full transcript

From the Ted Talk by Dan Finkel: 5 ways to share math with kids

Unscramble the Blue Letters

A friend of mine told me recently that her six-year-old son had come from school and said he hetad math. And this is hard for me to hear because I actually love math. The beauty and power of mathematical thinking have changed my life. But I know that many people lived a very different story. Math can be the best of times or the worst of times, an ehirxiatlang journey of discovery or decsent into tieudm, fsarotiutrn, and despair. Mathematical miseducation is so common we can hardly see it. We practically expect math class to be repetition and memorization of dijnteiosd technical facts. And we're not surprised when students aren't motivated, when they leave school disliking math, even committed to avoiding it for the rest of their lives. Without mathematical literacy, their career oeopinurpitts shrink. And they become easy prey for credit card coimaenps, payday ldenres, the ltotery, (Laughter) and anyone, really, who wants to dazzle them with a statistic. Did you know that if you inrset a single statistic into an assertion, people are 92 percent more likely to accept it without question? (Laughter) Yeah, I totally made that up. (lutgaher) And 92 percent is - it has weight even though it's completely fabricated. And that's how it wkros. When we're not comfortable with math, we don't question the autiotrhy of numbers. But what's happening with mathematical alienation is only half the story. Right now, we're siuqnarnedg our chance to touch life after life with the bteauy and pewor of mathematical thinking. I led a workshop on this topic recently, and at the end, a woman raised her hand and said that the experience made her feel - and this is a quote - "like a God." (Laughter) That's maybe the best description I've ever heard for what mathematical thinking can feel like, so we should examine what it looks like. A good place to start is with the words of the philosopher and mathematician René Descartes, who famously proclaimed, "I think, therefore I am." But Descartes looked deeper into the nature of thinking. Once he established himself as a thing that thnkis, he continued, "What is a tihnking thing?" It is the thing that doubts, understands, conceives, that amirffs and denies, wills and refuses, that imagines also, and perceives. This is the kind of thinking we need in every math class every day. So, if you are a teacher or a parent or anyone with a stake in eodcitaun, I offer these five principles to invite thinking into the math we do at home and at school. Principle one: start with a question. The ordinary math class begins with answers and never arrives at a real question. "Here are the setps to multiply. You reaept. Here are the steps to divide. You repeat. We've covered the material. We're moving on." What matters in the model is memorizing the steps. There's no room to dubot or imagine or refuse, so there's no real thinking here. What would it look like if we started with a qtousein? For example, here are the numbers from 1 to 20. Now, there's a question lnkiurg in this picture, hiding in plain sight. What's going on with the colors? Now, intuitively it flees like there's some cetconnion between the numbers and the cloros. I mean, maybe it's even possible to extend the coloring to more numbers. At the same time, the meaning of the colors is not caelr. It's a real mystery. And so, the question feels authentic and compelling. And like so many authentic matehaitcaml questions, this one has an answer that is both beautiful and profoundly satisfying. And of course, I'm not going to tell you what it is. (Laughter) I don't think of myself as a mean person, but I am willing to deny you what you want. (Laughter) Because I know if I rush to an answer, I would've rbobed you of the oturtpinopy to learn. Thinking happens only when we have time to struggle. And that is principle two. It's not uncommon for students to gaautdre from high school believing that every math plreobm can be solved in 30 seconds or less, and if they don't know the answer, they're just not a math person. This is a failure