[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:
- Communication
- Representation
- Connection
- 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?
Work in Pencil 
[…] I would think extra development on process standards would benefit […]
I too have been struggling to come up with good “standards” for algebra. Your improvements to the NCTM ones still leave me struggling to get my arms around how I would assess them consistently.
I also wrestle with skills vs concepts. Similar skills end up being stretched as they are used across a series of increasingly challenging concepts. The list of “general skills” might stay small, but the list of concepts will read sort of like a syllabus.
As a result, students who assess well on a skill as applied to one concept could assess poorly for the same/a similar skill taken up a notch to be used with a different concept. I have read about some clever ways of addressing condensing all this into a grade, but I am struggling more with the contents of the matrix of skills & concepts.
I had been thinking of core skills along the lines of:
– restates or diagrams the problem and states the objective in their own words, converts it into algebraic notation, solves it in a way others can follow, checks the solution, and expresses the solution in terms of the original problem.
– simplify or expand algebraic expressions or equations involving all operators and functions introduced to date, using all appropriate properties of the operator or function, and rewriting one operator or function in terms of another as needed.
I have not thought of others yet, but would hope to expand on or break up the above into no more than six core skill sets I could apply to topics and projects tackled during the year.
Whit
http://mathmaine.wordpress.com
> As a result, students who assess well on a skill as applied to one concept could assess poorly for the same/a similar skill taken up a notch to be used with a different concept.
I’m seeing this all the time. And I have to confess I have made little progress on assessing any inquiry standards at all. The only move in this direction I’ve made is that to get a “4” on any standard a student must demonstrate the ability either to clearly explain it or to check their work. But it’s a far cry for what I’m looking for. The two you mentioned are certainly core skills, although the “simplify or expand” skill might get tangled up in the “solves in a way others can follow” skill.
I’m glad someone else is searching for this!
I’ve been looking for a method to fold assessment feedback into my fledgling WCYDWT attempts, and these overarching standards seem like a good place to start. Thanks for sharing!
M>G>