The Pros and Cons? That’s quite a controversial heading, isn’t it? I’ve probably already lost all potential readers. Some, because how could there be any Pros? The others, because suggesting that there might be Cons automatically puts me in the category of those who just want to destroy those Standards. I see it quite differently. I share enough of the goals to want the Standards to take hold and succeed. I see enough that I don’t like in them to fear that those goals may not be achieved. I see the Standards as a work in progress, as an opportunity to discuss, disagree, exchange ideas, and improve on them, all in the spirit of respect for the motives of those you disagree with. After all, we are not on Capitol Hill.

The subtitle to *Teaching to Intuition* is “Constructive Implementation of the Common Core Standards in Mathematics.” “Constructive” as in “constructive criticism,” and the details of my criticism are prominently expressed in the book. So let me focus first on the Pros.

Sometimes, less is more. To best understand what the CCSS for Mathematics are really trying to achieve, as formulated in the Standards for Practice, I propose we stop half way through most sentences, that we read the verbs but skip the nouns they refer to. Let’s apply that reading strategy to the first paragraph of those standards:

*“Make sense…persevere…make conjectures…evaluate…change course if necessary…explain…conceptualize…plan a solution pathway rather than simply jumping to a solution attempt…check answers…use a different method…continually ask: ‘Does that make sense?’…understand…identify correspondences between different approaches…”*

Revealing, isn’t it? Essentially, as I see it, the CCSS Mathematics want to change an approach to learning/teaching/knowing math from a passive regurgitation of facts and procedures to *an activity of the mind. *The Standards recognize that knowing math does not consist in Mathematical truths stored in memory that students have the ability to blurt out when Pavlov rings the appropriate bell or when the test produces the appropriate question. I was thrilled when I realized that the Standards are really about math as an activity instead of an accumulation of knowledge—about discovering, working things out, understanding, formulating, communicating, modeling and applying. If done well, this should make mathematics more approachable and practical; both easier and more useful; a routine characteristic of an educated nation.

This approach to mathematical knowledge echoes and amplifies what professional organizations have long recommended—it wasn’t born in a vacuum–, but it represents a dramatic change to what actually happens in most classrooms. Mathematical knowledge seen as an activity of the brain instead of as passive and easily testable facts would represent a change in culture, and that is never easy to achieve. Change is threatening and moves at the speed of a Greenland glacier before climate change. So I welcome Standards that encourage concrete steps in that direction. Significant change is urgently needed. The alternative cannot be tacit satisfaction with the status quo, or some vague assumption that we can keep on doing the same thing just a little bit better. The future economic standing and technological leadership of this nation is at stake. That is something that makes us all stakeholders: parents, teachers, leaders in all fields of activities, and of course, students and the adults they become. The CCSS are the best realistic opportunity for seeing significant change happening, and some early tentative indications show encouraging first steps in that direction.

But it is not a given that the aims of the CCSS will be achieved, not only because of opposition from the outside, or because of all the peripheral but essentially unrelated agendas such as the uses and misuses of tests, but also from the flaws in the Standards themselves. A cultural change of this magnitude cannot be formulated by a few and delivered to a grateful world in a few in-service sessions. It needs to be offered as a work in progress worthy of gradual implementation and experimentation and extensive long term interactive training, where teachers learn, but also communicate their perplexities and offer their suggestions. In fact, to put it bluntly, the Standards for Content convince me that the writers of those Standards themselves do not have all the answers as to how to implement the goals set forth in the Standards for Practice.

A case in point is the heavy emphasis put on the commutative and associative properties of operations offered as a strategy for adding in the Standards for grades 1 to 5. Those properties of operations are a clumsy and inefficient way of moving numbers around in a series of additions and subtractions —in particular because they don’t apply to subtractions. But they could be interpreted as offering an explanation, so the deeply ingrained procedural mindset still present in all of us takes over and the thinking goes: Providing an explanation is what we want when we say that students should understand, those properties are some kind of an explanation, so let’s offer those properties. They allow us to check the box that we have met the requirement of asking students to understand. The deeply ingrained procedural approach to mathematical knowledge, instead of being cast aside, is transferred to meeting the requirements of the new goals, and essentially, on this topic, nothing has changed.

