mathematics for human flourishing
These are the notes at the end of the book, grouped by chapter, with hyperlinks to references.
Epigraph. Simone Weil, Gravity and Grace, trans. A. Wills (New York: G. P. Putnam’s Sons, 1952), 188.
1. From a letter Simone wrote to Father Perrin, which can be found in a collection of her writings: Waiting for God, trans. Emma Craufurd (London: Routledge & K. Paul, 1951), 64.
2. An excellent summary of Simone’s connection between mathematics and her spirituality can be found in Scott Taylor, “Mathematics and the Love of God: An Introduction to the Thought of Simone Weil,” available at http://colby.edu/~sataylor/SimoneWeil.pdf.
3. See Maurice Mashaal, Bourbaki: A Secret Society of Mathematicians (Providence: American Mathematical Society, 2006), 109–13.
4. Simone and André Weil’s relationship is explored in the memoir by Sylvie Weil, André’s daughter: At Home with André and Simone Weil, trans. Benjamin Ivry (Evanston, IL: Northwestern University Press, 2010).
5. Throughout 2018, the top four public companies by market capitalization were technology companies: Apple, Alphabet (the parent company of Google), Microsoft, and Amazon. In addition, the top ten included three more: Tencent, Alibaba, and Facebook.
6. Michael Barany, “Mathematicians Are Overselling the Idea That ‘Math Is Everywhere,’” Guest Blog, Scientific American, August 16, 2016, https://blogs.scientificamerican.com/guest-blog/mathematicians-are-overselling-the-idea-that-math-is-everywhere/.
7. See, for instance, Andrew Hacker, “Is Algebra Necessary?,” editorial, New York Times, July 28, 2012, https://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html; E. O. Wilson, “Great Scientist ≠ Good at Math,” editorial, Wall Street Journal, April 5, 2013, https://www.wsj.com/articles/SB10001424127887323611604578398943650327184. Both of these rely on flawed understandings of what mathematics really is.
8. Two of the more recent examples are the reports A Common Vision for Undergraduate Mathematical Sciences Programs in 2025 (2015), published by the Mathematical Association of America and available at https://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf, and Catalyzing Change in High School Mathematics: Initiating Critical Conversations (2018), by the National Council of Teachers of Mathematics, available for purchase at
9. See Christopher J. Phillips, The New Math: A Political History (Chicago: University of Chicago Press, 2015).
10. See, for example, Robert P. Moses and Charles E. Cobb Jr., Radical Equations: Civil Rights from Mississippi to the Algebra Project (Boston: Beacon, 2002), ch. 1.
11. For a sobering assessment, read Cathy O’Neil, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy (New York: Crown, 2016).
12. Erin A. Maloney, Gerardo Ramirez, Elizabeth A. Gunderson, Susan C. Levine, and Sian L. Beilock, “Intergenerational Effects of Parents’ Math Anxiety on Children’s Math Achievement and Anxiety,” Psychological Science 26, no. 9 (2015): 1480–88.
13. “Definitions,” trans. D. S. Hutchinson, in Plato, Complete Works, ed. John M. Cooper (Indianapolis: Hackett, 1997), 1680.
14. Some notable examples include Ubiratan D’Ambrosio, who emphasized the sociocultural dimensions of math education and initiated the study of ethno-mathematics in “Socio-cultural Bases for Mathematical Education,” in Proceedings of the Fifth International Congress on Mathematical Education, ed. Marjorie Carss (Boston: Birkh.user, 1986), 1–6; Reuben Hersh, who expounds a humanist philosophy of math in What Is Mathematics, Really? (Oxford: Oxford University Press, 1997); and Rochelle Gutiérrez, who addresses structures, policies, and practices that have dehumanized mathematics education, especially for students of color, in “The Need to Rehumanize Mathematics,” in Rehumanizing Mathematics for Black, Indigenous, and Latinx Students: Annual Perspectives in Mathematics Education, ed. Imani Goffney and Gutiérrez (Reston, VA: National Council of Teachers of Mathematics, 2018), 1–10.