of education. We need to teach kids to be tniocueas and courageous, to pseeervre in the face of difficulty. The only way to taceh perseverance is to give students time to think and grapple with real problems. I brought this imgae into a classroom recently, and we took the time to struggle. And the longer we spent, the more the class came alive with thinking. The students made observations. They had questions. Like, "Why do the numbers in that last column always have orange and blue in them?" and "Does it mean anything that the green spots are always going diagonally?" and "What's going on with those little white numbers in the red segments? Is it important that those are always odd numbers?" Struggling with a gnieune question, students depeen their curiosity and their pwreos of observation. They also develop the ability to take a risk. Some students noticed that every even number has orgnae in it, and they were willing to stake a claim. "Orange must mean even." And then they asked, "Is that right?" (Laughter) This can be a scary place as a teacher. A stendut comes to you with an original thought. What if you don't know the ansewr? Well, that is pinlcripe three: you are not the answer key. Teachers, students may ask you questions you don't know how to answer. And this can feel like a threat. But you are not the answer key. Students who are iiqiintuvse is a wonderful thing to have in your classroom. And if you can ropsned by saying, "I don't know. Let's find out," math becomes an arvedntue. And parents, this goes for you too. When you sit down to do math with your children, you don't have to know all the answers. You can ask your child to explain the math to you or try to figure it out together. Teach them that not kniowng is not failure. It's the first step to understanding. So, when this group of students asked me if orange means even, I don't have to tell them the answer. I don't even need to know the answer. I can ask one of them to explain to me why she thinks it's true. Or we can tohrw the idea out to the class. Because they know the answers won't come from me, they need to convince themselves and argue with each other to determine what's true. And so, one student says, "Look, 2, 4, 6, 8, 10, 12. I cehkced all of the even nrubems. They all have orange in them. What more do you want?" And another student says, "Well, wait a minute, I see what you're saying, but some of those numbers have one orange piece, some have two or three. Like, look at 48. It's got four orange pieces. Are you telling me that 48 is four times as even as 46? There must be more to the story." By refusing to be the answer key, you create space for this kind of mathematical conversation and debate. And this draws everyone in because we love to see poelpe disagree. After all, where else can you see real thinking out loud? sntedtus doubt, affirm, deny, unadrntesd. And all you have to do as the teacher is not be the answer key and say "yes" to their ideas. And that is principle four. Now, this one is difficult. What if a student comes to you and says 2 plus 2 elquas 12? You've got to correct them, right? And it's true, we want students to understand certain basic facts and how to use them. But saying "yes" is not the same thing as saying "You're right." You can accept ideas, even wrong ideas, into the debate and say "yes" to your students' right to papricitate in the act of thinking mathematically. To have your idea dismissed out of hand is disempowering. To have it accepted, studied, and dveprosin is a mark of rpecest. It's also far more convincing to be sowhn you're wrong by your peers than told you're wrnog by the teacher. But allow me to take this a step further. How do you actually know that 2 plus 2 doesn't equal 12? What would happen if we said "yes" to that idea? I don't know. Let's find out. So, if 2 plus 2 equaled 12, then 2 plus 1 would be one less, so that would be 11. And that would mean that 2 plus 0, which is just 2, would be 10. But if 2 is 10, then 1 would be 9, and 0 would be 8. And I have to aimdt this looks bad. It looks like we broke mathematics. But I actually understand why this can't be true now.

