At Alex Overwijk and Mary Bourassa’s morning session at Twitter Math Camp last summer, we spent a great deal of time diving into what an activity-based, fully spiraled curriculum could look like — dispensing with units, and instead using activities to build understanding of a variety of standards, over time, with increasing depth. There are a ton of moving pieces to making this type of curriculum work, and I’m not in a place where I can implement a full-on spiraled curriculum and be confident in its success. I have taken some of their ideas and fit them into my current curriculum structure.
One idea Alex introduced was “unloading” content early in a course. Instead of treating content as sequential and in silos separate by unit tests, Alex introduces all of the relevant content early in the semester. There is no expectation that students instantly learn everything — instead, the course is spent circling back and deepening student understanding from multiple perspectives. Students know from the beginning where they are going and what the mathematics is building toward, and each activity asks as another round of formative assessment.
This tweet from Andrew Gael struck me as a great way to think about unloading content.
Quick aside: here is a study excerpted from Make It Stick on the benefits of spaced and interleaved practice in retention:
Two groups of college students were taught how to find the volumes of four obscure geometric solids (wedge, spheroid, spherical cone, and half cone). One group then worked a set of practice problems that were clustered by problem type (practice four problems for computing the volume of a wedge, then four problems for a spheroid, etc). The other group worked the same practice problems, but the sequence was mixed (interleaved) rather than clustered by type of problem. During practice, the students who worked the problems in clusters (that is, massed) averaged 89% correct, compared to only 60 percent for those who worked the problems in a mixed sequence. But in the final test a week later, the students who had practiced solving problems clustered by type averaged only 20% correct, while the students whose practice was interleaved averaged 63%. The mixing of problem types, which boosted final test performance by a remarkable 215 percent, actually impeded performance during initial learning (49-50).
I’ve been working on doing this at the level of an individual unit. I recently had two opportunities to teach a unit on exponential functions — one to my Algebra-II class, the other to a Pre-Calc class. Both units had more or less the same goals for student understanding:
- An exponential function models a situation where quantities are being multiplied by the same factor for each unit of time; the two parts of an exponential function are the initial value, and that common ratio
- Exponential functions sometimes introduce new inverse operations; a logarithm is the inverse of exponentiation and is used to solve for a variable in the exponent
I teach 90 minute block classes, and chose four tasks that I felt effectively introduced the content for the unit and I could do in one day.
$6400 Question (Dan)
This task takes a scene from Parks & Rec. Two characters are playing pool and bet $25. The losing player bets double or nothing. The scene cuts to several minutes later, when he has lost $6400. This task begs the question “how many games did they play”. While it is an effective opportunity to make use of an exponential model, it is also a great chance to model with a table, and experience that representation of a function.
Cookies Task (linked here)
There are 100 cookies on a plate. Each minute, students eat 5% of the remaining cookies. This introduces exponential decay, and deliberately contrasts it with a linear function. This task also uses percents, which are the most common situation where an exponential function is useful. Students are asked to model the situation with a function, and use that function to find when half the cookies will be gone. This task effectively addresses misconceptions about exponential vs linear change, and motivates logarithms, as the process of making a table is likely to be frustrating and inefficient in this situation.
Video Game Tournament (Illustrative Mathematics)
This task is again exponential decay, although a fractional ratio is more likely to be useful than a decimal in this task. Students are asked to interpret function notation including p(-1), and move between the abstract and concrete representations multiple times.
Power Task (Kate)
Introduces logarithms by asking students to look for patterns in several solved logarithms, labeled “power” instead of “log”, and then fill in the blanks in several more. I’ve found this a less confusing way to introduce logarithms, and give students some practice moving back and forth between logarithms and exponents in multiple ways.
I didn’t teach this, but I think it could work effectively, maybe in place of the Video Game Tournament task, next time.
These tasks cover almost all of the content of the unit. Two tougher pieces of content that I introduced later were solving for the base of an exponent using fractional exponents, and solving for when two exponential functions are equal. It’s worth noting that I cut natural logs and rules for manipulating logarithms from this unit to be able to focus more specifically on exponential functions and their applications. They’re worthwhile for study, but I wanted to focus this unit more specifically on key concepts.
