Research Article

Preservice Teachers’ Learning to Respond on the Basis of Children’s Mathematical Understanding

Table 4

Examples of responses coded as robust, limited, and lack of evidence.
 Code Inquiry into Student Thinking—selected task Rationale Tutoring assignment—selected task Rationale Robust evidence I would create Joint-Result-Unknown (JRU) or Separate-Result-Unknown (SRU) problems for the students. A JRU example would be “Sunny has ____ fish, and then she buys ____ more. How many fish does she have now?” Number choices would include {(10, 50) (20, 30) (10, 41) (15, 25)}. This type of problem would be good for all of the students. Sunny and Daniella struggle to count by tens past the numbers 20 and 30, and this problem challenges them to do so. Emilio would be challenged to count by tens and keep track of the ‘one’ in 41. Both Jack and Emilio would be challenged by the last number choice, as both understand the concept of counting by tens, but they would have to extend their understanding to non‐zero ending numbers. Cornor has ___Wii games in his cupboard. He found ___more Wii games under his bed. How many Wii games does Cornor have?Number Choices: (7, 3) (4, 6)James has ___Wii games at his house. Cornor let James borrow ___more Wii games. How many Wii games does James have at his house now? Number choices: (5, 5) (2, 8) After my initial interview with the students, I knew they did not have a clear understanding on how to count on from a number other than one. When I gave each student that question in the interview none could count on from the number I had given them …I chose to do a Joint Result Unknown story problem because I wanted the problems to begin with a number other than 1 Limited evidence If Quinn had 89 pieces of pizza, and 10 pieces of pizza made a whole pizza, how many whole pizzas can Quinn make? This is a Separate Result Unknown problem. I chose 89 because the students have the concept of base 10 down; they are able to do it with the easy numbers, now I want to challenge them with bigger numbers, hoping they would use the manipulative[s] and not their fingers. I would hope students could lay out the manipulative[s] and see easily that they can make 8 pizzas. If students understand this concept they should have no problem with this problem. The student will be given visual balance with numbers in blocks. One black on the right side will be blank. My number choices are 6 and 2 on the left sides and a blank and 4 on the right side I plan to work on commutative property and relational thinking to help with those problems. Latter I plan to focus on his subtraction skills so that he will be willing to use them in other problems …The purpose of this exercise was to have Mathew begin thinking in terms of something balancing or equaling something else in a horizontal format. The balance scale is meant to be a visual tool to eventually lead to understanding of number sentences. Lack of any evidence If I were to teach the next lesson to these students, one problem I could give them would be a: Jack has 45 crackers. Sunny gives him 10 more. How many crackers does Jack have? I choose this Separate Result Unknown problem because I wanted the students to continue using addition. These are the problems they have been used to and need to keep getting trying to understand. I chose the numbers 45 and 10 because the students need to continue using large numbers so they can’t just count by ones and learn to use going by 5’s or 10’s as a first choice. The student will be presented with these problems one at a time and they determine whether the problem is true or not.4 + 2 = 63 + 3 = 64 + 2 = 3 + 3 These equations allow the student to look at the two different equations and see that although the numbers are different they equal the same thing.