Every year I start a calculus course, I try to get students to really grapple with the weirdness of rates. They’re often unintuitive, and the misconceptions surrounding rates are similar to those misconceptions that pertain to Newtonian physics, units, and Graphs (so clearing them up is a double-win)
A quick primer on misconceptions analysis, or at least my arm-chair researcher version of it, which I pretty much just stole from Veritasium and then spun with a bit of my own experience.
- You have to tee up the misconceptions. This does not look like simply saying the misconception followed by the “correct” model. This does look like getting the students to display their dizzying array of pre(mis)conceptions. Use whiteboards, video tape each other and compile it, and come to a consensus–even if it’s the wrong consensus.
- You have to provide an experience that frustrates and confuses. This is based on some awesome neuroscience; basically, the human brain is not like a computer file system. When you edit the file on a computer, the magnetic dipole or electronic domain are flipped with an application of current. Human memories are repetitive, sensory-correlated, and plastic. You have to have the file open, and you have to chisel at it repetitively from different angles, and the misconceptions will often spend some time living in bizarre discord with the more appropriate model. This is ok, if not somewhat drawn out.
- You have to measure the extensibility of the new model. Can the student take the new model on the road? Is their understanding of rates limited to iterations of meters and seconds, or can they take that and apply it to dollars per volume? Have you measured their abstraction level?
The core misconception students have about rates is that the value of a function and the value of its rate are correlated. That is, if a function has value 7.3, the rate will also be something like 7.3.
Here are some humble ways I attempt to tee-up, and frustrate with regard to this misconception:
Break out the cheap-o whiteboards and have the students pick randomly two items from the following list. The first they pick is the independent variable, the second is the dependent. (we use a D20, cuz winning)
- Twitter Followers
- Time in Weight Room
- Holes in Swiss Cheese
- Smartphones Sold
- Coolness of the acronym “YOLO”
- Squid Tentacles
- Chance at State Championship
- Time Wearing the One Ring
- Surface Area
- Goose Droppings
The students then have to draw a curve and tell the story of their curve. The conversation then shifts to what the rates of those curves mean, are there points we care about? Is the situation getting better, worse?
I then have them draw and tell the story of the speed that their quantities are changing at. This is much harder, and generally raises the flag of calculus and its need to be studied.
If you’re really feeling good on Day 1, you can always broach causation vs. correlation, just for giggles.
Finally, we need to be really clear about unintuitive moments, places where the number of walruses are really high, but the rate of gaining walruses is stagnant or even in a state of loss. Put those numbers in a table on the board for everyone to see and grapple with.
I’m going to try an entire semester of astronomy-based calculus this year, and this graph inspired me:
Students Everyone find astronomy interesting, but not everyone has a extragalactic view on how things are structured. Rates are rampant and fairly unintuitive here.
For instance, the universe appears to be expanding spatially, and that expansion appears to be accelerating. Acceleration is another classic rates misconception, and its study in this context is another win-win for physics and math. What’s more, is that our view and measurement of these accelerations are filtered by this goofy speed-of-light delay that we have as we look farther back in time at the objects that should be moving the fastest away from us.
Then there’s the rates-related joy of redshift!
If that doesn’t immediately ring a thousand bells, here’s a fun idea: get the Arduino’s out, create a sketch that will output a pitch based on the speed (use the GPS shield, and a small speaker) of the Arduino that negates the Doppler effect. In other words, you hear the same pitch no matter how fast you move the object (we put it on the front of a car)
Have a fun year. Go make something!