## 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

## 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.

## Parent Resources on Tools Page

Click on Tools (above) to see links to resources aimed at helping parents better understand the Common Core.

## 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.

## Toward Greater Focus and Coherence: an Illustrative Mathematics Common Core Conference

Register today to reserve your spot for IM&E/Illustrative Mathematics’ next Common Core Conference!

Towards Greater Focus and Coherence May 26-28, 2013 at the University of Arizona This is a great way to start the summer, while you are still in the classroom flow!

We are looking forward to meeting people who care about math education and collaborating with math coaches, classroom teachers, mathematicians, district math specialists, and mathematics educators.

Highlights of the conference include:

1. Perspective from Bill McCallum, lead writer of the Common Core
2. Activities that can be immediately used in your classroom, and a plan for creating similar Common Core aligned activities for students in the future.
3. Breakout sessions from classroom teachers modeling the focus of the Common Core by digging into a particular standard or cluster.
4. Highlights of the focus and coherence of different grade bands and the mathematics behind the standards.
5. Online resources to support the Common Core.

You don’t want to miss this opportunity. Reserve your spot by March 31st for the best rates by registering online.

## Attend to the verbs in the Mathematical Practices

Editor’s note: This is a guest post by Dev Sinha, a mathematician from the University of Oregon who is working with Illustrative Mathematics. Dev recently attend a meeting of Illustrative Mathematics devoted to elaborating the practice standards at different grade levels, where he made a very important point about verbs in the practice standards, so I invited him to submit this post.

Mathematical objects are key components of content standards.  Practice standards on the other hand describe student actions.  Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs.

For  MP7 Look for and make use of structure, “Look for” is a key phrase. Consider the task 2.NBT Making 124, which in brief asks students to decompose 124 into hundreds, tens and ones in all ways they can find (e.g. 6 tens and 64 ones).  In order to be efficient or complete, students will need to use exchanges—of a ten for ones or of a hundred for tens—systematically.  That is, there is a structure of systematic exchanges which students must look for and make use of in order to be highly successful.  We can say that this task implicitly invites students to engage in MP7, through both its “look for” and “make use of” halves.  If the systematic exchanges are suggested by the task or by the teacher before students have had a chance to search themselves, then the practice would not be fully be engaged.

For MP8 Look for and express regularity in repeated reasoning, “express” is an important verb.  Consider this instructional sequence from the progression on Progression on Ratios and Proportional Relationships in which students are to consider equivalent-tasting mixtures of juice.  While students may immediately notice some regularity it is the process of expression, going say from observations about a table to statements like “if we increase the grape juice by 1 cup we must increase the peach juice by 2/5 of a cup to taste the same” and ultimately to writing the equation $y = 2/5 x$, which constitutes the bulk of the mathematical work of the task.  MP8 provides language to discuss this kind of expressive mathematical work.

Development of tasks and lessons involves consideration of the mathematical work students are invited to do.  Content standards provide nouns to be employed in describing this work, while practice standards provide verbs.

Posted in Uncategorized | 5 Comments

## To B or not to B

Once every few months or so I receive a message about the following standard:

6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas $V=lwh$ and $V=bh$ to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

See if you can guess what people think the problem is before reading on. Continue reading

Posted in Uncategorized | 9 Comments

## Mathematics teaching community website

I just attended in interesting talk at the Joint Math Meetings in San Diego about a project started by Sybilla Beckmann, the Mathematics Teaching Community. It’s a place where you can post and answer questions about all aspects of mathematics teaching, both K–12 and college. I thought people on this list might be interested in checking it out.

## Draft progressions on high school Algebra and Functions

I’m pleased to be able to give you the draft progressions on Algebra and Functions. These progressions are somewhat different from the K–8 progressions. Since the high school standards are not arranged into courses, the progressions are really more like descriptions than progressions; they are not in any particular curricular order. Furthermore, because each one covers a topic that occupies a large part of the high school curriculum, it gives less detail about how each standard might be addressed or how different standards might be arranged into various different curricular implementations.

Comments as always are welcome in the relevant forums: Algebra or Functions.

Posted in Progressions | 13 Comments

## Mathematics Common Core in the Classroom: March 1-3, 2013 Syracuse, New York

Save the date and register to reserve your spot now!

IM&E/Illustrative Mathematics’ next Common Core Math Conference is coming to the east coast; Syracuse, New York!

Mathematics Common Core in the Classroom March 1-3, 2013 at the Doubletree Hotel, Syracuse New York

We are looking forward to meeting people who care about math education from across the country and collaborating with math coaches, classroom teachers, mathematicians, district math specialists, and mathematics educators. Highlights of the weekend include:

1. Perspective from Bill McCallum, lead writer of the Common Core, on “What’s different about these standards?”
2. Activities that can be immediately used in your classroom, and a plan for creating similar Common Core aligned activities for students in the future.
3. Breakout sessions from classroom teachers modeling the focus of the Common Core by digging into a particular standard or cluster.
4. Mathematicians discussing the focus of different grade levels and the mathematics behind the standards.
5. Online resources to support the Common Core.

You don’t want to miss this opportunity. Save your spot today by registering online.  OCM-BOCES teachers register with your district office.  All others register on the IM&E website.

Posted in Professional development | Tagged , , | 2 Comments