Ever since I started to get a real picture of what mathematics is about I’ve viewed middle school and high school mathematics education like a bit of a snob. I’ve read the treatise of Paul Lockhart, “A Mathematician’s Lament,” on the dystopia of cultural attitudes toward mathematics in pre-collegiate education. I’ve written responses to articles in the Atlantic authored by economists who, after getting PhDs in quantitative economics, still talk about math as if it’s just a bag of tricks. I’ve even taught guest lectures at high schools and middle schools to prove by example that an engaging, thought-provoking mathematics education is possible for 8th graders and up. I regularly tell my calculus students that half the things we make them do are completely pointless for their lives, while trying very hard to highlight the truly deep concepts and the few tools they might have reason to use.

But it’s generally agreed that something’s wrong with mathematics education in the US. There are a lot of questions to ask about why: are teachers not trained? Are students too focused on sports? Are Americans becoming intellectually lazy?

These all have their place in the debate, but the question I want to focus on in this article is: are policy makers designing good standards? I often hear about fantastic teachers who are stifled by administrators and standardized testing, so the popular answer is no. Moreover, though I haven’t done a principled study of this (again, my snobbishness peeking out), my impression is that even the fantastic math teachers at the most prestigious schools are still forced to hold the real mathematical learning in extracurriculars like math circles, or math symposiums.

And so the next natural step in analyzing the state of mathematics education in the US is to look at the standards in detail from a mathematical perspective. There was a nice article in the Washington Post detailing one ludicrous exam given to first graders in New York (exams for 6 year olds!). Here’s a snapshot:

This prompted me to actually look at the text of the Common Core State Standards in Mathematics, which is the currently accepted standard for most states. I have heard a lot about the political debate over efficacy and testing and assessment, but almost nothing about the mathematical content of the standards. Do they actually promote critical thinking skills and mathematical problem solving, as they claim? Does it differ enough from Lockhart’s dystopia?

My conclusion is:

While the Common Core indicates movement toward the right attitude on mathematics education, the attitudes aren’t reflected in the content of the standards themselves.

The big distinction I want to make in this article is the (perhaps counterintuitive notion) that mathematical thinking skills are largely unrelated to knowledge of mathematical facts, or ability to perform mechanical computations. The reason we teach mathematics to gain critical thinking skills is that mathematics gives examples of when they’re needed that are as simple and boiled down to their true essence as possible. And the text of the Common Core mostly ignores this.

I cannot claim that the writers of the standard don’t understand the mathematics deeply enough to realize this, and it would be too pompous even by my standards to imply that I know better than the thousands of educators that worked on this document. It could be the case that it’s instead the result of bureaucracy and partisanship, and the designers of the Common Core felt they could only make progress in certain areas. But even so, all we are left with is the document itself, and I want to give a principled (but more or less unstructured) inspection of its technical content.

There are some exceptions to my conclusion, and I will detail them as they come up, but the general picture is still this: the intent is better than it used to be but the implementation is still wrong. At the end, I’ll discuss why this is important and what I think should be done instead.

Preliminaries

Now, before we jump in to the text of the standards themselves, I want to point out a few resources provided for teachers and discuss them briefly. First, there is a 3-minute intro video by the Common Core about why the standards are important

Putting aside the animation style, this video sends some disturbing messages. First, that getting a lot of money is what defines and creates success. Not having a deep understanding of the problems you’re facing, not having strong relationships, not helping people, but money. All of this focus on competition and money suggests that the standards are primarily business oriented. I would argue that this is not a useful position for education, but that would digress. Suffice it to say, the Common Core people should know that the most successful mathematician of all time, Paul Erdős, was homeless and had but a few hundred dollars to his name at any given time. He instead survived (indeed, excelled to legendary status) on his deep understanding of problem solving and his strong relationships with other mathematicians. This is an extreme example but it makes my point clear: collaboration, not competition, breeds success.

The second misconception expressed in the video is that mathematics (indeed, all learning) is like a staircase, and you have to learn the concepts in, say, 6th grade before you can learn anything in 7th. Beyond basic technical proficiency, this is simply not true for the kinds of skills we want to develop in our students. And the standards for basic technical proficiency in mathematics have never really changed: be competent in arithmetic and know what a variable is by the end of grade school; be competent in basic algebraic manipulation by the end of middle school or freshmen year of high school. Then the rest of high school is about being exposed to other areas, most often geometry, trigonometry, calculus, and stats, none of which depends on another too heavily (at the level taught in high school). The details of exactly what technical skills are required for high school students is unclear. Why? Because as one goes from elementary school to middle school to high school, the focus on learned skills should gradually change from mechanical abilities to big ideas, and that transition arguably begins some time in middle school.