Formative Assessment & Deliberate Practice
So now I’m in a pretty interesting position. I spent less than 90 minutes introducing some pretty complex content. As you can imagine, I have students in a variety of places. In Algebra-II, many students are still struggling to write exponential functions from a context, and the connection between logarithms and exponents is tenuous. In Pre-Calc, some students know their logarithms cold; others are great at writing and interpreting exponential functions, but are struggling with logarithms. The rest of the unit consists of a whole bunch of tasks, with more explicit instruction in bits and pieces as necessary, all chosen in response to students’ proficiency on each task. Here are the tasks I’ve used (although I’m still wrapping up the unit with my Algebra-II class):
Representing Linear and Exponential Growth (Shell Centre)
This task has an awesome card sort looking at different examples of simple and compound interest and linking different representations (table, graph, equation, description), as well as an intro worksheet that I found useful. I didn’t use the last part of the lesson — just the intro to simple vs compound interest, and the card sort.
Incredible Shrinking Dollar (Dan)
Can be a tricky example of exponential decay because students are given the percent each successive step is of the step before it, rather than the amount it decays by. Also a great time to use some estimation and intution.
Domino Skyscraper (Dan)
Students watch a series of dominos, each 1.5 times bigger than the last, get knocked over. How many would they need to knock over the Empire State Buidling? (and lots of other fun extensions)
Fry’s Bank (Dan)
Fry from Futurama goes to get money out of an account he hasn’t touched in 1000 years. How much money will he have? I didn’t do a ton to emphasize applications involving interest, so this was an important one.
Algal Blooms (Illustrative Mathematics)
Construct two exponential growth functions from fairly dense information, evaluate those functions, and use logarithms to solve problems.
Celebrity Gossip (Illustrative Mathematics)
This is a relatively straightforward task asking students to model a situation of exponential growth and provides an opportunity to practice using logarithms.
Snail Invasion (Illustrative Mathematics)
This is a fascinating task about smuggled Giant African Land Snails, asking students to write an exponential function from two points, use logarithms to make a calculation, and interpret the function in context.
DDT (Illustrative Mathematics)
This task asked students to interpret an exponential decay function in context with an interesting twist on what the input represents, and asks students to interpret the function using function notation.
Comparing Exponentials (Illustrative Mathematics)
Students analyze the structure of two different accounts earning interest, and find when they will be the same.
Uniqueness of Functions (Illustrative Mathematics)
Students construct linear, quadratic and exponential functions from two points and use these to compare the structures of the three types of functions.
Bouncing Tennis Balls (Shawn Cornally)
I used the image from his post — freeze frames of a tennis ball bouncing — and asked students how high the next bounce would be. It works well as an exponential decay function, and requires some measurement and careful calculation to come up with a model.
Log War (Kate)
Play war with logarithms! The best way I know to practice working with logs.
St Matthew Island (Stuart McMillen)
A fascinating comic about population growth and carrying capacity. This isn’t strictly a math problem, although we calculated the growth rates for the two periods (1944-1957 and 1957-1963) in the comic book to compare. I chopped up the comic into a set of slides, and tacked on some slides of Easter Island, which is an interesting parallel.
Logarithm Practice (linked here)
More practice connecting logarithms to exponents and evaluating them.
Loan Ranger (Mathalicious)
Examines how credit cards work, and how much you can end up paying if you only pay the minimum balance each month.
This is not meant to be exhaustive — there are plenty more great resources floating around that I could use. It’s not in any specific order. It’s also missing a lot of the connective tissue that makes these activities cohere into a curriculum — explanations, sketches, discussions, and whiteboard-based practice to supplement all of the activities.
I do have two conclusions to draw from this:
- I’ve been thinking a lot about curriculum recently, and I’m skeptical that curricular coherence is largely a function of the order I go through a unit. Going in a carefully scaffolded progression from easy to difficult may help students have an easier time in class, but I think it may also hide gaps in learning, create short term gains that disappear. Instead, I am thinking of coherence in terms of the focus — do the activities all focus on the big ideas of a unit, and get at them from a variety of perspectives to deepen students understanding. I think this set of lessons meets that second criteria, though I did not make an effort to meet the first.
- I’ve written a bunch recently about my ambition to build a free, modular, open-source curriculum. Going through this unit has made me a bit skeptical of my ability to do that. I’ve been thinling in terms of Elizabeth’s framework for learning from How People Learn, but I don’t know if the “introductory task” or “transfer task” is very likely to be the same from one teacher to another, or even from one class to another. I want to keep thinking about that, but there are definitely significant challenges to building any kind of consensus around what a lesson or unit progression looks like.