Mindset Mathematics: Visualizing and Investigating Big Ideas, Grade 2

Jo Boaler, Jen Munson, Cathy Williams

• Publisher: Jossey-Bass
• Year: 2021
• Print ISBN: 9781119358633
• eBook ISBN: 9781119358640
• Edition: 1
• Topic: Education
• Subtopic: Education Curricula

Engage students in mathematics using growth mindset techniques

The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low-floor, high-ceiling tasks that will help you do just that, by looking at the big ideas in second grade through visualization, play, and investigation.

During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message?that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So, the authors designed Mindset Mathematics around the principle of active student inquiry, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to support student learning, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that:

• There is no such thing as a math person and anyone can learn mathematics to high levels.
• Mistakes, struggle, and challenge are opportunities for brain growth.
• Speed is unimportant, and even counterproductive, in mathematics.
• Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics.

With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.

BIG IDEA 1
Partitioning Shapes into Equal Parts

If you have picked up this book, I am going to assume that you want to introduce your students to a mathematics that is beautiful and creative and that encourages them to explore and be curious. After many years of working in mathematics education, researching and teaching different groups of students, I have come to realize that there are two completely different versions of mathematics. For many people in the world, mathematics is a subject of rules and procedures. But the mathematics we will explore together with your students is one that is open, creative, and beautiful. When your students meet this mathematics, I predict that they will be engaged and inspired, and they will learn the mathematical ideas deeply.

In this big idea, we help students understand two concepts that they will use for many years in mathematics and that are often misunderstood by students. The first is the meaning of equality; the second is the meaning of a fraction.

In students’ mathematical journeys, they will meet the idea of “equals” often and in many different ways; in this activity, we introduce the students to the idea of equality between shapes. Researchers have identified that many problems occur for students when they develop the idea that equals means “do something,” a kind of instruction or operation (Powell, 2012). This comes about when students see expressions such as 4 + 5 = ? They think the equals sign is telling them to add the numbers. But the equals sign is not an instruction; it is a relational symbol, showing that the numbers or shapes on either side of the symbol are in an equal relationship. This is an important and big idea for students. In the Visualize activity, students will discuss the meaning of equals as they consider different shapes.

As students compare shapes, they will also get an opportunity to combat a common misconception in fraction learning. Many students believe—incorrectly—that for fractions to be equal, they have to have the same shape. This misconception comes about because students are so often shown fractions that have been partitioned into shapes that are equal in area, as well as congruent. But when we think about equal fractions, we only need to think about the area; the shape of the fraction is not relevant. In this activity, students will see that different shapes fill a half of an object, as they have the same area even when the shape is different. As students play with spatial puzzles, and think about their qualities, they will learn that shapes, like numbers, can be understood flexibly.

In the Visualize activity, students consider the question, What does it mean for two or more shapes to be equal? They also think about ways to partition shapes into equal parts. Students will look at squares and cut squares to decide and then explain how they know that parts are equal. They will begin to use the language of fractions, and they will be naming shapes as well and justifying how they know whether they are equal or not.

The Play activity builds off the Visualize activity as students work to partition squares into fourths in as many different ways as possible, folding and cutting. Students are encouraged to find equal areas that are not congruent. Students should play with the shapes, rotating and flipping them, realizing that they can transform shapes and make new ones.

In our Investigate activity, students partition a rectangle into rows and columns where the pieces of the partition are equal. This activity starts to build toward a visual representation of multiplication and should be connected to area. Students investigate how to partition rectangles into rows and columns of square units, which should be challenging, giving students lots of opportunity for brain strengthening. It will be helpful and interesting to collect the different strategies they use.

Jo Boaler

Reference

Powell S. R. (2012). Equations and the equal sign in elementary mathematics textbooks. Elementary School Journal, 112, 627–648. http://doi.org/10.1086/665009.

Equal or Same?

Snapshot

Students explore squares that have been partitioned into different shapes to develop ideas about what it means for parts of a whole to be equal.

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Connection to CCSS

2.G.3

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Agenda

Activity	Time	Description/Prompt	Materials
Launch	5–10 min	Show students the Partitioned Square image and ask, Are the parts equal? How do you know? Surface different ideas about what it means to be equal.	Partitioned Square sheet, to display
Explore	20–25 min	Provide partners with the Partitioned Squares sheet. Partners explore t...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Dedication
• Introduction
• Note on Materials
• Activities for Building Norms
• BIG IDEA 1: Partitioning Shapes into Equal Parts
• BIG IDEA 2: Making and Using Equal Groups
• BIG IDEA 3: What Is 100?
• BIG IDEA 4: Composing and Decomposing Numbers
• BIG IDEA 5: Using Patterns in Place Value
• BIG IDEA 6: Thinking on the Number Line
• BIG IDEA 7: Rulers and Clocks Are Number Lines
• BIG IDEA 8: Using Units to Estimate
• BIG IDEA 9: Using Data to Visualize and Wonder about Our World
• Appendix
• About the Authors
• Acknowledgments
• Index
• End User License Agreement