Over at Lani’s place, Max talks about expanding CGI beyond CGI.

“Another grand challenge could be to extend the kind of work that went into CGI [Cognitively Guided Instruction] to high school Algebra, Geometry, Probability, Statistics, and Calculus concepts, so that teachers teaching, say, right-triangle trig, or similarity proofs, or solving quadratic equations, or standard deviation, had a map of learning progressions in that space and ideas about kids’ informal intuitions and misconceptions, multiple representations that connected to various aspects of a fully operational mathematical understanding, and then linkages among those multiple representations.”

Max talks about detailing how kids learn *topics*. But I’m not sure that this is exactly how I read CGI, though I can see it through that lens. The question is, do we seek learning progressions mapping how students think about *problems *or how students think about *concepts*? Meaning, do we want knowledge about how kids think about addition or about how kids think about addition problems?

I don’t think that this distinction really causes issues at the elementary level — to know addition is to know how to solve addition problems, and we all agree on what addition problems are — I wonder if it’ll be harder to nail down on the secondary level.

I was working on something I noticed in a trigonometry unit I recently ran with 9th Graders. Following CME Geometry, I see a road into trigonometry concepts that begins with comparing the steepness of different right triangles.

Here’s the different sort of thinking that I saw students doing in “comparing steepness” problems during the unit:

- Comparing steepness visually, or by measuring the angles of the ramps
- Comparing steepness by comparing the sides of the triangles
- Comparing steepness by looking at both the height and width of the triangles, but not distinguishing between ratio of sides, difference of sides, etc.
- Comparing steepness using the height/width ratio of each triangle
- Comparing steepness using the ratio of
*any* two sides of the right triangles

Here are some samples of student thinking to prove that I’m not *totally *pulling this out of my elbow.

I think it would be wrong to say that this is a way that my students thought about trigonometry. It’s certainly how my students thought about problems that are deeply related to trigonometry. It’s definitely how my students thought about “steepness comparison” problems.

Where does this leave us, if we’re interested in providing something of value to other trigonometry teachers?

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Hmm… I think the most productive way for me to think about this is to try some examples (thought experiments?). This reminds me a bit of my suspicion of prompts that ask students to explain a process for adding 2-digit numbers (in the abstract, not to explain how they solved a particular problem), or that ask them to somehow explain addition. Those are silly questions because they aren’t well-defined; on some level, mathematically, there is no knowing of addition besides knowing how to get right answers to addition problems. Because operations aren’t really defined other than by how they map R^2 onto R — a fancy way of saying, by the answers to their problems. On the other hand, there is lots more to know about addition if you’re in elementary school: what questions it answers, when to use it and when not to use it, how to check if you’re right, lots of different ways to solve lots of different problems using it depending on the numbers and the context, etc. Those are all things we can assess, and the best way to assess those things is to give kids contexts and context-less addition problems and attend to their choices about whether and when and how to use addition, and how they carry it out, and if they get the right numbers.

It seems like right triangle trig could also have a similar set of problems-to-be-solved. In a way, it’s a super simple domain: find missing side lengths or angles in right triangles. You can break that down into missing angles, missing sides, and further into missing angle given opposite and adjacent sides, missing angle given… blah blah blah, you get the point. And ultimately, that is what you assess. But like addition there are other questions: what questions it answers, when to use it and when not to use it, how to check if you’re right, lots of different ways to solve lots of different problems using it depending on the givens and the context, etc. But I feel like there’s another layer here… and now that I think about it, there’s another layer underneath addition too. What are the underlying concepts (and representations? and conventions?) that are underneath being able to know when and how to use it, how to check you’re right, how to be strategic, etc.

For addition, the huge underlying concept that causes trouble with the most people is place value. Our way of recording numbers is mysterious and also extremely useful, and leads to particular efficiencies in addition that are important to master — and their mastery both requires understanding and is built on other understandings within addition. For addition, besides place value which in many ways is learned through doing lots of adding and thinking about the results, the big ideas are super basic: counting (with one-to-one correspondence), knowing that the last number in the count gives the total number of objects, knowing the names for numbers, knowing the symbols for recording numbers. If kids know those things, they can solve any contextual addition problem as long as they understand the story.

For trig, I think the equivalent to counting is being able to solve similar triangles. Like how some kids don’t know that they can solve addition problems by counting, I think most kids don’t know that all solving right triangles is just comparing two similar triangles, and that if they can do that, they can solve any right triangle trig problem… IF they know how to look up the ratios using some tool or another. Teaching kids to look up trig ratios could be the equivalent of teaching them how to write numbers, the names of numbers, etc. It’s a convention that they need to internalize to be successful solving problems; on some level there’s nothing to understand about the words sine, cosine, tangent, etc. But like with place value in the number system, there’s also a big concept that kids are at the beginning or middle of learning about when they encounter right-triangle trig, and that’s functions. Trig is a chance to meet or reacquaint ourselves with functions that take inputs and give a well-defined output. Learning/making sense of notation like sin(30) = 1/2 is sort of analogous to learning/making sense of a notation like 20 + 3 = 23. There’s a lot of convention there, but there’s also a powerful mathematical tool that’s going to keep being useful. The more you know about functions, the more you know about trig, and the more you know about trig, the easier it is to make sense of the notation/expression sin(30).

