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Heuristic Refutation Feedback – �Affordances for Proof Comprehension

Jason Cooper & Alon Pinto

Weizmann Institute of Science, Israel

Symposium ITEM 2022

Innovation on Teaching Mathematics at HEI: Experiences on Classroom

Tenerife, March 15th – 18th, 2022

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The Plan

  • First, a story
  • Theoretical framing
  • Heurisict Proof Refutation
  • Some more examples
  • Implications

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An illustrative story

  • Prove: Any non-empty subset of the integers has a mínimum.
  • Proof by induction on the size of the subset:
    • Trivial for a subset with a single element.
    • For n+1, find the minumum of any n elements
    • Now find the minumum of a set of two.
  • This cannot be correct!
    • Same proof “works” for ‘maximum’
    • False implication

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What is this an example of?

Feedback to develop proof-comprehension

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Feedback on students’ flawed proofs

  • Mathematics professors expect students to comprehend proofs.
    • What is wrong with proof?
    • How to fix it?
    • Why?
  • When feedback is prescriptive, students can rarely explain what is wrong with their proof or why it needs to be revised.
  • When feedback is not prescriptive, students can rarely revise proofs as expected.
  • A new kind of feedback is called for.

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Heuristic refutation in �mathematics education

  • Mathematical knowledge can grow through cycles of proof and refutation.
  • Didactic affordances of heuristic refutation have been studied.
  • Refutation feedback: Argument (possibly incomplete) that “proves” the proof is invalid
    • Argument consists (at least) of claim, datum, warrant.
  • Can refutation feedback on flawed proof engage students in heuristic refutation for the development of proof-comprehension?

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A framework of Proof Comprehension

Local  

1. Meaning of terms and statements

1.1 Identify examples [and non-examples] that illustrate a term or statement in the proof [or in the feedback].

1.2 State [or recognize] a given term or a statement in a different but equivalent form.

1.3 Identify trivial implications of a statement in the proof [or in the feedback].

2. Proof framework / Logical status of statements

2.1 Identify the type of proof [or refutation] framework.

2.2 Identify the purpose of a sentence within a proof [or refutation] framework.

3. Justification of claims

3.1 Make explicit an implicit warrant in the proof [or in the refutation].

3.2 Identify the specific data supporting a claim in the proof [or in the refutation].

3.3 Identify the specific claims that are supported by specific data in the proof [or in the refutation].

Holistic

 

4. Modular structure

4.1 Identify the purpose [or limitations] of a module.

5. Transferring ideas or methods

5.1 Transfer the method to a different context.

5.2 Identify the method of the proof.

5.3 Appreciate the scope of the method.

6. Illustrating with examples

6.1 Illustrate sequence of inferences with examples.

7. Summarizing via high-level ideas

7.1 Identify or provide a good summary of the proof.

7.2 Identify or provide a good summary of a key sub-proof.

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Introducing: �Heruistic Refutation Feedback

  • Proof comprehension grows through heuristic engagement with feedback.
    • Making sense of the refutation argument
    • Reviewing the proof in light of the refutation to locate where/why it fails
    • Local or holistic revision?
    • Attempt to revise proof accordingly.

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Some detailed examples

  • Proposition
  • Common flaws in proof
  • Variety of RF, and their heuristic nature
  • Facets of proof comprehension that are implicated

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Example 1

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Example 1 – RF, flaw 1

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Example 1 – RF, flaw 2

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Example 1 – facets of proof comprehension

  • Local comprehenstion
    • 1.1 Meaning of 1-1 map
    • 1.3 Ill-defined map is not 1-1
    • 2.1 Figure out the refutation framework
    • 2.2 Figure out the refutation claim that is implied by provided datum
    • 3.1 Complete missing warrant in refutation argument
    • 3.2 Construct counter-example
    • 3.3 Reconstruct refutation claim
  • Holistic comprehension
    • 6.1 Illustrate failure of proof with example

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Example 2 – Refutation feedback on warranted claim

Statement: Root 18 is closer to 4 than 5 because 18 is closer to 16 than to 25.

    • Student’s claim: Root 18 is closer to 4 than 5
    • Student’s datum: 18 is closer to 16 than to 25
    • Student’s warrant: missing.
  • Refutation feedback: The same argument would show that √20.4 is closer to 4 than it is to 5, yet it is not!
  • The refutation implicitly attributes a warrant to the proof
    • For every 16≤x≤25, if x is closer to 16 than 25, then √x is closer to 4 than to 5.

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Example 2 – Refutation feedback on warranted claim

  • Engagement with proof comprehension
    • 2.1 What is the refutation “framework”?
    • 3.1 What warrant is attributed to the proof?
    • 3.2 Data (counter-example 20.4) could have been omitted
    • 5.1 What if x is closer to 25 than it is to 16?

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