Essays from the Noyce-Dana project: clarifying the mathematical underpinnings of secondary school

In 2008–2009 Dick Stanley and Phil Daro, with the help of Vinci Daro and Carmen Petrick, convened a group of mathematicians and educators to write essays clarifying the mathematical underpinnings of secondary school mathematics in the United States. At the urging of Dick Stanley I am publishing these essays here.

The confusion over Appendix A

A number of people have gotten in touch with me recently about Appendix A, so I wanted to clarify something about its role. States who adopted the standards did not thereby adopt Appendix A. The high school standards were intentionally not arranged into courses in order to allow flexibility in designing high school courses, and many states and curriculum writers have taken advantage of that flexibility. There was a thread about this on my blog 3 years ago, and there is a forum on the topic here.

Appendix A was provided as a proof of concept, showing one possible way of arranging the high school standards into courses. Indeed, on page 2 of the appendix it says:

The pathways and courses are models, not mandates. They illustrate possible approaches to organizing the content of the CCSS into coherent and rigorous courses that lead to college and career readiness. States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own.

States will of course be constrained by their assessments. But Smarter Balanced consortium does not have end of course assessments in high school, leaving states and districts free to arrange high school as they choose. And although PARCC does have end of course assessments, they do not follow Appendix A exactly. See the footnote on page 39 of the PARCC Model Content Framework , which says

Note that the courses outlined in the Model Content Frameworks were informed by, but are not identical to, previous drafts of this document and Appendix A of the Common Core State Standards.

Furthermore, there are plenty of states not using either the PARCC of SMARTER Balanced assessments.

I hope this helps clear things up.

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.

Lesson Plans to Accompany Published Tasks

This is a guest post by Morgan Saxby, a fifth grade teacher in Chesterfield County, Virginia, who works with Illustrative Mathematics.  Morgan has begun writing lesson plans to accompany published mathematics tasks.

A clear step after developing high-quality mathematical tasks is to develop accompanying lesson plans.  I wrote seven lesson plans to accompany published tasks, all of which I tested in my classroom.  My goal was to write lesson plans that guided students to the level of thinking required by both the standards and the practices.

One example is the lesson plan for the task What is a Trapezoid?, aligned to standard 5.G.B.4.  A student who is able to successfully complete the task not only knows the relevant content, but can also skillfully construct viable mathematical arguments (Practice 3).  The obvious question to teachers is, “How do we get students there?”  The lesson plan Plane Figure Court is one possible way.  In it, students serve as “lawyers,” charged with proving or disproving a particular mathematical statement.  For example, the statement, “A square is a rhombus” has a lawyer arguing that this is true, and a lawyer arguing that this is false.  I required that students create justifications, even if they knew their justification was wrong.  The other students (the jury) decided the case based on the mathematical arguments made, not on what they thought was correct.  My end goal here was to help students to recognize valid (e.g., a square is a rhombus because it has four congruent sides) and invalid mathematical arguments (e.g., a square is a rhombus, because if you turn it a little it looks like one).

The format for the lesson plans is consistent through each one.  The first section includes the objective(s), an overview, and the standards to which the lesson are aligned. The second section includes a detailed lesson plan, as well as suggestions for assessment and differentiation.  The third section includes commentary and relevant attachments, such as worksheets or diagrams.  Some lesson plans, like Cooking Time 1, include student work.

The initial seven lesson plans are listed below, and others will be added in the future.  Tasks with lesson plans will be tagged “Lesson Plan Included”, and are accessible under the “Resources” heading.

5.NF How Much Pie? / Cooking Time 1

5.NF How many servings of oatmeal? / Cooking Time 2

5.NF Making Cookies / Cooking Time 3

5.NF Salad Dressing / Cooking Time 4

5.OA Video Game Scores / The Order of Operations

5.MD Cari’s Aquarium / What is Volume?

5.G What is a Trapezoid? / Plane Figure Court

We’ve also been working on developing review criteria for lesson plans that develop Illustrative Mathematics tasks into full-blown lessons.  The criteria are available here.

If you’ve taken a look at the lesson plans and the criteria, we’d love to hear your feedback by September 1.

