the balls into groups of 10.  Each group of 10 had 170 cotton balls (10 x 17 = 170).  Then he added the three groups (170 + 170 + 170).  In addition to the groups of 10 bowls, there were another 6 bowls with 17 cotton balls in each (6 x 17).  To simplify that calculation, James thought of each bowl as having 10 white and 7 gray cotton balls, which yielded 60 white balls (6 x 10) plus 42 gray balls (6 x 7) for a total of 102 cotton balls in the 6 bowls.  He used the distributive property to find the answer: (10 + 10 + 10 + 6) x 17= 170 + 170 + 170 + 102 = 612.

3.  They took this problem to a teacher seminar and had the teachers work in pairs to find ways to approach the problem.  Again they looked at a wrong answer to make more discoveries.  800-(4 x 20) – (3 x 40) = 600.  As a result, they discovered that the answer was off by 12.

4.  Finally they drew an array, to think though the problem more clearly.  There were able the see their error and came to a better understanding of the procedure FOIL and why you multiply the first terms, outer terms, inner terms and last terms.

Analysis: I realized that as a teacher it is important to view math as a realm of ideas to be investigated rather than a set of facts, procedures and definitions.  This article made me reflect on a teacher I had in Junior High who helped me to love math and directed me in the mathematical course I should follow.   She helped us understand the important math ideas.