Sunday, 8 April 2018

Mathematical Mindsets Book Study Questions

This week, staff at Holly Meadows are embarking on a book study on "Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching" by Jo Boaler.  

How can we help all children know that they have vast mathematics potential? How can teachers instruct in a way that brings this belief to life? Jo Boaler- Stanford researcher, professor of math education, and expert on math learning- has studied why students don't like math and often fail in math classes. She's followed thousands of students through school to study how they learn and to find the most effective ways to unleash the math potential in all students. Her work has identified gap between what research has shown to work in teaching math and what happens in schools and at home. Boaler translates Carol Dweck's work around 'mindset' into math teaching, showing how students can go from self-doubt to strong self-confidence, which is important to math learning. The message behind this book pushes us to understand the power of mindsets in relation to math learning and how our approach as educators can help students realize the joys of learning and understanding math and improve outcomes for students. (adapted from Boaler, 2016)

I will be posting our weekly questions here as a central location for staff to access the weekly content,  as a forum for them to make comments and to share with others.

Chapter 1 The Brain and Mathematics Learning 
(April 12 reading and discussion)

Chapter Summary

As we encounter new learning, synaptic connections within the brain create lasting pathways and connections.  The more complex and challenging the learning, the deeper these connections will be. This is the foundation for our work with students, providing opportunities for mistakes to be made, feedback to be provided and learning to occur.

Chapter Questions
  • What flaws can be pointed out around the idea of being "developmentally ready" for math? 
  • What vital math experiences are some students lacking in schools? 
  • How can educators work with parents to promote the use of language that develops growth mindset in learning? 

Chapter 2 The Power of Mistakes and Struggle 
(April 12 reading and discussion) 

Chapter Summary

"Every time a student makes a mistake in math, they grow a synapse" (Boaler, p. 10)

Making a mistake is a good thing. Research has found that student's brains respond with increased activity when they make mistakes compared to when their answers are correct. This truth is even greater for individuals that display a growth mindset.

Chapter Questions

  • How as educators, can we create a safe and trusting environment where mistakes are acknowledged, valued, celebrating and used as a foundation for new learning? 
  • How can we encourage our students to persevere through challenges? 
  • Which strategy for valuing mistakes resonates with you? What are some other ways to show the value of mistakes?

Additional Reading- The man who's fighting girls' "mathematophobia"

Chapter 3 The Creativity and Beauty in Mathematics 
(April 19 reading and discussion)

Chapter Summary

"Math is too much answer time and not enough learning time." 

This chapter discusses the assertion that math is different from other subjects and that this is due to some widely  held misconceptions about math:
  • that math is only about right and wrong answers
  • that math is all about numbers 
  • that being good at math means being fast at math 
Many parents  and even teachers had negative math experiences in school and struggled to learn the subject but still believe that the same teaching practices must continue because that is the only way they know. 

This chapter shares studies that indicate that when students are given opportunities to pose real math problems, transition from real world to math models to perform calculations, communicate their thinking and then return to the real world model to see if their question was answered, they become  more deeply engaged and  perform at higher levels.

Additional Reading- Yes, I Can! Paying Attention to Well-Being in the Mathematics Classroom (Ministry of Education Capacity Building Document) 

Chapter Questions
  • What aspects of mathematics do you find to be creative or beautiful? 
  • What messages do we send about math when we give written diagnostic assessments the first day? What are some other ways we can gather diagnostic data from our students? 
  • Many students  only think of math simply as a series of answers. What sort of changes can we make in our classrooms to encourage the important mathematical aspects of communication of deep thinking and making connections? 
  • Mathematics is not solely about calculation.  Looking at the chart on page 27 based on the work of Wolfram, how much time do students in your class spend on stage 3 (calculating) vs. the other stages?  How might we change this?

