[Edited on 8/29 for the SBG Gala 2]
Shawn is the kick-ass physics teacher I want to be if I ever get to teach physics. In his SBG-routine for physics, along with his content standards he includes throughout his course a number of inquiry standards, viz.,
- Student can formulate a testable question (__/40)
- Student can design a valid experiment (__/40)
- Student can form reasonable hypothesis from theory (__40)
- Student can statistically (averages, stdev, & chi) analyze data (__/40)
- Student can draw reasonable conclusions (__/40)
I am trying to do the same for mathematics; this year that means Algebra II and Calculus. The NCTM offers these process standards:
- Problem Solving
- Reasoning and Proof
I have no quarrel with these as important parts of doing math, but I don’t love these as course objectives because I don’t know how to assess them in this form. I’m trying to develop a more observeable set of general math practices, and came up with:
- questioning (coming up with questions, conjectures, &c)
- visualizing (translating words or equations into diagrams, this spans communication and representation)
- abstracting (or maybe “mathematizing.” Translating a constraint or question into a mathematical representation we can use tools on. Maybe this is NCTM’s “representation,” but I’m focused here on going from the vernacular into mathematical representations more than on moving from one math rep to another)
- strategizing (or, as Sam says, “take what you don’t know and turn it in to what you do know.” Spans connecting and problem solving)
- generalizing (which spans problem solving and reasoning and proof)
- explaining (which spans proof and communication)
I was also thinking about something like “critiquing” or “debunking” or “skepticism” or something – spotting the errors in a line of argument. But maybe that’s just another way to demonstrate the ability to explain?
After I came up with these I got some training from College Preparatory Math (thanks for the steer, Riley). For their Algebra 2 course they explicitly call attention to five “Ways of Thinking” which are (from the opening page of the text),
- Justifying (explaining and verifying your ideas)
- Generalizing (predicting results for any situation)
- Choosing a Strategy (deciding which solution methods make sense)
- Investigating (gathering information and drawing conclusions) and
- Reversing (solving problems backwards and forward).
I think they have different ways of thinking emphasized in different courses. CPM doesn’t seem to call out any specific “ways of thinking” for Calculus, but I think these same 5 would be a good first cut.
I also recently checked out the Common Core, which builds on NCTM these with:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
The CPM Ways of Thinking seem to work. They don’t cover the common core standards completely (modeling, precision and structure aren’t stressed), but I can defend grading on them from the common core standards (or from the NCTM standards), and at least in the Algebra book, they are discussed (always in bold) throughout the text so students have support and examples for what I mean by them. So I am tentatively adopting these into my standards list for both courses, and we’ll see how it goes.
On the other hand, I notice that Shawn doesn’t seem to use anything equivalent to inquiry standards in his own math class. Maybe he knows something I don’t?