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Contemplate then calculate#

See also: teaching, teaching-mathematics, productive-failure

Resources: originators site, primer, PD PowerPoint

Mathematics pedagogical approach aiming to help students apply structural thinking. Avoiding the automatic application of procedures.

Simple summary/routine

  • Notice
  • Find calculation shortcut
  • Share and study shortcuts
  • Reflect on learning

Routine#

  1. Launch routine

    • Explain what, how, and why we're learning
    • Provide students with a calculation, expression, relationship to consider
    • Project the object for only 5-10 seconds and ask what do you notice

  2. Notice

    • Shift focus away from calculation to interpreting the object
    • Perhaps an orienting question
    • solo thinking (5-10 seconds) and some initial sharing (15-30 seconds)
    • then shared with the group (2-3 minutes)

  3. Develop shortcut

    • working in pairs students try to develop a shortcut (2-5 minutes)

  4. Discuss shortcuts

    • shortcuts shared, discussion (5-10 minutes)
    • one pair explain
    • record the strategy to show evidence of properties, rules of operation, or relationships
    • prompt others to listen/look for how the pair used structure to find a short cut
    • provide short thinking time

    • Ask questions

      • How did this pair think about the task?
      • What did they pay attention to? How did that help them?
      • What properties, rules of operations, or relationships did they surface/levarge? How?

    • Ask others to explain why it's valid

  5. Reflect on your thinking

    • Re-introduce why and what was done here - explain we're doing a write/pair/share

    • Write (1-2 minutes) prompting students to think/write about what they've learned structurally

      Observe writing, encourage specificity

      • One way to surface hidden structure is to...
      • You can find calculation shortcuts by...
      • The next time I will... before I calculate because...
      • A property that comes in handy when developing a shortcut is.. because..

    • Pair

      • read written to a partner
      • identify reflections to share

    • Share

Possible problems#

  • Reliance on noticing

    There is a line of thought amongst some that this is somewhat fraught/problematic. But that's perhaps where prompting comes in

  • Preparation and enculturation

    Isn't going to happen over night. Leading up to it.