**The Common Core Mathematical Practices Cubed**

just

But what do these Practice Standards

As I pondered the definitional paragraphs of the Common Core Mathematical Practice Standards, two things occurred to me. First, the details of the Practice Standards deliberately overlap—like colored spotlights at a stage play. Yet there are differing affinities among the Practice Standards—some relate closely, others not so much.

Then it occurred to me to deploy the 8 practice standards to the 8 vertices of a cube picturing how, and how closely, each Standard would relate to neighboring Standards in several practical dimensions. When I then noticed that the

Next it occurred to me to have each of those eight definitional paragraphs illuminate a different iconic mathematical master—exactly one master standing in each defining spotlight. Obviously Euclid was the poster-child for carefully constructed mathematical argument and critical reasoning (Standard 3). Obviously Alan Turing, founding philosopher and implementer of programmable computing machines, was the poster child for strategic use of mathematical tools (Standard 5). And it didn’t take me too long to get plausible candidates for the other corners of my milk-carton.

____All of us aspire to do and teach those great new Mathematical Practice Standards--because theyjust

__sound__so good:Make sense of problems and persevere in solving them;__Standard 1:__Reason abstractly and quantitatively;__Standard 2:__Construct viable arguments and critique the reasoning of others;__Standard 3:__Model with mathematics;__Standard 4:__Use appropriate tools strategically;__Standard 5:__Attend to precision;**Standard 6:**Look for and make use of structure;__Standard 7:__Look for and express regularity in repeated reasoning.__Standard 8:__

But what do these Practice Standards

__mean__exactly? These standards interlock almost like a Rubik’s cube! Yes, the Common Core defines each Practice Standard at paragraph-length, and those paragraphs are dense and compelling. But what__is__mathematical Modeling vs. Quantitative reasoning, Structure vs. Regularity, attending to Precision vs. Strategic Tool use?As I pondered the definitional paragraphs of the Common Core Mathematical Practice Standards, two things occurred to me. First, the details of the Practice Standards deliberately overlap—like colored spotlights at a stage play. Yet there are differing affinities among the Practice Standards—some relate closely, others not so much.

Then it occurred to me to deploy the 8 practice standards to the 8 vertices of a cube picturing how, and how closely, each Standard would relate to neighboring Standards in several practical dimensions. When I then noticed that the

__faces__of the resulting cube seemed to represent 6 over-arching mathematical disciplines, I actually got excited!Next it occurred to me to have each of those eight definitional paragraphs illuminate a different iconic mathematical master—exactly one master standing in each defining spotlight. Obviously Euclid was the poster-child for carefully constructed mathematical argument and critical reasoning (Standard 3). Obviously Alan Turing, founding philosopher and implementer of programmable computing machines, was the poster child for strategic use of mathematical tools (Standard 5). And it didn’t take me too long to get plausible candidates for the other corners of my milk-carton.

**Eight** **Gifted**** ** **Mathematic** **Practitioners** ** and**** ** **Their** **Special** **Gifts**

The more I worked with my eight chosen mathematical masters, the more each master seemed to illuminate his Mathematical Practice definition rather than be illuminated by it.

So much for prologue concerning how I came to the thoughts I’ll be presenting here. In my presentation, I want to accomplish two things.

First, I want my Praxis Cube and the relationships it embodies to become a visual framework for grasping and teaching the 8 Practice Standards. Much of the meaning of the 8 Standards lies in the

__relations__among the Standards. Those relations are not a topic I’ve seen addressed much in other presentations, and those relations are what my Praxis Cube is all about.

Second, I want my 8 iconic mathematical masters to do more than merely

__stand for__certain mathematical practices. I want those iconic math practitioners to

__inspire__best mathematical practices in ourselves, our students, and our teaching. Yes, we can teach our 10th graders why and how it mattered to Euclid that a floppy compass is constructionally equivalent to a lockable compass--even if that’s not a part of the Math Content Standard. Yes, we can teach our 9th graders why and how to convert a repeating decimal back to a rational fraction--even if that’s not in the Math Content Standard. (Well, it’s almost in the Standard.)

My point is: best mathematicians and best mathematical practices teach us what’s possible, not just what to know, learn or teach in school. Perhaps the Eight Mathematical Practices of the Common Core overshot their modest goals in a very good way!

Are the Eight Mathematical Practices of the Common Core a high-water mark for the Common Core, beyond math education reform? Can teachers outside mathematics--English teachers, Social Studies teachers, etc.--decide that Common Core Mathematics Practices help reclaim for mathematics pedagogy its liberal arts heritage?

We math teachers can't convince our colleagues in other disciplines that mathematics is historically the most liberal of the liberal arts. But perhaps our students can.

