This is the third post in a continuing series featuring Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement by Steven Leinwand.

As teachers, we all know that students learn in different ways – in fact, we all learn in different ways. Call them learning styles, learning modalities, multiple intelligences…each of us takes in information better in some forms than in others.

In this chapter, Steve Leinwand talks about the fact that teachers that don’t use pictures in their teaching are missing a major opportunity to help students visualize what is going on in mathematics. This brings us to…

**Instructional Shift 3:** Use multiple representations of mathematical entities.

Leinwand starts off the chapter having the reader think about picturing three quarters. How would you picture it? In my mind, I think of three twenty-five cent coins. Some would think about a circle divided into four parts with three of those shaded in. Others would think of a pizza pie with one quarter of it missing. Some might think of the numerical representation with three over four. Even others may think of a set of four objects with three of them circled or colored in – but this is the least likely of the options given.

Leinwand says this contributes to the fact that multiplication and division of fractions is so difficult for students. If we show students that 10 people can be served when we have 5 pizzas and each person gets one half of a pizza, that supports understanding of why 5 ÷ ½ = 10. Simply telling students that the rule is to invert and multiply gives them no understanding of why that works.

Yes, some students can gain understanding through abstractions, but others need concrete examples like pictures, models, and manipulatives. This doesn’t apply only to fractions, but to just about every concept we teach from addition and subtraction to such complex things as absolute value and positive and negative integers.

Along with using these multiple representations as teachers, frequent opportunities for students to draw or show and then describe what is drawn or shown should be included in classroom instruction. This alone can transform the dynamics of a math classroom. Students who are generally “weaker” can often draw on their natural sense of how math works to provide some strong concrete examples, while students who are generally “stronger” can help inform the rest of the group with their divergent thinking. Not only that, but valuing alternative approaches in the classroom contributes to a respectful community of learners where students feel comfortable taking risks in their own learning.

Can you visualize what this would look like in your classroom??