One of the common core “big ideas” we’ll look at later is that of similarity in geometric figures. But it’s clear that you can go through your entire life’s work without thinking about similar triangles, and it’s hardly relevant to most disciplines. Why then, should we teach it? Well, there’s a very good reason, but it’s at the heart of what the Common Core standards are missing, so we’ll save the explanation for later.

The video does make one good point: standards is not learning, and learning only happens with great teachers. But really, meeting standards isn’t learning either! It’s just evidence to learning, which is extremely hard to measure. But even worse is meeting the wrong standards, and that’s what I’m afraid Common Core is promoting.

Speaking of which, here are the actual standards themselves for the reader to peruse. The Common Core website has the full standard, also available as a pdf, and the Chicago Public School administration released a lengthy pdf explaining the standards in detail more or less specific to the school system. CPS also provides example lessons on their website, and I’m pleasantly surprised that a handful of the lessons try to flush out the more insightful ideas I find lacking in the Common Core itself.

So let’s dive right on in.

Common Core: Too Narrow and Too General

The standard starts out by describing a set of general guidelines which I generally agree with. They are:

Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for an make use of structure. Look for and express regularity in repeated reasoning.

But even as they are true, the descriptions of these tenets are either far too narrow or far too general. Take, for example, this excerpt from the description of the (arguably MOST mathematical) skill “Look for an express regularity in repeated reasoning,” also known as “Reasoning about patterns.”

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. … Noticing the regularity in the way terms cancel when expanding and might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain an oversight of the process while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Yes, a mathematically proficient student will be able to infer a general pattern here. But it’s phrased in a way that makes it seem like expanding products of polynomials is the central goal while the real mathematical skill is just “looking for a shortcut” in calculations. Moreover, the sentence that follows is equally applicable to any profession. Indeed, while working to cook a meal, a proficient chef maintains an oversight of the process while attending to details, and continually evaluates the reasonableness of their intermediate results. Finding shortcuts is not mathematical, nor is it culinary. But reasoning about those shortcuts is, whether or not they’re correct. I have plenty of calculus students who “find shortcuts” that simply aren’t true, but don’t bother to think about them, and are hence exercising no mathematical abilities. It’s a fine distinction that the Common Core seems to ignore at some times and embrace at others.

The best example of this is in “Construct viable arguments and critique the reasoning of others.” Here they wonderfully lay out the kind of logical reasoning students should learn in mathematics. How I wish that all mathematics education was based around this sole principle! The problem is that none of these thoughts are reflected in the standards themselves! Instead, the standards generally simply request that a student “knows” a particular argument, not that they generate original ideas or critique the ideas of others. The generation of new mathematical questions and arguments is, without a doubt, the best way to learn mathematical thinking.

Indeed, let’s take a closer look at the standards themselves.

Inspecting the Standards Themselves

The list of Common Core Standards breaks mathematical abilities down by grade level and by area. I think the Washington Post article gives a very good critique of the lowest grade standards, so let’s focus on high school level. This is where I claim the true big ideas must shine through, if they come up anywhere at all.

The high school standards are broken up into areas by subject:

Number and Quantity

Algebra

Functions

Modeling

Geometry

Statistics and Probability

So far so good, I guess. I’m quite pleased to see statistics recognized here. Let’s start with “Number and Quantity.” This section is broken into “The Real Number System,” “Quantities,” “The Complex Number System,” and “Vector and Matrix Quantities.”

The first one, “The Real Number System,” already shows some huge red flags. There are three standards here, and I quote:

Extend the properties of exponents to rational exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Use properties of rational and irrational numbers Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

This is supposed to indicate that someone understands real numbers? Number 1 only shows that one knows how to do arithmetic with exponents, asking the student to know a very specific argument, and number 2 is just memorizing some basic properties of rational numbers. There are some HUGE questions left unasked. Here are a few

What is a real number? How does it differ from other kinds of numbers? Is infinity a real number? If so, how does it fit with the definition of a real number? If not, is it some other kind of number? What does it mean to be rational and irrational? What are some examples of irrational numbers? Why are those examples irrational? Are there more irrational numbers than rational numbers? Vice versa? Are they “equal” in size? If we can’t know it exactly, how would you estimate a the value of a number like ?

To be fair, the grade 8 standards address some of these questions, but in an odd way. Rather than say that students should know that real numbers can be (sort of) defined by a finite integer part and an infinite decimal expansion, it says

Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Does that mean that a number can have multiple decimal expansions? Does every decimal expansion also correspond to a number? Or can a decimal expansion represent multiple numbers? What about the number whose “decimal expansion” has an infinite number of 1’s before the decimal point? Does that mean that infinity is a real number? As you can see, these are some very basic questions about real numbers, which are arguably more stimulating and important than being able to convert back and forth between decimal expansions and rational numbers (as the 8th grade standard requires, but nobody actually does for numbers harder than 1/3).