**Our Eight Imperatives for Doing Good Mathematics--Parsimoniously Restated **

__Explain to yourself the meaning of a problem and look for entry points to its solution. Analyze the givens, constraints, relationships and goals. Conjecture the form and meaning of a solution before jumping in. Consider analogous problems, special cases and simpler versions of the problem to gain insights into the solution. Monitor and evaluate your progress—change course if necessary. Apply correspondences between equations, verbal descriptions and tables. Draw diagrams of important features. Seek regularities and trends. Use concrete objects or pictures to conceptualize and solve the problem. Check your answer by another method—continually ask, “Does it make sense?” Understand approaches of others to the problem; identify correspondences among multiple approaches.__

**1. Solve Tenaciously!**__Make sense of quantities and relationships in problems.__

**2. Quantify****Contextually!***Decontextualize*—represent a problem situation symbolically and manipulate symbols as if they have a life of their own.

*Contextualize*--pause as needed during manipulation to probe the referents for the symbols. Consider the units involved. Attend to meaning of quantities, not just how to compute them. Know and flexibly use multiple properties of objects and operations.

__Use stated assumptions, definitions and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore__

**3. Argue Logically!**the truth of conjectures. Analyze situations by breaking them into cases. Recognize and use counterexamples. Justify conclusions, communicate them to others, and respond to the arguments

of others. Reason inductively about data; make plausible arguments that take into account the context from which the data arose. Compare the effectiveness of two plausible arguments. Distinguish correct logic or reasoning from flawed; if there is a flaw in an argument, explain the flaw. Construct arguments using concrete objects, drawings, diagrams, and actions. Make correct sense even if the argument is not generalized or formal. Determine the domains to which the argument applies. Listen to and read arguments of others, and decide whether their arguments make sense. Ask useful questions to clarify or improve arguments of others.

__Apply mathematics to solve problems arising in everyday life, society, and the workplace. Write an addition equation to describe a simple situation. Apply proportional reasoning to plan a school event or analyze a community problem. Use geometry to resolve a design problem. Use functions to describe how a quantity you care about depends on other quantities. Be comfortable making simplifying assumptions/approximations to apply what you know to a complicated situation. Realize simplifications may need revision later. Identify important quantities in a practical situation. Use diagrams, tables, graphs, flowcharts and formulas to model relationships among important quantities. Analyze the model to draw conclusions about the practical situation. Reflect whether the conclusions make sense. Improve the model if it has not served its purpose.__

**4. Model****Re****alistic****ally!**__Consider available tools when solving a problem: pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, a computer algebra system, statistical package, or dynamic geometry application. Decide when tools are helpful, recognizing insights to be gained and limitations. Analyze graphs of functions and solutions on a graphing calculator. Detect errors using estimation and related knowledge. Make models that use technology to visualize results of varying assumptions. Explore consequences and compare predictions with data. Use external resources such as digital content on a website to pose and solve problems. Use technology to explore and deepen understanding of concepts.__

**5. Tool Incisively!**__Use clear definitions in your reasoning and when discussing with others. State the meanings of symbols you choose. Use the equal sign consistently and appropriately. Specify units of measure, and label axes with quantities in a problem. Calculate accurately and efficiently. Express numerical answers with a degree of precision appropriate to the problem. Give carefully formulated explanations, and examine claims of others through explicit definitions.__

**6. Communicate Precisely!**__Discern patterns and structures. For example, note that three and seven more is the same amount as seven and three more. Sort a collection of diverse shapes according to how many sides. Distribute 8 × 7 into 8 × 5 + 8 × 2. See 9 as 2 + 7 and 14 as 2 x 7 in the expression x^2 + 9x +14. Recognize the significance of an existing line in a geometric figure and draw auxiliary lines to solve problems. Step back for an overview and shift perspective. See algebraic expressions as single objects or compositions of objects. For example, see 5 – 3(__

**7. Structure Hierarchically!***x*–

*y*)^2 as 5 minus 3-times-a-square, and realize the expression can never exceed 5 for any real numbers x and y.

__Notice when calculations repeat. Look for general methods and shortcuts. When dividing 25 by 11, notice the calculation repeats endlessly. When finding points on a line through (1, 2) with slope 3, notice the relationship (__

**8. Iterate Recursively!***y*– 2)/(

*x*– 1) = 3. When expanding (

*x*– 1)(

*x*+ 1), (

*x*– 1)(

*x^2*+

*x*+ 1), and (

*x*– 1)(

*x^3*+

*x^2*+

*x*+ 1), notice the nth product equals x^(n+1) - 1. Maintain oversight of the problem-solving process while attending to details. Evaluate reasonableness of intermediate results.

HERE’S HOW I CREATED my imperative re-statements of the Eight Math Practice Standards. Everywhere I dropped the droning phase “Mathematically proficient students can ….” I restated bare imperatives—know, make sense, use, apply, consider, specify, discern, notice etc.

I find my imperatives to be denser and more compelling than the originals! My imperatives make the originals easier and clearer to re-read! However no turns of phrase, no sequences of ideas, and none of the mathematical examples were harmed in the making of my imperatives!

