# VISIBLE THINKING IN MATH: USING MULTIPLE REPRESENTATIONS AS RECORDS of students' strategic thinking

Representations can be useful tools to reveal the mathematical thinking that students engage in as they reason through a problem. In one of the research lessons, the teacher had students act out the problem by having the students hold books representing lockers. When the students came to the front of the class to become “lockers,” it helped some of the students visually represent the “change in state” for the locker problem.

The use of the manipulatives and “acting it out” provided students’ access to the problem, but students quickly realized they needed a systematic way to record what was going on with each change of state. The technology tool shown below and the paper number chart were nice ways to use the affordance of keeping count of the changes to each locker. The resulting record showed the number of factors of each locker number and was more efficient to keep track of changes than manipulatives such as chips and cards. The teachers had students present their approaches, helping students to make connections and see patterns between the multiple representations of the problem. This gave them more evidence to verify their conjectures about the numbers and the state of the lockers. The use of a number line and the number chart also helped students record the pattern while solving the problem.

Using this thinking sheet helped students talk about the problem more deeply. They noticed how every number has two factors, 1 and itself. Therefore, every locker is opened on the first pass and shut on the pass where the student number equals the locker number. In addition, all numbers (lockers) *except perfect squares* have factors that occur in pairs, so that every locker except those whose number is a perfect square has its state changed an even number of times: It gets changed and then changed back, or opened and shut again. Only perfect squares have a duplicate factor pair like 3 x 3 = 9, so that the state of these lockers is changed an odd number of times or opened and left open.

**Figure 4.2 Student uses cards to open and close to act out the locker problem. ***Source:* Authors.

As teachers planned for the lesson, every member of the lesson study team was asked to work out the problem for homework. One of the teachers created the PowerPoint shown below that was used in his research lesson.

The PowerPoint helped to organize and keep track of the information. The tool was used to help the students to see numbers and number sense as a mathematical topic with order and within a built-in system of relationships. These patterns can be used to enhance the conceptual understanding and student thinking about fundamentals of number theory including communicating about the classification of numbers as primes, composites, connecting with the concept of factors and multiples, and

**Figure 4.3 Locker problem recording sheet to keep track of the change of state. *** Source:* Authors.

**Figure 4.4 Locker PowerPoint simulation.**

realizing how this understanding of common factors connects to reducing fractions, thus extending the conceptual understanding from patterns of whole numbers to fractions. The lesson was extended in some grades by visiting the Sieve of Eratosthenes; prime and composite numbers in context; arithmetic sequencing; common multiples; figurate numbers, and perfect squares (see Resource List at the end of the chapter) to continue the investigation of patterns in numbers and factors.

**Posing purposeful questions. **The teachers developed some probing questions that elicited students to think more deeply about number relationships.

1. Given the numbers of several lockers that were touched by:

a. two students. How are these numbers related or similar? What are they called?

b. any odd number of students. How are these numbers related or similar?

c. any even number of students. How are these numbers related or similar?

d. three students. How are these numbers related?

e. four students. How are these numbers related?

- 2. How can you determine how many students have touched a specific locker
- 3. Given the first locker touched by:

a. both Student 4 and Student 6? How do you know?

b. both Student 5 and Student 13? How do you know?

c. both Student 12 and Student 30? How do you know?

d. two students of your choice. Show your work.

- 4. Given two students’ numbers, how can you determine which locker will be the first touched by both students? (Select and record two students’ numbers and show your thinking.) How can you determine which locker will be the last touched by both students?
- 5. Given two lockers, how can you determine which students touched both?
- 6. Which lockers are still open after the twentieth student is finished? Which locker or lockers changed the most?

**Think about it!**

**Но» does the locker problem provide ail opportunity to explore number relationships, particularly the relationship between factors and multiples, prime and composite and square numbers?**