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

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.

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.

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.

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.

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.

The most recent message said quite sternly that $V=bh$ was NOT correct, and that it MUST BE $V = Bh$. This point of view is one of the starkest examples I know of the obstacles we must overcome in restoring the culture of mathematics in schools. The notion that certain letters MUST always stand for the same thing across different formulas is itself a mathematical error, a profound misunderstanding of how symbols are used. It’s like thinking the word blue must always be written in blue.

And the misconception is not harmless. Students who come to college with it do not fare well amid the profusion of symbols in their science classes, unable to see that the function $f(x) = \sin(ax)$ in their calculus class is the same as the function $A(t) = \sin(\omega t)$ in their physics class. Part of the power of algebra is that you can choose any letter you like to represent a quantity, as long as you specify what the letter stands for, and you can re-use that letter with a different meaning in a different problem.

That said, the standard does not dictate the use of any specific letters; indeed, the core meaning of the standard is not about formulas at all, but rather about finding the volume of a rectangular prism by multiplying its length, width, and height, or by multiplying the area of its base by its height. So teachers and curriculum materials can use whatever letters they like, including $V = Bh$, or no letters at all. Indeed, formulas should always have words associated with them. Naked formulas like $V = bh$ mean nothing by themselves without surrounding words, such as in the sentence

The volume $V$ in cubic inches is given by $V = bh$, where $b$ is the area of the base in square inches and $h$ is the height in inches.

Changing the two occurrences of $b$ to $B$ changes the meaning of this sentence not one whit.

[Thanks to Jason Zimba for suggesting the title of this post.]

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

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.

How can mathematicians aid teachers in learning this mathematics, in collaboration with others responsible for teacher education?

Current research and experience are synthesized to answer these questions in the new report The Mathematical Education of Teachers II (MET II) from the Conference Board of the Mathematical Sciences. This report updates The Mathematical Education of Teachers (published in 2001) and extends its scope from preparation to professional development in the context of the Common Core State Standards for Mathematics.

The audience for the report includes all who teach mathematics to teachers—mathematicians, statisticians, and mathematics educators—and all who are responsible for the mathematical education of teachers—department chairs, educational administrators, and policy-makers at the national, state, school-district, and collegiate levels.

The report may be downloaded free at the Conference Board of Mathematical Sciences web site.  Printed copies may be ordered from the American Mathematical Society.

The Conference Board of the Mathematical Sciences (CBMS) is an umbrella organization consisting of sixteen professional societies all of which have as one of their primary objectives the increase or diffusion of knowledge in one or more of the mathematical sciences. Its purpose is to promote understanding and cooperation among these national organizations so that they work together and support each other in their efforts to promote research, improve education, and expand the uses of mathematics.

For further information, contact CBMS director Ronald Rosier, 410-730-1426; 202-293-1170.

October 2012 Berkeley Conference page Participants made classroom activities based on a particular task or set of 2-3 tasks

May 2012 New Orleans Conference page Participants made short PD units or classroom activities around a particular standard, group of standards, or cluster

February 2012 Tucson Conference page Participants made PD modules around a particular standard, cluster, or domain

Please register quickly as our previous workshops have sold out!

I will keep the current email notification feature for those who prefer to keep using that.

[29 July 2012] This thread is closed. You can ask questions here.

ccss_progression_g_k6_2012_06_27

[30 July 2012] This thread is now closed. Please go here to discuss this progression.

Here is the information page and registration.

I know somebody is going to ask about printing tasks. We don’t have that yet, but are working on it!

Here is the information page and registration.

9_11 Scope and Sequence_traditional1

9_11 Scope and Sequence_traditional2

9_11 Scope and Sequence_integrated1

9_11 Scope and Sequence_integrated2

High School Units-All-03feb12

1. Registered users can rate tasks by voting them up or down.
2. Registered users can comment on tasks.

You can edit and delete your comments, although it will leave a placeholder note that you deleted your comment in case someone replies to your comment (so their comment won’t be deleted also).

We have more improvements in store, and illustrations are being added every day, so keep checking in.

A Grecian urn

In the Common Core State Standards, individual statements of what students are expected to understand and be able to do are embedded within domain headings and cluster headings designed to convey the structure of the subject. “The Standards” refers to all elements of the design—the wording of domain headings, cluster headings, and individual statements; the text of the grade level introductions and high school category descriptions; the placement of the standards for mathematical practice at each grade level.

