## 2. Reason abstractly and quantitatively.

How are **fourth** and **fifth** grade teachers helping their students reason abstractly and quantitatively?

How can students be helped to:

- make sense of quantities and their relationships in problem situations?
- decontextualize — to abstract a given situation?
- represent a problem symbolically?
- manipulate the representing symbols as if they have a life of their own?

*The CCSS states:*

Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

It’s important to encourage students’ to use of a variety of tools and strategies. These math tools need to be on hand for students to pick up and use at anytime. The primary thing is to provide students with real world scenarios, which allows them to think about how the problem and the solution fit together. Then you are taking students concrete knowledge and getting them to think abstractly.

As teachers, we need to help students develop reasoning, both abstractly and quantitatively. To do this, we provide word problems or real-life situations where students can make a personal connection as they solve their problem. While they’re doing this, they use manipulatives or hands-on tools to help them work through the problem. As students solve their problems, they are able to use the precise vocabulary and clearly communicate the process they went through. They need to be comfortable with sharing their process to get to the answer, and not just what their answer is. As teachers, we allow, and encourage students to share a variety of strategies that they used to help come to the solution of solving their problem. This would then lead into a discussion between students about the diverse ways of thinking about the same problem.

We often provide story problems that students can easily relate to the real world. Before they begin to solve it, we brainstorm together a variety of strategies that we have practiced to solve a problem. Students are expected to solve a problem in more than one way. Students critique each other through pair share. Finally, whole class sharing provides students the opportunity to learn through each other through friendly debate.

One area I noticed students struggling with was subtraction. When we’d look at a “subtraction” word problem, they’d often represent the problem with a missing addend equation. They would think of the problem as addition. I spent time in class talking about fact families and exploring the inverse relationship between addition and subtraction. Students were encouraged to use a counting-up strategy to solve subtraction, we worked on recognizing subtraction situations as different than addition situations. We can go back to the basic fact families to help students remember the structure of fact families for the larger numbers.