Ways of thinking and ways of doing

Somewhere back in days of Facebook fury about the Common Core there was a post from an outraged parent whose child had been marked wrong for something like this:
$$6 \times 3 = 6 + 6 + 6 = 18.$$
Apparently the child was supposed to do
$$6 \times 3 = 3 + 3 + 3 + 3 + 3 +3 = 18$$
because of this standard:

3.OA.A.1. Interpret products of whole numbers, e.g., interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as $5 \times 7$.

The parent had every right to be upset: a correct answer is a correct answer. Comments on the post correctly pointed out that, since multiplication is commutative, it shouldn’t matter in what order the calculation interpreted the product. But hang on, I hear you ask, doesn’t that contradict 3.OA.A.1, which clearly states that $6 \times 3$ should be interpreted as 6 groups of 3?

The fundamental problem here is a confusion between ways of thinking and ways of doing. 3.OA.A.1 proposes a way of thinking about $a \times b$, as $a$ groups of $b$. In other words, it proposes a definition of multiplication. It could have proposed the other definition: $a \times b$ is $b$ groups of $a$. The choice is arbitrary, so why make it? Well, there’s an interesting discovery to me made here: the two definitions are equivalent. That’s how you prove that multiplication is indeed commutative. It’s not obvious that $a$ groups of $b$ things each amounts to the same number of things as $b$ groups of $a$ things each. At least, not until you prove it, for example by arranging the things into an array:

You can see this as 3 groups of 6 by looking at the rows,

and as 6 groups of 3 things each by looking at the columns,

Since it’s the same number things no matter how you look at it, and using our definition of multiplication, we see that $3 \times 6 = 6 \times 3$. (We leave it as an exercise to the reader to generalize this proof.)

None of this dictates the way of doing $6 \times 3$, that is, the method of computing it. In fact, it expands the possibilities, including deciding to work with the more efficient $3 \times 6$, as this child did. The way of thinking does not constrain the way of doing. If you want to test whether a child understands 3.OA.A.1, you will have to come up with a different task than computation of a product. There are some good ideas from Student Achievement Partners here.

When the Standard Algorithm Is the Only Algorithm Taught

Standards shouldn’t dictate curriculum or pedagogy. But there has been some criticism recently that the implementation of CCSS may be effectively forcing a particular pedagogy on teachers. Even if that isn’t happening, one can still be concerned if everybody’s pedagogical interpretation of the standards turns out to be exactly the same. Fortunately, one can already see different approaches in various post-CCSS curricular efforts. And looking to the future, the revisions I’m aware of that are underway to existing programs aren’t likely to erase those programs’ mutual pedagogical differences either.

Of course, standards do have to have meaningful implications for curriculum, or else they aren’t standards at all. The Instructional Materials Evaluation Tool (IMET) is a rubric that helps educators judge high-level alignment of comprehensive instructional materials to the standards. Some states and districts have used the IMET to inform their curriculum evaluations, and it would help if more states and districts did the same.

The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:

If the only computation algorithm we teach is the standard algorithm, then can we still say we are following the standards?

Provided the standards as a whole are being met, I would say that the answer to this question is yes. The basic reason for this is that the standard algorithm is “based on place value [and] properties of operations.” That means it qualifies. In short, the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.

Additional mathematics, however, is required. Consistent with high performing countries, the elementary-grades standards also require algebraic thinking, including an understanding of the properties of operations, and some use of this understanding of mathematics to make sense of problems and do mental mathematics.

The section of the standards that has generated the most public discussion is probably the progression leading to fluency with the standard algorithms for addition and subtraction. So in a little more detail (but still highly simplified!), the accompanying table sketches a picture of how one might envision a progression in the early grades with the property that the only algorithm being taught is the standard algorithm.

The approach sketched in the table is something I could imagine trying if I were left to myself as an elementary teacher. There are certainly those who would do it differently! But the ability to teach differently under the standards is exactly my point today. I drew this sketch to indicate one possible picture that is consistent with the standards—not to argue against other pictures that are also consistent with the standards.

