Mindset Mathematics: Visualizing and Investigating Big Ideas, Grade 6

Jo Boaler, Jen Munson, Cathy Williams

• Publisher: Jossey-Bass
• Year: 2019
• ISBN:
  9781119358671
• Edition: 1
• Topic: Bildung
• Subtopic: Mathematik unterrichten

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 sixth-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
Taking Apart Prisms and Polygons

The mathematical concepts at the heart of this big idea are area and volume. Although these ideas call for students to learn through objects, holding them in their hands and exploring with them, many students are asked only to memorize formulas and so do not develop an understanding of area, volume, or the differences between them. In our Youcubed summer camp, we gave the students an activity with sugar cubes; they were invited to build different sized larger cubes with the sugar cubes. When we interviewed the students a year after they attended the camp, one of the boys told us that he now thinks about the sugar cubes every time he learns about volume, as they gave him a physical representation of a 1 × 1 × 1 cube. His experience holding the cubes and building with them contributed to a deep understanding of volume that was powerful and enduring for him. In our Investigate activity, we invite the students to build with very similar cubes—snap cubes. In all of the activities, students are asked to build with two‐ and three‐dimensional shapes.

In the Visualize activity, we ask students to find different ways to take apart two‐dimensional complex shapes as they work to find area. We have used shapes that require students to reason about how to determine the area when its boundary doesn’t fit exactly on a square grid. As students reason through determining the area, they also need to break the shape into other shapes they are familiar with, such as triangles and rectangles. This type of thinking is foundational for later work in geometry and calculus.

In the Play activity, we ask students to determine the area of a complex piece of artwork that is made from different polygons. We like to connect mathematics and art in our books, as it is important for students to see that mathematics can be beautiful, creative, and applied to all sorts of different real‐world situations. Because of the uneven border of the shape, students will need to come up with different creative ways to find the area. This lesson also provides students opportunities to discuss estimation. In studies of mathematics in the world, estimation has been found to be one of the most used concepts and one that is undertaught in schools. We are sure your students will enjoy making their own piece of mathematical art.

In the Investigate activity, students build off the Visualize activity as they imagine complex two dimensional shapes as the bases of buildings. Students are asked to use multilink cubes to construct the buildings, giving them an important opportunity to understand volume. In this activity, we also provide an opportunity for students to work with rational numbers. Students in sixth grade are learning to expand their number system, yet questions in traditional textbooks often ask the students only to work with whole numbers. We have provided problems that use rational numbers, fractions, to support students’ growth in understanding of both volume and rational numbers. Students are asked to visualize fractions of multilink cubes as they work to determine volume and connect the idea of volume to the idea of area for the shapes that they worked with in the Visualize activity. Students often have trouble understanding the difference between area and volume because they have not had enough experience spending time connecting their numbers with visual two‐ and three‐dimensional models. We hope that this set of activities will provide time for fun and challenge together, and that students will get an enjoyable opportunity to struggle and to use their creativity in finding different ways to see and solve problems.

Jo Boaler

How Big Is the Footprint?

Snapshot

Students develop methods for finding the area of irregular polygons by exploring ways to decompose two‐dimensional figures and reason about partial square units.

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Connection to CCSS

6.G.1

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Agenda

Activity	Time	Description/Prompt	Materials
Launch	5–10 min	Show students the Quadrilateral in Question sheet on the document camera and ask, How might we find t...

Table of contents

• Cover
• Table of Contents
• Introduction
• Note on Materials
• Activities for Building Norms
• BIG IDEA 1: Taking Apart Prisms and Polygons
• BIG IDEA 2: Folding and Unfolding Objects
• BIG IDEA 3: Expanding the Number Line
• BIG IDEA 4: Finding and Using Unit Rates
• BIG IDEA 5: Reasoning with Proportions
• BIG IDEA 6: Visualizing the Center and Spread of Data
• BIG IDEA 7: Using Symbols to Describe the World
• BIG IDEA 8: Generalizing
• Appendix
• About the Authors
• Acknowledgments
• Index
• End User License Agreement