15. Joshua Wilkerson, “Cultivating Mathematical Affections: Developing a Productive Disposition through Engagement in Service-Learning” (PhD thesis, Texas State University, 2017), 1, https://digital.library.txstate.edu/handle/10877/6611.
Epigraph 1. Maryam Mirzakhani, quoted in Bjorn Carey, “Stanford’s Maryam Mirzakhani Wins Fields Medal,” Stanford News, August 12, 2014, https://news.stanford.edu/news/2014/august/fields-medal-mirzakhani-081214.html.
Epigraph 2. Eugenia Cheng, How to Bake Pi (New York: Basic Books, 2015), 2.
1. John Joseph Fahie, Galileo: His Life and Work (New York: James Pott, 1903), 114.
2. See, for instance, Blaine Friedlander, “To Keep Saturn’s A Ring Contained, Its Moons Stand United,” Cornell Chronicle, October 16, 2017, http://news.cornell.edu/stories/2017/10/keep-saturns-ring-contained-its-moons-stand-united, and “Giant Planets in the Solar System and Beyond: Resonances and Rings” (Cornell Astronomy Summer REU Program, 2012), http://hosting.astro.cornell.edu/specialprograms/reu2012/workshops/rings/.
3. A more well-developed example appears in Paul Lockhart’s “A Mathematician’s Lament” (2002), available on the blog Devlin’s Angle: Keith Devlin, “Lockhart’s Lament,” March 2008, https://www.maa.org/external_archive/devlin/devlin_03_08.html.
4. Achi and other games from Africa can be found in physical form in the MIND Research Institute’s South of the Sahara game box, available at https://www.mindresearch.org/mathminds-games.
5. I have not seen any definitive source that resolves these ambiguities.
6. See Claudia Zaslavsky, Math Games & Activities from Around the World (Chicago: Chicago Review Press, 1998).
7. Fawn Nguyen, “These Twenty Things,” Finding Ways (blog), December 19, 2016, http://fawnnguyen.com/these-twenty-things/.
8. See, for instance, Kevin Hartnett, “Mathematicians Seal Back Door to Breaking RSA Encryption,” Abstractions Blog, Quanta Magazine, December 17, 2018, https://www.quantamagazine.org/mathematicians-seal-back-door-to-breaking-rsa-encryption-20181217/; Rama Mishra and Shantha Bhushan, “Knot Theory in Understanding Proteins,” Journal of Mathematical Biology 65, nos. 6–7 (December 2012): 1187–213, available at https://link.springer.com/article/10.1007/s00285-011-0488-3; Chris Budd and Cathryn Mitchell, “Saving Lives: The Mathematics of Tomography,” Plus Magazine, June 1, 2008, https://plus.maths.org/content/saving-lives-mathematics-tomography.
9. The Art of Problem Solving website is a good resource: https://artofproblemsolving.com/.
10. Ben Orlin, Math with Bad Drawings (New York: Black Dog & Leventhal, 2018), 10–12.
11. You may remember wayfinding being featured in Disney’s movie Moana (2016). “Star of gladness” refers to the star Arcturus.
12. See Richard Schiffman, “Fantastic Voyage: Polynesian Seafaring Canoe Completes Its Globe-Circling Journey,” Scientific American, June 13, 2017, https://www.scientificamerican.com/article/fantastic-voyage-polynesian-seafaring-canoe-completes-its-globe-circling-journey/.
13. Linda updated this quote from her interview with Cheryl Ernst, “Ethnomathematics Makes Difficult Subject Relevant,” Mālamalama, July 15, 2010, http://www.hawaii.edu/malamalama/2010/07/ethnomathematics/.
Epigraph 1. Sónya Kovalévsky: Her Recollections of Childhood, trans. Isabel F. Hapgood (New York: Century, 1895), 316.