The important point I want to make here is that the truly “Big Ideas” underlying this topic are as follows, and they’re only halfway related to numbers themselves:

Understand the importance of precise definitions, and be able to apply those definitions to simple questions, such as “Prove that 1/3 is a real number,” and “Argue why infinity is not a real number.” Understand that we can define notation as we see fit, e.g. , and more deeply that mathematical concepts are invented via definitions. Understand the concept of a correspondence between two collections. Know how to argue that a correspondence is or is not bijective (by any other name). Understand basic proofs of impossibility. Understand the concept of approximation, and understand how to quickly get rough approximations of quantities. Extend this to estimate concrete real-world quantities, like the number of pianos in your hometown.

And these ideas are the actual ideas that the Common Core is looking for, the ones that apply across all of mathematics and actually relate to real critical thinking skills. But it seems that the only place in the standard where they address the idea of a correspondence is in a Kindergarten “Counting” standard:

CCSS.Math.Content.K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

“Number and Quantity” goes on to describe the importance of consistently using units and rounding measurements to the right number of digits; memorizing properties of complex numbers (with regards to which any introductory college professor will start over anyway); and more rote manipulation of vectors and matrices that few high school students have any reason to know. Other big ideas apply here, ideas from geometry and the idea of correspondence in particular, but the standards still focus on mechanical abilities.

The Common Core folks argue in quite a few places that knowing the reasons for why certain mathematical facts are true is what constitutes a true understanding of those facts. And yes, I agree. But there’s much more to the story. If we want students to know why we define as we do, to make a nice extension of the rules of exponent arithmetic, it’s certainly a deeper understanding than just memorizing how to do the arithmetic itself. But it’s just another kind of memorization! It’s memorization of a specific mathematical reason for a specific mathematical fact. It’s a better kind of memorization than we used to require, but is it critical thinking or problem solving? It’s hard to say whether or not it requires more of the mental faculty we want it to, but if it’s not then we lose, and if it is, then this is a pretty indirect way of going about it.

Again, my big point here is that the requirements of the Standard overlook the deep underlying mathematical thinking skills that we hope are being developed when we ask them to know whatever it is we want them to know. These big concepts like correspondence and impossibility and approximation should be the central focus. The particular rules of exponents and the specific properties of irrational numbers, these are tools and sidenotes that accentuate fluency in the big concepts as applied to solving problems. Almost nobody needs to know facts about irrational numbers in their careers, but relating things by correspondence is a truly useful mathematical skill.

Taking a Step Back

So let’s pause for a moment and give some counterpoint. I could just be focusing super narrowly on one or two topics that I feel the Common Core misrepresents, and using that gripe to claim the entire Common Core is crap when it actually has lots of merit in other areas.

While I do think that the standard addresses a few topics well (more on that later), I claim the pattern of “Number and Quantity,” is endemic. Take for example the section on Geometry, Measurement & Dimension. I was really hopeful here that one of the standards would be “Understand what dimension means,” but no dice. Instead it’s the same old memorization of formulas for volumes of geometric shapes.

Even worse, when asked to do derive these formulas, the standard says

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

I’d be surprised if many of my readers had even heard of Cavalieri’s principle before reading this, but this is the pattern striking again. The standard expects the students to know something very specific: not just how to use Cavalieri’s principle, but how to apply it just to these special objects, while ignoring the underlying principles at work. A true understanding of measurement and volume would be:

Reason about the volume of a solid you’ve never seen before.

But more deeply, Cavalieri’s principle is just another kind of correspondence argument applied to geometry. And I see this right away without muddling through that terribly written Wikipedia page because I have a solid understanding of the notion of a correspondence and I can recognize it at work.

The “dissection” argument is another deep principle that is mostly ignored in the standard, so let me spell it out. One way to solve problems is to break them down into simpler problems you know how to solve, and to figure out how to piece them back together again. Any high school student can understand this technique because by high school they’ve learned to put their pants on one leg at a time. But this idea alone accounts for a wide breadth of mathematical solutions to problems (the day it was applied to signal processing is often credited as the day the Age of Information began!). So they should be seeing (and using) this technique applied to many problems, be they about algebra or geometry or cats. It shouldn’t be hidden away in a single (perhaps memorized) geometric argument.

And finally, the “limit argument” (called exhaustion in some educational circles, but I don’t like this name) is an application of approximation. The idea is that a sequence of increasingly good lies will eventually give you the truth, and it’s one of the deepest thoughts that mankind has ever had! So indeed, the “Big Ideas” across this standard are big ideas, but the writers of the standard neglect to point out their significance in favor of very specific and arguably pointless factual requirements.

