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Chapter Introduction

A Math Journey

Visual Strategies for Understanding Mathematics

Think about a world without math. What would it look like? How would it be different?

Math - What is it good for?

Ask someone about their experience learning math and chances are you’ll hear more negative responses than positive. Many of us struggle with math and have sad memories of not understanding math in school. 

Some people believe that only some are blessed with a math brain–you either have it or you don’t.

Two human heads: one with math in its brain and one without.

Not true! In fact, from the moment we’re born, we all have the ability to learn and enjoy math, whether we become mathematicians or not.

This website is a journey through the landscape of ideas that build number sense and lead to deeper understanding of mathematics. During our travels, you’ll be asked to look at math in new ways and possibly more deeply than you have before. You’ll see how and why numbers work and how they’re used to solve everyday problems.

Visual Math

From counting simple objects to identifying patterns to graphing sets of data, math is all about seeing and exploring the relationships of numbers and structures.

Look at the following arrangements of tiles. What pattern do you see?

1: 3 blocks 2: 8 blocks 3: 15 blocks

What will the 4th arrangement look like? How many tiles will it have? Use the tiles below to build the 4th arrangement.

Engaging the Curious Mind

Problems like the tile pattern above encourage us to play with math. They can be explored by people of all ability levels and solved in a variety of ways.

Learning math is rarely an independent activity. When we listen to the ideas of others, we strengthen our own understanding. For instance, how does your strategy for making the 4th tile arrangement compare to the following?

A large square with one missing

A smaller square with some more on the top and side

A rectangle with some tile on the side

Learning math is also more productive and enjoyable when it inspires a sense of wonder. While building the 4th arrangement, did you find yourself thinking about questions like these?

  • How many tiles will be in the 5th arrangement?
  • Which equation will help me predict the total number of tiles in any arrangement?
  • Can I make another tile pattern with the same number of tiles in each arrangement that looks entirely different?

What questions do you wonder about?

Understanding How We Learn

In recent years, brain research has expanded our understanding of how we learn in two ways: 

  1. Intelligence is far less static than neurologists once thought. Our brains are constantly changing—growing, developing, rewiring, and recovering—in response to our experiences and surroundings.
  2. What one believes about their own ability to learn, their mindset, has a tremendous impact on their learning.

One's approach to learning exists on a spectrum between two different mindsets: Fixed and Growth.

People who act with a fixed mindset think their potential is determined at birth and their abilities are set at fixed limits. They’re either skilled at something or not.

People who work from a growth mindset believe they can learn just about anything and their potential is limitless. They see that learning anything is possible through effort and persistence.

An illustration of two head profiles; one with a lock inside and one with a plant growing.
Fixed vs. Growth Mindset

Explore statements those with fixed and growth mindsets might have about mathematics

This math looks tough. I think I’ve found my limit.

This math looks tough but this part looks familiar. I’ll start there.

I really don’t want to make a mistake and look silly.

A mistake helps me learn.

As long as I get the right answer, I don’t care about the strategies that others use.

The ideas of others make me a stronger mathematician.

Math is fun when I get the right answer without too much work.

Math is fun when it's difficult and I stretch what I know and can do.

I just don’t have a math mind and I’m ok with that.

Sometimes I have to work really hard, but my math mind is continually growing.

The good news is that mindsets aren’t permanent. You can explore new ways to learn math. Not being a math person today doesn’t mean you can’t be a math person in the future.  

We hope this journey of numbers engages your sense of wonder and helps you see that math can be visual, flexible, and enjoyable. We encourage you to travel with us using a growth mindset, and along the way, discover a new sense of numbers and desire to learn more about mathematics in our world.

Learn More

The Stereotypes That Distort How Americans Teach and Learn Math

When it comes to math, does speed matter? Is there such a thing as a "math person"? Is meaning more important than memorization? Read an Atlantic article by Jo Boaler to learn more about common generalizations and misconceptions when it comes to learning math.

Most Americans Actually Liked K-12 Math Class

A majority of U.S. adults (58%) enjoyed math classes in grades K-12, and most who liked them say the subject matter was the main reason.

Positive Attitude Toward Math Predicts Math Achievement in Kids

A Stanford study finds that a positive attitude toward math boosts the brain’s memory center and predicts math performance independent of factors such as a child’s IQ.

Mindset Works: The Impact of a Growth Mindset

This website is a rich resource for learning about growth mindset and how one can “change” their mindset.