From my (relatively limited) exposure to CGI (I’ve read Children’s Mathematics, but I’ve never taken a course in CGI or read the historical research base or taught using what I’ve learned from CGI, for example) it feels like it’s not really an attempt to explore explicitly about how addition teaches place value or how addition problems assess place value understanding or even really how counting develops into understanding of addition or how kids make sense of addition as an operation… And maybe because there isn’t really a need to, because the point of CGI is to listen to kids and notice how they think about addition problems, and you learn that (if they understand the story) they make sense of problems, choose appropriate operations, develop efficient strategies that happen to make use of place value, etc. CGI isn’t exactly a theory about how counting and place value and addition fit together conceptually, it’s a description of how kids develop in their use of counting and place value when faced with problems that require addition.

Could there be a book called Children’s Trigonometry? I don’t know. Can kids “invent” trigonometry if they understand the usefulness of similar triangles? Probably. They might even invent the idea of a function though I bet they wouldn’t use anything that looked like sin(30) = 1/2. Would the invent the idea of trig tables or a calculator storing (or recalculating) trig ratios for them. Probably not, is my guess… that’s an innovation that takes concerted effort of a math community. So perhaps a CGI that simply observes the natural development of trigonometric thinking among students who had worked a lot with similar triangles and were now trying to find heights of flagpoles based on their shadows would be not as useful as Children’s Mathematics was. Although, we’ll never know until we try! People probably thought CGI was a silly project at first!

Also, a lot of the work that went into CGI at first was probably profoundly conceptual and based on a sense of what makes a problem easy, hard, connected to, or disconnected from, children’s previous understandings, besides just the idea of “missing addend” being a problem type. Isn’t 19 + 63 easier for most kids than 17 + 64? Because of the strategy of using 20 and adjusting? And the adjustment is easier? Did the researchers LEARN that those problems were different and invited different strategies or required different comfort levels from listening to kids, or did they go in purposely using different types of numbers? The researchers definitely had specific context set-ups in mind to elicit different representations and strategies? What work went into that?

If we were going to extend CGI into Trigonometry (a topic, like Arithmetic or Addition), wouldn’t we want to know:

– the contexts that elicit trigonometric tools for solving problems

– what a trigonometric tool for solving a problem really is

– the prior knowledge required to make progress in those contexts

– situations and given quantities that are likely to elicit different tools

Also, Michael, given the questions you have about this, and your excitement about Lesh and Zawojewski’s article on Problem Solving, I would HIGHLY recommend the book Lesh and others wrote called Beyond Constructivism, A Models and Modeling Approach.

I think Lesh’s team would say that what I’m really asking for when I ask for a CGI-like approach is a set of contexts (mathematical or “real world” or imaginary world) that require trigonometry to model precisely, and a sense of the multiple models that children naturally bring to them, and how their models are refined as they encounter other models, examples that break their models, examples that their models aren’t precise enough to deal with, etc.

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I want to add on to your compare/contrast of addition with trig in relation to the CGI project. The 1979-1981 CGI studies were about word problems, and the initial papers were about carving the landscape of addition word problem land. (http://eric.ed.gov/?id=ED188892) Solving problems such as 23+46, I think, was beyond the scope of these initial studies, but even if it wasn’t there’s a difference between problems that develop the concept and mastery of the formal operation.

We can separate the conceptual development and work with the formal operation in trigonometry as well, where “find the missing sides of this triangle” is a lot like “23+46” and “compare the steepness of these ramps” is a lot like an addition word problem.

This analogy is helping me think through some of the difficulties of something like their project for trigonometry.

We got lucky with early arithmetic in that there is broad agreement as to what sorts of word problems addition/subtraction are good for, and everyone agrees that they need to be taught. There’s nothing like that sort of consensus with trigonometry (or anything else in high school). Imagine a world where everyone agreed that kids simply

needto learn trig so that they can go about their daily life systematically comparing the steepness of differently-sized ramps. In such a world, I think we could talk about the ability of children to solve trigonometry problems without instruction and we’d be handing out ramp problems left and right.In our real world, there’s no such consensus. That’s why, in high school math, we’re always trying to solve curricular problems and we’re constantly searching for problems that can help us do the work of conceptual development that will end in mastery of formal skills.

CGI is broadly useful because there’s broad agreement as to the sorts of non-formal problems that children ought to be able to solve with arithmetic. In trigonometry, though, there is no such agreement, and teachers interested in using problem solving as part of their instruction in the concepts of trig pick different problems. So any articulation of the way children think while working on the non-formal problems of trigonometry is likely to be idiosyncratic. So anything like CGI for trig would not be globally useful for all teachers of trig. It would only be useful if you share a curricular orientation.

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First, thanks for the book recommendation. Can’t wait to get my hands on it.

Your post is long, thoughtful and subtle. I’m going to try to summarize it briefly, sloppily and bluntly: the worry is that there’s no way to talk about developing trigonometric thinking without providing curricular details. These curricular details would be idiosyncratic, not universal.

Does that sound right?

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