Standards for a Grecian Urn

The analogy with the urn only goes so far; the Standards are a policy document, after all, not a work of art. In common with the urn, however, the Standards were crafted to reward study on multiple levels: from close inspection of details, to a coherent grasp of the whole. Specific phrases in specific standards are worth study and can carry important meaning; yet this meaning is also importantly shaped by the cluster heading in which the standard is found. At higher levels, domain headings give structure to the subject matter of the discipline, and the practices’ yearly refrain communicates the varieties of expertise which study of the discipline develops in an educated person.

Fragmenting the Standards into individual standards, or individual bits of standards, erases all these relationships and produces a sum of parts that is decidedly less than the whole. Arranging the Standards into new categories also breaks their structure. It constitutes a remixing of the Standards. There is meaning in the cluster headings and domain names that is not contained in the numbered statements beneath them. Remove or reword those headings and you have changed the meaning of the Standards; you now have different Standards; you have not adopted the Common Core.

Sometimes a remix is as good as or better than the original. Maybe there are 50 remixes, adapted to the preferences of each individual state (although we doubt there are 50 good ones). Be that as it may, a remix of a work is not the same as the original work, and with 50 remixes we would not have common standards; we would have the same situation we had before the Common Core.

Why is paying attention to the structure important? Here is why: The single most important flaw in United States mathematics instruction is that the curriculum is “a mile wide and an inch deep.” This finding comes from research comparing the U.S. curriculum to high performing countries, surveys of college faculty and teachers, the National Math Panel, the Early Childhood Learning Report, and all the testimony the CCSS writers heard. The standards are meant to be a blueprint for math instruction that is more focussed and coherent. The focus and coherence in this blueprint is largely in the way the standards progress from each other, coordinate with each other and most importantly cluster together into coherent bodies of knowledge. Crosswalks and alignments and pacing plans and such cannot be allowed to throw away the focus and coherence and regress to the mile-wide curriculum.

Another consequence of fragmenting the Standards is that it obscures the progressions in the standards. The standards were not so much assembled out of topics as woven out of progressions. Maintaining these progressions in the implementation of the standards will be important for helping all students learn mathematics at a higher level. Standards are a bit like the growth chart in a doctors office: they provide a reference point, but no child follows the chart exactly. By the same token, standards provide a chart against which to measure growth in childrens’ knowledge. Just as the growth chart moves ever upward, so standards are written as though students learned 100% of prior standards. In fact, all classrooms exhibit a wide variety of prior learning each day. For example, the properties of operations, learned first for simple whole numbers, then in later grades extended to fractions, play a central role in understanding operations with negative numbers, expressions with letters and later still the study of polynomials. As the application of the properties is extended over the grades, an understanding of how the properties of operations work together should deepen and develop into one of the most fundamental insights into algebra. The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.

This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events.

The intent of the Illustrative Mathematics website is to present sets of tasks that, when taken together, illustrate a particular standard. Eventually, we want to have sets of tasks for each standard that:

• Illuminate the central meaning of the standard and show connections with other standards,
• Clarify what is familiar about the standard and what is new with the Common Core,
• Include both teaching and assessment tasks, and
• Reflect the full range of difficulty expected.

At the moment we have quite a few standards that only have one or two tasks. For the new contest, we are asking people to help us fill in some of the sets that are not yet complete. People can send a task for any standard that already has at least one task, and in addition to sending a task, we are asking people to explain what “gaps” it helps fill. For example, there is one task that illustrates some of the expectations in A-CED.2 ”Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.” However, a task that asks students to graph equations that represent relationships between quantities would help round out the set needed to fully illustrate this standard.

Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, March 12th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Feb 13 – Mar 12, 2012.”

We have also created a permanent page with general information about the Illustrative Mathematics Contest that all who are thinking about submitting a task should check out.

Lastly, we would like to encourage any groups doing PD on the Common Core Standards to consider using this contest as a way to approach thinking about the standards and the depth of understanding that they represent. Tasks are reviewed by other professionals in mathematics education and we work with contributors to refine their submission. Listed below are tasks that have been through the review process and are now available on the Illustrative Mathematics website.