Open Cloze

A friend of mine told me recently that her six-year-old son had come from school and said he _____ math. And this is hard for me to hear because I actually love math. The beauty and power of mathematical thinking have changed my life. But I know that many people lived a very different story. Math can be the best of times or the worst of times, an ____________ journey of discovery or _______ into ______, ___________, and despair. Mathematical miseducation is so common we can hardly see it. We practically expect math class to be repetition and memorization of __________ technical facts. And we're not surprised when students aren't motivated, when they leave school disliking math, even committed to avoiding it for the rest of their lives. Without mathematical literacy, their career _____________ shrink. And they become easy prey for credit card _________, payday _______, the _______, (Laughter) and anyone, really, who wants to dazzle them with a statistic. Did you know that if you ______ a single statistic into an assertion, people are 92 percent more likely to accept it without question? (Laughter) Yeah, I totally made that up. (________) And 92 percent is - it has weight even though it's completely fabricated. And that's how it _____. When we're not comfortable with math, we don't question the _________ of numbers. But what's happening with mathematical alienation is only half the story. Right now, we're ___________ our chance to touch life after life with the ______ and _____ of mathematical thinking. I led a workshop on this topic recently, and at the end, a woman raised her hand and said that the experience made her feel - and this is a quote - "like a God." (Laughter) That's maybe the best description I've ever heard for what mathematical thinking can feel like, so we should examine what it looks like. A good place to start is with the words of the philosopher and mathematician René Descartes, who famously proclaimed, "I think, therefore I am." But Descartes looked deeper into the nature of thinking. Once he established himself as a thing that ______, he continued, "What is a ________ thing?" It is the thing that doubts, understands, conceives, that _______ and denies, wills and refuses, that imagines also, and perceives. This is the kind of thinking we need in every math class every day. So, if you are a teacher or a parent or anyone with a stake in _________, I offer these five principles to invite thinking into the math we do at home and at school. Principle one: start with a question. The ordinary math class begins with answers and never arrives at a real question. "Here are the _____ to multiply. You ______. Here are the steps to divide. You repeat. We've covered the material. We're moving on." What matters in the model is memorizing the steps. There's no room to _____ or imagine or refuse, so there's no real thinking here. What would it look like if we started with a ________? For example, here are the numbers from 1 to 20. Now, there's a question _______ in this picture, hiding in plain sight. What's going on with the colors? Now, intuitively it _____ like there's some __________ between the numbers and the ______. I mean, maybe it's even possible to extend the coloring to more numbers. At the same time, the meaning of the colors is not _____. It's a real mystery. And so, the question feels authentic and compelling. And like so many authentic ____________ questions, this one has an answer that is both beautiful and profoundly satisfying. And of course, I'm not going to tell you what it is. (Laughter) I don't think of myself as a mean person, but I am willing to deny you what you want. (Laughter) Because I know if I rush to an answer, I would've ______ you of the ___________ to learn. Thinking happens only when we have time to struggle. And that is principle two. It's not uncommon for students to ________ from high school believing that every math _______ can be solved in 30 seconds or less, and if they don't know the answer, they're just not a math person. This is a failure of education. We need to teach kids to be _________ and courageous, to _________ in the face of difficulty. The only way to _____ perseverance is to give students time to think and grapple with real problems. I brought this _____ into a classroom recently, and we took the time to struggle. And the longer we spent, the more the class came alive with thinking. The students made observations. They had questions. Like, "Why do the numbers in that last column always have orange and blue in them?" and "Does it mean anything that the green spots are always going diagonally?" and "What's going on with those little white numbers in the red segments? Is it important that those are always odd numbers?" Struggling with a _______ question, students ______ their curiosity and their ______ of observation. They also develop the ability to take a risk. Some students noticed that every even number has ______ in it, and they were willing to stake a claim. "Orange must mean even." And then they asked, "Is that right?" (Laughter) This can be a scary place as a teacher. A _______ comes to you with an original thought. What if you don't know the ______? Well, that is _________ three: you are not the answer key. Teachers, students may ask you questions you don't know how to answer. And this can feel like a threat. But you are not the answer key. Students who are ___________ is a wonderful thing to have in your classroom. And if you can _______ by saying, "I don't know. Let's find out," math becomes an _________. And parents, this goes for you too. When you sit down to do math with your children, you don't have to know all the answers. You can ask your child to explain the math to you or try to figure it out together. Teach them that not _______ is not failure. It's the first step to understanding. So, when this group of students asked me if orange means even, I don't have to tell them the answer. I don't even need to know the answer. I can ask one of them to explain to me why she thinks it's true. Or we can _____ the idea out to the class. Because they know the answers won't come from me, they need to convince themselves and argue with each other to determine what's true. And so, one student says, "Look, 2, 4, 6, 8, 10, 12. I _______ all of the even _______. They all have orange in them. What more do you want?" And another student says, "Well, wait a minute, I see what you're saying, but some of those numbers have one orange piece, some have two or three. Like, look at 48. It's got four orange pieces. Are you telling me that 48 is four times as even as 46? There must be more to the story." By refusing to be the answer key, you create space for this kind of mathematical conversation and debate. And this draws everyone in because we love to see ______ disagree. After all, where else can you see real thinking out loud? ________ doubt, affirm, deny, __________. And all you have to do as the teacher is not be the answer key and say "yes" to their ideas. And that is principle four. Now, this one is difficult. What if a student comes to you and says 2 plus 2 ______ 12? You've got to correct them, right? And it's true, we want students to understand certain basic facts and how to use them. But saying "yes" is not the same thing as saying "You're right." You can accept ideas, even wrong ideas, into the debate and say "yes" to your students' right to ___________ in the act of thinking mathematically. To have your idea dismissed out of hand is disempowering. To have it accepted, studied, and _________ is a mark of _______. It's also far more convincing to be _____ you're wrong by your peers than told you're _____ by the teacher. But allow me to take this a step further. How do you actually know that 2 plus 2 doesn't equal 12? What would happen if we said "yes" to that idea? I don't know. Let's find out. So, if 2 plus 2 equaled 12, then 2 plus 1 would be one less, so that would be 11. And that would mean that 2 plus 0, which is just 2, would be 10. But if 2 is 10, then 1 would be 9, and 0 would be 8. And I have to _____ this looks bad. It looks like we broke mathematics. But I actually understand why this can't be true now.

