Both here and elsewhere, I have been part of a lively and robust discussion of the role of the *real-world* in mathematics instruction. One concern that is often expressed of instruction that is based in real-world problems is that of time and efficiency.

But sometimes it is really easy to turn the tables and make the standard word problem the motivational setup. An example.

In my College Algebra class, I started class with the following image the other day.

And I asked *What is there to wonder about here? What questions can we ask?* Here is what my students came up with.

- What is APY?
- What is the difference between Rate and APY? And what does this have to do with interest?
- How much does the principal grow over time in response to the rate?
- How does the amount of the deposit affect the growth over time?

These are precisely the questions I would hope for. But *they* asked them, not me.

In order to get to the answer to the question about the difference between APR and APY, we have to consider this one:

If you offer a 1.54% APR, how should you go about compounding interest monthly?

I have no idea how to get my students to the place where *they* can ask this question, so I asked it myself. Their ideas:

- Figure a year’s worth of interest in dollars, then divide that by 12 and add to the account each month, and
- Divide the interest rate by 12. Apply this new interest rate each month.

Brilliant! Next year, I’ll use (1) as part of a homework assignment-how do we run the computations when money gets added to account midway through the year? It gets very tricky very quickly.

But (2) is how we have agreed to run the computations in the financial world.

And it’s what accounts for the difference between APR and APY.

Now they were ready to hear about and derive interest formulas and to do some computations on their own.

A small bit of preparation put my students in a much better frame of mind for the material.

Sometimes it *is* that easy.

But not always.

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This reminds me of “Preparation for future learning through invention”, by Bransford and Schwartz. They did this in statistics as well as other math areas. In statistics, I think they showed students data of where baseball pitchers were pitching on a grid? I think they just asked the question, “Which pitchers were most consistent?” and students went about inventing solutions. These students were much better prepared to learn about standard deviation. But they could easily change this to, “What questions could we ask about this data?”

Thanks for the reference. The article is by Schwartz and Martin.

Direct link: http://www.jstor.org/stable/3233926

I have only read the summary, but I am also noticing that there may be relevance to a recent discussion over on Dan Meyer’s blog about

efficiencyof this sort of instruction.Efficiency must be considered in deciding to go for understanding. But surely the alternative is unacceptable: keep up a cracking pace and make sure that students are completing the assigned work and can remember the right procedures long enough to pass each exam. How DO we manage to develop understanding in as many students as possible, whilst simultaneously covering required content? I don’t have the answer, but it’s worth pursuing, I believe.

Yes. But we cannot throw the term

efficientaround as though we all agree on what it means. We do not, I think. Does it meanminimal wasting of time? If so, then efficiency is more of a classroom management issue than a curricular/instructional one. Is efficiency closely related totime on task? Does it refer to the ratio of topics: time? How are we to measure efficiency in the classroom?I have this conversation regularly with my close colleagues. I always come down on the side of

fewer big ideasas the guiding principle for instructional planning.Having taught Calculus last semester, I have been thinking a lot about what students really need to take away from College Algebra/Pre-Calculus in order to be successful in Calculus.

Perhaps this is worth a full-blown post, but it turns out that (IMHO), students really need to draw on a very small set of ideas once they get to Calculus.

Changeand various ways ofdescribing changealgebraically would be one of these. So students need experience with finite differences, ratios (in the case of exponentials) and the difference quotient before Calculus. They need to know the algebraic characteristics of the various function families. And they need to know quite a bit about the mathematical idea ofinverse.But they don’t need to solve systems of equations in three unknowns by hand. And they don’t need the Rational Roots Theorem, nor any of a dozen other little topics that consume these courses.

So for me the first step in balancing understanding with coverage is to make challenging and thoughtful choices about what to “cover”. And to ask ourselves hard questions about what “understanding” we hope to develop.

I agree with you, Peter, that these are fundamentally important questions. Thanks for asking them. I’m curious about your own thoughts.

Chris, as you put it, “fewer big ideas as the guiding principle for instructional planning,” is also one of the big messages from the NRC’s “Taking Science to School”, and is their central criticism of “standards-based” curriculum reform. We can’t demand that student must know 170 random fragmented science ideas.

But using fewer big ideas (in meaningful way) is difficult for many for many reasons. One reasons is that it requires a different view of the discipline than is usually afforded by education. You have to see the discipline as not a collection of topics, or chapters, or units, or whatever. And just because you see the discipline that way doesn’t mean you can meaningfully organize learning around that view.