Epigraph 2. Jorge Luis Borges, This Craft of Verse (Cambridge, MA: Harvard University Press, 2002), 22.
1. For an amusing video of a similar situation, do an online search for news coverage of President Obama’s car getting stuck as it rolled over an exit ramp at the US Embassy in Dublin in May 2011.
2. Henri Poincaré, Science and Hypothesis, trans. William John Greenstreet (New York: Walter Scott, 1905), 141.
3. Jo Boaler, “Memorizers Are the Lowest Achievers and Other Common Core Math Surprises,” editorial, Hechinger Report, May 7, 2015, https://hechingerreport.org/memorizers-are-the-lowest-achievers-and-other-common-core-math-surprises/.
4. Robert P. Moses and Charles E. Cobb Jr., Radical Equations: Civil Rights from Mississippi to the Algebra Project (Boston: Beacon, 2002), 119–22.
5. See Cassius Jackson Keyser, Mathematics as a Culture Clue, and Other Essays (New York: Scripta Mathematica, Yeshiva University,1947), 218.
6. William Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton: Princeton University Press, 2007).
7. This definition was popularized by the mathematician Keith Devlin in Mathematics: The Science of Patterns (New York: Scientific American Library, 1997) and may have originated in Lynn Steen, “The Science of Patterns,” Science 240, no. 4852 (April 29, 1988): 611–16.
Epigraph 1. Martin Buber, Pointing the Way: Collected Essays, ed. and trans. Maurice S. Friedman (New York: Harper & Row, 1963), 21.
Epigraph 2. Attributed to Sophie Germain by Count Guglielmo Libri-Carducci in his eulogy for her. See Ioan James, Remarkable Mathematicians: From Euler to Von Neumann (New York: Cambridge University Press, 2002), 58.
1. Johan Huizinga, Homo Ludens: A Study in the Play-Element of Culture, translated from the German [translator unknown] (London: Routledge & Kegan Paul, 1949).
2. G. K. Chesterton, All Things Considered (London: Methuen, 1908), 96.
3. Huizinga, Homo Ludens, 8.
4. Paul Lockhart, “A Mathematician’s Lament” (2002), 4, available at Devlin’s Angle: Keith Devlin, “Lockhart’s Lament,” March 2008, https://www.maa.org/external_archive/devlin/devlin_03_08.html.
5. For a description of the modeling cycle, see, for instance, GAIMME: Guidelines for Assessment and Instruction in Mathematical Modeling Education, 2nd ed., ed. Sol Garfunkel and Michelle Montgomery, Consortium for Mathematics and Its Applications and the Society for Industrial and Applied Mathematics (Philadelphia, 2019), available at
6. Blaise Pascal, Pensées, trans. W. F. Trotter (New York: E. P. Dutton, 1958), 4, no. 10.
7. To answer the question, it will be helpful to first convince yourself that only the last two digits of both numbers affect the last two digits of the product (think about how multiplication is done). So it’s enough to check the square of an ending to see if it is stubborn. To check if 21 is stubborn, square 21 and see if its last digits are 21. They are not. Then it’s helpful to realize that you don’t need to test all hundred possibilities for two-digit endings. That’s because a stubborn two-digit ending must have a stubborn one-digit ending. But there are only four such endings: 0, 1, 5, 6. So you need to check only the two-digit endings that end in 0, 1, 5, 6.
8. It may surprise you that among all 10^15 possible fifteen-digit endings, there are only four that are stubborn! Here they are:
. . . 000000000000000,
. . . 000000000000001,
. . . 259918212890625,
. . . 740081787109376.
Now, what do you notice about them? What do you wonder? Are there patterns?
9. These numbers are known in the mathematical literature as automorphic numbers. They are also related to p-adic numbers when the base is prime.
