Mathematical Modeling and Word Problems
Operations, modeling, translation
This entry focuses on the non-mathematical strategies that many students use to select an operation (or operations) to use when solving a word problem.
|Problem: A bag of snack food has 4 vitamins and weighs 228 grams. How many grams of snack food are in 6 bags?|
Student 1: You’d probably divide because there’s a big number and a little number.
Student 2: 228 + 4
Student 3: I was thinking about the other ones…adding, subtracting, multiplying. They didn’t work.
Interviewer: What do you mean they didn’t work?
Student 3: They wouldn’t work because the answer should be a lot lower than this [pointing to a number in the problem].
Students have multiple issues transitioning from informal problem solution methods to using more formal algebraic processes and developing symbolic representations of the problem. Information on this can be found in the Encyclopedia entry Algebraic Relations: Translating Word Problems into Equations.
When translating word problems into operational notation, students often struggle to identify the correct operation(s) to use (e.g., addition, subtraction, multiplication, or division). And even when they are able to identify the correct operation(s) to use, their strategies for choosing operations are limited and superficial. Examples of such strategies are below:
1. Random guess. Students make a guess at the operation to be used without considering all reasonable options.
2. Guess based on recent classroom material. Students make a guess on the operation to be used based on which operation was covered or discussed most recently in class.
3. The numbers determine the operation. Students determine the operation based on which operation seems most likely to be used on two numbers in the problem. For example, if there is a 78 and a 3, then the operation is probably division (As with Student 1).
4. Try all and choose the best. Students try all four operations with two numbers in the problem and choose the answer that seems most reasonable or is simplest to compute (As with Student 3).
5. Look for key words or phrases. Students determine an operation based on key words or phrases. For example “all together” means the numbers should be added.
6. Add/Multiply for larger, subtract/divide for smaller. Students reason about whether or not the answer should be larger or smaller than the numbers given in the problem. If the answer should be larger, the student tries addition and multiplication and then chooses the more reasonable answer. If the answer should be smaller, the student tries subtraction and division and then chooses the more reasonable answer.
All of the above strategies are mathematically immature, yet students use them because (1) they work often enough, (2) teachers tend not to push students to use a more sophisticated approach, (3) teachers may not question students about why they used a particular strategy, and (4) these are the only strategies students have at their disposal. However, students that have a well-developed mathematical understanding of the operations will select operations based on which operation fits the story best. The goal is for all students to choose operations based on their understanding of both the problem situation and of the meanings of the operations.
It is important to acknowledge that correct answers do not necessarily indicate good mathematical thinking. Teachers should require students justify their choice of an operation, in addition to requiring that students give an answer to the problem. Teachers could have students consider questions like “How did you know to [divide] here?” or "What words made you think addition was the appropriate operation to use?"– on individual written work, in small group discussion, and in whole class discussion. Teachers can then emphasize the benefits of choosing an operation based on understanding the meaning of the operation.
Shortcomings of limited or unsophisticated strategies can be highlighted in class discussion after having students solve problems that involve extraneous information, fractions or decimals, or multi-step solutions.
Bell, A., Swan, M., & Taylor, G. (1981). Choice of operation in verbal problems with decimal numbers. Educational Studies in Mathematics, 12, 399-420.
Johanning, D. I. (2004). Supporting the development of algebraic thinking in middle school: A closer look at students’ informal strategies. Journal of Mathematical Behavior, 23, 371-388.
Sowder, L. (1988). Children's solutions of story problems. The Journal of Mathematical Behavior, 7, 227-238.
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?