Whatever one thinks of the details in the table, I would think that if the culminating standard in grade 4 is realistically to be met, then one likely wants to introduce the standard algorithm pretty early in the addition and subtraction progression.

Writing about algorithms is very difficult. I ask for the reader’s patience, not only because passions run high on this subject, but also because the topic itself is bedeviled with subtleties and apparent contradictions. For example, consider that even the teaching of a mechanical algorithm still has to look “conceptual” at times—or else it isn’t actually teaching. Even the traditional textbook that Garelick points to as a model attends to concepts briefly, after introducing the algorithm itself:

Brownell et al., 1955

This screenshot of a Fifties-era textbook is as old-school as it gets, yet somebody on the Internet could probably turn it into a viral Common-Core scare if they wanted to. What I would conclude from this example is that it might prove difficult for the average person even to decide how many algorithms are being presented in a given textbook.

Standards can’t settle every disagreement—nor should they. As this discussion of just a single slice of the math curriculum illustrates, teachers and curriculum authors following the standards still may, and still must, make an enormous range of decisions.

This isn’t to say that the standards are consistent with every conceivable pedagogy. It is likely that some pedagogies just don’t do the job we need them to do. The conflict of such outliers with CCSS isn’t best revealed by close-reading any individual standard; it arises instead from the more general fact that CCSS sets an expectation of a college- and career-ready level of achievement. At one extreme, this challenges pedagogies that neglect the key math concepts that are essential foundations for algebra and higher mathematics. On the other hand, routinely delaying skill development until a fully mature understanding of concepts develops is also a problem, because it slows the pace of learning below the level that the college- and career-ready endpoint imposes on even the elementary years. Sometimes these two extremes are described using the labels of political ideology, but I have declined to use these shorthand labels. That’s because I believe that achievement, not ideology, ought to decide questions of pedagogy in mathematics.

Jason Zimba was a member of the writing team for the Common Core State Standards for Mathematics and is a Founding Partner of Student Achievement Partners, a nonprofit organization.

Common Core Math Parent Handouts by Tricia Bevans and Dev Sinha

In the transition to the Common Core, we have focused more on supporting teachers and administrators, through tools to help improve their own understanding and to help work more fruitfully with their students.   But parents can also use help in this transition.  They have many legitimate questions and concerns such as having difficulty in helping their child with homework or wondering how the Common Core is designed to support their child’s mathematical development.    As parents ourselves we certainly empathize with others who are looking for clear, accessible knowledge.
We have written these parent handouts at the link below to help begin conversations which address these questions and concerns.  They are meant to be used for example at curriculum nights for parents.  We limit ourselves to one page of discussion and one page of an example (mostly taken from Illustrative Mathematics) at each grade, both for ease of use and so as to not overwhelm people with too much information at first.  Locally, we have been involved in discussions of deeper learning opportunities for parents, with these handouts as a starting point.
Edit:  Some people have asked for this document in a Spanish translation.  If you want to translate the document we would be happy to share the Spanish version here.

Learning about the standards writing process from NGA news releases. Take 2.

About a year ago I noticed there was a lot of misinformation being spread about the process for writing the standards, so I came up with the brilliant idea of pointing people to the historical source documents that chronicled the process: the NGA press releases about the Common Core during 2009–2010. That will solve the problem, I thought; people will just read the press releases and figure it out. Boy was I ever wrong. In this post I’ll try to give a clearer timeline of the process. Along the way I’ll point out the involvement of testing organizations, since I think that one of the reasons the misinformation has survived for so long is a narrative, compelling to some, that the testing industry dominated the process. (Spoiler alert: they didn’t.)

First, here is the list, with an additional one from July 2009 that I missed last time (which has been the source of much confusion):

Notice that there seem to be duplicate announcements of the Work and Feedback Group and duplicate releases of the standards. What’s going on here is that there were two documents. First, in summer of 2009, the people listed in the July 2009 release worked on the document that was announced in September of 2009. That document, which was actually entitled College and Career Readiness Standards for Mathematics, was confusingly referred to as Common Core State Standards in the title of the September 2009 press release. If you take a look at it you will see that it is a draft description of what students should know by the end of high school.