10. Simone Weil, Waiting for God, trans. Emma Craufurd (London: Routledge & K. Paul, 1951), 106.
11. G. H. Hardy, A Mathematician’s Apology (Cambridge: Cambridge University Press, 1940).
12. See the prime minister’s remarks at Sarah Polus, “Full Transcript: Prime Minister Lee Hsien Loong’s Toast at the Singapore State Dinner,” Washington Post, August 2, 2016, https://www.washingtonpost.com/news/reliable-source/wp/2016/08/02/full-transcript-prime-minister-lee-hsien-loongs-toast-at-the-singapore-state-dinner/.
13. “Republic,” trans. Paul Shorey, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns (Princeton: Princeton University Press, 1961), 768 (7.536e).
Epigraph 1. “Autobiography of Olga Taussky-Todd,” ed. Mary Terrall (Pasadena, California, 1980), Oral History Project, California Institute of Technology Archives, 6; available at http://resolver.caltech.edu/CaltechOH:OH_Todd_O.
Epigraph 2. Quoted in Donald J. Albers, “David Blackwell,” in Mathematical People: Profiles and Interviews, ed. Albers and Gerald L. Alexanderson (Wellesley, MA: A. K. Peters, 2008), 21.
1. “Interview with Research Fellow Maryam Mirzakhani,” Clay Mathematics Institute Annual Report 2008, https://www.claymath.org/library/annual_report/ar2008/08Interview.pdf, 13.
2. Semir Zeki, John Paul Romaya, Dionigi M. T. Benincasa, and Michael F. Atiyah, “The Experience of Mathematical Beauty and Its Neural Correlates,” Frontiers in Human Neuroscience 8 (2014): 68.
3. G. H. Hardy, A Mathematician’s Apology (Cambridge: Cambridge University Press, 1940); Harold Osborne, “Mathematical Beauty and Physical Science,” British Journal of Aesthetics 24, no. 4 (Autumn 1984): 291–300; William Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton: Princeton University Press, 2007).
4. Paul Erdös, quoted in Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth (London: Fourth Estate, 1998), 44.
5. Martin Gardner, “The Remarkable Lore of the Prime Numbers,” Mathematical
Games, Scientific American 210 (March 1964): 120–28.
6. Erdős reportedly quipped, “You don’t have to believe in God, but you should believe in The Book” (Hoffman, Man Who Loved Only Numbers, 26). In homage to Erdős, Martin Aigner and Günter Ziegler’s collection of elegant proofs of various theorems is playfully titled Proofs from THE BOOK (New York: Springer, 2010).
7. Sydney Opera House Trust, “The Spherical Solution,” https://www.sydneyoperahouse.com/our-story/sydney-opera-house-history/spherical-solution.html.
8. Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking (New York: Penguin, 2014), 436–37.
9. Albert Einstein, Ideas and Opinions (New York: Crown, 1954), 233.
10. Erica Klarreich, “Mathematicians Chase Moonshine’s Shadow,” Quanta
Magazine, March 12, 2015, https://www.quantamagazine.org/mathematicians-chase-moonshine-string-theory-connections-20150312/.
11. Simon Singh, “Interview with Richard Borcherds,” The Guardian, August
12. C. S. Lewis, The Weight of Glory (New York: Macmillan, 1949), 7.
13. Barbara Oakley, “Make Your Daughter Practice Math. She’ll Thank You Later,” editorial, New York Times, August 7, 2018, https://www.nytimes.com/2018/08/07/opinion/stem-girls-math-practice.html.
Epigraph 1. Bernhard Riemann, “On the Psychology of Metaphysics: Being the Philosophical Fragments of Bernhard Riemann,” trans. C. J. Keyser, The Monist 10, no. 2 (1900): 198.
Epigraph 2. Network of Minorities in Mathematical Sciences, “Tai-Danae Bradley: Graduate Student, CUNY Graduate Center,” Mathematically Gifted and Black, http://mathematicallygiftedandblack.com/rising-stars/tai-danae-bradley/.
