Mindset Mathematics

Visualizing and Investigating Big Ideas, Grade 5

Jo Boaler, Jen Munson, Cathy Williams

• Publisher: Jossey-Bass
• Year: 2018
• Print ISBN: 9781119358718
• eBook ISBN: 9781119358763
• 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 fifth-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 mathematics 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
Thinking in Cubes

The new brain science shows that five different pathways are involved when people think about mathematical ideas, and two of these are visual. When we make mathematics visual for students, we help them learn and hold ideas in powerful ways in their brains, as the introduction to this book explains. Similarly, we now know that movement really helps with mathematical ideas and is important for brain development. When students move with mathematics, it means that the mathematical ideas are held in the sensory-motor portions of the brain, which helps students understand the ideas powerfully. We see evidence of people holding mathematical ideas in these parts of their brains when they gesture to illustrate an idea; when people talk about circles, for example, they often draw a circle in the air. This big idea gives students opportunities to touch and feel mathematical ideas, and that is meaningful to students of any age.

In our Visualize activity, students will build with cubes and develop connections between two- and three-dimensional representations of solids. They will be asked to think about the outside and inside of cubes, which is important geometric thinking. As they physically model and also draw, they will build significant brain connections.

In the Play activity, students will construct cities of cubes that match views that we give them, again using brain pathways that will develop mathematical thinking. Students also will build their own cities, which will be engaging and exciting for them, enhancing the learning potential of the activities. As students think visually and also bring in numerical thinking, their brains will develop pathways between the areas that are used for these different types of thinking.

In our Investigate activity, students will again have the opportunity to feel cubes and consider their size physically and with numbers, encouraging brain connections. They also get to work with some constraints that will guide their thinking and learning. Students will be asked to investigate the volume of rectangular solids by packing little boxes into larger boxes of their own design. Any time that students are asked to bring their own ideas into mathematics, such as when they make their own designs, they are working with agency, which will help them enjoy mathematics and also see it as an active subject that they should think deeply about. When students work with agency, their work is closer to that of a mathematician, and inviting students to combine their own ideas with formal mathematical ideas is a really worthwhile goal. The activities that make up this big idea provide plenty of opportunities for students to combine their own thinking with major mathematical ideas and principles.

Jo Boaler

Solids, Inside and Out

Snapshot

Students build connections between two- and three-dimensional representations of solids by using views of a rectangular solid to construct a model with cubes. Students investigate what the inside looks like and compare results.

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Connection to CCSS

5.MD.3

5.MD.4

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Agenda

Activity	Time	Description/Prompt	Materials
Launch	5 min	Show students the two-dimensional views of a rectangular solid constructed out of 60 cubes. Challenge students to build this solid.	Rectangular Solids Sheet, to display for the class
Explore	30 min	Partnerships try to build a rectangular solid from 60 cubes so that it matches the views provided. Students then consider what the inside looks like and figure out how to construct and draw a model of the cubes that cannot be seen.	Rectangular Solids Sheets, one per partnership Snap or multilink cubes for each partnership: 15 each of red, green, yellow, and blue Drawing Solids Sheet, one per partnership
Discuss	20 min	Students compare their results and discuss how they used the views to construct the solid. Students discuss the differences between their models of the inside.
Extend	30–60 min	Partnerships construct their own rectangular solid puzzles and swap with other groups to solve.	Drawing Solids Sheet, at least one per partnership Cubes Colors Baskets, trays, or bags for students' puzzles

To the Teacher

Students often struggle when moving between two-dimensional representations of solids objects, like the ones shown in Figure 1.1, and three-dimensional representations. Two-dimensional drawings of solids force us to imagine the parts we cannot see, an...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Introduction
• Activities for Building Norms
• Big Idea 1: Thinking in Cubes
• Big Idea 2: Estimating with Fractions
• Big Idea 3: Using Fraction Equivalence
• Big Idea 4: Exploring the Coordinate Plane
• Big Idea 5: Seeing and Connecting Patterns across Representations
• Big Idea 6: Understanding Fraction Multiplication Visually
• Big Idea 7: What Does It Mean to Divide Fractions?
• Big Idea 8: Thinking in Powers of 10
• Big Idea 9: Using Numbers and Symbols Flexibly
• Appendix
• About the Authors
• Acknowledgments
• Index
• End User License Agreement