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?

Personally, I would show them how to punch 8% into their calculators and let the technology do the calculation in order to work on the key concept for the lesson. I would make a note of those students who were panicked about using percents and urge those students and anyone else who wanted or needed a brush-up on percents to come into the school’s math help centre for extra help. For those who did not show up, the next step would be a phone call home to advise parents about the situation.

Whew! Good analysis of the student understanding in this lesson. I’m going to try to respond, though it feels like you are ahead of me in this thinking:

1. On one level, it’s clear the students should not be learning compund interest when percent understanding is missing. On the surface level, though, they need to do the requested objective, and most teachers do not have the choice. So doing what you did, raising it as a place where percents are used is a good 2nd best, if you’re willing to talk where the students are at, instead of forcing the discussion to your objective.

2. You are differentiating! Providing the content needed in real time. I do like Gradual Release of Responsibility demonstrating, too. Providing demonstrations as needed, etc. Maybe there were students who just needed refresher, and students who needed the concept.

3. Tech is where my mind went with this. Take out the computation, and let students experiment with linear vs exponential, different starting amounts, different interest rates. Spreadsheets seem relevant here. Maybe that goes back to differentiation?

4. This is such a good context for getting the feel for exponential growth, especially the self-similarity part of it. Interest on your interest makes sense. Multiplicative thinking is so hard to people thinking additively, we need as many of these as we can get.

5. My play here is not to reteach percents, but share my understanding of them in the new context. “8%, or 8/100, means $8 for every $100 you have. …”

6. Teach the students we have. Such a hard lesson, that I seemingly need to work on it year in and year out. Since the students keep changing. :-)

As always, a great post Max. This semester, I am teaching a “low-level” 9th grade Algebra class, with students who are headed towards the PA Keystone exam in May. I’ve found that any most attempts to reach them in a non-worksheet-packet, non-rigid-notes method meets with acceptance and real questioning.

I absolutely endorse the use of technology for students to at least wrap their heads around a possible pattern. My suspicion is that many students are exposed to percents as a one-shot problem: i.e. “what perecent of 20 is 15?”. With just two data points, patterns aren’t developed. This isn’t a problem to the middle-school teacher, but manifests itself later when we ask students to understand a percent decrease as an eventual geometric progression. This is where departments which have a longitudinal view of math succeed in their goals.

Site like graphingstories.com and visualpatterns.com have been valuable as openers. I want my students to communicate math concepts in a non-threatening manner, and be able to share what they see.

I think your last point is the most important here. We can only go as far as the students we have in front of us are ready. If it doesn’t feel like they are ready for the next idea, even if it is in your plan, then you can’t go there. Conversely, we need to be ready to put our foot on the gas pedal if the kids seek more than what our plans provide.

Max —

Wait… were you teaching MY class today??? Because that’s what *I* was trying to teach too!!!!!!!!!!!!!!

I have more to say about this, but it will wait until tomorrow (I’m falling asleep). But I had to comment on the coincidence (conspiracy?) of our teaching the same exact problem at the same exact time.

- Elizabeth (@cheesemonkeysf)

As always, I agree with everything that John Golden said. ;)

My biggest problem, and I suspect yours too, is that there are basically only three approaches to dealing with gaps like this one: (1) you can avoid dealing with it; (2) you can spackle over it; or (3) you can tackle it by reteaching in some way that will help students to notice and become more aware of the gap and of their own need to take responsibility for filling it.

I have trouble accepting #1 or #2, so I find myself working this morning on ways to implement #3.

There were a bunch of things going on in my class (and I suspect in yours too) that were causes for confusion. The first was the percent-to-decimal conversion, which includes the decoding of the problem (from reading the symbols “8%” to a multiplier or factor of .08). The second was the “over and over again”-ness of the compounding process. The third was the weakness of their functional thinking process (input-output-input-output-input…).

Because I think these three pieces are fundamental to a conceptual understanding of exponential functions, I’m going to do a brief warm-up as a review on Monday to give students another swing at this. Here’s a link to my activity sheet (printed two-up to fit in an INB):

http://msmathwiki.pbworks.com/w/file/fetch/76107542/remediation%20-%20compound%20interest%202-UP.pdf

My goal was/is to set students up to see — and physically to experience — the over-and-over-again-ness of multiplicative reasoning. Based on my own experience as a math learner and on watching my students struggle, I think this is a bridge many learners need to be encouraged over.

- Elizabeth (@cheesemonkeysf)

Would it have been possible to create a bell ringer or warm up with 1-3 percent questions that would highlight the difference between .8 and .08 or common percent misconceptions? Could that be considered scaffolding up to the main concept of the lesson?

My number one way of reteaching concepts students should know but don’t is through my daily bell ringers. They are all middle school concepts that I present to my Geometry and Algebra II students at the beginning of class. I find this does 2 things. One, it’s a refresher of concepts they haven’t seen in 2-3 years. Two, it highlights concepts that have not been taught in middle school that should be. And of course the added bonus of something they can immediately start on (without me) at the beginning of class.