1. The word law is sometimes also used in reference to mathematical ideas— usually either empirically observed patterns that have been validated by a theorem (e.g., the law of large numbers) or axioms that are being assumed as a foundation for knowledge (e.g., the commutative law, the law of the excluded middle).
2. David Eugene Smith, “Religio Mathematici,” American Mathematical Monthly 28, no. 10 (1921): 341.
3. Morris Kline, Mathematics for the Nonmathematician (New York: Dover, 1985), 9.
4. Read more about The Art of Gaman, curated by Delphine Hirasuna, in Susan Stamberg, “The Creative Art of Coping in Japanese Internment,” NPR, May 12, 2010, https://www.npr.org/templates/story/story.php?storyId=126557553.
5. An eighty-one-step solution, starting from a slightly different starting configuration, was printed in Martin Gardner, “The Hypnotic Fascination of Sliding Block Puzzles,” Mathematical Games, Scientific American 210 (February 1964): 122–30. A solution starting from Matsumoto’s starting configuration can be found in this book’s Hints & Solutions to Puzzles.
6. George Orwell, 1984 (Boston: Houghton Mifflin Harcourt, 1949), 76.
Epigraph 1. John 18: 38 (Good News Translation).
Epigraph 2. Blaise Pascal, Pensées, trans. W. F. Trotter (New York: E. P. Dutton, 1958), 259, no. 864.
1. Hannah Arendt, “Truth and Politics,” New Yorker, February 25, 1967, reprinted in Arendt, Between Past and Future (New York: Penguin, 1968), 257.
2. See Michael P. Lynch, True to Life: Why Truth Matters (Cambridge, MA: MIT Press, 2004).
3. Errors sometimes lead to further exploration. The correct calculation is 777 x 1,144 = 888,888. Interesting! But the mistyped calculation 777 x 144 = 111,888 also has a noteworthy pattern. What’s going on here?
4. Gian-Carlo Rota, “The Concept of Mathematical Truth,” Review of Metaphysics 44, no. 3 (March 1991): 486.
5. Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Mathematics 13 (1960): 14.
6. Kenneth Burke, “Literature as Equipment for Living,” collected in The Philosophy of Literary Form: Studies in Symbolic Action (Baton Rouge: Louisiana State University Press, 1941), 293–304.
7. Quoted in David Brewster, The Life of Sir Isaac Newton (New York: J. & J. Harper, 1832), 300–301.
Epigraph 1. Simone Weil, Waiting for God, trans. Emma Craufurd (London: Routledge & K. Paul, 1951), 107.
Epigraph 2. Martha Graham, “An Athlete of God,” in This I Believe: The Personal Philosophies of Remarkable Men and Women, ed. Jay Allison and Dan Gediman, with John Gregory and Viki Merrick (New York: Holt, 2006), 84.
1. Alasdair MacIntyre, After Virtue: A Study in Moral Theory, 3rd ed. (South Bend, IN: University of Notre Dame Press, 2007), 188.
3. See Eric M. Anderman, “Students Cheat for Good Grades. Why Not Make the Classroom about Learning and Not Testing?,” The Conversation, May 20, 2015, https://theconversation.com/students-cheat-for-good-grades-why-not-make-the-classroom-about-learning-and-not-testing-39556.
4. Carol Dweck, “The Secret to Raising Smart Kids,” Scientific American, January 1, 2015, https://www.scientificamerican.com/article/the-secret-to-raising-smart-kids1/.
5. Teachers will find an excellent resource on how mindsets affect learning in mathematics, as well as practical suggestions for how to change mindsets, in Jo Boaler, Mathematical Mindsets (San Francisco: Jossey-Bass, 2016).
6. “Interview with Maryam Mirzakhani,” Clay Math Institute Annual Report 2008, https://www.claymath.org/library/annual_report/ar2008/08Interview.pdf.