Solution

  1. power
  2. laughter
  3. clear
  4. equals
  5. principle
  6. wrong
  7. beauty
  8. companies
  9. education
  10. repeat
  11. feels
  12. numbers
  13. disjointed
  14. admit
  15. works
  16. image
  17. disproven
  18. problem
  19. exhilarating
  20. teach
  21. inquisitive
  22. respect
  23. frustration
  24. orange
  25. lurking
  26. thinks
  27. hated
  28. student
  29. descent
  30. steps
  31. adventure
  32. thinking
  33. persevere
  34. respond
  35. powers
  36. knowing
  37. checked
  38. shown
  39. lottery
  40. mathematical
  41. throw
  42. affirms
  43. genuine
  44. answer
  45. robbed
  46. people
  47. graduate
  48. participate
  49. lenders
  50. squandering
  51. doubt
  52. understand
  53. deepen
  54. question
  55. students
  56. opportunities
  57. opportunity
  58. insert
  59. tenacious
  60. tedium
  61. authority
  62. colors
  63. connection

Original Text

A friend of mine told me recently that her six-year-old son had come from school and said he hated math. And this is hard for me to hear because I actually love math. The beauty and power of mathematical thinking have changed my life. But I know that many people lived a very different story. Math can be the best of times or the worst of times, an exhilarating journey of discovery or descent into tedium, frustration, and despair. Mathematical miseducation is so common we can hardly see it. We practically expect math class to be repetition and memorization of disjointed technical facts. And we're not surprised when students aren't motivated, when they leave school disliking math, even committed to avoiding it for the rest of their lives. Without mathematical literacy, their career opportunities shrink. And they become easy prey for credit card companies, payday lenders, the lottery, (Laughter) and anyone, really, who wants to dazzle them with a statistic. Did you know that if you insert a single statistic into an assertion, people are 92 percent more likely to accept it without question? (Laughter) Yeah, I totally made that up. (Laughter) And 92 percent is - it has weight even though it's completely fabricated. And that's how it works. When we're not comfortable with math, we don't question the authority of numbers. But what's happening with mathematical alienation is only half the story. Right now, we're squandering our chance to touch life after life with the beauty and power of mathematical thinking. I led a workshop on this topic recently, and at the end, a woman raised her hand and said that the experience made her feel - and this is a quote - "like a God." (Laughter) That's maybe the best description I've ever heard for what mathematical thinking can feel like, so we should examine what it looks like. A good place to start is with the words of the philosopher and mathematician René Descartes, who famously proclaimed, "I think, therefore I am." But Descartes looked deeper into the nature of thinking. Once he established himself as a thing that thinks, he continued, "What is a thinking thing?" It is the thing that doubts, understands, conceives, that affirms and denies, wills and refuses, that imagines also, and perceives. This is the kind of thinking we need in every math class every day. So, if you are a teacher or a parent or anyone with a stake in education, I offer these five principles to invite thinking into the math we do at home and at school. Principle one: start with a question. The ordinary math class begins with answers and never arrives at a real question. "Here are the steps to multiply. You repeat. Here are the steps to divide. You repeat. We've covered the material. We're moving on." What matters in the model is memorizing the steps. There's no room to doubt or imagine or refuse, so there's no real thinking here. What would it look like if we started with a question? For example, here are the numbers from 1 to 20. Now, there's a question lurking in this picture, hiding in plain sight. What's going on with the colors? Now, intuitively it feels like there's some connection between the numbers and the colors. I mean, maybe it's even possible to extend the coloring to more numbers. At the same time, the meaning of the colors is not clear. It's a real mystery. And so, the question feels authentic and compelling. And like so many authentic mathematical questions, this one has an answer that is both beautiful and profoundly satisfying. And of course, I'm not going to tell you what it is. (Laughter) I don't think of myself as a mean person, but I am willing to deny you what you want. (Laughter) Because I know if I rush to an answer, I would've robbed you of