A Common Core Math Problem With a Hint

Ironically, confusion of the CCSS with methods for testing mathematical proficiency is a prime example of the kind of imprecise thinking that the CCSS intend to rectify.
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"Teaching math according to the Common Core principles means today's students could master all the mathematical thinking ability they need for college and life by focusing only on elementary arithmetic and algebra."

The Common Core State Standards (CCSS) for mathematics continue to generate a lot of online debate. As a mathematician and math educator, I usually follow the latest missives that do the rounds on social and traditional media, and in almost all cases the post or story has little to do with the CCSS, but rather methods for standardized testing. That was definitely the case with comedian Louis CK's recent series of tweets on the topic, as he himself later acknowledged in subsequent posts and TV interviews.

Ironically, confusion of the CCSS with methods for testing mathematical proficiency is a prime example of the kind of imprecise thinking that the CCSS intend to rectify. (I will henceforth focus entirely on the mathematics standards. The CCSS also cover English language arts and literacy.)

Since politicians and policy makers are generally not immune to popular opinion, and many such may not be well-versed in mathematics education, a public debate that is so widely off target has the potential to do dangerous harm to the nation's future.

The CCSS were created to ensure that all students who graduate from an American high school do so with the skills and knowledge necessary to succeed in college, career, and life in the Twenty-First Century, regardless of where they live.

The reference to the current century is significant. The last 50 years (particularly the last twenty) have seen such dramatic changes in the way human beings live in the developed world that skills important for many centuries have rapidly become largely irrelevant, their place being taken by new skills that the grandparents -- and in some cases the parents -- of today's students never required.

For a good explanation of why some CCSS-oriented math homework problems can, as a result, have parents baffled, see this excellent blog post by a practicing teacher.

At their heart, the CCSS comprise a set of eight basic principles of "Mathematical Practice":
MP1. Make sense of problems and persevere in solving them.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.

(The CCSS document provides a paragraph that elaborates each principle.)

If we do not continue to evolve our education system with those goals in mind, we will fall drastically behind in the Flat World of the New Millennium. (See here and here.)

Though I played no role in drafting he CCSS, every single one of those principles have formed the heart of my mathematics teaching for a career of over 40 years. They are the very heart of what I (and others) call mathematical thinking.

Anyone who opposes the CCSS needs to say which of these eight guiding principles they believe should not be followed, and why.

Beyond the guiding principles, we start to get into implementation issues. (All of the opinions I have seen expressed about the "Common Core" have focused on implementation.)

The CCSS document takes those general principles and breaks them down into sets of specific items that should be mastered at each grade level. These items (there are many) provide learning goals that teachers should seek to achieve within a specific school year.

For example, one of the standards for Grade 5 algebra says "Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols."

This breakdown of the overall learning goals into specific items is done to provide the kind of nationwide uniformity required in an equitable society that provides equal opportunities to all students. It provides classroom teachers with a list of specific topics to cover during the course of a school year.

The order in which to address those specific goals, and the methods used, are left to the individual teachers, who can obtain or design lesson plans suited to their particular students.

In particular, the CCSS are not a curriculum; they do not prescribe any particular method or approach. Teachers may choose to cover items individually or in connected groups. Because, at the end of the day, those eight guiding principles are what the Common Core is about.

A mathematically knowledgeable teacher given the goal of producing mathematically able students well-equipped for Twenty-First Century life and career(s), without the need to meet detailed systemic metrics (a position I have the good fortune to be in but the vast majority of teachers do not) would need look no further than the eight guiding principles.

The guiding principles do such a good job of articulating the requirements of good mathematical thinking, that pretty well everything a student really needs to master in terms of mathematical thinking (not just for graduation but for life!) can be developed by following the eight principles and focusing entirely on elementary arithmetic and algebra. Nothing else! That would equip a student to acquire any specific mathematical skill required later in life with relative ease, as and when required.

The reason why many people find that hard to believe is that they were only ever exposed to the algorithmic-skills math instruction developed for earlier times -- a form of teaching that is hopelessly inadequate for life in today's world. (A fact ably demonstrated every time a parent -- taught in a previous era of rules and procedures -- is unable to help their child with what is inevitably a very simple mathematical homework problem of the kind relevant to today's world.)

So why are there good mathematic teachers who I know (and know of) and respect, who express opposition to the CCSS?

Like all seasoned math teachers, I propose leaving that final question for you, the reader, to answer.

But I will give some hints that may help you find a solution. (Guiding Principles MP1, MP3, and MP6 in particular should help here.)

By my reading of what they write, those teachers are opposed not to the CC principles themselves, rather to what they suspect (likely with good reason) will be required of them in the classroom when the Standards become codified into contractual obligations.

The CCSS were created by highly qualified and credentialed experts in both mathematics and mathematics education. They are intended to be operationalized in the classroom by highly qualified and credentialed mathematics teachers. In between the two, are a whole host of politicians, system administrators, and educational materials providers, the vast majority of whom are not expert in either mathematics or mathematics learning. Where do you think the weak link is likely to be, and how can that weak link be addressed? Is there more than one weak link? Can you suggest solutions?

By way of comparison, do the same problems arise in, say, medicine or aviation, where there is also a need for domain experts to work with and through politicians, administrators, and suppliers? If not, why not? (See that teacher's blog I referred to earlier.)

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