This is a series about “developing the question” in math class.
Here is a resolution: ask your students for a sketch first.
I’ve been a bit obsessed with “Barbie Bungee,” a lesson on linear regression which you’ll find all over the Internet. It’s the kind of lesson that doesn’t seem to have any original mother or father, only descendants. (Here is NCTM’s version as well as a video from the Teaching Channel.)
Search the Internet for “Barbie Bungee handouts. I have. Invariably, the handout asks students to collect data for how far Barbie falls given a number of rubber bands tied around her ankles and then graph the results precisely. Often times those handouts include a blank graph with precise units and labeled axes.
Developing the question means starting from a more informal place. It means asking the students, “What do you think the relationship looks like between the number of rubber bands and Barbie’s distance? Sketch it.”
Asking students to sketch the graph serves so many useful purposes.
- It helps us clarify assumptions. What do we mean by “distance”? Barbie’s distance off the ground? The distance Barbie has fallen?
- Predicting the relationship makes it easier to answer questions about it later. This is from Lisa Kasmer’s research. It’s productive for students to decide if they think the relationship is linear, constant, increasing, decreasing, etc. What is its general shape? How do these quantities covary? As rubber bands increase, what happens to distance? Later, when students start to graph data precisely, the fact that the shape of their data matches their sketch will help confirm their results.
- It’s great formative assessment. Do your students even know what a graph represents? Find out by asking for a sketch. If they can’t sketch a graph, their later precise graphing is likely only going to be mechanical and instrumental. (ie. “First number right, second number up.”)
- Comparing informal sketches, which may vary widely, will likely make for better debate than comparing precise graphs, which will largely look the same. And controversy generates interest.
Which would make for a more interesting classroom debate? These three precise graphs?
Or these three imprecise sketches?
If the answer is “make a precise graph of a real-world relationship,” then developing the question means asking for a sketch first. That’s my resolution.