## From Al Cuoco, a commentary on formal algebra and bookkeeping

March 2, 2015 Leave a comment

Some interesting thoughts by Al Cuoco about polynomial arithmetic and how Taylor series can be used to solve some interesting problems.

the blog of the Mathematical Practices Institute

March 2, 2015 Leave a comment

Some interesting thoughts by Al Cuoco about polynomial arithmetic and how Taylor series can be used to solve some interesting problems.

December 2, 2014 Leave a comment

After a brief conversation, Paul Goldenberg, lead author of EDC’s Think Math! curriculum, had this to say on the debate between the phrases “units digit” and “ones digit”.

Yes, anything can be a unit. There are lots of arguments one could make either way, I suppose.

A simple case *for* “ones” is that it is more parallel to “tens” and “tenths” — a specific magnitude named after a specific number.

The number 1 (ok, and –1) is, of course, a “unit” (in the world we care about) but that’s a different use of the word. Unity and oneness are also, though synonymous, slightly different uses than the number-name “one.” CCSS chooses to call teen numbers “one ten and some extra ones.” It feels slightly weirder to say “one ten and some extra units” but only (again) because of the not-quite-parallel language. And, educationally, “one” is a more familiar word than “unit.”

There! I’ve defended “one” about as much as I can. But is that a case for “ones digit” over “units digit”? Only the “familiarity” argument honestly holds any weight with me, and not much weight. After all, the idea of choosing a unit and basing everything else on that unit seems valuable, and “1” is the obvious candidate in the integers. When we’re measuring distance between cities (in any sensible country), we definitely choose km and not m as the purpose-specific unit but the entire naming system is based on m as The Unit (km and cm are just modifications of m).

Hmm, so maybe 10 should be the unit in the integers, since our decimal (that’s its name) system is based on powers of 10. So there! I’ve wound up defending “one” a bit more by showing how slippery “unit” can be.

I’m actually not sure what led to the choice of “units digit” for all of Think Math, but I do know why we chose the symbol *u* to label that column (because I had to answer it many times, which might well have been the source of the naming we chose): *o* looks too much like 0.

Is there a strong case to be made *against* “units digit,” given the various ambiguities we get for either choice? I’m not convinced there’s anything *mathematical* in this discussion, since the meanings of any of these terms depends so much on context. After all, in the numeral 235, the two’s place is in the hundreds’ place. (Oh, should those have apostrophes or not?) Pedagogically, I’m convinced that it doesn’t matter at all. Oh, and one more argument for “ones digit” (or ones’ digit): if vast majority usage goes one way, then bucking the usage is ornery and unhelpful. But, sigh, there are so many places to put one’s (or unit’s) energy. I can’t get worked up much about this unit.

January 21, 2014 Leave a comment

Over the last few months, the Common Core State Standards have hit the headlines in a big way, with commentaries from educators, elected officials, columnists, and even talk-show personalities. Some argue that the Core is rotten, a liberal tactic to usurp local control. Others express faith in the Core’s ability to transform K-12 education, providing a lever needed to budge the stubborn inertia that stalls improvement. Sides have formed and the simmering rhetoric has come to a full, rolling boil.

As a former high school mathematics teacher — 25 years in the Massachusetts Public Schools — and a current curriculum developer and teacher educator, I’ve read the mounting criticism of and praise for the Common Core State Standards in Mathematics with great interest. Much of the criticism that I read is oversimplification; some of it exhibits a genuine ignorance of the daily work of teaching and learning. Very little shows familiarity with the actual content of the standards, and much of it is removed from a focus on the two groups that matter most: teachers and students.

When I first heard of the Common Core — I supplied some background research to the writing team — I saw a promise that it might become one of many tools to help us tackle some of the pervasive problems that have plagued our schools for generations. The authors delivered on that promise. And there’s one problem in particular for which these mathematics standards hold great potential: closing the vast gap between *school mathematics* and *mathematics as a scientific discipline*.

Many adults have suffered through school mathematics. At the high school level, it’s a disconnected zoo of special-purpose methods and meaningless topics that most mathematicians wouldn’t recognize. There’s the “box method” for setting up equations. There’s the “y = mx+b” method for finding equations of a line through two points. And there are many more examples of paraphernalia that has no existence outside of school. There’s also incredible flatness: everything is as important as everything else. I’ve seen a half-period wasted discussing whether or not the positive *y*-axis is in the first quadrant.

Common Core calls for something different. Its content standards are devoid of schoolish nonsense and free of the standards-speak that has made so many state documents unreadable. Anyone who uses mathematics professionally can read the Common Core and recognize what’s there, a direct path from the mathematics students learn in school to the mathematics used in so many fields today. There is a practice of mathematics, just as there is a practice of medicine and a practice of teaching. Before the advent of the Common Core, that practice was largely invisible in state standards documents.

The Common Core reflects the reality that the discipline of mathematics is as much a web of *habits of mind* — key ways of thinking employed by proficient users of mathematics — as it is a vast edifice of results. And, unlike current state standards, the Common Core specifies Standards for Mathematical Practice that give us a lens through which we can look at the content standards. The Standards for Mathematical Practice can help bring some desperately needed coherence and parsimony to school mathematics. The Standards for Mathematical Practice are not, as some claim, the “most important part” of the Common Core. Rather, as Jason Zimba, one of the lead writers for the Common Core, puts it: “Practices are an engine for focusing, and a reward for doing so.”

