discussion & resources

FOR REFLECTION & DISCUSSION

These first three questions are ones you may wish to think about before reading the book, and then return to after you finish it, to see how your answers compare.

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1. What is mathematics? How would you describe it to a friend, in a sentence or two? What do you feel is the purpose of learning mathematics, for yourself or others?​

2. What connections do you see between doing mathematics and being human?​

3. Describe any virtues you have acquired as a result of doing mathematics. (Think of virtues as aspects of character that mathematics might build, such as habits of mind, that shape the way you approach life.)
    

4. Have you ever compared yourself to others mathematically? Describe how that made you feel. Was it motivating or discouraging?

5. ​Should mathematical opportunities be available for all who want them? Are they available for all who want them? Discuss.

6. What value is there is studying math if you'll never use what you're learning?

7. ​Why is "leaving math to the mathematicians" a bad idea?

8. ​What does "mathematical affection" look like for you? How might mathematical affection manifest itself in others differently than in you?

9. ​"Every single one of us is, whether we realize it or not, a teacher of math." In what ways is this statement true for you?​

10. Math is often taught as a bunch of procedures, without sense-making or understanding. Why do you think that is?​

11. "The skills society needs from math may change, but the virtues needed from math will not." What are some examples of math skills that were important 30 years ago but are less important to learn now, and why are they less important now?

12. Teachers: how can you communicate to your students that you are not just helping them learn skills, but that you are also helping them to build virtues, like creativity and persistence? How can your assessments reflect the importance of those virtues?​

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RESOURCES

• Readers may be interested in the original speech on which this book is based. The audience for the speech was a group of mathematicians rather than the general public.

• A recent call for change in mathematics education is worth reading closely: ​A Common Vision for Undergraduate Mathematical Sciences Programs in 2025, from the Mathematical Association of America (2015).​

• Ben Braun has compiled a list of readings that make the ideas in this book actionable.

• The book's endnotes also contain many related readings.

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​FOR REFLECTION & DISCUSSION

1. Think of a time when you were captivated by exploring something (e.g., a location, an idea, a game). What analogies can you draw between doing math and doing this exploration?​

2. The rings of Saturn exhibit a pattern that possesses a mathematical explanation. What other patterns have you encountered that cause you to think they might have mathematical explanations?​ Why so?

3. "Even wrong ideas soften the soil in which good ideas can grow." Describe some examples from your own experience that embody this truth.

4. Consider this statement: “The wayfinders were mathematical explorers of their society, using attentive study, logical reasoning, and spatial intuition to solve the problems they encountered in their cultural moment.” Choose any cultural practice and reflect on ways that mathematical thinking might present itself in that practice.

5. Teachers: ​What are some ways that you can train your students to expect enchantment?

6. ​Teachers: What are some ways you can build exploration into how you teach and into the homework you give students?  For instance, can you change a 'compute' problem to an exploratory problem?​

7. Teachers: Math problems often have multiple solutions. In what ways can you be flexible, and allow students to find their own path to a solution?

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RESOURCES

• Exploratory learning is the heart of inquiry-based learning. The Academy of Inquiry Based Learning provides resources and training for teachers at all levels to facilitate this type of learning.

• Tracy Johnston Zager helps K-8 teachers see and teach math the way professional mathematicians think about it: full of inquiry, exploration, taking risks, asking good questions, in Becoming The Math Teacher You Wish You'd Had.

• Everyone should read Paul Lockhart’s “A Mathematician’s Lament.” There is also a book.

• Achi and other games from Africa can be found in physical form in the MIND Research Institute’s South of the Sahara game box. Claudia Zaslavsky has two books: Math Games & Activities from Around the World and a sequel.

• The Art of Problem-Solving website is an excellent resource for interesting problems. 

FOR REFLECTION & DISCUSSION​

1. “Mathematical ideas are metaphors.” Reflect on one mathematical idea that you’ve now seen in multiple situations, and how the meaning of that idea was enhanced in each encounter.

2. The chapter draws a similarity between the growth of the meaning of words in poetry and the growth of meaning of ideas in math. What other similarities do you see between math and poetry?

3. Teachers: reflect on Poincaré's statement that "an accumulation of facts is no more a science than a heap of stones is a house." What do you take away from this analogy for how you teach mathematics?

4. Teachers: is story-building part of your teaching? How do you show students connections between ideas?

5. How does abstraction enrich the meaning of an idea? Describe one example from your own experience.

6. How does the experience of joy (from successful attempts at sense-making) cultivate persistence?

7. “Mathematics is... the art of engaging the meaning of patterns.” Consider this statement in light of a scientific discovery in which mathematics played a part. In what sense is it an art?