Chapter 4 Creating Mathematical Mindsets: The Importance of Flexibility with Numbers  
(April 19 reading and discussion) 

Chapter Summary

As children and infants, we engage with mathematical play as a natural part of development.  Stacking blocks, noticing and creating patterns and exploring and experimenting with the world around us.  Traditionally in North America, when students begin school, this natural curiosity is replaced by memorization of a dry set of methods that are associated with dislike of the subject which fails to make meaningful connections.  To truly and meaningfully engage with math, students should not take a single method and practice it over and over again.  This doesn't teach them the knowledge, concepts and relationships that are the building blocks of mathematical thinking.  Rather, students need to be given opportunities to work on mathematics as a whole, considering a variety of ideas and applications, making connections, solving real problems and communicating reasoning.  The most important start we can give to our students is to allow and encourage them to explore, experiment  and play with  numbers, shapes and patterns, communicating their thoughts, questions and ideas throughout the process.

Chapter Questions
  • Research by Gray and Tall (p. 35) showed that students who simply followed pre-taught memorized algorithms (vs. those who experimented with different methods and used math more flexibly) achieved lower on math tasks.  What are your beliefs on facts vs. flexibility and how do our actions in class align (or not align) with these thoughts?
  • Our FDK program in Ontario has already made many changes to align with the premises in this book.  What lessons might educators of other grades learn from this revised approach to teaching and learning? 
  • Consider the following quote, "There are some math facts that are good to memorize, but students can learn math facts and commit them to memory through conceptual engagement with math. Unfortunately, teachers and parents think that because some areas of math are factual, such as number facts, they need to be learned through mindless practice and speed drills.  It is this approach to early learning about numbers that causes damage to students, makes them think that being successful at math is about  recalling facts at speed and pushed them into a pathway that works against their development of a mathematical mindset." (p. 37)  What are some tasks you have used in your classroom recently that encourage flexible thinking vs. memorization of facts?  How are these tasks different than those you may have used at another point in your career and what prompted you to make the change in your practice? 

Additional Reading and Next Step

As we read the next five chapters, consider this OAME link on creating rich math assessment tasks and be prepared to share some tasks you have created and changes in practice you have adopted as a result of our book study work.

Chapter 5 Rich Mathematical Tasks
(April 26 reading and discussion) 

Chapter Summary

Teachers have a great influence on students. They have the power to create excitement in the classroom and share positive messaging that supports the development of students' feelings about school and the  subjects they are learning. The types of tasks, materials and resources that teachers  provide their students to engage with can mean the difference between inspiration and disengagement in their students.

This chapter suggests six questions that support the creation of rich mathematical tasks:

  • Can you open the task to encourage multiple pathways, methods and representations?
  • Can you make it an inquiry task?
  • Can you ask the problem before teaching the method? 
  • Can you add a visual component? 
  • Can you make the task low floor, high ceiling?
  • Can you add the requirements to convince and reason? 

Chapter Questions
  • Discuss a rick task that you have used recently in your class.  What about the task made it a rich task?   How did it go and what might you have done differently?
Additional Resource

Here is a link to a conversation tool for worthwhile math tasks.  When working with grade partners to co-plan, this is a fantastic tool to support discussion around tasks that support the purpose of your lesson.

Chapter 6 Mathematics and the Path to Equity 
(April 26 reading and discussion) 

Chapter Summary

"Mathematics has the greatest and most indefensible differences in achievement and participation for students of different ethnicities, genders and socioeconomic levels of any subject taught" (p. 93)

To achieve higher levels of achievement and more equitable outcomes in math, we must recognize the elitist role that math plays in society.  Society often puts math on an intellectual pedestal, believing that people that can calculate more quickly are somehow more intelligent than others.  We need to challenge this thinking in order to open the study of math up to all students and truly empower all students to succeed.