Because the centuries-old practice of mathematics has not appeared in an explicit way until now, the Standards for Mathematical Practice are a source of confusion among educators and school administrators. In Massachusetts, the Department of Elementary and Secondary Education has addressed this challenge head on: it partnered with our group at Education Development Center to create a 45-hour course designed to provide experience with and examples for how the practice of mathematics can work in concert with specific content.

The course revisits everyday content that has traditionally been difficult to teach and learn, and it develops that content with a focus on practices such as using precision, making viable arguments, and abstracting from examples. These habits of mind are essential to mathematics outside of school, and they provide teachers and students with tools to make school mathematics a coherent and sensible body of knowledge. Teams of teachers who have participated in the course are excited about mathematics and are now teaching it around and outside Massachusetts. Most importantly, we’ve seen that debates about the value of the Common Core disappear when teachers are immersed in concrete methods for bringing the standards to life.

So, there is promise. By raising the practice of mathematics to the same level of importance as results, I see hope that students will no longer be subjected to the numbing stranglehold of school mathematics. But Common Core is not, in itself, able to transform education. The problems go deeper than standards—they go down to the kind of mathematics teachers learn in their preparation and professional development. They go down to the oppressive working conditions that encourage quick fixes, shallow treatments of mathematics, and a profession that feels under siege. They go down to tyrannical assessments of low-level skills that cripple any chance of a coherent approach to school mathematics. And they go down to the outdated foundations on which schooling was built that advantage certain groups over others. Common Core is our first step, a blueprint for what school mathematics can be. Everyone involved in education and policy should get past this first step. Stop quibbling about the standards themselves — they are a refreshing and positive first step — and mobilize to focus resources on the rest of the picture.

**BRIEF BIO INFO**

Al Cuoco, Distinguished Scholar at Education Development Center, Inc. in Waltham, Mass., taught high school mathematics in the Woburn, Mass. public schools for 24 years. He received his doctorate in mathematics from Brandeis University. He is a widely published author and presenter on issues such as effective mathematics curriculum and mathematical habits of mind. He is the lead author on the 9-12 *CME Project *mathematics curriculum, an NSF-funded program founded on mathematical habits of mind, deeply interconnected with the Standards for Mathematical Practice.

The author thanks Eden Badertscher for our many discussions and her deep insights into the ideas in this essay.

For more information about the Massachusetts DESE course, see http://www.doe.mass.edu/candi/institutes/2013/math.html?section=3 and http://mpi.edc.org.

**Links:**

http://profkeithdevlin.org/2013/06/19/faulty-logic-in-the-new-math-wars-skirmish

http://mathsugaroff.wordpress.com/2013/06/12/practice-standard-3-construct-viable-arguments-and

December 18, 2013 2 Comments

*Due to an error, a commentary by Al Cuoco on the Common Core was mistakenly posted ahead of schedule. The commentary will be reposted in its entirety in January. Our apologies. The following is a short blurb from the commentary.*

Over the last few months, the Common Core State Standards have hit the headlines in a big way, with commentaries from educators, elected officials, columnists, and even talk-show personalities. As a former high school mathematics teacher — 25 years in the Massachusetts Public Schools — and a current curriculum developer and teacher educator, I’ve read the mounting criticism of and praise for the Common Core State Standards in Mathematics with great interest. Much of the criticism that I read is oversimplification; some of it exhibits a genuine ignorance of the daily work of teaching and learning. Very little shows familiarity with the actual content of the standards, and much of it is removed from a focus on the two groups that matter most: teachers and students.

September 12, 2013 Leave a comment

Al Cuoco, the director of the Center for Mathematics Education at EDC, has written some essays that show how mathematical practices can be used in high school. The essays explore examples of Common Core’s Standards for Mathematical Practice. We’ll post all the essays here in the coming weeks.

For more information about EDC programs around the Mathematical Practices, visit http://mathpractices.edc.org and http://mpi.edc.org/dmp-hs-sampler.

The second essay is on the use of formal algebra for bookkeeping, and you can use the methods to solve at least one of this site’s “We Got A Problem” problems. (Click on the link to view PDF.)

September 4, 2013 3 Comments

We have two different programs at EDC devoted to the Standards for Mathematical Practice., Implementing the Standards for Mathematical Practice (IMPS) and Developing Mathematical Practice in High School (DMP). For more information, see each program’s website:

http://mpi.edc.org/dmp-hs-sampler

These programs involve workshops for teachers. Al Cuoco, the director of the Center for Mathematics Education at EDC, has written some essays that show how mathematical practices can be used in high school. The essays were written for these projects, but we think the essays might be useful to others working to implement the Common Core. We’ll post the essays here in the coming weeks.

The first essay is on perseverance. (Click on the link to view PDF.)

July 28, 2013 Leave a comment

At my son’s recent 5th birthday party, he got 15 birthday cards. Sure enough, he got more than one of the same card.

Suppose everyone buys their cards at Foyerjohn, and randomly picks one of the cards. How many *different* cards would need to be on sale for there to be a 50% chance that all 15 people pick a distinct card?

*Harder:* how many different cards would need to be on sale for there to be a 50% chance that my son doesn’t get *three* of the same card?