## Levels of Comprehension, and Math!

19 Jan

I had a lightbulb during class this week that I wanted to share.  In context, we’re in the middle of a unit on rates and ratios, but our math team is simultaneously working with the kids on creating data displays and writeups for a big science project.  I decided to extend our rates unit into graph/table/equation relationships, because it would dovetail nicely into the science discussion.

The students solved a rate problem involving calculator prices, and I tasked them with creating a line graph.  Many students produced something like this.

Pretty straightforward line graph.

Some students finished early while others were still working, and the work extended over a second day.  Due to procrastination on my part, I ended up at my desk five minutes before students were to come in, asking myself, “okay, what do I have the fast workers do while the other kids are finishing their graphs??”  So I decided to have everyone write their own questions that could be answered from their graphs.  Remembering a little from staff meetings past, I asked kids to write some Level 1, Level 2, and Level 3 questions.  They shared their questions in whole-class discussion.

Level 1:  These questions could be answered right from the graph. The answers were literally right there.  They wrote questions such as:
– How much do 20 graphing calculators cost?
– How much is 1 fraction calculator?
– How many scientific calculators could you buy for \$80?
– What is on the x axis?

Level 2:  These questions could be answered with a little problem-solving if you used the graph to start with.  They definitely had a right answer, but they weren’t right in front of you as the Level 1 questions were.  They had to be inferred.  Students wrote:
– How much would 100 graphing calculators cost?
– What’s the difference in price between 20 graphing calculators and 20 fraction calculators?
– How much would 10 graphing calculators cost in pesos?
– How much would you spend on 10 fraction and 5 scientific calculators?

Level 3:  These questions didn’t necessarily have a right answer.  They are opinion or judgment questions that make real-world connections with the graph, or they extend the thinking beyond the graph.  Students had a lot of fun with these and came up with some clever and off-the-wall questions.
– Who is my audience for this graph?
– What kind of features do graphing calculators have that make them so expensive?
– Why isn’t there a calculator that does all three functions?
– Who would buy 20 graphing calculators?
– What happens if you buy half a calculator?
– Do infinity graphing calculators cost more than infinity fraction calculators?
– Why don’t you go to the dollar store?
– Why not just use the app on your phone?
– Can I have a discount if I buy more than 20 calculators?
– If we save money by getting scientific calculators instead of graphing calculators, can we use the saved money for donuts?

I considered how this would blend into their science report task ahead of them, and I like where it’s headed.

In a science fair report, you generally have your data table / graph, a “results” writeup, and a “conclusions” writeup.  I know now how I am going to structure the kids’ writing for “results” and “conclusions”.

“Results”:  This section should be Level 1 and some Level 2 observations about their science data.  They should start with specific, low-level observations, and then move into general patterns they notice by problem-solving with their graph and data table.

“Conclusions”:  In this section, you start with Level 2 inferences about how well your data supports your hypothesis, and you move into Level 3 references about what your data means in a real-life context, what your sources of error could be, and what you should study in the future.

That five-minute “oh shoot what do I next” moment ended up giving me the gift of some pretty good structure for the kids’ science lab reports.  After they create their graphs, they can craft some statements in each level, and then weave them into a narrative.  Hooray!