Geometry Progression

[Tad Watanabe]

Bill,

I’m finally starting to read the geometry progression document.  I have only read through Grade 2 and just begun to read Grade 3.  But, I am puzzled on a couple of issues.

First, it is not quite clear at what grade students are supposed to understand the formal definition of various 2-D shapes.  For example, in Grade 2 section, the document says, “Students learn to name and describe the defining attributes of two-dimensional shapes, including circles, triangles, squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral.”  Does this mean students should understand the formal definitions – for example, a rectangle is a 4-sided figure with 4 right angles and a circle as the collection of points that are equidistant from a given point?

In the beginning of Grade 3 section, on the right side margin, the document include different types of quadrilaterals.  One category is parallelograms.  However, for both trapezoids in Grade 2 and parallelograms in Grade 3, I don’t understand what understanding of trapezoids and parallelograms we are expecting from students if they have yet to learn about parallelism as a property.  It seems like the possible types of quadrilaterals students can identify are rectangles, squares, and rhombi.

Also in Grade 3 section, the document says, “they (students) could solve the problem of making a shape with two long sides of the same length and two short sides of the same lengths that is not a rectangle.”  However, if are expecting them to “draw shapes with prespecified attributes,” isn’t this beyond Grade 3 students’ knowledge?  I can see them doing this by actually using physical objects – 2 pairs of sticks, for example.

Thanks.

[Bill McCallum]

Tad, I’ve asked Doug Clements to take a look at this and hopefully he will get to it soon. On the issue of “defining attributes”, I will say that being able to describe defining attributes is not the same thing as being able to give a formal definition. You might want to look at the discussion here to see if it answers some of your questions.

[Bill McCallum]

More from Doug Clements about defining attributes:

… the phrase (perhaps unfortunate) of “defining attribute” is for the adult to understand it is defining, but for the child, it is not expected that they know a formal definition, only that they understand “rectangles need to have all right angles” but that their color doesn’t matter. (Until middle school or later, of course, we don’t expected them to know what’s truly defining, such as needing only to say a parallelogram needs “at least one right angle” to be a rectangle, etc.).

[Tad Watanabe]

Bill,

Thanks for relaying Doug’s response.  However, his response is very unsatisfying.

How do children know a quadrilateral is a parallelogram?  What are the defining characteristics of parallelograms?   In theory, we can define parallelograms as quadrilaterals with opposite sides being equal.  I can also see a possible investigation by Grade 3 students by focusing on the lengths of sides of quadrilaterals.  They can identify rhombi as having 4 equal sides, parallelograms and kites having two pairs of equal sides, etc..  But, this definition, though useful in identifying parallelograms, will be difficult (if not impossible) for 3rd graders to use to draw parallelograms.  Trapezoids are even more problematic.  When we introduce trapezoids as a type of shape, exactly what are we wanting students to know as “defining attributes” without parallelism?

I think I can rephrase my question like this: at what grade do we say (for example) it is NOT ok for children to say an ellipse is a “circle”?  I think it is perfectly fine for K & Grade 1 students to call an ellipse a circle – because they are distinguishing it from the figures that are made of straight segments.  But, is it ok for Grade 3 students?  Grade 5?

I certainly do not think we should be treating geometry very formally in elementary school.  However, we should be introducing formal definitions (stated in language that is appropriate for elementary school students) gradually.  Otherwise, elementary school geometry instruction becomes just a series of vocabulary lessons (a bit of over simplification, I realize).  I’m afraid when we are so loose about the definitions of figures, we are developing more students who think squares are not rectangles.  Children need to understand what makes some shapes a particular type before they can start investigating the relationships of classes of shapes – as they do with 4-sided figures in Grade 3.

[Bill McCallum]

Tad, the passage I quoted from Doug was not intended to be a complete response to your question; it arose out of a discussion between me, him, and Cathy Kessel about the meaning of the term “defining attribute” as it is used in the standards, and I thought it was helpful.

On the issue of parallelograms and trapezoids you raise a good point. I agree students would have to have some incipient notion of parallelism to talk about these figures; and it’s true that this notion is not introduced in the standards until Grade 4. So the formal notion of a parallelogram or a trapezoid would have to wait for Grade 4. I don’t think this excludes students seeing and talking about them in Grade 3, and having a discussion about how you would describe the attribute that sets them apart (e.g., “the two opposite sides point in the same direction”). We’ll try to elaborate on that in the final draft.

[Tad Watanabe]

Bill,

I’m perfectly ok with the idea of children informally discussing about shapes.  But, as we look at a K-12 curriculum, shouldn’t there be a time where formal definitions are expected to be understood and students will use the terms accurately – isn’t it a part of mathematical practice?  Doug’s quote seems to suggest that point to be middle school or even later, without specifying the timing.  As I quickly glance at Geometry standards in middle grades, they don’t seem to specify formal definitions – yet in many ways, formal definitions may be needed for some of the standards.  So, I remain puzzled – but that seems to be pretty much a normal state for me 🙂

[Jim]

I feel like I’m missing something, but why is the first “Difficult Distractor” in the rectangles box not a rectangle?

[Jim]

Also, there’s a typo: “kindergarden” instead of “kindergarten.”

[Jim]

From the progressions document:

“Students build these competencies, often more slowly, in the domain of three-dimensional shapes. For example, students may intentionally combine two right triangular prisms to create a right rectangular prism, and recognize that each triangular prism is half of the rectangular prism. 1.G.3”

It seems to me that composing three-dimensional shapes falls under the standard 1.G.2, not 1.G.3 as indicated.  I see that there is the concept of two halves making a whole, a concept that a student might transfer from learning 1.G.3, but looking at 1.G.3 reveals that it limits itself to the realm of 2D shapes (circles and rectangles).

I’d like to see 1.G.3 addressed more fully in the Progressions document.

[Bill McCallum]

Jim, all good points, and ones that will inform the revision, thanks. (It looks like a rectangle to me, too.)

[LeandraC]

Quick question:  Page 8 of the geo progression at the bottom of the page it says, “They also show recognition of the composite shape of “arch.”

What does this mean?

[Cathy Kessel]

Yes, that sentence could be less terse.

I think it helps if you read it in the context of the paragraphs that precede it. In particular, for two-dimensional shapes students  “develop competencies that include . . . creating and maintaining a shape as a unit, and combining shapes to create composite shapes that are conceptualized as independent entities (MP2).” Students do similar things for three-dimensional shapes but more slowly. “Arch” is one such shape.

[Author]

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