I also dropped elementary, middle and high school tags from the examples because all the practices apply equally to adult mathematics. True, kindergartners make arguments using concrete objects as referents--but so does NASA!

Finally, I needed eight even shorter verbal glosses of the Practice Standards—thumbnail statements or

slogans for each practice.

I created one-word headlines and two-sentence stories about each Standard, as follows.

**Eight Imperatives for Doing Good Mathematics--In a Nutshell **

**1. Solve!****Analyze your problem’s givens, constraints, relationships, goals, special cases and simpler versions.**

Monitor your progress and check your answers by another method.

__Manipulate symbols as if they have a life of their own, but pause to probe referents for the symbols.__

**2. Quantify****!**Consider the units involved.

**3. Argue!****Use established results and logical statements to explore conjectures. Communicate your conclusions**

and justify them to others.

__Apply mathematics to everyday life and society. Use functions, diagrams and formulas to model__

**4. Model****!**real quantities and draw practical conclusions.

__Consider pencil and paper, concrete models, computers, spreadsheets and calculators.__

**5. Tool!**Use external websites to pose and explore problems.

**6. Communicate!****Formulate explanations and examine claims using stated symbols and explicit definitions.**

Specify units of measure and use appropriate numerical precision.

**7. Structure!****Discern significant lines in a geometric figure and draw important auxiliary lines. Shift perspective**

to view an algebraic expression as one object or a composition of significant objects.

**8. Iterate Recursively!****When your calculations repeat, seek general methods and cumulative shortcuts. Investigate**

reasonableness of your problem-solving process while attending to its details.

So how can you tell if I've gotten these slogans and thumbnails right? Try the following exercise. Reverse-read each math practice from the nutshell, back to the parsimony, back to the canonical original. If anything in the reverse-progression jars you, contact me and correct me! Also, I'd love to hear translations of these progressions into other languages more stylish or allusive or precise or compelling than English, at least where various key turns of phrase are concerned.

**What the Colors of the Math Practices Mean**

Associating bright colors with each of the math practices is a subjective conceit but not a capricious extravagance. I think my color scheme can help you focus on the primary meaning of each practice and distinguish between closely related math practices.

Solving is

Quantifying has two colors: gray and brown.

Arguing is blue-eyed-sky-

Modeling has two colors: silver and gold.

Computing is

Communicating precisely is traffic-light-

Structuring is

Iterating is dark

YOU can build a foldable 3D Praxis Cube of your own! Just download and print out the .pdf page below.

Solving is

**ORANGE**because of the ardent flaming intensity needed to persevere in problem solving.Quantifying has two colors: gray and brown.

**GRAY**is the color of math calculations that measure physical quantities, and**B****ROWN**is the color of the the physical quantities themselves.Arguing is blue-eyed-sky-

**BLUE**. Perfect logical argument is the color we see, or look through, when we perceive mathematical beauty bare.Modeling has two colors: silver and gold.

**SILVER**is the color of the mathematical models we build, and**GOLD**is the color of the reality we model. (Modeling is the big brother of Quantifying.)Computing is

**RED**because of the cheery, focused power of every math tool from the straightedge to the home computer.Communicating precisely is traffic-light-

**GREEN****because it allows good mathematical ideas to move forward in the world.**Structuring is

**PURPLE**because the exacting constraints of finite mathematical relationships produce such amazing mathematical insights.

Iterating is dark

**B****ROWN**because of the density of the real continuum. Iterative procedures allow us to begin to appraise the mathematical relationships of the integers to the real numbers, the infinitesimals, and infinity.**But Wait! There's More!**YOU can build a foldable 3D Praxis Cube of your own! Just download and print out the .pdf page below.

The downloadable file below folds to make a 3D Praxis Cube about the size of a baseball.

Just print the page, cut it into two strips, fold the edges, and apply a gluestick.

The result is a Praxis Cube you can live with in 3 dimensions!

A somewhat flimsy paper Praxis Cube you say? Wait! There's more!

You can make a Sonobe Module Origami cube that

__exactly__provides an armature for this Praxis Cube! The Sonobe cube uses 6 sheets of 8.5 x 11 paper, no glue involved. The resulting Praxis Cube is

__very__satisfying to the touch. Weighs over an ounce...

How do you make a Sonobe Cube anyway? Do I have to explain

__everything__? Google it, damnit!

There's more! The first 5 callers who respond in the next million minutes will receive a complete full-color Sonobe-Structured Praxis Cube ABSOLUTELY FREE OF CHARGE! Additional callers pay shipping and handling.

I have one more question. If you make a Sonobe Origami Cube from six unit-square sheets of paper, what is the exact length of the resulting cube's edge, in square roots?

Actually I have another question. Are you a high school geometry teacher whose students are interested in crowd-sourcing Praxis Cubes for the common good?

Get back to me, and we'll talk.

Peter F. Lang, Cleveland OH

peter-lang-01@hotmail.com

praxis_cube_foldable_3d.pdf |