7. David Richeson, “A Conversation with Timothy Gowers,” Math Horizons 23, no. 1 (September 2015): 10–11.
8. Laurent Schwartz, A Mathematician Grappling with His Century (Basel: Birkhauser, 2001), 30.
Epigraph 1. Quoted in Stephen Winsten, Days with Bernard Shaw (New York: Vanguard, 1949), 291.
Epigraph 2. Augustus de Morgan, quoted in Robert Perceval Graves, The Life of Sir William Rowan Hamilton, vol. 3 (Dublin: Dublin University Press, 1889), 219.
1. See Isidor Wallimann, Howard Rosenbaum, Nicholas Tatsis, and George Zito, “Misreading Weber: The Concept of ‘Macht,’” Sociology 14, no. 2 (May 1980): 261–75.
2. Andy Crouch, Playing God: Redeeming the Gift of Power (Downers Grove, IL: InterVarsity Press, 2014), 17.
3. Thanks to my friend Lew Ludwig for pointing this out to me.
4. Dave Bayer and Persi Diaconis, “Trailing the Dovetail Shuffle to Its Lair,” Annals of Applied Probability 2, no. 2 (May 1992): 294–313.
5. These details can be found in Karen D. Rappaport, “S. Kovalevsky: A Mathematical Lesson,” American Mathematical Monthly 88, no. 8 (October 1981): 564–74.
6. Erica N. Walker, Beyond Banneker: Black Mathematicians and the Paths to Excellence (Albany: SUNY Press, 2014).
7. Cathy O’Neil, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy (New York: Crown, 2016).
8. Parker J. Palmer, The Courage to Teach: Exploring the Inner Landscape of a Teacher’s Life, 10th anniversary ed. (San Francisco: Jossey-Bass, 2007), 7.
Epigraph. Simone Weil, Gravity and Grace, trans. A. Wills (New York: G. P. Putnam’s Sons, 1952), 188.
1. E.g., Timothy Keller, Generous Justice: How God’s Grace Makes Us Just (New York: Penguin, 2012).
2. Take the tests at https://implicit.harvard.edu/implicit/.
3. Victor Lavy and Edith Sands, “On the Origins of Gender Gaps in Human Capital: Short- and Long-Term Consequences of Teachers’ Biases,” Journal of Public Economics 167 (2018): 263–79.
4. Michela Carlana, “Implicit Stereotypes: Evidence from Teachers’ Gender Bias,” Quarterly Journal of Economics (forthcoming): https://doi.org/10.1093/qje/qjz008.
5. In 2004, about one-third of students who entered US universities intended to major in a STEM field. Of those, the six-year completion rate was about 45 percent for white and Asian students, and 25 percent for others. There’s a lot of interesting data in Kevin Eagan, Sylvia Hurtado, Tanya Figueroa, and Bryce Hughes, “Examining STEM Pathways among Students Who Begin
College at Four-Year Institutions,” paper commissioned for the Committee on Barriers and Opportunities in Completing 2-Year and 4-Year STEM Degrees (Washington DC: National Academies Press, 2014), https://sites.nationalacademies.org/cs/groups/dbassesite/documents/webpage/dbasse_088834.pdf.
6. See Jennifer Engle and Vincent Tinto, Moving beyond Access: College Success for Low-Income, First-Generation Students (Washington DC: Pell Institute, 2008), https://files.eric.ed.gov/fulltext/ED504448.pdf.
7. For instance, among US citizens who earned math PhDs in 2015, 84 percent were white and 72 percent were men. See William Yslas Vélez, Thomas H. Barr, and Colleen A. Rose, “Report on the 2014–2015 New Doctoral Recipients,” Notices of the AMS 63, no. 7 (August 2016): 754–65.
8. See “Finally, an Asian Guy Who’s Good at Math (Part Two),” Angry Asian Man (blog), January 4, 2016, http://blog.angryasianman.com/2016/01/finally-asian-guy-whos-good-at-math.html.
