Mathematical Mindsets: The Highlights {Part 5}

This book I would say has changed my thoughts on math, teaching, and teaching math more than any other I've read in my seven year career. I will recommend it and link it forever. I will have to post my highlighted notes from it in several posts because no one would ever scroll through all of it otherwise! There is just so much to process and that I will need to read over and over again- so many opportunities for growth and change!

It's only $10.71 for the paperback and $7.99 for the Kindle version. You NEED this book. But until you get your own, this should be enough to make you want more.


Part 1 {here}
Part 2 {here}
Part 3{here}
Part 4 {here}

Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching
Jo Boaler

Chapter 9: Teaching Mathematics for a Growth Mindset

I believe in every one of them, that there is no such thing as a math brain or a math gene, and that I expect all of them to achieve at the highest levels. I love mistakes. Every time they make a mistake their brain grows. Failure and struggle do not mean that they cannot do math—these are the most important parts of math and learning. I don't value students' working quickly; I value their working in depth, creating interesting pathways and representations. I love student questions and will put these onto posters that I hang on the walls for the whole class to think about.

Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss, and critique.

To run a participation quiz, choose a task for students to work on in groups, then show them the ways of working that you value.

Once you have shown these to students, you can start them working. As they work together in groups, walk around the room watching group behavior, writing down comments.

As you circulate and take notes, quote students' actual words when they are noteworthy.

But it is even more important to communicate positive beliefs and expectations to students who are slow, appear unmotivated, or struggle. It is also important to realize that the speed at which students appear to grasp concepts is not indicative of their mathematics potential (Schwartz, 2001).

The most productive classrooms are those in which students work on complex problems, are encouraged to take risks, and can struggle and fail and still feel good about working on hard problems.

We must also resist valuing “effortless achievement”—praising students who are fast with math. Instead, we should value persistence and hard thinking.

When students fail and struggle it does not mean anything about their math potential; it means that their brains are growing, synapses are firing, and new pathways are being developed that will make them stronger in the future.

Instead of saying “You are so smart,” it is fine to say to students something like “It's great that you have learned that,” or “I love how you are thinking about the problem.”

My undergraduates have really worked on this and now praise people for having good thinking and for being accomplished, learned, hard working, and persistent.

There is always some logic in students' thinking, and it is good to find it, not so that we avoid the “failure” idea, but so that we honor students' thinking.

I recently read about a second-grade teacher, Nadia Boria, who offers this response to students when they ask for help: “Let's think about this for a minute. Do you want my brain to grow or do you want to grow your brain today?” (Frazier, 2015).

Mathematics tasks should offer plenty of space for learning. Instead of requiring that students simply give an answer, they should give students the opportunity to explore, create, and grow.

Open up math tasks:

  • Instead of asking students to answer the question 1/2 divided by 1/4, ask them to make a conjecture about the answer to 1/2 divided by 1/4 and make sense of their answer, including a visual representation of the solution. As I described in Chapter Five , when Cathy Humphreys asked students to solve 1 ÷ 2/3 she started by saying, “You may know a rule for solving this question, but the rule doesn't matter today, I want you to make sense of your answer, to explain why your solution makes sense.” 
  • Instead of asking students to simplify 1/3(2x + 15) + 8, a common problem given in algebra class, ask students to find all the ways they can represent that are equivalent.
  • Instead of asking students how many squares are in the 100th case, ask them how they see the pattern growing, and to use that understanding to generalize to the 100th case

Ask students to discuss:

  • Ways of seeing the mathematics 
  • Ways of representing ideas 
  • The different pathways through the problem and strategies 
  • The different methods used: “Why did you choose those methods? How do they work?”

Encourage students to propose different methods to solve problems and then ask them to draw connections between methods, discussing for example, how they are similar and different or why one method may be used and not another. This could be done with methods used to solve number problems, such as those shown in Figure 5.1 , in Chapter Five .

Ask students to draw connections between concepts in mathematics when working on problems.

In my own teaching of mathematics, I encourage student creativity by posing interesting challenges and valuing students' thinking. I tell students I am not concerned about their finishing math problems quickly; what I really like to see is an interesting representation of ideas, or a creative method or solution. When I introduce mathematics to students in this way, they always surprise me with their creative thinking.

Teachers can encourage students to use intuition with any math problem simply by asking them what they think would work, before they are taught a method.

The teacher provocatively took the students' ideas and made incorrect statements for the students to challenge, and the class considered together all of the possible relationships of angles that preserve the definitions.

In the lesson in China, the teacher did not ask complete-this-sentence questions; she listened to students' ideas and made provocative statements in relation to their ideas that pushed forward their understanding. Her statements caused the students to respond with conjectures and reasons, thinking about the relationships between different angles.

If you give students the opportunity to extend problems, they will almost always come up with creative and rich opportunities to explore mathematics in depth, and that is a very worthwhile thing for them to do.

He points out that students currently spend 80% of the time they spend in math classrooms performing calculations, when they should instead be working on the other three parts of mathematics—setting up models, refining them, and using them to solve real problems.

In my own teaching experience, when I have asked students in classrooms to consider a situation and pose their own question, they have become instantly engaged, excited to draw on their own thinking and ideas. This is an idea for math classrooms that is very easy to implement and needs to be used only some of the time. Students should be able to experience this in school so that they are prepared to use it later in their mathematical lives.

It is so important that employees describe their mathematical pathways to others, in teams, because others can then use those pathways in their own work and investigations and can also see if there are errors in thinking or logic. This is the core of mathematical work; it is called reasoning.

When students act as a skeptic, they get an opportunity to question other students without having to take on the role of someone who doesn't understand.


  1. I wasn't even halfway through your post when I went on kindle and bought the book. Looking forward to binge reading it before the school year starts!

    1. Worth every penny! Better get your highlighting finger ready :)