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Further Beyond Sudoku: Using Logic Puzzles to Develop Mathematical Reasoning

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Further Beyond Sudoku:Using Logic Puzzles to Develop Mathematical Reasoning

Breedeen Pickford-Murray

The Bay School of San Francisco

A proof, that is, a mathematical argument, is [like] a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly…

-Paul Lockhart

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

How Do You Know What to Do?

How Do You Record Your Thinking?

How Do You Convince Your Peers?

- New Ideas
- Engaging Format
- Intuitive Yet Not Obvious

UnsolvedSolved

What is the goal of this puzzle?

What are the rules?

http://www.nikoli.com/en/puzzles/yajilin/

How Do You Know What to Do?

Make sense of problems and persevere in solving them.

Look for and make use of structure.

UnsolvedSolved

http://www.nikoli.com/en/puzzles/yajilin/

UnsolvedSolved

What is the goal of this puzzle?

What are the rules?

http://www.nikoli.com/en/puzzles/slitherlink/

How Do You Record Your Thinking?

Reason abstractly and quantitatively.

Look for and express regularity in repeated reasoning.

UnsolvedSolved

http://www.nikoli.com/en/puzzles/slitherlink/

Five NCTM participants ran a race.

- Sadie came two places behind Christopher, but did not come in last.
- Justin lost to Ashli, but beat Shauna.
- Christopher did not come in first, Sadie did not come in last.
- What place did each person come in?
- Explain how you solved this puzzle, and prove that your answer is the only one that will work, using the statements above to support your argument.

How Do You Convince Your Peers?

Construct viable arguments and critique the reasoning of others.

Five NCTM participants ran a race.

- Sadie came two places behind Christopher, but did not come in last.
- Justin lost to Ashli, but beat Shauna.
- Christopher did not come in first, Sadie did not come in last.
- What place did each person come in?
- Explain how you solved this puzzle, and prove that your answer is the only one that will work, using the statements above to support your argument.

How Do You Know What to Do?

How Do You Record Your Thinking?

How Do You Convince Your Peers?

The Life-Cycle of Mathematics—Avery Pickford

Indirect reasoning

Induction

Two-column

Parity

Visual

Axiomatic

Be skeptical

Respectfully challenge

Reflect

Hunt for counter-examples

Estimate

Bound

Contextualize

Collect data

Pattern-sniff

Record results

How Do You Convince Your Peers?

How Do You Record Your Thinking?

How Do You Know What to Do?

Wild Guess

Educated Guess

Conjecture

Proof

Theorem

- Mathematical reasoning isn’t simple.
- Start with something intuitive and engaging—but not obvious!
- Push students to explain their thinking—at all levels.
- Help students develop ways to record and organize their thinking.
- Build formal written structures separately from introducing new mathematical content.

Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.

-Benjamin Peirce

BreedeenPickford-Murray

The Bay School of San Francisco

@btwnthenumbers

The Space Between The Numbers

betweenthenumbers.wordpress.com

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