## Introduction

Models help promote mathematical thinking by facilitating an understanding of key concepts and mathematical structures. By seeing and moving objects, students engage their senses to better understand and reason with abstract concepts, or to make sense of — and solve — problems.

It is important that students of all ages — from elementary school through to high school — are able to create and use models. As the Common Core Math Practice Standard #4 (Model with Mathematics) explains, students in the early grades might use modeling to write an addition equation to describe a situation. In middle grades, students might apply proportional reasoning to plan a school event, and in high school, students might use geometry to solve a design problem.

## Ways to Differentiate Instruction

The ability to create and use models is particularly valuable to struggling students and those with disabilities. Teachers can support students who struggle with abstraction by using models to differentiate instruction. For example, you can help your students understand how different models can be used in different ways to illustrate a concept or solve a problem. You can also help them understand the relationship between different models (and how each model can be used to solve the same problem), and you can help them use drawings, manipulatives, number lines, bar models, and virtual versions of physical models to deepen their understanding and strengthen their ability to process mathematical problems.

Identifying your students’ strengths and needs allows you to more effectively differentiate instruction. Different strategies — such as encouraging students to ask questions, share their thinking, and respond to the feedback you provide — can help you understand what students know and how they process information. You can ask them to explain why they chose the model and/or particular technology tool they selected, which will help them to understand why their choice was the appropriate or inappropriate model/tool for the task. If necessary, you can redirect them to more appropriate models and technology tools. Follow-up questions can focus on reasons for simplifying or elaborating a model’s features. You can also encourage students to consider the possibility of using other models or tools that you — or other students in the class — have suggested.

Having students explain how and why they chose a particular model or tool has many benefits:

- It provides evidence of what the student understands about the modeling process and highlights any incorrect assumptions.
- It encourages students to “stop and think” about their decisions and become more strategic in decision making.
- It gives other students an opportunity to ask questions respectfully if a student selects the wrong model. This helps all students to refine their reasoning abilities and helps foster a collaborative learning environment (see UDL Checkpoint 8.3: Foster Collaboration and Community).

## Using Technology Tools

A variety of technology tools can support students, including web-based applets, virtual manipulatives, graphing software, drawing tools, calculator, dynamic geometry software, and spreadsheets. Virtual models provide a different sensory experience, which can help students who have difficulty with physical objects.

The video below provides an overview of virtual manipulatives, which can support students in developing modeling skills.

## In One Classroom

Ms. Howard’s Grade 4 class has reviewed multiplication facts up to 10 x 10 and products of one digit with multiples of 10 up to 90 (e.g., 3 x 40, where one digit — 40 — is a multiple of 10). Ms. Howard plans to begin teaching students the distributive property of multiplication over addition, using various supports to model the situation. The specific objective of her lesson is to have students use a model to represent multiplication of a one-digit and two-digit number. This will lay the groundwork for the multiplication algorithm.

Her lesson objective aligns with two Mathematics Common Core State Standards:

- CCSS.Math.4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
- CCSS.Math.M4 Model with mathematics.

Mrs. Howard makes use of the technology she has available in her classroom. For this lesson, she will use her interactive whiteboard to communicate visually with the class and interact with numbers. She will also engage students in a virtual manipulative as an alternative to base-ten blocks or other models.

She is planning to employ two formative assessment strategies. First, she will solicit explanations about the model in order to verify students’ understanding of the representation. Second, she will observe students’ discussions and strategies, intervening when necessary to ask prompting questions.

The section below outlines her lesson plan. It is organized into three sections to show how she will (1) launch the lesson, (2) carry out the learning task, and (3) bring closure to the lesson.

## Lesson Plan

### Launch

- Explain the goal of the lesson.
- Introduce the problem that students will work on during the lesson.
- Give students a few minutes to consider the problem and then introduce a model of the situation.
- Discuss the model with the class.
- Continue the discussion, connecting the model to abstract mathematical concepts.

### Learning Task

- Have students work on the learning task.
- Circulate around the room, pausing to listen to each pair’s discussion.
- Guide students toward appropriate manipulatives.
- As students move on to another problem, observe the models they choose.
- Check in with the students who did not understand the model at the beginning of the lesson.

### Closure

- Bring the class together and ask pairs to share their strategies.
- Lead discussion of students’ strategies.
- Give each student an exit ticket (assessment).

## Online Teacher Resources on Modeling

This article draws from the PowerUp WHAT WORKS website, particularly the Modeling Instructional Strategy Guide . PowerUp is a free, teacher-friendly website that requires no log-in or registration. The Instructional Strategy Guide on Modeling includes a brief overview that defines modeling and an accompanying slide show; a list of the relevant Mathematics Common Core State Standards; evidence-based teaching strategies to differentiate instruction using technology; short videos; and links to resources that will help you use technology to support mathematics instruction. There is also a section on formative assessment that you can use to further explore the strategies described in the article. If you are responsible for professional development, the PD Support Materials provide helpful ideas and materials for using the modeling resources. Want more information? See www.PowerUpWHATWORKS.org .