the opportunity to learn. Thinking happens only when we have time to struggle. And that is principle two. It's not uncommon for students to graduate from high school believing that every math problem can be solved in 30 seconds or less, and if they don't know the answer, they're just not a math person. This is a failure of education. We need to teach kids to be tenacious and courageous, to persevere in the face of difficulty. The only way to teach perseverance is to give students time to think and grapple with real problems. I brought this image into a classroom recently, and we took the time to struggle. And the longer we spent, the more the class came alive with thinking. The students made observations. They had questions. Like, "Why do the numbers in that last column always have orange and blue in them?" and "Does it mean anything that the green spots are always going diagonally?" and "What's going on with those little white numbers in the red segments? Is it important that those are always odd numbers?" Struggling with a genuine question, students deepen their curiosity and their powers of observation. They also develop the ability to take a risk. Some students noticed that every even number has orange in it, and they were willing to stake a claim. "Orange must mean even." And then they asked, "Is that right?" (Laughter) This can be a scary place as a teacher. A student comes to you with an original thought. What if you don't know the answer? Well, that is principle three: you are not the answer key. Teachers, students may ask you questions you don't know how to answer. And this can feel like a threat. But you are not the answer key. Students who are inquisitive is a wonderful thing to have in your classroom. And if you can respond by saying, "I don't know. Let's find out," math becomes an adventure. And parents, this goes for you too. When you sit down to do math with your children, you don't have to know all the answers. You can ask your child to explain the math to you or try to figure it out together. Teach them that not knowing is not failure. It's the first step to understanding. So, when this group of students asked me if orange means even, I don't have to tell them the answer. I don't even need to know the answer. I can ask one of them to explain to me why she thinks it's true. Or we can throw the idea out to the class. Because they know the answers won't come from me, they need to convince themselves and argue with each other to determine what's true. And so, one student says, "Look, 2, 4, 6, 8, 10, 12. I checked all of the even numbers. They all have orange in them. What more do you want?" And another student says, "Well, wait a minute, I see what you're saying, but some of those numbers have one orange piece, some have two or three. Like, look at 48. It's got four orange pieces. Are you telling me that 48 is four times as even as 46? There must be more to the story." By refusing to be the answer key, you create space for this kind of mathematical conversation and debate. And this draws everyone in because we love to see people disagree. After all, where else can you see real thinking out loud? Students doubt, affirm, deny, understand. And all you have to do as the teacher is not be the answer key and say "yes" to their ideas. And that is principle four. Now, this one is difficult. What if a student comes to you and says 2 plus 2 equals 12? You've got to correct them, right? And it's true, we want students to understand certain basic facts and how to use them. But saying "yes" is not the same thing as saying "You're right." You can accept ideas, even wrong ideas, into the debate and say "yes" to your students' right to participate in the act of thinking mathematically. To have your idea dismissed out of hand is disempowering. To have it accepted, studied, and disproven is a mark of respect. It's also far more convincing to be shown you're wrong by your peers than told you're wrong by the teacher. But allow me to take this a step further. How do you actually know that 2 plus 2 doesn't equal 12? What would happen if we said "yes" to that idea? I don't know. Let's find out. So, if 2 plus 2 equaled 12, then 2 plus 1 would be one less, so that would be 11. And that would mean that 2 plus 0, which is just 2, would be 10. But if 2 is 10, then 1 would be 9, and 0 would be 8. And I have to admit this looks bad. It looks like we broke mathematics. But I actually understand why this can't be true now.

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
mathematical thinking 5
math class 3
answer key 3
credit card 2
real thinking 2
authentic mathematical 2
takes courage 2

