My biggest professional breakthrough this last year was to understand that every idea in mathematics can be appreciated, understood, and practiced both formally and also informally.
In this activity, students first use their informal home language to describe how the red point turns into the blue point. Then, more formally, I ask them to predict where I’ll find the blue point given an arbitrary red point. Finally, and most formally, I ask them to describe the rule in algebraic notation. Answer: (a, b) -> (a/2, b/2).
It’s always harder for me to locate the informal expression of a idea than the formal. That’s for a number of reasons. It’s because I learned the formal most recently. It’s because the formal is often easier to assess, and easier for machines to assess especially. It’s because the formal is often more powerful than the informal. Write the algebraic rule and a computer can instantly locate the blue point for any red point. Your home language can’t do that.
But the informal expressions of an idea are often more interesting to students, if for no other reason than because they diversify the work students do in math and, consequently, diversify the ways students can be good at math.
The informal expressions aren’t just interesting work but they also make the formal expressions easier to learn. I suspect the evidence will be domain specific, but I look to Moschkovich’s work on the effect of home language on the development of mathematical language and Kasmer’s work on the effect of estimation on the development of mathematical models.
- Before I ask for a formal algebraic rule, I ask for an informal verbal rule.
- Before I ask for a graph, I ask for a sketch.
- Before I ask for a proof, I ask for a conjecture.
- David Wees: Before I ask for conjectures, I ask for noticings.
- Before I ask for a calculation, I ask for an estimate.
- Before I ask for a solution, I ask students to guess and check.
- Bridget Dunbar: Before I ask for algebra, I ask for arithmetic.
- Jamie Duncan: Before I ask for formal definitions, I ask for informal descriptions.
- Abe Hughes: Before I ask for explanations, I ask for observations.
- Maria Reverso: Before I ask for standard algorithms, I ask for student-generated algorithms.
- Maria Reverso: Before I ask for standard units, I ask for non-standard units.
- Kent Haines: Before I ask for definitions, I ask for characteristics.
- Andrew Knauft: Before I ask for answers in print, I ask for answers in gesture.
- Avery Pickford: Before I ask for complete mathematical propositions, I ask for incomplete propositions.
- Dan Finkel: Before I ask for the general rule, I ask for a specific instance of the rule.
- Dan Finkel: Before I ask for the literal, I ask for an analogy.
- Kristin Gray: Before I ask for quadrants, I ask for directional language.
- Jim Murray: Before I ask for algorithms, I ask for patterns.
- Nicola Vitale: Before I ask for proofs, I ask for conjectures, questions, wonderings, and noticings.
- Natalie Cogan: Before I ask for an estimation, I ask for a really big and really small estimation.
- Julie Conrad: Before I ask for reasoning, I ask them to play/tinker.
- Eileen Quinn Knight: Before I ask for algorithms, I ask for shorthand.
- Bill Thill: Before I ask for definitions, I ask for examples and non-examples.
- Larry Peterson: Before I ask for symbols, I ask for words.
- Andrew Gael: Before I ask for “regrouping” and “borrowing,” I ask for grouping by tens and place value.
At this point, I could use your help in three ways:
- Offer more shades between informal and formal for the blue dot task. (I offered three.)
- Offer more SAT-style analogies. sketch : graph :: estimate : calculation :: [your turn]. That work has begun on Twitter.
- Or just do your usual thing where you talk amongst yourselves and let me eavesdrop on the best conversation on the Internet.
BTW. I’m grateful to Jennifer Wilson and her post which lodged the idea of a secret algebraic rule in my head.
David Wees reminds us that the van Hiele’s covered some of this ground already.