*Classroom Instruction That Works*(Marzano, Pickering, & Pollock, 2001) and

*How to Differentiate Instruction in Mixed-Ability Classrooms*(Tomlinson, 2001) provide different but overlapping visions of good beginnings.

#### Version 1: Classroom Strategies That Work Approach.

**As a result of this unit, I will be able to**

Use what I have learned about division to make calculations quickly and accurately.

Explain how and why I solve division problems the way I do.

Understand key terms: divide, divisor, dividend, quotient, remainder, fraction, decimal.

Use problem-solving strategies to solve challenging word problems about division.

Recognize when division is a useful strategy and apply it to everyday situations.

**Personal goals during this unit: I will...**

(Adapted from Marzano, Pickering, & Pollock, 2001)

#### Version 2: Differentiated Instruction Approach

**During our study of division, we want to provide opportunities for you to learn about the topic in ways that work best for you. Number your choices from 1 to 6 (1 = easiest, 6 = most difficult).**

__using manipulatives

__observing demonstrations

__sketching out the math situation

__reading

__comparing your work with a partner

__solving complex problems in a team

**What do you already know about division?**

What does “division” mean?

How would you explain doing division to someone in a younger grade?

What are some examples of division that you know how to do?

Where do people use division? And why?

(Adapted from Tomlinson, 2001)

#### Version 3: Combined Approach

**In this unit, you will learn**

How to use division to calculate quickly and accurately.

How to connect and apply division to everyday situations.

How to develop good explanations about how and why division works.

How to use problem-solving strategies, especially diagrams, to solve challenging word problems.

**What's your interest?**

We will be investigating a number of situations in which division is applied to situations in the real world. Rank these in order of interest, from 1 (most) to 4 (least):

___ sharing food fairly;

___ figuring out how fast a car is going;

___ making money;

___ spending money.

**What do you already know about division?**

What does division mean?

Can you give an example of divisor, dividend, quotient, and remainder?

What is division for? Why do we need it?

What are some examples of division that you know how to do?

What makes division difficult?

- Do different students possess different styles of mathematical learning?
- What is the relationship between the research on effective strategies and student styles of mathematical learning?
- What constitutes a fair and thoughtful assessment of student progress in mathematics, given students' personal differences when it comes to learning mathematics?

*Classroom Instruction That Works*and in

*How to Differentiate Instruction in Mixed-Ability Classrooms*. We call this hybrid “differentiation that works.”

## What Is Mathematical Learning Style?

- The Mastery style: People in this category tend to work step-by-step.
- The Understanding style: People in this category tend to search for patterns, categories, and reasons.
- The Interpersonal style: People in this category tend to learn through conversation and personal relationship and association.
- The Self-Expressive style: People in this category tend to visualize and create images and pursue multiple strategies.

In interviews we conducted with more than 200 students over the past seven years, we consistently found these patterns of thinking. For instance, we presented students with the following problem:Nineteen campers are hiking through Acadia National Park when they come to a river. The river moves too rapidly for the campers to swim across it. The campers have one canoe, which holds three people. On each trip across the river, one of the three canoe riders must be an adult. There is only one adult among the 19 campers. How many trips across the river are necessary to get all the children to the other side?

- Students favoring the Mastery style learn most easily from teaching approaches that emphasize step-by-step demonstrations and repetitive practice. Students in this group struggle with abstractions, explanations, and non-routine problem solving. They define mathematics as proficiency in calculation and computation.
- Students favoring the Understanding style learn most easily from teaching approaches that emphasize concepts and the reasoning behind mathematical operations. These students struggle with work that emphasizes collaboration, application, and routine drill and practice. They define mathematics primarily in terms of explanations, reasons, and proofs.
- Students favoring the Interpersonal style learn most easily from teaching approaches that emphasize cooperative learning, real-life contexts, and connections to everyday life. Students in this group struggle with independent seatwork, abstraction, and out-of-context, nonroutine problem solving. They define mathematics primarily in terms of applications to everyday life.
- Students favoring the Self-Expressive style learn most easily from teaching approaches that emphasize visualization and exploration. These students struggle with step-by-step computation and routine drill and practice. They define mathematics primarily in terms of nonroutine problem solving.

Barb Heinzman, a 5th grade teacher, describes the connection between style-based differentiation and curriculum design for mathematics in this way:What I saw right away was that not only did different students approach mathematics using different learning styles, but real mathematical power required using all four styles. Think about it: If you can't compute accurately, explain your ideas, discover solutions, and apply math in the real world—you don't know math. Miss even one of these and you miss the boat. The problem with most math programs is they emphasize just one of these and leave out the rest. By building every unit so it includes all four styles of learning, I support all my students, and I stretch them into areas where they wouldn't naturally go. (cited in Strong, Silver, & Perini, 2001)

## The Style-Strategy Connection

- Rotate strategies: Identify learning goals, then deliberately use multiple strategies over the course of a unit to guarantee that all students believe that the variety of strategies both validates their dominant style and challenges them to work in less preferred styles.
- Use flexible grouping: Identify a common purpose, such as developing accurate explanations in a unit on time, rate, and distance problems, and then divide students into different style groups that use alternate strategies. Style groups can validate or challenge students' dominant styles, depending on whether groups are style-alike or style-diverse.
- Personalize learning: When students struggle or need an extra challenge, shift the strategy to individualize their instruction. For example, a middle school math student with a penchant for creativity and imaginative exercises was having trouble internalizing the steps in the operation-solving process. His teacher then used a Self-Expressive strategy to challenge him to create a metaphor—he chose digestion—for the operation-solving process.

## Fair and Thoughtful Tests

## Constructing a Differentiated Mathematics Classroom

- Include all four dimensions of mathematical learning—computation, explanation, application, and problem solving—in every unit we teach;
- Help students recognize their own mathematical learning styles—Mastery, Understanding, Interpersonal, or Self-Expressive—along with their strengths, their weaknesses, and where they need to grow;
- Use a variety of teaching strategies to explore mathematical topics; and
- Create or revise our assessments to reflect all four dimensions of mathematical learning and all four learning styles that students use to approach those dimensions.