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RESOURCES

• Mathematical ideas are not isolated concepts. For instance, you can see how the Common Core state standards are related to one another in this coherence map. There's also a set of progressions documents from the University of Arizona that show the development of topics across grade levels.

• Bob Moses' book Radical Equations: Civil Rights from Mississippi to the Algebra Project is important reading for any math educator.

• I really enjoyed thinking through the insights found in William Byers' book: How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics.

FOR REFLECTION & DISCUSSION​

1. Think of an activity that you associate with play, i.e., that feels playful to you. (Try to think of something that isn't a game.) Make a list of all the characteristics of that activity that makes it feel playful. Are there mathematical activities that share those characteristics?

2. Teachers: inquiry and justification are both essential parts of the process of doing math, and play is an important component of both. What activities do you do in your class that encourage inquiry or justification? 

3. Some people seem to have patience and hopefulness when trying to solve a problem, and they will persist for a long time in thinking about it. Others seem to give up quickly. How does math play build hopefulness and patience? Compare this to the discipline of learning a sport.

4. Describe how hopefulness (that problems can be solved) and (patience in waiting for a solution) might carry over to non-mathematical problems we face. 

5. Math play “asks you to change perspective, to look at a problem from different viewpoints.” In what ways is this virtue useful in life?

6. In what ways does the quest for mathematical achievement (eg, chasing external recognition/validation) skew or limit the way we experience math?

7. Teachers: how can you build a class environment that encourages students to play with the ideas they are learning?

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RESOURCES

• Sunil Singh and Chris Brownell show that play is central to math teaching in their recent book: Math Recess: Playful Learning in an Age of Disruption.

FOR REFLECTION & DISCUSSION

1. Think of (non-mathematical) examples of: (a) something you find beautiful whose beauty took awhile to appreciate, and (b) something you find beautiful at first glance. In each instance, what preparation made those experiences of beauty possible?

2. Describe any experience you’ve had with sensory, wondrous, insightful, or transcendent mathematical beauty. How did that experience make you feel?

3. Think about all your educational experiences—for instance, classes you’ve taken in different subjects (mathematics or otherwise). Which ones implicitly acknowledged the human desire for beauty?

4. Where do you find mathematical beauty in the world?

5. List some features of art that you find beautiful, and discuss how those features compare (or contrast) to what you find beautiful about math. 

6. Have you had the experience of seeing the same beautiful idea pop up in lots of different places? What feelings did that evoke in you?

7. Teachers:what experiences of mathematical beauty have you had, and what can you do to experience more?  How can you enable similar experiences of beauty for your students?

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RESOURCES

• A very accessible collection of beautiful "proofs without words" can be found in the series by Roger Nelsen: Proofs without Words I, II, III.

• You can visit the Mathematical Art Galleries from the Bridges Organization.

• Kelsey Houston-Edwards and Tai-Denae Bradley hosted an excellent video series about mathematics called PBS Infinite Series.

• Martin Aigner and Günter Ziegler’s collection of elegant proofs of various theorems is playfully titled Proofs from THE BOOK.

FOR REFLECTION & DISCUSSION

1. What mathematical laws, truths, or ideas do you rely on in your daily life?

2. How is mathematics a refuge? For whom is it a refuge?

3. Many things in the universe change over time (and let’s not forget— the subject of calculus was developed to study such change). Do you find it surprising that the laws of mathematics do not change over time?

4. "Each theorem is a nostalgic souvenir of a past adventure that stretched one's capabilities." Can you think of a mathematical idea that once seemed hard, but now seems second nature? Reflect on how this idea has become a marker of progress for you.

5. Teachers: Think of a puzzle (mathematical or otherwise) that kept you engrossed for hours. What aspects of that puzzle captivated you? How can you find or design mathematical experiences and assignments for your students that will keep them engrossed?

6. What are some things in your life whose permanence (or enduring nature, at least) you trust in?

7. Is mathematical reasoning always trustworthy? How might humans wittingly or unwittingly take advantage of the trust that people have in mathematics?

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RESOURCES

• You can learn more about the art collection that included the Matsumoto puzzle in “The Creative Art of Coping in Japanese Internment,” an NPR story that aired May 12, 2010.

FOR REFLECTION & DISCUSSION

1. Reflect on a time when shallow knowledge (in any subject) has led you astray. How did that make you feel? How is deep knowledge an antidote?

2. Sometimes parties on two sides of an argument have different perspectives on the same event. Both views may be true, but each may be just part of the picture. How is knowing the whole truth a better place to be? Similarly, in mathematics, what does knowing the whole truth look like?

3. How can mathematical thinking equip you to converse with and respect people who hold different views?

4. Give an example of a truth that you once thought was simple, but now you feel is more complex and layered than you realized. Use that example to explain why deep investigation of any truth is essential.

