responsive)teaching)in)elementary) mathemacs:)the)case)of...
TRANSCRIPT
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Responsive Teaching in Elementary Mathema6cs: The Case of Frac6ons
Susan Empson, The University of Texas at Aus6n Vicki Jacobs, University of North Carolina at Greensboro
Funded by the National Science Foundation
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• “Effec6ve mathema6cs teaching elicits evidence of student thinking to assess progress toward mathema6cal understanding and to adjust instruc6on con6nually in ways that support and extend learning”
– NCTM, Principles to Ac.on
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• Problem posed without demonstra6ng a specific strategy – To learn about students’ thinking and what they understand
The zoo keeper has 8 bananas to feed to the 6 monkeys. If she wants to use up all the bananas and give the same amount to each monkey, how much should she give each monkey?
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The zoo keeper has 8 bananas to feed to the 6 monkeys. If she wants to use up all the bananas and give the same amount to each monkey, how much should she give each monkey?
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Teaching that is responsive to children’s mathema6cal thinking
• In-‐the-‐moment decisions about what to pursue and how to pursue it, in response to students’ thinking
• Instruc6onal prac6ces centered on students’ thinking
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Responsive Teaching
Knowledge of Children’s Mathema:cal Thinking
Genera:ve Instruc:onal Prac:ces
No6cing children’s mathema6cal thinking (before responding)
Suppor6ng children’s mathema6cal thinking before the correct answer
Extending children’s mathema6cal thinking a2er the correct answer
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Responsive Teaching
Knowledge of Children’s Mathema:cal Thinking
Genera:ve Instruc:onal Prac:ces
No6cing children’s mathema6cal thinking (before responding)
Suppor6ng children’s mathema6cal thinking before the correct answer
Extending children’s mathema6cal thinking a2er the correct answer
![Page 8: Responsive)Teaching)in)Elementary) Mathemacs:)The)Case)of ...soe.uncg.edu/rtem/wp-content/uploads/2014/02/nctmHouston_2014.… · Responsive)Teaching)in)Elementary) Mathemacs:)The)Case)of)Frac6ons)](https://reader033.vdocument.in/reader033/viewer/2022052019/6032fec92babd52a6f73d8db/html5/thumbnails/8.jpg)
Responsive Teaching
Knowledge of Children’s Mathema:cal Thinking
Genera:ve Instruc:onal Prac:ces
No6cing children’s mathema6cal thinking (before responding)
Suppor6ng children’s mathema6cal thinking before the correct answer
Extending children’s mathema6cal thinking a2er the correct answer
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• Professional no6cing of children’s mathema6cal thinking (Jacobs et al., 2011) – AXending to details – Interpre6ng the understanding – Deciding how to respond
• Invisible prac6ce • Drives responsive teaching
No:cing
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Solve it!
On a field trip to the museum, 12 kids were given 16 pizzas to share equally. How much should each kid get?
…the way you think an elementary student might solve it
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Knowledge of Children’s Thinking: Types of Strategies
Beginning Understanding: Students are not ini6ally thinking about how to par66on the shared items with the number of sharers On a field trip to the museum, 12 kids were given 16 pizzas to share equally. How much should each kid get?
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Emergent Understanding: Coordina6on between par66ons of shared items with the number of sharers at the beginning of the strategy (addi6ve rela6onship) On a field trip to the
museum, 12 kids were given 16 pizzas to share equally. How much should each kid get?
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Advanced Understanding: Coordina6on between par66ons of shared items with the number of sharers at the beginning of the strategy (mul6plica6ve rela6onship) On a field trip to the
museum, 12 kids were given 16 pizzas to share equally. How much should each kid get?
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No6cing students’ thinking in wriXen work
Take some 6me to look over the student work from Brandon’s classmates and think about the following ques6ons: • How does each child solve the problem? • What does the strategy tell you about the child’s understanding?
• Bonus – What frac6on concepts could you address with this student work?
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The zookeeper had 8 bananas to feed 6 monkeys. If she wants to use up all of the bananas and give each monkey the same amount, how much should she give each monkey?
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No6cing allows a teacher to…
• Introduce and build conceptual understanding of frac6ons by building on children’s thinking – Frac6ons as quan66es – Wholes can be par66oned into parts – Parts are related to wholes – Frac6ons are related to division
• Address mul6ple content standards
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Responsive Teaching
Knowledge of Children’s Mathema:cal Thinking
Genera:ve Instruc:onal Prac:ces
No6cing children’s mathema6cal thinking (before responding)
Suppor6ng children’s mathema6cal thinking before the correct answer
Extending children’s mathema6cal thinking a2er the correct answer
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Suppor:ng and extending children’s thinking
• Ques6oning before and a2er a correct answer
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The zoo keeper has 8 bananas to feed to the 6 monkeys. If she wants to use up all the bananas and give the same amount to each monkey, how much should she give each monkey?
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Some possible ques6ons for Brandon
• How did you solve this problem? • What are these liXle squares? • Where are the last two bananas in your drawing? • Why did you decide to split the last two bananas each into
three parts? • How big is one of those parts?
– How much of a banana is it? – Is it more or less than ½ banana?
• How much does each monkey get? • Would you say this is true or false?
1⅓ = 1 + ⅓
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“S6ck with the kid”
These suppor6ng and extending ques6ons are centered on Brandon’s thinking. They… – Invite him to ar6culate and reflect on his ideas – Nudge him to advance his ideas
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How does the teacher respond to Ryan’s mathema6cal thinking?
There are 5 pizzas for 8 kids to share equally. How much pizza could each kid get?
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Teaching moves • Allowed plenty of 6me for Ryan to think
– Did not insist that he answer • In response to ini6al incorrect response, explored his thinking,
rather than redirect it • Invited Ryan to an6cipate final share at a strategic point:
– Do you know how much each person is going to get?
• Invited Ryan to compare his final share to a benchmark and reason about the rela6onship between ½ and ⅝ – Is it enough for each kid to have a whole pizza? How do you know? – Is it enough for each kid to have ½ pizza? How do you know? – How much more than ½ is ⅝? What would that frac.on size be?
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Suppor6ng and Extending: Exploring the details of a child’s strategy
Ø Press for detailed explana6on (even on incomplete strategies)
Ø Invite to an6cipate next step (at a strategic growth point)
Ø Invite to reflect on quan66es in the solu6on Ø compare to benchmarks Ø reason about rela6onships
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Suppor6ng and Extending During Instruc6on
• 4th/5th grade classroom • Spring of school year • Instruc6on: – Problems posed
• Differen6ated number selec6on, problems
– Teacher circulates – Student strategies shared and discussed as group
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Video Example
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Teaching moves • Asked a starter ques6on – Tell me about your picture
• Invited child to connect representa6on to quan66es in the problem – You’re saying that’s how much of that cup she drank? How far has she gone when she drinks that much?
– Do you want to put something down there so you can keep track of your miles?
• Invited child to reflect on quan66es in the strategy – Would she have any le2?
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Suppor6ng and Extending: Exploring the details of a child’s strategy
Ø Ask a starter ques6on Ø Press for detailed explana6on (even on incomplete strategies)
Ø Invite to connect representa6on to quan66es in the problem
Ø Invite to reflect on quan66es in the solu6on Ø compare to benchmarks Ø reason about rela6onships
Ø Invite to an6cipate next step
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The power of ques6oning that s6cks with the kid