9. Rochelle Gutiérrez, “Enabling the Practice of Mathematics Teachers in Context: Toward a New Equity Research Agenda,” Mathematical Thinking and Learning 4, nos. 2–3 (2002): 147.
10. See, for instance, National Council of Teachers of Mathematics, Catalyzing Change in High School Mathematics: Initiating Critical Conversations (Reston, VA : The National Council of Teachers of Mathematics, 2018); Jo Boaler, “Changing Students’ Lives through the De-tracking of Urban Mathematics Classrooms,” Journal of Urban Mathematics Education 4, no. 1 (July 2011): 7–14.
11. William F. Tate, “Race, Retrenchment, and the Reform of School Mathematics,” Phi Delta Kappan 75, no. 6 (February 1994): 477–84.
Epigraph 1. Helen Keller, The Story of My Life (New York: Grosset & Dunlap, 1905), 39.
Epigraph 2. Eleanor Roosevelt, You Learn by Living (New York: Harper & Row, 1960), 152.
1. To learn some of his shortcuts, see Arthur Benjamin and Michael Shermer, Secrets of Mental Math (New York: Three Rivers, 2006).
2. This shortcut for multiplying by 11 will require a “carry” if the sum of the digits is 10 or more. For instance, to compute 75 x11, you should add 7 and 5 to get 12, put the 2 between the 7 and the 5, and then carry the 1 by adding it to the 7, to get 8. Thus, the answer is 825. If you know some algebra, you can use it to show why the shortcut works: the number 10a + b is the number with digits a and b. Then (10a + b) x 11 = 110a + 11b = 100a + 10(a + b) + b. This last expression does indeed suggest adding the two digits and putting their sum between them.
3. Georg Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite,” trans. William Ewald, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. Ewald (New York: Oxford University Press, 1996), vol. 2, 896 (sec 8). Italics in the original.
4. Evelyn Lamb, “A Few of My Favorite Spaces: The Infinite Earring,” Roots of Unity (blog), Scientific American, July 31, 2015, https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-infinite-earring/.
5. J. W. Alexander, “An Example of a Simply Connected Surface Bounding a Region Which Is Not Simply Connected,” Proceedings of the National Academy of Sciences of the United States of America 10, no. 1 (January 1924): 8–10.
6. Robert Rosenthal and Lenore Jacobson, “Teachers’ Expectancies: Determinants of Pupils’ IQ Gains,” Psychological Reports 19 (1966): 115–18. It is worth noting that this study has attracted controversy. An interesting account of this study, including critiques and follow-up studies, can be found in Katherine Ellison, “Being Honest about the Pygmalion Effect,” Discover Magazine, October 29, 2015, http://discovermagazine.com/2015/dec/14-great-expectations.
7. bell hooks, Teaching to Transgress: Education as the Practice of Freedom (New York: Routledge, 1994), 3.
Epigraph 1. Bill Thurston, October 30, 2010, reply to “What’s a Mathematician To Do?,” Math Overflow, https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do.
Epigraph 2. Deanna Haunsperger, “The Inclusion Principle: The Importance of Community in Mathematics,” MAA Retiring Presidential Address, Joint Mathematics Meeting, Baltimore, January 19, 2019; video available at https://www.youtube.com/watch?v=jwAE3iHi4vM.
1. Parker Palmer, To Know as We Are Known (New York: Harper Collins, 1993), 9.
2. See Gina Kolata, “Scientist at Work: Andrew Wiles; Math Whiz Who Battled 350-Year-Old Problem,” New York Times, June 29, 1993, https://www.nytimes.com/1993/06/29/science/scientist-at-work-andrew-wiles-math-whiz-who-battled-350-year-old-problem.html. Wiles’s error was fixed a couple of years later with the help of Richard Taylor.