5. In what ways is mathematics "equipment for thinking"?

6. Why is it important to think for oneself and not believe every statement that's given to you by an authority? Teachers: how should the importance of thinking for oneself influence the way you teach math?

7. When you learn a new truth, how will you seek multiple ways of knowing it? Reflect on a mathematical example as well as a non-mathematical example.

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RESOURCES

• Gian Carlo Rota's “The Concept of Mathematical Truth".

• Eugene Wigner's “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.

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FOR REFLECTION & DISCUSSION

1. Describe an activity you enjoy, and make a list of all the internal and external goods you can think of that are associated with that activity. Now think of an activity you don’t enjoy, and make a similar list. What do you notice about these lists?

2. What internal goods does mathematics offer? Discuss how these goods multiply when you share them with others.

3. What external goods does mathematics offer? How can you tell they are external and not internal goods?

4. Teachers: how can you incentivize students to value the process of struggle and not just the outcome?

5. How is looking up an answer to a math problem, before struggling with it, ultimately detrimental to your learning?

6. "A declining society---with deep inequity, lack of opportunity, or corrupt leadership---incentivizes people to acquire external goods dishonestly, because they see their leaders unfairly distributing them." Discuss.

7. "When people appreciate the inherent value of internal goods, they see the struggle to grow as a means of flourishing---even within an unjust system---and as a means of fighting against it." Describe any examples you know that illustrate this point.

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RESOURCES

• Eric M. Anderman's article discusses making mastering the goal of student learning: “Students Cheat for Good Grades. Why Not Make the Classroom about Learning and Not Testing?”

• I like to have my students read this article by Carol Dweck to learn about growth mindset: “The Secret to Raising Smart Kids.”

• There's lots to ponder in Alasdair MacIntyre's After Virtue: A Study in Moral Theory.

FOR REFLECTION & DISCUSSION

1. Think of a recent challenging math problem you explored. Which of the powers of mathematics did you develop or use in that exploration (interpretation, definition, quantification, abstraction, visualization, imagination, creation, strategization, modeling, multiple representations, generalization, structure identification)?

2. Teachers: do students explicitly see how you are helping them grow in the powers of mathematics (mentioned above)? How can you articulate to them how these powers are important in almost any career or life path?

3. Discuss creative power and coercive power that you have witnessed in mathematical settings.

4. Your social media feed is curated by algorithms. Are these a creative or coercive use of mathematical power? How do you know? In any given situation, how will you tell if mathematics is being used creatively or coercively?

5. Teachers: what responsibility do we have as math educators to help our students wield mathematical power well?

6. Teachers: how do you affirm your students’ dignity as creative human beings in the way that they do mathematics?

7. What relationship does (creative or coercive) mathematical power have to mathematical truth?

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RESOURCES

• A great book to learn about the math of card shuffling, as well as magic tricks that use mathematical ideas, is Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks by Persi Diaconis and Ron Graham. That's creative power being used to inspire and entertain.

• A sobering look at a coercive use of mathematical power is Cathy O'Neil's Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy.

FOR REFLECTION & DISCUSSION

1. If people have realized that the way we teach math needs to change, why hasn’t it changed yet? Who benefits from keeping it the same way it has always been?

2. All of us unwittingly harbor bias, so how can we mitigate bias in mathematical spaces? Who is harmed by bias in mathematical spaces, and why?

3. What inequities do you notice in mathematical spaces? Who is harmed by those inequities? Think deeper than the obvious answers.

4. With whom do you share the "secret menu" in mathematics, and why? How might that change now that you've read this chapter?

5. How does one's view of the purpose of mathematics affect one's view of who should be pursuing it? Compare several different views of the purposes of mathematics (eg, for human flourishing, for getting a good job, for producing elite mathematicians) and discuss their ramifications for who belongs in mathematics.

6. Teachers: make a list of ways that mathematical learning may involve cultural experiences. How can you mitigate the effects of cultural barriers?

7. Teachers: have you ever felt tempted to discourage a particular student from studying math? Why? 

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RESOURCES

• See what your implicit biases are with the online tests at https://implicit.harvard.edu/implicit/.

• A collection of essays that reflect on inequities in the way we teach math can be found in: Rehumanizing Mathematics for Black, Indigenous, and Latinx Students: Annual Perspectives in Mathematics Education, edited Imani Goffney and Rochelle Gutiérrez.

• Although this chapter focuses on justice concerning who does math, one might also consider how math can be used to address social justice concerns. Here are two recent books: Mathematics for Social Justice: Resources for the College Classroom (by Gizem Karaali and Lily Khadjavi), and High School Mathematics Lessons to Explore, Understand, and Respond to Social Injustice (by Robert Berry, Basil Conway, Brian Lawler, John Staley).