This chapter offers six equitable strategies to make math more inclusive:
  • Offer all students high level content
  • Work to change ideas about who can achieve in math
  • Encourage students to think deeply about math 
  • Teach students to work together 
  • Give additional encouragement to groups that have been traditionally marginalized 
  • Change the nature of homework 

Chapter Questions
  • What inequities are present in our math classrooms and how can we work to overcome these?
  • Based on the discussion of homework on pages 107-109, what are your thoughts about homework?

Chapter 7 From Tracking to Growth Mindset Grouping 
(May 3 reading and discussion)

Chapter Summary

If students spend time in classes where they are given access to high-level content, they achieve at higher levels. Research shows that the brain has the capacity to grow and rewire at any time.  Streaming students into higher and lower level classes not only delivers a fixed mindset message, but  denies the opportunity to learn at a high level to all students and limits their achievement.   Additional studies show that countries with the highest and most equal achievement in math (Korea, China and Finland) reject the concept of ability grouping.  Conversely, countries that have the strongest correlation between achievement and socioeconomic status have a tradition of streaming.

Additional Resource

During last weeks meeting, we discussed Graham Fletcher's "Progressions" videos. Here is a link to the videos. I encourage you to take a minute and watch one or two that would  be appropriate to the grade you  teach.

Chapter 8 Assessment for Growth Mindset 
(May 3 reading and discussion)

Chapter Summary

We want students to be engaged and intrinsically motivated in their learning.  Since math is often taught as a performance subject, students who are most motivated are those who are extrinsically motivated, getting incentive from high grades.  The result is that those who are already achieving at high levels continue to be motivated, but the rest of the students are de-motivated.

Shifting from a grade to a feedback focused classroom is an amazing gift that teachers can give to students, freeing them from defining themselves as solely a grade and shifting the focus on what they can do to improve. This shift toward giving the students the information they need to learn well, accompanied with growth mindset messaging has the power to dramatically change the classroom environment and students attitudes toward and achievement in math.

Some strategies suggested for supporting students to develop self-awareness include:
  • Self assessment
  • Peer assessment 
  • Traffic lighting (p. 159)
  • Jigsaw groups 
  • Exit tickets 
  • Online forms 
  • Doodling
  • Students designing their own questions

Task for May 3 (in lieu of Chapter questions for Chapters 7 and 8) 

Bring a "traditional" (textbook task) or closed task at your grade level to the meeting.  We will work together using the conversation tool  to discuss how the task might be improved and discuss how we might we assess each of these tasks.

Chapter 9 Teaching Mathematics for a Growth Mindset 
(May 10 reading and discussion)

Chapter Summary

This chapter discusses the importance of setting norms in the classroom that clearly place value on the development of growth mindset.  The 7 norms discussed in this chapter include:

  • Everyone can learn math to the highest levels.  Students are encouraged to believe in themselves and work hard. 
  • Mistakes are valuable and help your brain grow. It is good to struggle and make mistakes.
  • Questions are really important. We should always be asking and answering questions.
  • Math is about creativity and making sense.  Math is about visualizing patterns and creating solution paths that others can see, discuss and critique. 
  • Math is about connecting and communicating. Represent math in different forms- words, pictures, numbers, graphs, equations...the possibilities are endless!
  • Depth is more important than speed.
  • Math is about learning, not performing. 

Chapter Questions

  • When working with students, how can we ensure that we are not  doing the mathematical thinking for them? 
  • Which one of the norms discussed in this chapter resonates most with you? Why?
  • The conclusion of this book stated, "Now it is time for you to invite others onto the pathways you have learned..." (p. 208).  Moving forward, how might we share the messaging from this book with other staff members? 

Additional Resources 
For those that want to delve deeper into this topic, here is a link to some fantastic "Math Mindset" resources from YouCubed (including posters of the norms discussed in this chapter).  Additionally, here is another link from YouCubed including a teaching guide to the 5 Mathematical Mindset Practices and videos to accompany each.


Boaler, J., (2016). Mathematical Mindsets: Unleashing students potential thorough creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.  

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