And, as recompense for the long wait, we have added a bonus grade level, Grade 6. Here it is:

ccss_progression_gk6_2014_12_27

[Updated 2014/12/27]

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## About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.

I thought it would be in French! Thanks for the English version!

This week I am teaching a Grade 3 – 5 CCSSM course, and Geometry and Measurement will be the focus on Friday, so your timing is impeccable.

Thank you!

Tres bien. Merci.

Bill,

First – THANK YOU to you and the whole team who worked to put this together for us!

Here’s the first major question this progression has raised for me and my understanding of the standards:

“Students learn to name and describe the defining attributes of categories of two-dimensional shapes, including circles, triangles, squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral. They describe pentagons, hexagons, septagons, octagons, and other polygons by the number of sides, for example, describing a septagon as either a “seven-gon” or simply “seven-sided shape” (MP2).2.G.1” (page 10, first paragraph).

This description really expands the shapes listed in the standard itself (triangles, quadrilaterals, pentagons, hexagons, and cubes). Those shapes are all families named specifically by their numer of sides (of faces for a cube)… and the list does not include “septagons, octagons, and other polygons by the number of sides,” though those at least seem to follow what I thought the intent of the standard to be.

More problematic than this slight expansion of the standard is the preceding sentence that now asks students to “describe the defining attributes of categories of two-dimensional shapes, including circles, triangles, squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral.” This MUST be an accident? That has to be reserved for 4.G.2 and 5.G.3 & 4, right? Second grade students haven’t yet learned anything about about angles or parallel in order to be able to “describe the defining attributes”! And the defining attributes of circles?! Possible with sixth graders… but the CCSS doesn’t even deal with circles until 7th grade.

Does this paragraph in the Progression accurately describe the expectations around 2.G.1, or was this paragraph misplaced?

Thanks,

Brian

Brian, you are quite right that the gap between the standards and the progression is larger here than for the progressions that describe the main focus of K-5. The underlying issue here is the relationship between the standards and curriculum. This requires a much longer essay than I have time for, but here are a few thoughts.

The progressions describes a lot of things that teachers might do in the classroom; the standards describe the key mathematical achievements that should result from this. For the core progressions in K–5 there’s not much difference between the two, but for the progressions in geometry and in measurement and data the difference is bound to be greater, since the standards in those progressions were intentionally restricted to a thin stream in order to make room for the core focus. In practical terms, this means that curriculum materials for geometry might well have students engaging in some activities which are not mentioned in the standards, but not thereby forbidden. Talking about 7-sided figures might be such an activity. In some classes there will be time for this, in others there won’t. Classes that don’t have the time will not be penalized in assessments for focusing on the core progressions in number and operations.

For your question about attributes, I’m going to ask Doug Clements to see if he can find the time to clarify (but he’s busy, so don’t hold your breath!).

Brian and Bill, there’s quite a bit of discussion about the meaning of “defining attribute” earlier in the Progression, beginning with p. 2. Part of the footnote on page 3 says:

“Attributes” and “features” are used interchangeably to indicate any characteristic of a shape, including properties, and other defining characteristics (e.g., straight sides) and nondefining characteristics (e.g., “right-side up”).

Page 8 talks about:

differentiate between geometrically defining attributes (e.g., “hexagons have six straight sides”) and nondefining attributes (e.g., color, overall size, or orientation).

Do these quotes help to clear things up?

Thank you for this! This was perfect timing for me as well as I needed to work on the Geometry standards this week.

Also, while I was working on the Kindergarten standards yesterday, I have the same questions Brian has above. Maybe we need to place these questions in the “General Questions about the Mathematics Standards” section?

Thanks,

Jessica

“In practical terms, this means that curriculum materials for geometry might well have students engaging in some activities which are not mentioned in the standards, but not thereby forbidden.”

My favorite quote for the day. Perfect.

Bill,

Thank you for your explanation. This helps me recover a bit from the shock I suffered when reading the geometry progression – which quickly became filled with highlighting and question marks on statements that I though went beyond the grade-level standards.

In a related response you left to a question in the Progression for SP, you mentioned what has sort of become your signature catchphrase – “that which is not mentioned in the standards is not thereby forbidden,” and added a new twist – “that which is mentioned in the progressions is not thereby required.” Both are great and make perfect sense. My concern is that, as a reader of the standards and of the Progressions, I can’t always discern for myself what is “required” and what is simply “not forbidden” without asking… which doesn’t lend itself to “common” understandings of the standards.

I know the people working on these Progressions are some of the busiest among us, but I would like to offer two possible ideas for embedding this clarity in future iterations of the Progressions. Either:

• organize each grade level’s narrative into two sections: 1) required by the standards at this grade level, and 2) related to the standards at this grade level; or

• add those two categories as footnote tags and simply superscript a 1 or a 2 after each paragraph or sentence that needs to be clarified such that the field understands the intent and the scope of the particular standard.

I realize this is easier said than done, but it is certainly a much more doable way of achieving the clarity sought by some of your frequent posters (ex., Lane, Turtle, Jessica, and me) to increase the likelihood that curriculum writers, test writers, and teachers share a “common” understanding of what needs to be taught at each grade.

Thanks for all of your time and support,

Brian

Brian, I’m not sure about this for two reasons. First, the obvious one, we don’t have the time. Second, more seriously, I’m not sure it’s a great idea. I do understand the desire for clarity, but in the end I think there’s no way around the sort of discussions we are having on this blog, where people discuss questions that arise from a close reading of the standards. Ultimate authority over how the standards are interpreted shouldn’t come from me or the other lead writers or the progression authors, although of course all those people have some insights to contribute. It should come from discerning expert communities of teachers, mathematicians, and educators, such as the one that is emerging from this blog.

Wohoo! Thanks Bill – You Rock!

Bill,

Question on 4.G.2:

The sentence “Classify two-dimensional figures based on the presence of absence of… angles of a specified size” leads me to believe that students need to be able to identify acute, right, and obtuse triangles. However, the sentence immediately following it, “Recognize right triangles as a category, and identify right triangles” leads me to believe that students do not need to be able to recognize/identify acute or obtuse triangles, only right triangles. The first paragraph of the grade 4 section of the Progression (p. 14) supports the first interpretation. Is that the intent? If so, what is the intent of the second sentence, which wouldn’t seem to say anything that the first sentence doesn’t?

The same paragraph of the progression interprets the same standard to also state that “[Students] can use side length to classify triangles as equilateral, equiangular, isosceles, or scalene…” I cannot find a way to interpret this standard that would include that statement. Is this actually “required” by the standard, or just a natural “extension” that is “not forbidden”?

Thanks,

Brian

Brian, I think the second sentence is a specific instance of the first sentence that we want to make sure is included. As for your second question, students in Grades 2 and 3 have already been identifying figures on the basis of congruent side lengths and faces (among other bases). So it is a natural extension of that progression to look various sorts of triangles here, particularly since we are already looking at right triangles. Also, I suppose you could interpret “angles of a specified size” as including the statement “all angles have the same size.”

Bill,

Thanks for your help with the first question – I had misinterpreted that due to the second sentence of 4.G.2, so I am glad the Progression and this blog exist to clarify!

To be clear, classifying triangles by side lengths would be a “natural extension,” but not required by the standard (with exception to “equiangular,” for the reason you state)?

Brian