Making Mathematical Thinking Visible with The Explanation Game

In Minnesota, third graders need to be able to represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting (3.1.2.3).

For students aged 8- and 9-years-old, this mathematics benchmark is quite dense. Even the "I Can" statement, written in kid-friendly language, is still fairly complex and abstract.

To help her students understand the concept and be able to successfully demonstrate the skill described in this benchmark, one third grade teacher used the Thinking Routine The Explanation Game (from Making Thinking Visible by Ritchhart, Morrison, and Church.)

This teacher wanted her students to look closely at features and details of the different ways to show multiplication facts, so The Explanation Game seemed like the perfect thinking routine to elicit that type of thinking from her students.

During the Explanation Game, students take a close look at something they're trying to understand and:

Name it. Name a feature that you notice.
Explain it. What could it be? What role or function might it serve? Why might it be there?
Give reasons. What makes you say that? Or why do you think it happened that way?
Generate alternatives. What else could it be? And what makes you say that?

With her students, the teacher made six posters, each with a different way to represent multiplication facts. The teacher purposefully left off the name of the particular multiplication strategy shown on the poster.

The students then had to answer these questions:

• What is the multiplication equation shown? 
• What strategy are you using to show that equation? Explain the strategy to a partner. 
• Write another equation and solve it using that strategy. 

Finally, the students labeled each poster with the name of the particular multiplication strategy. By allowing the students to independently construct their own understanding of the multiplication strategy first and then naming each strategy, the teacher was guiding her students' learning with inquiry, a powerful vehicle for learning in the IB Primary Years Program.

Below are the posters after the learning engagement.

Teachers in other grade levels could use this same Thinking Routine and lesson structure to address mathematics benchmarks at their particular grade level, particularly when the benchmark requires students to represent a number or a mathematical operation in a variety of ways. The list of benchmarks below is not an exhaustive list, but is only provided to show the variety of settings in which this Thinking Routine and lesson structure could be used.

Kindergarten (K.1.1.2): Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes. 

1st grade (1.1.2.1): Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.

2nd grade (2.1.2.1):Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the 
commutative and associative properties.

4th grade (4.1.2.1): Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. 

5th grade (5.1.1.1):  Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal. 

6th grade (6.1.1.4): Determine equivalences among fractions, decimals and percents; select among these representations to solve problems.