If you hang out with educators on a regular basis, then surely you have heard someone say a variation on, “My kids come to me not knowing stuff, and that makes it hard for me to teach them enough new stuff.” If you hang out with helpers-of-educators regularly, then you hear these helpful helpers saying things like, “Tough, you have to teach the kids you have.” And “Don’t remediate, accelerate.” and “You can’t go back and teach everything; if they’re behind it’s their responsibility to get extra help,” and “You have to differentiate and make your lesson work for everyone,” and a whole other mess of conflicting things. Because (duh) this is a hard problem. Personally, when I have a hard problem I like to look at a specific case, and guess what! One happened to me the other day. So let’s play “What would *you* do?”

I was guest-teaching a class that had been studying exponential growth (doubling and tripling, mostly), writing recursive and explicit rules, making tables, and graphing. Today, we were introducing the concept of compound interest as an exponential scenario, and the teacher’s goal was to have the kids recognize just that — compound interest can be modeled with exponential functions (and is cool and important, because free money!).

I was expecting that the hard bits would be the vocabulary of compound interest, the concept of investing, and the concept of the bank giving you more money each month because you have more money each month. And all of those things were hard for some subset of kids, but what was really hard for the majority of kids was doing mathematical operations with the quantity 8%.

Here’s the lesson plan:

- Pair kids and ask them to discuss, “You won the lottery and there are two prize options: would you rather have $10,000 now or $20,000 in ten years? And what would you do with the money?”
- Individuals answer the above question by making a graph of the money they’d have over the next 10 years based on their choice.
- Share a few different graphs on the document cam and discuss the story they tell.
- Pretend to get a call from a banker offering to hold the $10,000 in a CD earning 8% interest, compounded annually.
- Ask kids “What would you do now? Take $10,000 now, $20,000 in ten years, or invest the $10,000 in the 8% interest rate account for 10 years?”
- Define the question: will earning 8% interest on $10,000 over 10 years earn you more money than just waiting and letting the lottery folks give you $20,000 in ten years.
- Turn students loose to use a table (strongly encouraged), graph, or rule to help them answer that question, using their knowledge of percents.
- Come together as a class to compare & discuss tables, and convert tables to graphs and tables and graphs to rules to help us understand what’s going on in compound interest situations.
- Practice on another compound interest situation.

So it turned out that the students right away thought of investing. On their warm-up graphs, some saw investment as a linear scenario (I’ll get $1000 a year from investing in the stock market) and others saw their growth rate increasing as their money grew (exponential graphs).There was engaged discussion about spending vs. saving vs. investing, and more than one student knew what compounded annually meant when the banker called.

Here was the part I wasn’t quite prepared for. When 8% was mentioned, I right away heard fear about the use of percents, and students asking tentatively, “doesn’t that mean we have to divide something by 8 or zero-point-eight?” Maybe half the students knew that 8% was the decimal 0.08 (most went right away to 0.8, which sounds a lot like point-zero-eight but is a lot different!), and only one could state clearly that 8 percent was 8/100 which was 0.08, since 0.8 was 8/10 or 80/100. And worse, even once we’d established that 8% could be typed on a calculator as 0.08, more students wanted to divide than multiply (I guess because they wanted a smaller number)?

So the conversation about making the table wasn’t around what I was hoping — identifying those students who’d found 8% of the principal but forgot to add it to the principal and so who got 10,000, 800, 64, for the balance, and identifying those students who’d found that she made $800 in interest the first year and so reasoned she’d get $10,000; $10,800; $11,600; $12,400; etc. Then we could have talked about linear vs. exponential growth and what that has to do with compounding, and we could have talked about 8% vs. 108%, and gotten into some key ideas that would let us make sense of an exponential rule.

Instead we got bogged down in 8% vs. 80%, and number that didn’t make any sense, like $10,000 / 8 or .8 or .08. Even though we talked through those issues, I lost a lot of kids in talking through those issues, and we didn’t have very good conversations about the other, more connected-to-the-concept-of-exponential-growth conversations.

So, what would you have done, with 20/20 hindsight? Here are my questions:

- Can a class of students, the majority of whom don’t understand how to calculate with percents, learn compound interest? Should they be asked to? If so, how? If not, what should we do instead?
- Would there have been a way to differentiate this lesson so that more students got through the right amount of it at their own pace? What could I have expected everyone to have gotten? What could I have done to support the most struggling? The most advanced?
- Could technology have helped students focus on the concepts, rather than calculations? How? What technology?
- Is there a way to present this concept more conceptually, so we avoided getting bogged down in the calculations? Were the students conceptually ready but lacked fluency, or were there underlying conceptual issues I needed to address?
- How long does it take a good understanding of percents to sink in? Did I need to stop and teach a conceptual foundation for percents and then come back to compound interest? What would that do to my pacing guide (already pretty far behind with snow and 4 rounds of state testing instead of the 1-2 we used to do)?
- And finally, does thinking about this particular story give us any more insight into the general question of teaching kids who come in with gaps in their understanding, fluency, or both?