3. See Dennis Overbye, “Elusive Proof, Elusive Prover: A New Mathematical Mystery,” New York Times, August 15, 2006, https://www.nytimes.com/2006/08/15/science/15math.html.
4. See Thomas Lin, “After Prime Proof, an Unlikely Star Rises,” Quanta Magazine, April 2, 2015, https://www.quantamagazine.org/yitang-zhang-and-the-mystery-of-numbers-20150402/.
5. Jerrold W. Grossman, “Patterns of Collaboration in Mathematical Research,” SIAM News 35, no. 9 (November 2002): 8–9; also available at https://archive.siam.org/pdf/news/485.pdf.
6. I’ll mention just a few programs here 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 (http://www.mathcircles.org/). BEAM (Bridge to Enter Advanced Mathematics; https://www.beammath.org/) 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 (http://www.mathpath.org/), 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 (https://www.ias.edu/pcmi) has a three-week summer program for math teachers (and other groups in the math community) to reflect on math teaching and leadership.
7. See, for instance, Talithia Williams, Power in Numbers: The Rebel Women of Mathematics (New York: Race Point, 2018); 101 Careers in Mathematics, ed. Andrew Sterrett, 3rd ed. (Washington DC: Mathematical Association of America, 2014).
8. Simone Weil, letter to Father Perrin, collected in Waiting for God, trans. Emma Craufurd (London: Routledge & K. Paul, 1951), 64.
9. See the MAA Instructional Practices Guide (2017), including references, from the Mathematical Association of America, available at https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide.
10. See Darryl Yong, “Active Learning 2.0: Making It Inclusive,” Adventures in Teaching (blog), August 30, 2017, https://profteacher.com/2017/08/30/active-learning-2-0-making-it-inclusive/.
11. Ilana Seidel Horn’s book Motivated: Designing Math Classrooms Where Students Want to Join In (Portsmouth, NH: Heinemann, 2017) contains ideas on how to do so.
12. See Justin Wolfers, “When Teamwork Doesn’t Work for Women,” New York Times, January 8, 2016, https://www.nytimes.com/2016/01/10/upshot/when-teamwork-doesnt-work-for-women.html.
13. See Association for Women in Science–Mathematical Association of America Joint Task Force on Prizes and Awards, “Guidelines for MAA Selection Committees: Avoiding Implicit Bias” (prepared August 2011, approved August 2012), Mathematical Association of America, https://www.maa.org/sites/default/files/pdf/ABOUTMAA/AvoidingImplicitBias_revisionMarch2018.pdf.
14. Karen Uhlenbeck, “Coming to Grips with Success,” Math Horizons 3, no. 4 (April 1996): 17.
Epigraph 1. 1 Corinthians 13:1 (Good News Translation).
Epigraph 2. The Papers of Martin Luther King, Jr., ed. Clayborne Carson, vol. 1, Called to Serve: January 1929–June 1951, ed. Ralph E. Lucker and Penny A. Russell (Berkeley: University of California Press, 1992), 124.
1. See, for example, Hannah Fry, The Mathematics of Love: Patterns, Proofs, and the Search for the Ultimate Equation (New York: Simon & Schuster, 2015).
2. Simone Weil, letter to Father Perrin, collected in Waiting for God, trans. Emma Craufurd (London: Routledge & K. Paul, 1951), 64.
3. I tell this story in Francis Edward Su, “The Lesson of Grace in Teaching,” in The Best Writing on Mathematics 2014, ed. Mircea Petici (Princeton: Princeton University Press, 2014), 188–97, also available at http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html.
4. Simone Weil, “Reflections on the Right Use of School Studies with a View to the Love of God,” in Waiting for God, trans. Emma Craufurd (London: Routledge & K. Paul, 1951), 115.
Simone Weil’s original quote is “L’attention est la forme la plus rare et la plus pure de la générosité.” See Weil and Jo. Bousquet, Correspondance (Lausanne: Editions l’Age d’Homme, 1982), 18.