Units, a Unifying Idea in Measurement, Fractions, and Base Ten

Think about a 4-by-5 rectangle. The rectangle contains infinitely many points$—$you could never count them. But once you decide that a 1-by-1 square is going to be “one unit of area,” you are able to say that a 4-by-5 rectangle amounts to twenty of these units. A choice of unit makes the uncountable countable.

Or think about two intervals of time. One is the period of Earth’s rotation about its axis; the other, the period of Earth’s revolution around the sun. Both intervals are infinitely divisible$—$a continuum of moments. But once you decide that the first period is going to be a “unit of time,” you are able to say that the second period of time amounts to 365 of these units. A choice of unit makes the uncountable countable.

More abstractly, think of a number line. The line is an infinitely divisible continuum of points. Zero is one of them. Now make a mark to show 1. The mark shows 1 as the indicated point; it also defines 1 as a quantity whose size is the interval from 0 to 1. This interval is the unit that makes the uncountable countable. We mark off these units along the line as 2, 3, 4, 5 and so on. (Later, we go the other way from zero marking off units: $–$1, −2, −3, −4, −5 and so on.) It is not for nothing that mathematicians call 1 “the unit.”

To a physicist, measurement is an active idea about using one empirical quantity (such as Earth’s rotation period) to “measure” (divide into) another empirical quantity of the same general kind (such as Earth’s orbital period). Mathematically, one can see that measurement is linked to division: how many units “go into” the quantity of interest.

Another way to say it is that measurement is linked to multiplication, in particular to a mature picture of multiplication called “scaling,” (5.OA) in which we reason that one quantity is so many times as much as another quantity. The concept of “times as much” enters the Standards in Grade 4 (4.OA.1, 4.OA.2, 4.NF.4), limited to whole-number scale factors.

The number line picture of this is that we are progressing from concatenating lengths to stretching them too. Thus, 2 is simultaneously “one more than one-more-than-0” and “twice as much as 1.” These two perspectives on 2 are linked by the distributive property, which defines the relationship between addition and multiplication. $1 + 1 = 1\times 1 + 1\times 1 = (1 + 1)\times 1 = 2\times 1$.

By Grade 5, scale factors and the quantities they scale may both be fractional; the flow of ideas extends into Grade 6, when students finally divide fractions in general. At that point, we may consider a deep measurement problem such as, “$\frac{2}{3}$ of a cup of flour is how many quarter-cups of flour?”

The roots of all this in K$–$2 are 1.MD.2 and 2.MD:1–7, and 2.G:2,3.

When we reflect back on the geometry in K$–$2 from this perspective, we see that some of what is going on is learning to “structure space” by, for example, seeing a rectangle as decomposable into squares and composable from squares. Researchers show interesting pictures of the warped grids that students make until they get sufficient practice.

Already by Grade 3 we are dissatisfied with measurement. We want to know what happens when the unit “doesn’t go evenly into” the quantity of interest. So we create finer units called thirds, fourths, fifths, and so on. This is the intuitive concept of a unit fraction, $\frac{1}{b}$: a quantity whose magnitude is equal to one part of a partition of a unit quantity into $b$ equal parts. (3.NF) We reason in applications by thinking of the unit quantity as a bucket of paint, or an hour of time. We reason about fractions as numbers by thinking of the unit quantity as that portion of the number line lying between 0 and 1. Then $\frac{1}{b}$ is the number located at the end of the rightmost point of the first partition.

Because you can count with unit fractions, you can also do arithmetic with them (4.NF:3,4). You can reason naturally that if Alice has $\frac{2}{3}$ cup of flour (two “thirds”) and Bob has $\frac{5}{3}$ cup of flour (five “thirds”), then together they have $\frac{7}{3}$ cup (seven “thirds,” because two things plus five more of those things is seven of those things). The meanings you have built up about addition and subtraction in K$–$2 morph easily to give you the “algorithm” for adding fractions with the same denominator: just add the numerators. (And don’t change the denominator $\dots$ after all, you would hardly change the unit when adding 3 pounds to 8 pounds.)

