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Ender’s Math Ian Underwood 2010 Numeracy and Financial Education Summer Institute

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Suppose you're looking at some problem, P. A small hummingbird flaps its wings 78 times per second. 1.Estimate how many times the hummingbird would flap its wings in one minute. 2. Estimate how many times the hummingbird would flap its wings in one hour. [PoW #9235]

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It's _already_ linked to curriculum and standards, which is why you're looking at it!

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What is the abstract problem, A? (What is this problem about?) It's about accumulation at a constant rate, with a change in units. Total |. total = rate * units | units

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What kinds of things, related to money, accumulate in this way? How about an account with regular deposits? Or the cost of some commodity (gasoline, cans of tuna)? Or the cost of an item, with inflation?

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We can instantiate the abstract problem with these other items, P -> A -> Q1: rate is deposit per week, unit is months Q2: rate is price per ounce, unit is gallon Q3: rate is change in price of burger per year, unit is decades and presto! We have some 'finance problems'.

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Note that this is reversible. We could start with a 'finance problem’, and turn it into a problem that is 'about something different’, but still really the same problem.

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But wait! There's more. To use the vocabulary of 'notice and wonder': It's a 'scenario' until you ask a question. Then it's a 'problem'.

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We can identify the abstract scenario, S, and generate a cluster of related problems by changing the unknown, P -> A -> Q1: rate is deposit per week, | unit is months v Q2: rate is price per ounce, unit is gallons S Q3: rate is change in price | of burger per year, v unit is decades Qa: Given rate and total, find units Qb: Given total and units, find rate and we end up with 2*3 = 6 separate problems, all about 'finance'. Or, we can just work with the scenarios directly.

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Why do this? There are lots of problems out there already. Why re-invent them? These problems are already linked to curricula and standards. This approach reinforces the idea that abstraction makes math powerful as a tool. If done properly, it could ameliorate the usual difficulty of cross-domain transfer. And if done correctly, the students can do the heavy lifting.

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Ender's Math? "Knocking him down won the first fight. I wanted to win all the next ones, too. So they'd leave me alone."

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Conclusion: There's no such thing as 'financial math'. There's just math, applied to finance.

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Shall we give it a try? Things to think about: Is this feasible? How would we make it easier? What would have to happen to make this something that the students could do directly?

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