So, if not the properties of operations, what can guide us as we seek to move numbers around in a series of additions and subtractions? Not all groupings are acceptable, so how can we tell? As shoppers, we all know the answer: As we put a $4.00 box of cereal in our shopping cart, we all know that this adds $4.00 to our final cost. As we use a $1.00 coupon, we all know that this reduces our cost by $1.00. As shoppers, we all act from knowledge that the order in which purchases and coupons are scanned is irrelevant. So students can be helped to think about their shopping experience and draw from it the mathematical understanding that it represents. Instead of looking at a series of additions and subtractions in terms of those properties of operations, where the signs are permanently stuck between two numbers and where those numbers can switch position (commute) if the sign is an addition but cannot if it is a subtractions, our students will recognize that they already practice a different understanding, one where each number is permanently attached to the operation sign in front of it. As we look at the label on the box of cereal, we don’t just have 4, we have a 4 that is added. As we cut out the discount coupon, we don’t just think: “$1.00.” We think of one dollar that is saved. It’s the sign and number combinations that matter and that can be moved around and grouped freely in our shopping cart, at the check-out counter, and in a sequence of additions and subtractions.

If we all know this, why don’t we teach it? 1^{st} graders can re-enact their understanding with independent price tags and discount coupons that they manipulate and group as strategies for adding and subtracting. They can transfer their understanding to a series of additions and subtractions by drawing bubbles around sign and number, and then move those bubbles around or group them by crossing out {+ 7} and {+ 3} anywhere in the sequence to make 10, or {+ 14) and {– 4}, or even {– 8} and {– 2} to make {– 10}. Try those empowering strategies with the properties of operations! I have seen 1^{st} graders who can calculate 9 + 7 – 4 + 1 – 6 + 3 before they know how to add 9 + 7.

Quite a few grades later students are taught operations on integers. This remains for many a significant hurdle to a smooth transition into algebra that, for some, lasts all the way to remedial classes in Community Colleges and Universities. The perspective learned from those shoppers can change a major obstacle into something almost as intuitive as operations on counting numbers, without any need to process the understanding in terms of cumbersome rules, additive inverse, or other technical terms. For years, students saw an addition and added; they saw a subtraction and subtracted. How could it be any different? And then they face -237 + 148. Seen as an operation, -237 + 148 is an addition that we evaluate by performing a subtraction. There is no longer a complete identity between operation (an addition) and calculation (a subtraction.) Hence the confusion and the rules that some master and that many don’t.

With the shoppers’ perspective, and a vertical number line as the standard model in our mind, we can intuitively perceive {-237} as moving from zero a total of 237 units down into negative territory. {+ 148} then moves us back up 148 units, clearly short of making it all the way back to zero and into positive territory. Both the need to subtract the distances and the negative nature of the answer can be intuitively perceived. 6^{th} graders get the point based on their previous experience of what we have called the shoppers’ perspective and their customary use of a vertical number line. There is no longer any need to drag the confusion and frustration from 6^{th} grade all the way to College remedial classes. On this topic, the simple introspection introduced in 1^{st} grade could have a statistically significant impact on the number of students who never satisfactorily make it through algebra.

This is just one example fleshed out in more details in *Teaching to Intuition*. On this and other entirely different topics the link with real life and the urge to understand that are at the heart of the Common Core Standards can be recognized by students in the mental structures and thinking patterns of their own mind before they are translated and generalized into the language of Mathematics. Mathematics then becomes for our students something that comes from within instead of something imposed from the outside. The truths of math are recognized instead of just being learned. The Common Core Standards may not quite implement their goals in this way, but I see an invitation to do so in their emphasis on understanding, on modeling real life, on Mathematical knowledge presented as an activity of the brain instead of a passive accumulation of facts and rules. But to achieve their goal, these Standards have to be accepted as a work in progress; one that will benefit and grow from criticism and contributions on one pedagogical topic after another until Math becomes for the next generation a natural and integral dimension of everyone’s perception of the world we live in.