Contest Winners

We are pleased to announce a few of the winners of our earlier contests whose tasks are available at the Illustrative Mathematics website!

Dan Meyer – Doctoral student, Stanford University, California. His task is titled “8.F High School Graduation”.

Travis Lemon – Teacher, American Fork Junior High School, Utah. He has two tasks in development and two posted titled “7.RP Cooking with the Whole Cup” and “8.EE Find the Change.”

James E. Bialasik and Breean Martin–Teachers, Sweet Home CSD, New York. Their task is titled “8.EE Cell Phone Plans.”

We have more winning entries making their way through the review process now and will post them here once they are published.

• K.MD Describe and compare measurable attributes.
• 1.MD Measure lengths indirectly and by iterating length units.
• 2.MD Measure and estimate lengths in standard units.
• 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
• 4.MD Geometric measurement: understand concepts of angle and measure angles.
• 5.MD Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
• 6.G Solve real-world and mathematical problems involving area, surface area, and volume.
• 7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
• 8.G Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
• G-GMD Explain volume formulas and use them to solve problems.
• G-GMD Visualize relationships between two-dimensional and three dimensional objects.
• G-MG Apply geometric concepts in modeling situations.

Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, February 13th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Jan 30 – Feb 13, 2012.”

We have also created a permanent page with general information about the Illustrative Mathematics Contest that all who are thinking about submitting a task should check out.

Contest Winners

We are pleased to announce three of the winners of our first Contest!

Dan Meyer – Doctoral student, Stanford University, California. His task is titled “8.F High School Graduation” and is available on the Illustrative Mathematics website.

Travis Lemon – Teacher, American Fork Junior High School, Utah. He has four tasks in development. We’ll announce it here when they are ready.

James E. Bialasik and Breean Martin–Teachers, Sweet Home CSD, New York. Their task is titled “8.EE Cell Phone Plans”.

We have other people whose names will appear here as their tasks get closer to completion.

• 2.MD.9
• 3.MD.4
• 4.MD.4
• 5.MD.2

Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, January 30th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Jan 17 – Jan 30, 2012.”

We have also created a permanent page with general information about the Illustrative Mathematics Contest that all who are thinking about submitting a task should check out.

Contest Winners

We are pleased to announce three of the winners of our first Contest!

Dan Meyer – Doctoral student, Stanford University, California. His task is titled “8.F High School Graduation” and is available on the Illustrative Mathematics website.

Travis Lemon — Teacher, American Fork Junior High School, Utah. He has four tasks in development. We’ll announce it here when they are ready.

James E. Bialasik and Breean Martin–Teachers, Sweet Home CSD, New York. Their task will appear soon!

We have other people whose names will appear here as their tasks get closer to completion.

• F.IF.4
• F.IF.5
• F.IF.6

The new theme for this week is focused on Expressions and Equations and we recommend reading the progressions document for 6 – 8 as you think about the task you would like to submit. People are invited to submit tasks for these specific standards:

• 6.EE.9
• 7.EE.3
• 8.EE.8
• A.CED (there are four standards under this heading)

Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, January 9th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Jan 2 – Jan 9, 2012.” If your task is accepted, we will notify you the week following the deadline. We may ask you to work with us to revise the task before we accept it. People may submit multiple tasks. Any questions about the contest should be sent to the same email address with subject line “Question about Illustrative Mathematics Task Writing Contest Jan 2 – Jan 9, 2012.”

How it Should Look

The Illustrative Mathematics team would like to encourage participants to review some of the tasks already published on the website (once you follow this link, click on e.g. K-8 Content Standards with Illustrations) for standards 6.RP (Shirt Sale), 7.RP (Tax and Tip), and F.BF (Kimi and Jordan) to get an idea for how tasks should look. Because the tasks are entered into the database in a particular way, the format is important. Also keep in mind that all task submissions must include at least one complete solution; here is a word_template that shows the fields that need to be included. We will give extra consideration to tasks written by pairs or teams of people, tasks that have natural connections to other tasks related to this stream, and tasks with insightful commentary. Please submit tasks in word format or LaTeX, along with a pdf if possible. To learn more about what makes a good mathematical task, read this article by Kristin Umland.

We look forward to another great set of tasks!

