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Mile wide –inch deep

       causes

        cures



                       2011 © New Leaders | 1
Mile wide –inch deep
  cause:
too little time per concept
  cure:
more time per topic
            = less topics
                              2011 © New Leaders | 2
Two ways to get less topics:
1. Delete topics
2. Coherence: when studied a little
   deeper, mathematics is a lot more coherent
  a) coherence across concepts
  b) coherence in the progression across grades
Why do students have to do
     math problems?

  A. To get answers because Homeland Security
     needs them, pronto

  B. I had to, why shouldn’t they?

  C. So they will listen in class

  D. To learn mathematics
Why give students problems
               to solve?
• To learn mathematics
• Answers are part of the process, they are not the product
• The product is the student’s mathematical knowledge and
  know-how
• The ‘correctness’ of answers is also part of the process:
  Yes, an important part
Wrong answers
• Are part of the process, too
• What was the student thinking?
• Was it an error of haste or a stubborn
  misconception?
Three responses to a math problem

1. Answer getting
2. Making sense of the problem/situation
3. Making sense of the mathematics by learning to
   work through the problem
Answers are a black hole:
      hard to escape the pull
• Answer getting short circuits
  mathematics, and lacks mathematical sense
• Very habituated in U.S. teachers versus
  Japanese teachers
• Devised methods for slowing
  down, postponing answer getting
Answer getting vs. learning
            mathematics
U.S.:
• How can I teach my kids to get the answer to
  this problem?
  Use mathematics they already know – this is
   easy, reliable, works with bottom half, good for
   classroom management
Japan:
• How can I use this problem to teach the
  mathematics of this unit?
Butterfly method
Use butterflies on this TIMSS item:



1/2 + 1/3 +1/4 =
Mile wide –inch deep

       causes

        cures



                       2011 © New Leaders | 13
Mile wide –inch deep
  cause:
too little time per concept
  cure:
more time per topic
            = less topics
                              2011 © New Leaders | 14
Two ways to get less topics:
1. Delete topics
2. Coherence: when studied a little
   deeper, mathematics is a lot more coherent
  a) coherence across concepts
  b) coherence in the progression across grades
Why do students have to do
     math problems?

  A. To get answers because Homeland Security
     needs them, pronto

  B. I had to, why shouldn’t they?

  C. So they will listen in class

  D. To learn mathematics
Why give students problems
               to solve?
• To learn mathematics
• Answers are part of the process, they are not the product
• The product is the student’s mathematical knowledge and
  know-how
• The ‘correctness’ of answers is also part of the process:
  Yes, an important part
Wrong answers
• Are part of the process, too
• What was the student thinking?
• Was it an error of haste or a stubborn
  misconception?
Three responses to a math problem

1. Answer getting
2. Making sense of the problem/situation
3. Making sense of the mathematics by learning to
   work through the problem
Answers are a black hole:
      hard to escape the pull
• Answer getting short circuits
  mathematics, and lacks mathematical sense
• Very habituated in U.S. teachers versus
  Japanese teachers
• Devised methods for slowing
  down, postponing answer getting
Answer getting vs. learning
            mathematics
U.S.:
• How can I teach my kids to get the answer to
  this problem?
  Use mathematics they already know – this is
   easy, reliable, works with bottom half, good for
   classroom management
Japan:
• How can I use this problem to teach the
  mathematics of this unit?
Butterfly method
Use butterflies on this TIMSS item:



1/2 + 1/3 +1/4 =

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Daro CC Alg presentation

  • 1. Mile wide –inch deep causes cures 2011 © New Leaders | 1
  • 2. Mile wide –inch deep cause: too little time per concept cure: more time per topic = less topics 2011 © New Leaders | 2
  • 3. Two ways to get less topics: 1. Delete topics 2. Coherence: when studied a little deeper, mathematics is a lot more coherent a) coherence across concepts b) coherence in the progression across grades
  • 4. Why do students have to do math problems? A. To get answers because Homeland Security needs them, pronto B. I had to, why shouldn’t they? C. So they will listen in class D. To learn mathematics
  • 5. Why give students problems to solve? • To learn mathematics • Answers are part of the process, they are not the product • The product is the student’s mathematical knowledge and know-how • The ‘correctness’ of answers is also part of the process: Yes, an important part
  • 6. Wrong answers • Are part of the process, too • What was the student thinking? • Was it an error of haste or a stubborn misconception?
  • 7. Three responses to a math problem 1. Answer getting 2. Making sense of the problem/situation 3. Making sense of the mathematics by learning to work through the problem
  • 8. Answers are a black hole: hard to escape the pull • Answer getting short circuits mathematics, and lacks mathematical sense • Very habituated in U.S. teachers versus Japanese teachers • Devised methods for slowing down, postponing answer getting
  • 9. Answer getting vs. learning mathematics U.S.: • How can I teach my kids to get the answer to this problem? Use mathematics they already know – this is easy, reliable, works with bottom half, good for classroom management Japan: • How can I use this problem to teach the mathematics of this unit?
  • 11.
  • 12. Use butterflies on this TIMSS item: 1/2 + 1/3 +1/4 =
  • 13. Mile wide –inch deep causes cures 2011 © New Leaders | 13
  • 14. Mile wide –inch deep cause: too little time per concept cure: more time per topic = less topics 2011 © New Leaders | 14
  • 15. Two ways to get less topics: 1. Delete topics 2. Coherence: when studied a little deeper, mathematics is a lot more coherent a) coherence across concepts b) coherence in the progression across grades
  • 16. Why do students have to do math problems? A. To get answers because Homeland Security needs them, pronto B. I had to, why shouldn’t they? C. So they will listen in class D. To learn mathematics
  • 17. Why give students problems to solve? • To learn mathematics • Answers are part of the process, they are not the product • The product is the student’s mathematical knowledge and know-how • The ‘correctness’ of answers is also part of the process: Yes, an important part
  • 18. Wrong answers • Are part of the process, too • What was the student thinking? • Was it an error of haste or a stubborn misconception?
  • 19. Three responses to a math problem 1. Answer getting 2. Making sense of the problem/situation 3. Making sense of the mathematics by learning to work through the problem
  • 20. Answers are a black hole: hard to escape the pull • Answer getting short circuits mathematics, and lacks mathematical sense • Very habituated in U.S. teachers versus Japanese teachers • Devised methods for slowing down, postponing answer getting
  • 21. Answer getting vs. learning mathematics U.S.: • How can I teach my kids to get the answer to this problem? Use mathematics they already know – this is easy, reliable, works with bottom half, good for classroom management Japan: • How can I use this problem to teach the mathematics of this unit?
  • 23.
  • 24. Use butterflies on this TIMSS item: 1/2 + 1/3 +1/4 =