174. Boaler, “Prove It to Me!”

175. “Painted Cube,” youcubed.org, n.d., https://www.youcubed.org/ tasks/painted-cube/.

176. Menon, “Brain Networks for Mental Arithmetic.”

177. Ericsson and Pool, Peak.

178. Ericsson and Pool, Peak.

179. Ericsson and Pool, Peak.

180. J. Hawkins, A Thousand Brains: A New Theory of Intelligence (New York: Basic Books, 2021).

181. L. Bofferding, “Negative Integer Understanding: Characterizing First Graders’ Mental Models,” Journal for Research in Mathematics Education 45, no. 2 (2014): 194–245; J. M. Tsang, K. P. Blair, L. Bofferding, and D. L. Schwartz, “Learning to ‘See’ Less Than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure,” Cognition and Instruction 33, no. 2, (2015): 154–97; S. L. Macrine and J. M. Fugate, eds., Movement Matters: How Embodied Cognition Informs Teaching and Learning (Chicago: MIT Press, 2022).

182. Amalric and Dehaene, “Origins of the Brain Networks”; R. A. Cortes, E. G. Peterson, D. J. Kraemer, R. A. Kolvoord, D. H. Uttal, N. Dinh… and A. E. Green, “Transfer from Spatial Education to Verbal Reasoning and Prediction of Transfer from Learning-Related Neural Change,” Science Advances 8, no. 31 (2022): eabo3555.

183. Menon, “Brain Networks for Mental Arithmetic.”

184. C. Kalb, “What Makes a Genius?” National Geographic, May 2017.

185. J. Park and E. M. Brannon, “Training the Approximate Number System Improves Math Proficiency,” Psychological Science 24, no. 10 (2013): 2013–19; Cortes et al., “Transfer from Spatial Education.”

186. Bofferding, “Negative Integer Understanding”; Tsang et al., “Learning to ‘See’”; Macrine and Fugate, eds., Movement Matters.

187. Bruce McCandliss, Stanford Graduate School of Education, n.d., https://ed.stanford.edu/faculty/brucemc.

188. M. Guillaume, E. Roy, A. Van Rinsveld, G. S. Starkey, Project iLe-ad Consortium, M. R. Uncapher, and B. D. McCandliss, “Groupi-tizing Reflects Conceptual Developments in Math Cognition and Inequities in Math Achievement from Childhood through Adolescence,” Child Development 94, no. 2 (2023): 335–47.

189. I. Benson, N. Marriott, and B. D. McCandliss, “Equational Reasoning: A Systematic Review of the Cuisenaire-Gattegno Approach,” Frontiers in Education (2022): 507.

190. M. Penner-Wilger, L. Fast, J.-A. LeFevre, B. L. Smith-Chant, S.-L. Skwarchuk, D. Kamawar, and J. Bisanz, “Subitizing, Finger Gnosis, and the Representation of Number,” Proceedings of the 31st Annual Cognitive Science Society 31 (2009): 520–25.

191. J. Boaler and L. Chen, “Why Kids Should Use Their Fingers in Math Class,” Atlantic, April 13, 2016, https://www.theatlantic.com/education/archive/2016/04/why-kids-should-use-their-fin-gers-in-math-class/478053/.

192. R. S. Siegler and G. B. Ramani, “Playing Linear Numerical Board Games Promotes Low-Income Children’s Numerical Development,” Developmental Science 11 (2008): 655–61.

193. Yeager et al., “Teacher Mindsets Help Explain.”

194. Anderson et al., “Achieving Elusive Teacher Change,” 98; Bui et al., “Systematic Review.”

195. Мы с соавторами Джен Мансон и Кэти Уильямс написали несколько книг по математике для программы К–8; смотрите “K–8 Curriculum”, youcubed.org.

196. Gray and Tall, “Duality, Ambiguity, and Flexibility”; H. Chang, L. Chen, Y. Zhang, Y. Xie, C. de Los Angeles, E. Adair… and V. Menon, “Foundational Number Sense Training Gains Are Predicted by Hippocampal – Parietal Circuits,” Journal of Neuroscience 42, no. 19 (2022): 4000–15.

197. L. Ma, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States, Studies in Mathematical Thinking and Learning Series (Oxfordshire, UK: Routledge, 2010).

198. “Online Student Course,” youcubed.org, n.d., https://www. youcubed.org/online-student-course/.

199. “WIM Videos,” youcubed.org, n.d., https://www.youcubed.org/ resource/wim-videos/.

200. J. Boaler and C. Humphreys, Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning (Portsmouth, NH: Heinemann, 2005).

201. “Videos,” youcubed.org, n.d., https://www.youcubed.org/resource/ videos/.

202. M. Cordero, “It’s (Not) Ours to Reason Why: A Comparative Analysis of Algorithms for the Division of Fractions” (honor’s thesis, Stanford University, 2017), 1.

203. Cordero, “It’s (Not) Ours to Reason Why,” 1.

204. D. D. Pesek and D. Kirshner, “Interference of Instrumental Instruction in Subsequent Relational Learning,” Journal for Research in Mathematics Education 31, no. 5 (2000): 524–40.

205. T. P. Carpenter and M. K. Corbitt, eds., “Results from the Second Mathematics Assessment of the National Assessment of Educational Progress,” Fraction Bars, n.d., https://fractionbars.com/ Research_Tch_Fracs/Results2nd.html.

206. “Online Student Course,” youcubed.org.

207. Merzenich, SoV-Wired, 2; Doidge, Brain That Changes Itself.

208. Dweck and Yeager, “Mindsets.”

209. Boaler, “Prove It to Me!”

210. Фон Нгуен дает множество примеров алгебраических паттернов на своем прекрасном сайте “Visual Patterns,” https:// www.visualpatterns.org/.

211. “Mathematical Mindset Algebra,” youcubed.org, n.d., https:// www.youcubed.org/algebra/.

212. A. Proehl, “For Bay Area Designer Diarra Bousso, Math + Art = Happiness,” KQED, June 1, 2023, https://www.kqed.org/arts/13929878/ for-bay-area-designer-diarra-bousso-math-art-happiness.

213. E. Farra, “A Senegal-Raised, Silicon Valley-Based Designer Shares Her Vision for a More Sustainable and Inclusive Future,” Vogue, June 5, 2020, https://www.vogue.com/article/ diarra-bousso-diarrablu-sustainable-made-in-senegal-collec-tion; H. Jennings, “Meet Diarra Bousso: One of Senegal’s Most Promising Designers,” CNN, April 19, 2021, https://www.cnn.com/style/article/diarrablu-diarra-bousso-senegal/index.html.

214. “Diarra Bousso, MA ’18 Stanford Teacher Education Program: Fusing Fashion and Math,” Stanford Graduate School of Education, December 12, 2022, https://ed.stanford.edu/about/commu-nity/diarra-bousso.

215. Proehl, “For Bay Area Designer Diarra Bousso.”

216. “What Is Notice and Wonder?” National Council of Teachers of Mathematics, n.d., https://www.nctm.org/noticeandwonder/.