Important Words

  1. ability
  2. accept
  3. accepted
  4. act
  5. admit
  6. adventure
  7. affirm
  8. affirms
  9. alienation
  10. alive
  11. answer
  12. answers
  13. argue
  14. arrives
  15. asked
  16. assertion
  17. authentic
  18. authority
  19. avoiding
  20. bad
  21. basic
  22. beautiful
  23. beauty
  24. begins
  25. believing
  26. blue
  27. broke
  28. brought
  29. card
  30. career
  31. chance
  32. changed
  33. checked
  34. child
  35. children
  36. claim
  37. class
  38. classroom
  39. clear
  40. coloring
  41. colors
  42. column
  43. comfortable
  44. committed
  45. common
  46. companies
  47. compelling
  48. completely
  49. conceives
  50. connection
  51. continued
  52. conversation
  53. convince
  54. convincing
  55. correct
  56. courageous
  57. covered
  58. create
  59. credit
  60. curiosity
  61. day
  62. dazzle
  63. debate
  64. deepen
  65. deeper
  66. denies
  67. deny
  68. descartes
  69. descent
  70. description
  71. despair
  72. determine
  73. develop
  74. diagonally
  75. difficult
  76. difficulty
  77. disagree
  78. discovery
  79. disempowering
  80. disjointed
  81. disliking
  82. dismissed
  83. disproven
  84. divide
  85. doubt
  86. doubts
  87. draws
  88. easy
  89. education
  90. equal
  91. equaled
  92. equals
  93. established
  94. examine
  95. exhilarating
  96. expect
  97. experience
  98. explain
  99. extend
  100. fabricated
  101. face
  102. facts
  103. failure
  104. famously
  105. feel
  106. feels
  107. figure
  108. find
  109. friend
  110. frustration
  111. genuine
  112. give
  113. god
  114. good
  115. graduate
  116. grapple
  117. green
  118. group
  119. hand
  120. happen
  121. happening
  122. hard
  123. hated
  124. hear
  125. heard
  126. hiding
  127. high
  128. home
  129. idea
  130. ideas
  131. image
  132. imagine
  133. imagines
  134. important
  135. inquisitive
  136. insert
  137. intuitively
  138. invite
  139. journey
  140. key
  141. kids
  142. kind
  143. knowing
  144. laughter
  145. learn
  146. leave
  147. led
  148. lenders
  149. life
  150. literacy
  151. lived
  152. lives
  153. longer
  154. looked
  155. lottery
  156. loud
  157. love
  158. lurking
  159. mark
  160. material
  161. math
  162. mathematical
  163. mathematically
  164. mathematician
  165. mathematics
  166. matters
  167. meaning
  168. means
  169. memorization
  170. memorizing
  171. minute
  172. miseducation
  173. model
  174. motivated
  175. moving
  176. multiply
  177. mystery
  178. nature
  179. noticed
  180. number
  181. numbers
  182. observation
  183. observations
  184. odd
  185. offer
  186. opportunities
  187. opportunity
  188. orange
  189. ordinary
  190. original
  191. parent
  192. parents
  193. participate
  194. payday
  195. peers
  196. people
  197. perceives
  198. percent
  199. perseverance
  200. persevere
  201. person
  202. philosopher
  203. picture
  204. piece
  205. pieces
  206. place
  207. plain
  208. power
  209. powers
  210. practically
  211. prey
  212. principle
  213. principles
  214. problem
  215. problems
  216. proclaimed
  217. profoundly
  218. question
  219. questions
  220. quote
  221. raised
  222. real
  223. red
  224. refuse
  225. refuses
  226. refusing
  227. rené
  228. repeat
  229. repetition
  230. respect
  231. respond
  232. rest
  233. risk
  234. robbed
  235. room
  236. rush
  237. satisfying
  238. scary
  239. school
  240. seconds
  241. segments
  242. shown
  243. shrink
  244. sight
  245. single
  246. sit
  247. solved
  248. son
  249. space
  250. spent
  251. spots
  252. squandering
  253. stake
  254. start
  255. started
  256. statistic
  257. step
  258. steps
  259. story
  260. struggle
  261. struggling
  262. student
  263. students
  264. studied
  265. surprised
  266. teach
  267. teacher
  268. teachers
  269. technical
  270. tedium
  271. telling
  272. tenacious
  273. thinking
  274. thinks
  275. thought
  276. threat
  277. throw
  278. time
  279. times
  280. told
  281. topic
  282. totally
  283. touch
  284. true
  285. uncommon
  286. understand
  287. understanding
  288. understands
  289. wait
  290. weight
  291. white
  292. wills
  293. woman
  294. wonderful
  295. words
  296. works
  297. workshop
  298. worst
  299. wrong
  300. yeah