Click here to see the complete report.

ccss_progression_sp_68_2011_12_26_bis

As usual, comments and suggestions are welcome.  [New file with corrections uploaded 12/26/11, 11:38 am MST.]

[5 August 2012] This thread is now closed for comment. Please ask questions about Grades 6–8 Statistics and Probability here.

Second, here is a report from a meeting organized by Paola Sztajn, Karen Marrongelle, and Peg Smith: http://www.nctm.org/uploadedFiles/Math_Standards/Summary_PD_CCSSMath.pdf

• 3.OA.9
• 4.OA.5
• 5.OA.3
• 6.EE.9
• 7.RP.1
• 8.F.4

Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task (not per author, sorry!) and must be emailed by Monday, December 19^th midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Dec 12 – Dec 19, 2011.” If your task is accepted, we will notify you the week following the deadline. We may ask you to work with us to revise the task before we accept it.  People may submit multiple tasks. Any questions about the contest should be sent to the same email address with subject line “Question about Illustrative Mathematics Task Writing Contest Dec 12 – Dec 19, 2011.”

How it Should Look

All task submissions must include at least one complete solution. We will give extra consideration to tasks written by pairs or teams of people, tasks that have natural connections to other tasks related to this stream, and tasks with insightful commentary. Please submit tasks in word format or LaTeX, along with a pdf if possible. Here is a word_template. 

Things You Should Know Before Submitting

Writing a great task is an art, and tasks often benefit from multiple revisions. It would be helpful to read some of the tasks that have already been accepted at http://illustrativemathematics.org. To learn more about what makes a good mathematical task, read this article by Kristin Umland.

We look forward to reading your tasks!

New Version of the CCSSM Clickable Map is now up on the Tools page!

http://ime.math.arizona.edu/commoncore/

IM&E is working as a part of a collaboration between NCTM, NCSM, ASSM, AMTE, CBMS, Achieve, AFT, IAS/PCMI, MfA, NAGB, and NEA to produce a day long professional development on the Common Core. The effort is authored by teachers across the country, will be field tested by teachers, and ultimately facilitated by teachers with teachers as the targeted audience. The toolkit will have activities hitting four main goals; to see structure in the standards, to understand the Standards of Mathematical Practice, to align tasks to the standards, and to understand the language used in the standards. Activities will be developed for Elementary, Middle School, and High School teachers. As we develop tools and activities for this project we will be posting them here in the Tools section of the blog so check back soon for PD tools to further explain the Common Core in Mathematics.  The toolkit should be ready for initial pilot testing this summer so stay tuned if your school or district might be interested in participating as a beta testing site for activities designed for the Toolkit.

Eventually the sets of tasks will include elaborated teaching tasks with detailed information about using them for instructional purposes, rubrics, and student work. Such a fully developed set of tasks will be what we call a Complete Illustration of the standard. Right now we are trying to build up our collection of Initial Illustrations of standards, which will have the following characteristics:

• A minimum of 4 tasks (although typically 5-6 or more depending on the standard).
• Most will be more like assessment tasks or brief teaching tasks. At least one will be the kernel of an instructional task that can eventually be more fully developed and elaborated with the help of teachers using it in classrooms.
• The tasks in the set will vary in difficulty. Some but not all will be scaffolded.
• A balance in computational/algorithmic and conceptual tasks.
• An appropriate number of contextual problems for the standard.
• Most of the tasks will illuminate the “center of mass” of the standard, and a few will light up the periphery.
• At least one task will bridge in some way to another standard, ideally across domains or grade levels.

The new site also allows users to register. This is not necessary to see the tasks, but if you register you will be eligible for news bulletins and various opportunities for involvement in the project that will arise over the next few months.

Go to illustrativemathematics.org to see the new goodies. (This is still a beta site, and you may encounter slowness or other problems from time to time.)

If you are reading it with Adobe Acrobat, you will need the latest version (10.1). If you are using a Mac, you can also use the native Mac pdf reader Preview, or the open source pdf reader Skim.

We hope to get drafts of the remaining K–8 progressions out soon; thank you for your patience.

[File updated 2/5/12 to fix printing problems.]

[31 July 2012] This thread is now closed. Please ask questions here.