Subsequently, as described in the November 2009 press release, a new process with new groups was started, to produce “K–12 standards.” These were to be a set of grade level recommendations that described a pathway to college and career readiness. For the K–12 process, there were about 50 people on the Work Team and about 20 people on the Feedback Group for mathematics, representing a wide range of professions, including teachers, mathematicians, policy makers, and one representative each from College Board and ACT … none representing for-profit providers of assessments. The members of this group are listed in a linked pdf in the press release.  This is the document that was released for public comment in March 2010, as described in the March 2010 press release, and released in final form as the Common Core State Standards on 6 June 2010, as described in the final press release.

As you can see from the list of members, I chaired the Work Team for the second document. Within the work team there was a smaller writing team consisting of myself, Jason Zimba, and Phil Daro (who had all been involved in the summer 2009 document, Phil Daro as chair for mathematics). We based the standards on narrative progressions of particular mathematical topics across grade levels that were solicited from the Work Team. We circulated many drafts to the Work Team, the Feedback Group, the 48 participating states, various national organizations such as AFT and NCTM, and, in March 2010, the public (see the March 2010 release). I personally made sure that we responded to and made considered decisions about all of the voluminous feedback we received.

When you hear people claim that “the standards were written by the testing industry,” they are probably referring to the first document, because of the greater involvement of College Board and ACT. Both organizations, along with Achieve, which was also represented, had conducted research into the requirements of college and career readiness. (All are non-profits, by the way.) The problem is that some people refer to the first document in a way that suggests they are talking about the second document (i.e., the actual K-12 standards adopted by states). That is an error and a misleading one.

The two documents are different in nature, of course, since one of them is just a picture of an endpoint while the other is a progression. Feel free to compare them. One influence of the first document on the second is that in the first document you can see the first draft of what became the Standards for Mathematical Practice. And the topic areas listed in the first document evolved into the high school conceptual categories in the second. All this evolution happened under the processes for the second document, with input from the various groups described above.

I think the second document is the work of the 70-odd people listed as the Work Team and Feedback Group in the November 2009 press release. But, just for fun, I put the teams for the two documents together and counted how many of them came from ACT and College Board (no other testing organizations were represented). It comes to a total of 81 people with 7 from ACT and College Board, about 9%. So even with this interpretation the claim that the process was dominated by the testing industry is false.

Drawings in the Common Core

There’s been a lot of talk recently about the Common Core and drawings. I won’t get into the politics of that here—that’s for my other blog—but, politics aside, there is an important question of interpretation, and possible misinterpretation, of the standards here. Let’s look at the wording of standards that talk about drawings.

K–5 Elaborations of the Practice Standards

Illustrative Mathematics, with the assistance of Mary Knuck, Deborah Schifter, and Susan Jo Russell, has been working on developing grade band elaborations of the Standards for Mathematical Practice. Here is a draft of the K–5 document.

As usual, please comment by starting a new thread in the forums. I’ve created a new forum for the practice standards there.

Join with me in support of the Common Core

I have tried to stay out of the politics swirling around the standards and focus this blog on helping people who are trying to implement them. And, after this post, I will keep it that way here at Tools for the Common Core.

But I’ve decided it’s time take a stand against the swirling tide of insanity that threatens our work, so I’m starting a new blog called I Support the Common Core. It will provide resources, links to articles, rebuttals, and discussion to help those who are fighting the good fight. If you sign up you will be getting emails and calls for action from me and others. Tools for the Common Core will remain available for those of you who prefer a quieter life and just want to get on with your jobs.

The success of this effort depends on you. If only 10 brave souls sign up I will thank them and close down the effort. If 1,000 of you join then we can get something done (and I promise there will be jokes).

Statement by CBMS presidents in support of the mathematics standards

I thought my readers might be interested in seeing this. The Conference Board on the Mathematical Sciences is an umbrella organization for the various societies in mathematics, applied mathematics, statistics, symbolic logic, and mathematics education.
Support Statement for CCSSMath

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.

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.