Math Relearning/Progressions/K-6 Geometry

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These are my comments on the Common Core Math Progressions document Geometry: Grades K-6 (PDF).

Yay! I was looking forward to geometry.


"Levels of Geometric Thinking" - These distinctions seem vague.

"Classification of Quadrilaterals" - Interesting topic for a book. I'm curious what benefits different definitions and classification schemes give us.

"do not learn limited concepts" - Even if we decide not to require much math, we still have to teach the parts we do carefully, and it's probably more than non-specialists expect (definitely not just arithmetic like this article suggests:

decomposing by covering - Interesting, I wouldn't have thought that would count.

composing with units of units - I wonder if there are canonical or typical sets of compositions, maybe progressions of units.

spatial structuring - Isn't this just mentally decomposing?

Grade 1

"geometrically defining attributes" - Why do we pick these and not other attributes?

I was going to dismiss the task of distinguishing between defining attributes and others, since I already basically know those, but then I thought I should examine the issue in my typical way, and I quickly arrived at that question, which I think will be enlightening to answer, so I'm glad I took another look.

I was thinking later about how satisfying it is to dig up new concepts like that, or at least the questions that would lead to them, and how I won't get to do that as much now since the conceptual math people have already done so much. But then I thought it might achieve a similar result if I came up with the kinds of questions that could have led to the insights of those researchers. So I might do some of that.

"like combining 10 ones ... foundations for later mathematics" - How far do the parallels go between shapes and counting or arithmetic? Area is a model for multiplication. The number line can model a lot. Geometry can be done through coordinates. What else?

Grade 2

"need not have the same shape" - It would be good to explore how this works. It's closely related to the area-perimeter relationship.

tangrams - I didn't know they were related to isosceles right triangles. I've wondered about the history/context of these puzzles, so I should look them up.

"transformed into" - This confused me for a second, but I think they mean decomposing the first shape and composing the pieces into the second. This could be an interesting general model of some kinds of transformation.

Grade 3

classification of shapes - This feels like an especially preparatory step, and it reminds me of the reason-orientedness of my project. The goal is largely to ask why we do things the way we do in math. Along those lines I can also ask why we learn the things we do. I know math is continually growing, like every other field, but are there key intermediate goals in math knowledge? Tasks we typically want to accomplish at certain points that require specific background knowledge?

Relatedly, how do we select what to teach? Are there other lower level concepts we don't teach because they don't matter to our later goals? Are there concepts on those levels that we haven't investigated because they haven't really interested anyone? Things like the Ulam spiral. It's a pretty basic idea, but we didn't come up with it until 1963.

"without making a priori assumptions regarding their classification ... may still need work building or drawing squares" - These progressions split up math tasks and skills very finely. Some of these distinctions might be worth pursuing for this project (concepts, problem types such as solving for various unknowns), and others would be less relevant (mental or physical abilities like drawing straight lines).

Grade 4

"turtle geometry" - It's interesting that a feature of a particular computer program gets a type of geometry named after it.

"connect what are often initially isolated ideas" - It's interesting that building shapes with the turtle can do this.

"triangular (isometric) grids" - Interesting. I hadn't heard of this one. Hopefully EngageNY will cover the different kinds of grids.

"shape is fixed by the side lengths" - Explore how this works.

Grade 5

A lot of this and grade 4 seems like repetition of concepts and skills found elsewhere in the progressions, maybe with more technical terminology and combined in slightly different ways. At least it gives me less to write about.

The relationship between spatial structuring and the coordinate plane does seem significant though. It's described as sort of a paradigm shift for the students.

Venn diagram of quadrilaterals - Very useful!

Grade 6

The concepts and skills are starting to rush past, so I'll wait to figure out what it all means until the curriculum. It's crossed my mind a few times whether the progressions would be adequate for pre-algebra, but I think I really will need the lesson plans.

Where the Geometry Progression is Heading

It's nice to know composition and decomposition aren't just training wheels for elementary school but are used past high school math. They do feel sort of like crutches, but it's reassuring that everyone uses them.

The progressions give the impression that Common Core geometry ends after grade 6, but the standards continue it through high school, thankfully. In fact, James Milgram criticizes the CC technique of teaching geometry using rigid transformations after that point, saying it may be the only rigorous method we have, but it's also too advanced for most teachers. Well, that isn't really relevant to my project, so for now I'm happy with whatever's in store.