FOR REFLECTION & DISCUSSION

1. Describe settings in which you’ve experienced any of these freedoms: the freedom of knowledge, the freedom to explore, the freedom of understanding, or the freedom to imagine.

2. Give an example of a mathematical problem that you solved easily because you knew multiple ways of approaching the problem. 

3. What is the strangest, most imaginative idea that you have encountered so far in mathematics, and why did it capture your fancy?

4. Who do you think may not be feeling welcome in mathematical spaces? In what ways can you extend the freedom of welcome in mathematics to those around you? Think of concrete actions you can take.

5. The virtue of "seeing setbacks as springboards" means not simply discarding wrong ideas, but exploring how they can lead us to good answers or push us off to new areas of investigation. Give some examples from your own experience when a misunderstanding was actually beneficial to your growth.

6. What things have you experienced in a math classroom that feel like freedom? What things feel like domination?

7. For teachers: how can you create an environment where your students feel free to explore, to brainstorm and ask questions, and don't feel tempted to pretend that they understand if they don't?

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RESOURCES

• A book that will make you think deeply about freedom and its role in education: bell hooks, Teaching to Transgress: Education as the Practice of Freedom 

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FOR REFLECTION & DISCUSSION​​

1. Why should hospitality, or excellence in teaching and mentoring, be central to doing mathematics well?

2. How can you build a community in the classroom or the home in which participants push one another to grow while not being overly focused on achievement?

3. What actions can you take to address feelings of not belonging in math communities?

4. How does an emphasis on building mathematical virtues counter the notion of math as a one-dimensional endeavor?

5. Why are we so tempted to rank people according to a singular mathematical ability?

6. More of us should see hospitality as a virtue that math communities should embody. Describe three concrete actions you can take to extend mathematical hospitality to others.

7. If a virtue is an "excellence of character that leads to excellence of conduct", explain why vulnerability is a virtue, especially in math communities.

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RESOURCES

• The next four bullets are math communities mentioned in the book endnotes that may appeal widely. There are more than two hundred “math circles” throughout the US that gather children periodically for discovery and excitement around low-threshold, high-ceiling problems and interactive exploration; you can find a group on the National Association of Math Circles website.

• BEAM (Bridge to Enter Advanced Mathematics) offers day and residential programs designed to help underserved students enter the scientific professions.

• In the past I’ve taught at a math camp called MathPath, which brings middle school kids together each summer for a mix of math and outdoor activities; programs like this exist at all educational levels. 

• The Park City Mathematics Institute has a three-week summer program for math teachers (and other groups in the math community) to reflect on math teaching and leadership.

• Anything my colleague Darryl Yong writes on math education is worth reading. See his blog post: “Active Learning 2.0: Making It Inclusive.”

• Ilana Seidel Horn’s book Motivated: Designing Math Classrooms Where Students Want to Join In has many tips for cultivating healthy math communities.

FOR REFLECTION & DISCUSSION

1. In what harmful ways do we use mathematics as “a showcase for flaunting talent rather than a playground for building virtue”?

2. In what ways do you see people using mathematical achievements as a "stamp of self-importance"?

3. What are some ways we can fight the temptation to compare ourselves to others mathematically? Teachers: how can you help students build identities in mathematics that do not depend on comparisons?

4. How can you honor each person you meet as a dignified mathematical thinker?

5. Reflect on a friendship you've had that deepened because you worked together on a mathematical problem. In what ways is getting to know someone through and because of mathematics different than other ways of knowing them?

6. "To love is to believe that everyone can flourish in mathematics." Why is believing that everyone can flourish in mathematics a form of unconditional love?

7. Who are the forgotten among you, mathematically speaking? Whom will you love, whom will you read differently?

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RESOURCES

• Parker Palmer's To Know as We Are Known is a wonderful book about teaching, and about love.

• I wrote an essay called The Lesson of Grace in Teaching that calls us away from the trappings of an achievement-oriented mindset.

FOR REFLECTION AND DISCUSSION​

1. Describe the ways that you see Christopher Jackson flourishing even in challenging circumstances.

2. Learn about mass incarceration and use mathematics to quantify the impact (however you want to define impact) that mass incarceration has had on communities of color.

3. Write an essay describing how this book has changed your understanding of what math is, who it's for, and why anyone should learn it.

4. The COVID-19 pandemic has presented the world with a challenge that it has never grappled with before. Consider the mathematical virtues presented in this book and explain carefully how such virtues can help us (you, epidemiologists, and others) to solve problems arising from the pandemic.