Chapter 7, page 148(^)
Problem 7.6
Understanding students’ thinking: Problem representation
Two high school juniors are solving a practice math problem for the SAT. Each creates a problem representation.
Evaluate the quality of each representation.
Problem:
Donna has three kinds of pets – cats, dogs, and hamsters. She has one more hamster than she has cats. She
has three times as many dogs as hamsters. Of the following, which could be the total number of these pets?
A) 15
B) 16
C) 17
D) 18
E) 19
Haruka’s Reponse: Amber’s Reponse:
One more hamster than cat:
H = 1 + c
Three times as many dogs as hamsters: H,d,c
3h = d
3 variables, 2 equations, can’t
solve. So try one number at a time.
Response. Haruka has created a complete problem representation that captures the two needed equations and
explicitly notes that there is no unique solution for 2 equations with 3 variables. Amber does not clearly set out a
problem representation. Amber has merely written down three variable labels. This is not a complete response.
Good problem representations often include inferences that go beyond the information that is initially
given. For example, one researcher asked history professors and high school students to analyze eight
documents. Some documents provided testimony by eyewitnesses about who fired the first shot of the
American Revolution at Lexington Green in 1775 (Wineburg, 1991). When reading these documents, the
history professors developed more elaborated mental representations of the situation than the high school
students did. This required drawing inferences that went beyond the information given in the eight
documents. One historian said,
One has to try to put themselves in the minds and the bodies of the British. They’re starting out early in the
morning, they must be walking quickly; I’d have to figure out how many miles between the barracks where