Likewise, multiplying a unit fraction by a whole number is a baby step from Grade 3 multiplication concepts. If there are seven Alices who each have $\frac{2}{3}$ cup of flour, it is a bit like when we reasoned out the product $7\times 20$ in third grade: seven times two tens is fourteen tens; likewise seven times two thirds is fourteen thirds. Again the meanings you have built up about multiplication in Grade 3 morph easily to give you the “algorithm” for multiplying a fraction by a whole number: $n\times \frac{a}{b} = \frac{n\times a}{b}$.

The associative property of multiplication $x\times (y\times z) = (x\times y)\times z$ is implicit in the reasoning for both $7\times 20$ and $7\times \frac{2}{3}$. So is unit thinking. In $7\times \frac{2}{3}$, the unit of thought is the unit fraction $\frac{1}{3}$. In $7\times 20$ and other problems in NBT, the units of thought are the growing sequence of tens, hundreds, thousands and ever larger units, as well as the shrinking sequence of tenths, hundredths, thousandths, and ever smaller unit fractions.

The conceptual shift involved in progressing from multiplying with whole numbers in Grade 3 to multiplying a fraction by a whole number in Grade 4 might be aided by the multiplication work in Grade 4 that extends the whole number multiplication concept a nudge beyond “equal groups” to a notion of “times as many” or “times as much” (4.OA:1,2). The reason this meshes with the problem of the seven Alices is that those seven Alices don’t exactly have among them seven “groups of things,” yet they do among them have seven “times as much” as one Alice.

The step in Grade 4 from “equal groups” to “times as much,” along with the coordinated step of multiplying a fraction by a whole number, represents the first major step toward viewing multiplication as a scaling operation that magnifies or shrinks. Multiplying by 7 has the effect of “magnifying” the amount of flour that a single Alice has. In Grade 5, we will “magnify” by non-whole numbers, for example by asking how many tons $4\,\frac{1}{2}$ pallets weigh, if one pallet weighs $\frac{3}{4}$ ton. We will find, during the course of that study, that a product can sometimes be smaller than either factor.

This kind of thinking about scaling is connected to proportionality, as when we use a “scaling factor” to get answers to compound multiplicative problems quickly. For example, 1500 screws in 6 identical boxes $\dots$ how many in 2 of the boxes? What would be a multi-step multiplication and division problem to a fifth grader becomes, for a more mature student, a proportional relationships problem: a third as many boxes, so scale the number screws of by a third. We quickly have the answer 500.

A year isn’t exactly 365 days$—$nor is it exactly 365.25 days. Could any rational number express the number of days in a year? Students of mathematics run up against a similar problem when they ask how many times the side of a square “goes into” its diagonal. By Grade 8, we learn without proof that the diagonal cannot be written as any rational multiple of the side. In this way, irrational numbers such as $\sqrt{2}$ enter the discussion, and likewise $\pi$ for the quotient of the circumference of a circle by its diameter. The Greeks called the diagonal and the side, or the circumference and the diameter, incommensurable quantities. This ancient idiom, meaning not measurable by a common unit, underscores the importance of measurement thinking to arithmetic.

This is an excerpt from a larger document (almost two years old now). It’s revised here with input from Phil Daro and William McCallum. Also see the Progression document on measurement in grades K$–$5. For those interested in the scholarly literature about these questions, I’m sure it is vast$\dots$but I’ll pass along one article I came across just the other day (Thompson and Saldhana, 2003). $–$J.

K–8 Publishers’ criteria released

Criteria for materials supporting implementation of the Common Core are now available at corestandards.org (go to the resources page to download the pdf, here is a direct link). They were written by the lead writers of the standards (Phil Daro, myself, and Jason Zimba). There’s an article about them by Erik Robelen at Education Week. A high school document, along with any revisions to the K–8 document resulting from public feedback, will be released early next year.

Phil Daro on learning mathematics through problem solving

Here is a video of Phil Daro calling for a change in classroom habits of how we teach students to solve problems. Great quote, from Phil’s daughter: “Daddy, I don’t have time to understand it, I just want to get it right on the test.”

Article by Roger Howe, “Three Pillars of First Grade Mathematics”

Roger Howe, a mathematician at Yale University who has thought a lot about elementary school mathematics education, has written Three Pillars of First Grade Mathematics, in which he expands on the CCSS Grade 1 mathematics standards. It’s well worth a read.