Progression on the Number System (Grades 6–8) and Number (HS)

Here is the latest progression hot off the presses. It includes the Grades 6–8 Number System domain and the Number part of the High School Number and Quantity category (Quantity part coming soon). As usual, please comment at the forum, here.

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Modeling Progression, Take 2

After talking to some teachers at PCMI on Tuesday and hearing from my fellow standards writer Jason Zimba I decided to do a quick fix on the modeling progression. The previous version ventured into territory that has been discussed on this blog: the different possible meanings of the word “model.” I decided this could be confusing, so edited it down so that it now sticks to the meaning of the word as used in the standards. The new version is here.

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Progressions preface and introduction, and updates to Algebra and Functions

A a couple of things today. First, a a draft of the front matter for the Progressions, including an introduction explaining the sources of evidence, organization, and terminology for the standards. It also lists the members of the work team that produced the Progressions, who have been sadly unacknowledged until now. I would like in particular to call attention to the work of our editor, Cathy Kessel, who is also an occasional contributor to this blog.

Second, thanks to work of Al Cuoco, we have updated versions of the Algebra and Functions Progressions.

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Draft of Modeling Progression

Isn’t summer wonderful? This has been sitting on my desk for a while, waiting to be typeset. Some teachers at PCMI this summer needed it for their c-TaP projects, so I finally got to it. As always, this is still only a draft. Please leave comments in the appropriate forum.

[Edited 4 July 2013. Please go here for the latest draft.]

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The three Rs in MP8. And the E. And the L.

Standard for Mathematical Practice number 8 is probably the hardest for people to wrap their heads around:

MP8. Look for and express regularity in repeated reasoning.

There are too many words in there: regularity, repeated, reasoning. I’ve seen a lot of people latching onto one or two of these. If it’s regular, it’s MP8! If it’s repeated, it’s MP8! If it’s both regular and repeated, it must really be MP8!! One thing that is fairly regular and repeated is generating coordinate pairs from an equation in two variables. So there are lots of fake MP8 lessons out there about generating points from a linear equation in two variables to draw the graph of the equation, a straight line. The more points, the better—it’s more repeated that way. And regular.

But that word reasoning is also important. There’s precious little reasoning involved in generating coordinate pairs from an equation. But if we turn the question around, there’s lots of reasoning. Instead of going from an equation to a line, let’s go from a line to an equation. Consider a line through two points in the coordinate plane, say (2,1) and (5,3). How do I tell if some randomly chosen third point, say (20,15), is on this line or not? Given any two points on a line in the coordinate plane, I can construct a right triangle with vertical and horizontal legs, using the line to form the hypotenuse, as shown here.

Why_a_line_is_straight

It is a wonderful geometric fact that all of these triangles are similar. (Exercise: prove this!) So, if (20,15) is on my line, then the triangle formed by (20,15) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:

$$
\frac{15-1}{20-2} = \frac{3-1}{5-2}?
$$

Oops. Not true. So (20,15) is not on the line. Let’s try (20,13) instead. If (20,13) is on the line, then the triangle formed by (20,13) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:

$$
\frac{13-1}{20-2} = \frac{3-1}{5-2}?
$$

Yes! Both sides are equal to $\frac23$. And in fact, to confirm, the reasoning works the other way: if the ratios are equivalent, then the triangles are similar, then the base angles are the same, so the hypotenuses of these two triangles are on the same line. (Exercise: prove all this, too!)

So we have a way of testing whether points lie on the same line. (This is Al Cuoco’s point tester; google it.)

After testing a lot of points, we look for some regularity in our repeated reasoning. Every one of our calculations looks the same. We can express the regularity by a general statement: to test whether a point $(x,y)$ is on the line, we check whether

$$
\frac{y-1}{x-2} = \frac{3-1}{5-2}.
$$

By our reasoning, every point on the line satisfies this equation, and no point off the line satisfies it. We have discovered the equation for the line by expressing regularity in our repeated reasoning.

All the words in MP8 are important: reasoning, repeated, regularity, and also express and look for. See this post by Dev Sinha for more discussion.

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New Common Core Discussion from ETS

Check out this new document from ETS, also linked on the tools page, that describes the Common Core Math and English standards through tasks.

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Learning about the standards writing process from NGA news releases

There’s a lot of misinformation going around these days about how the Common Core State Standards were written. It occurred to me that a simple way of learning about the process is through the press releases from the National Governors Association during 2009–2010. If you type Common Core into the search box you will find releases detailing the initial agreement of the Governors, the composition of the work teams, feedback groups, and validation committee, the state and public reviews, and various other pieces of information. It’s not a detailed history by any means, but I would encourage readers to check information they receive against this source.

[19 June] I noticed the search feature at NGA isn’t working today, so here are the main releases for 2009–2010:

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Summer Professional Development

Thinking about where to focus the Math Common Core PD for your school or district this summer?  Check out these two resources:

1. A report from Institute for Mathematics & Education suggesting places that might need some extra PD work.

2. Consider requesting trained teacher facilitators to deliver the Common Core Toolkit, a one-day add on to existing professional development, focused on the Common Core.  This is available for K-5th grade teachers, 6th-8th grade teachers, or high school teachers and is a project of an ad-hoc committee of the CBMS.

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EDC course on the mathematical practices for high school teachers

Here’s a note from Al Cuoco:

Friends,

For the past two years, we’ve been working with support from the MA department of education to create a course for high school teachers that helps them implement the Standards for Mathematical Practice. The approach of the design is to take examples suggested by the high school content standards—everyday, non-exotic content that is hard to teach and that causes students difficulty—and to develop that content in ways that are consistent with the practice of mathematics as it exists outside of high school, making the topics easier to teach, easier to learn, and more satisfying for everyone.

We field tested the course with over 100 teachers in two sessions over the past two summers at EDC. The a team of 10 colleagues (teachers who work with us) taught it in pairs in 5 sessions around the state at the end of last summer. All of this led to revisions, and we’re now publishing the course and offering it nationally. A sampler is at http://mpi.edc.org/dmp-hs-sampler

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Grant Wiggins on Granularity

Grant Wiggins has a great post about the dangers of breaking the standards down into statements of the finest possible grain size:

This problem of turning everything into “microstandards” is a problem of long standing in education. One might even say it is the original sin in curriculum design. Take a complex whole, divide into the simplest and most reductionist bits, string them together and call it a curriculum. Though well-intentioned, it leads to fractured, boring, and useless learning of superficial bits.

Read also his spirited defense of the standards a couple of days earlier.

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