## Four Problems, Four Interpretations

## A Window into Student Thinking

## Of Special Benefit to ELLs

## What Students Need to Know

## The Language of Mathematics

*even, odd*, and

*improper*, but these words have a different meaning when used in mathematics. Sometimes the same mathematical word is used in more than one way within the field itself. The word s

*quare*, for example, can refer to a shape and also to a number times itself (Rubenstein & Thompson, 2002). Also, mathematics language makes use of certain syntactic structures—such as

*greater than/less than, n times as much as, divided by*as opposed to

*divided into, if/then*, and so on (Chamot & O'Malley, 1994).

## The Unit Whole

*more*. How many oranges would Bill end up with?" It's unclear what whole is associated with the 5/7: the number of oranges that Bill currently has or the number of oranges that were in the bag originally. In either case, we can't answer the question because we don't know what the original number of oranges was. Asking "what fraction of a bag" instead of "how many oranges" would be more appropriate.

## A Reasonable Context

*big*is ambiguous because it could mean either the area or perimeter of the pan.

*multiply*their candy bars together, how much would they have?" But what does multiplying two candy bars really mean?

## Getting It Right

*ate 2/3 as much pizza as Julie did*. How much pizza did Janet eat?

## Tips for Teachers

- Start by writing several word problems that involve different mathematical concepts yourself (such as the four examples we described earlier) before you use this activity in class. This will give you insight into what you might expect in student responses.
- To avoid getting problems like "Jimmy needs to find out 2/3 × 1/4. What is it?" explain to students that they can't use the words
*multiply*or*times*or the multiplication symbol in their problem. - Discuss how to choose topics in small groups or as a whole class. Be sure to introduce different contextual situations (for example, in addition to dividing a pie or a pizza, mention running a leg of a relay race, planting a garden, or making a scale drawing). This will encourage the use of different modes of representations (such as a number line model or an area model) and different interpretations (such as part-of-part or scaling). If students have trouble coming up with a topic, provide situations to which they can respond.
- If appropriate, start with problems in which a fraction is to be multiplied by a whole number and then progress to using two fractions. This will help students familiarize themselves with writing strategies and ease the transition to problems involving fraction operations.
- When students present their word problems (which may be incomplete or incorrect), encourage them to clearly explain what they mean and solve the problem as part of their explanation. Alternatively, a classmate might solve the problem and give the student feedback. This gives students the opportunity to use their math reasoning strategies and apply mathematical concepts.
- Students can write word problems that involve other operations of fractions, such as division. Or students can write problems to enhance their conceptual understanding of important mathematical ideas that are often taught as rule-based, such as a procedure for finding a common denominator.