Mindset Mathematics

Visualizing and Investigating Big Ideas, Grade 4

Jo Boaler, Jen Munson, Cathy Williams

• Publisher: Jossey-Bass
• Year: 2017
• Print ISBN: 9781119358800
• eBook ISBN: 9781119358817
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

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 at the first-grade level 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 engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, 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 - anyone can learn mathematics to high levels.
• Mistakes, struggle and challenge are the most important times for brain growth.
• Speed is unimportant 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
Seeing Patterns inside Numbers

Numbers make up our world, and they are used throughout our lives, whatever our age, job, or level of interest. But many people develop a narrow relationship with numbers, seeing them as something to use in calculations, rather than as a fascinating set of ideas that can enrich their world. Our first big idea invites students to become captivated by numbers and to get to know numbers deeply. What is enchanting about numbers is that they are all made up of different arrangements, have different factors, can be seen differently, and have their own intricate system to be explored.

When we first came across Brent Vorgey's number visual (Figure 1.1), we were enthralled, as we immediately saw the creativity, beauty, and insights that the visual representations revealed.

Figure 1.1

In our Visualize activity, we invite students to explore this depiction of numbers and to see what patterns are uncovered by the visual representations. We invite them to see what primes look like and to see the different factors inside numbers. We invite them to investigate patterns among the numbers, seeing what their positioning on the diagram reveals. Also, we invite students to see numbers visually and to develop a realization that numbers contain all sorts of information that make them different from each other, special, and interesting.

In the Play activity, we extend students' time with the number visuals in a more playful setting. Students play a game with the number visual page as a game board, and move between visual and numerical representations. This, as with the other two tasks in this big idea, encourages important connections between different areas of the brain.

In the Investigate activity, we invite students to think carefully about number flexibility. One of the ways that numbers are different from one another is the number of factors they have and the degree of flexibility they give us when using them. For example, 24 is a very flexible number, as it can be broken up in all sorts of different ways. This makes it a useful number for packaging, for designing, and for measuring time. In this activity, we invite students to give value to different numbers according to their flexibility, helping them develop an appreciation for these numbers. The activity also invites students to make equal groups and gives teachers an opportunity to discuss whether students are thinking additively or multiplicatively and what those differences mean.

All three activities give students an opportunity to develop new insights into the numbers that they will use for the rest of their lives.

Brain science tells us that when students are engaging with numbers as symbols, such as the numeral 4, and with numbers as visuals, as shown in Figure 1.2, they are connecting between different areas of the brain, and such connections are critical for mathematics learning and achievement. The activities in this big idea will invite a lot of brain connecting, with students developing pathways that will help them as they go forward in their mathematical careers.

Jo Boaler

Figure 1.2

Visualizing Numbers

Snapshot

In this activity, students work with the number visual page to explore the patterns that they can see inside of numbers. In this activity, we open the door to understanding factors, multiples, and primes, as well as other number patterns.

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Connection to CCSS

4.OA.4

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Agenda

Activity	Time	Description/Prompt	Materials
Launch	5 min	Generate multiple ways that numbers can be represented and introduce the number visual page.	Number visual page reproduced for students and one to display
Explore	20+min	Students look for patterns inside the number visual page and color-code them.	Colors f...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Introduction
• Activities for Building Norms
• Big Idea 1: Seeing Patterns inside Numbers
• Big Idea 2: Building and Designing with Shapes and Angles
• Big Idea 3: Making and Naming Number Patterns
• Big Idea 4: Units Are a Relationship
• Big Idea 5: Modeling with Unit Fractions
• Big Idea 6: Exploring Fraction Equivalence
• Big Idea 7: Illustrating Multiplication and Division
• Big Idea 8: Using Operations Flexibly
• Big Idea 9: What Is a Decimal?
• Appendix
• About the Authors
• Acknowledgments
• Index
• End User License Agreement