Chapter 7, page 150
division key, arriving at the answer 450 hours, which he writes on his test paper. If the student had checked
his work, he would have realized that this answer is so far from being reasonable that it must be a mistake.
But he did not check to see whether the answer made sense.
Noticing commonalities and differences. Polya’s last step directs problem solvers to reflect on what
they can learn from the problem they have solved. Learning from problems is enhanced when problem
solvers construct general schemas for how to do problems (Reeves & Weisberg, 1994). One way to build
general schemas is to reflect on the current problem and compare it with other problems that have been
encountered (Novick & Bassok, 2005). Suppose a child is learning to solve word problems that involve
addition and he encounters two word problems:
- Ellen has 3 nickels, and Miranda has 5 nickels. How many nickels do they have together?
- Gina has 5 baseball cards, and Caleb gives her 4 more. How many baseball cards does she have now?
It would be useful if the child compares these problems and notices what they have in common at a general
level: in both cases, the problem statement describes objects being combined or put together, and then the
problem asks how many objects there are after they are put together. The child might notice that the two
problems have different wordings, and therefore that both wordings signal that objects are being combined
and that addition is called for. When students construct general schemas that can be used to describe
different problems within the same general category, students are better able to solve related problems in
the future (Gick & Holyoak, 1983).
Problem 7.7
Understanding students’ thinking: Problem solving strategies
A group of seventh graders are working on the following algebra problem: “Rachel is 2/3 as tall
as Mario, who is 6 feet tall. Mario weighs 142 pounds, which is 53 pounds more than Rachel.
How tall is Rachel?” Here is an excerpt from their conversation:
Student 1: I think this is one of those fraction multiplication problems.
Student 3: Yeah, like the one yesterday with the guy who was half as tall as his father.
Student 2: So in these problems, we’re supposed to make an equation with the fraction. And
we can forget about the weight. Weight doesn’t matter to find the height.
Student 4: [after a pause as everyone worked on the equation] What did you get?
Student 3: I used m for Mario’s height, and r for Rachel’s height, and I got 2/3r = m.
Student 2: But that would mean that Mario is two thirds as tall as Rachel. And that would mean
Rachel is 9 feet tall.
Student 1: Rachel has to be shorter than Mario.
Student 3: I see, so it has to be the other way. r = 2/3 m. So when you see two-thirds of
something, you multiply the number by two thirds.
Student 2: [after a pause as they quickly solve the equation] So Rachel is 4 feet tall. Right?
Student 4: That’s what I got. And 4 is two thirds of 6. So it works.Evaluate the students’ problem-solving strategies?Response: This conversation displays good use of several strategies:
Ɣ Problem representation (when the students work on setting up the correct equation, and
explaining to themselves how the equation should be set up, and when Student 2 notes that
weight can be excluded from their problem representation)
Ɣ Noticing commonalities and differences (when Student 3 notices that it is like a problem