[Edited 2011/7/09 to add pdf file.
Documents Edited 2011/12/01]

We will be releasing a draft of the Fractions progression before the end of June, and the Ratios and Proportional Relationships progression in early July.

[29 July 2012] This thread is now closed. You can ask questions here.

[29 July 2012] This thread is now closed. You can ask questions here.

[29 July 2012] This thread is now closed. You can ask questions here.

[2012/08/31. This thread is now closed. Please ask questions here.]

[29 July 2012] This thread is now closed. You can now ask questions on this progression here.

More information is available online at:
http://www2.edc.org/CME/mpi/workshop.html

“Articulating Research Ideas that Support the Implementation of the Professional Development Needed for Making the Common Core State Standards in Mathematics Reality for K-12 Teachers is a newly funded NSF project that will coordinate knowledge from different fields to develop recommendations for the design, implementation, and assessment of large scale professional development systems consistent with the mathematics of the CCSS. Research results from diverse perspectives (e.g, mathematics education, organizational theory, professional development) will be articulated into a coherent framework and a set of recommendations for successful large-scale, system-level implementation of mathematics professional development initiatives. The recommendations will be disseminated through the National Council of Teachers of Mathematics. Additionally, the Association of Mathematics Teacher Educators, the Mathematical Association of America, the National Council of Teachers of Mathematics, the National Council of Supervisors of Mathematics, and the Association of State Supervisors of Mathematics are partners in this effort.”

1. Read the standards, noting domains, clusters that are particularly  high priority for college and career readiness (don’t include individual standards unless you absolutely must).

2. Select the most important domain or cluster at each of the grades 6-8 and themes in high school (or just some of the grades and themes if you don’t have time for all).

3. [Optional] For each one pick one or two practices that are particularly  salient, and explain how it is exemplified.

4. Rule: Give higher priority to things that are harder to fix [if students come to college not having them], not things  that you hate to have to fix but that are easier.

5. Write a one page common agreement on priorities that both higher education and high school (1) accept as important (2) clearly understand.

[Post edited for clarification, 2/19/11]

Big question: what is going to be possible to do this time that it wasn’t possible to do before?

There are echoes of the new math in some of what is going on now, and there are lessons to be learned from more recent reform efforts.

Project at Michigan that is focusing on coherence across grade levels. The New Math had its take on coherence; NCTM Curriculum and Evaluation Standards, PSSM, and Focal Points all had their take. When I was here in California in the 1980s, California had a progressive framework, followed in the 90s by a framework which many saw as a U-turn. Teachers saw a radical shift. In the 10 years I’ve been at Michigan, teachers have had three different sets of standards. Some teachers are numb to it.

On the content side, the big thing that Common Core brings is understanding to the expectations. Banned in Michigan and many other states, because of assessment. I’m not sure that 50 years after the New Math we are any closer to figuring out how to assess understanding. We also have the standards for mathematical practice, but still not in the content. (Good to call them standards.)

To the extent that the assessment can be seen as driving attention to the practice standards, the conversations with teachers will be a lot easier. Not sure that the outline of content is any better than any other we have had. New Math and PSSM had unintended consequences (back to basics, math wars respectively).

The important issue is scalability. Teacher education is a state driven enterprise. Now that everybody is adopting the same set of standards. That allows for collaboration across institutions and across state lines. That’s a very exciting project. AMTE can be at the forefront of doing that work.

1. Clarify the meaning
2. Support stake holders
3. Prepare and support PK-16 teachers
4. Support the development of high quality formative and summative assessments
5. Promote research
6. Develop a governing structure

Big question on number 3: what do we need to do now?

• Assessment examples
• Capacity building and leadership development (interpreting the standards)
• Professional development materials (raise awareness, how to change what you are doing now)
• Dissemination efforts

Long term efforts:

• tools for teachers
• tools to help administrators recognize classroom implementation
• examples of student work
• think about professional development that is differentiated based on the needs of different teachers (novice, veteran, …)
• mathematics content courses that can help people understand how the mathematical practices can be implemented in content

“More testing isn’t necessarily better,” said Dr. Linn, who said her work with California school districts had found that asking students to explain what they did in a science experiment rather than having them simply conduct the hands-on experiment — a version of retrieval practice testing — was beneficial. “Some tests are just not learning opportunities. We need a different kind of testing than we currently have.”