The geometry section is full of other similar nonsense: using laws of sines and cosines, the same geometry proofs that Paul Lockhart derides (page 19), and memorizing minutiae about the equations of parabolas, ellipses, hyperbolas. They say the big ideas are similarity, transformations, and symmetry (indeed these are big ideas!) but then largely revert to the same old awful kinds of rote memorization and symbol pushing we’ve grown to hate about math.

A real course in geometry, measurement, and mathematical problem solving might even follow Lockhart’s book, Measurement. In fact, if students really absorbed the contents of this book, that would constitute an entire high school mathematics education. Why do I say that? Because this book focuses on developing mathematical thinking skills in a way that no high school education I’ve heard of has attempted. This book emphasizes methods and exploration over facts, and teaches readers to conjecture and reason without telling them how to do everything. It provides numerous exercises without clear answers and has no solution manual. And I claim that any facts required by the standards that are not covered there could be taught to a student who is comfortable with this book in one month or less. That is, all of the “standards” simply fall out of the more important deeper concepts, and we should be working forward from the deep ideas. We should use things like the law of sines as examples of these deep principles in action, but knowing or not knowing the law of sines or when to use it gives little indication of critical thinking.

I could continue with algebra, and the other sections, but I think my point is clear. The standards are filled with the same arbitrary choices of technical facts, and the deep ideas, the kinds of thinking we want to develop, are absent.

Modeling and Other Big Ideas

There is one aspect of mathematical problem solving that I think the Common Core addresses well, and that is modeling. That is, students need to be able to take a poorly defined problem, whether it’s “analyzing the stopping distance for a car” or asking what constitutes a number, and boil it down to its essence. This means making and questioning assumptions, debating the quality of a model, testing and revising, and interpreting results in a principled way. This is arguably the only kind of mathematics that non-mathematicians do outside of academia, and I feel that the description in the Common Core does justice to its importance. Even better, they admit there can be no “list” of facts the students are expected to know about specific models or tools. Here the Common Core gives in to the truth that discussion and original arguments are the key to developing fluency. In my imagination there were a select few key players lobbying for this to be included in the Core, and I say bravo to you, well done!

But there are some other big ideas that the Common Core misses entirely.

One of these is the idea of generalization. That is, one core part of mathematical thinking is to take a solution to a given problem and extend it to more general patterns and problems. I see only vague allusions to this concept in the Common Core (students are expected to know, for example, how to generate a sequence when given a pattern). But this is literally the core stuff of mathematical problem solving: if you can’t solve a hard problem, try to simplify it until you can solve it, and then try to generalize your solution back to the original problem. This is why it makes sense to think about matrices and polynomials as “generalizations of integers,” because natural facts about integers extend (or don’t extend as the case may be) to these more general settings. Students should be comfortable facing problems that may require simplification and mathematically “feeling around” for insights.

The second idea is that of the algorithm. I’m not talking about programming, that’s a different story. I’m talking about procedures that anyone might follow to get something done. People follow algorithms all day, and some of the most natural problems (and interesting problems for students to think about) are algorithmic in nature: how to guarantee you win a game, how to find the quickest way to get somewhere, how to win the heart of that cute guy or girl. Indeed, students are expected to follow algorithms all over the Common Core, from approximating irrationals by rationals to solving algebra and making inferences. The only non algorithmic aspect of the Common Core is modeling, and here they provide an algorithm for how to do it! And so it makes sense to study exactly what makes an algorithm an algorithm, when algorithms apply, and more deeply what makes an algorithm good.

The last point I want to make is that true mathematical understanding arises from trying to solve problems that you are not told how to solve ahead of time, and recognizing when these big ideas apply and when they do not. Students love to solve puzzles for their own sake, and they don’t need to be embedded in stupid “real world” applications like computing mortgage payments. Indeed, this is what Sergio Correa did in his financially destitute school in Mexico, and his students have made progress beyond belief (see, Common Core people? Money is not the problem or the solution!). It’s okay for problems to be left unsolved by students for days, weeks, or even years, and students need to be comfortable with identifying their own lack of understanding.

I want to expand on the idea a bit more. Taking it to the extreme, you could ask a more daring question: should students be exposed to problems they cannot possibly solve? My answer is yes! Emphatically, yes! A thousand times, yes! Student need to be exposed to many kinds of problems they cannot solve to be prepared for a world in which most problems don’t have known solutions (or else they wouldn’t be problems in the first place). Here are a few examples:

1. They should be exposed to problems that can be solved in principle but are too hard to solve with the techniques they know well.

For example: elementary level students who are just beginning to learn about variables should be asked to add up all the numbers between 1 and 100. They should be encouraged to try it by hand until they’re convinced it’s too hard, and they should be rewarded if they actually do manage to do it by hand. They should then be encouraged to think of other, cleverer, ways to solve the problem. No idea is crazier than adding it up by hand, and so much time (at least a full class session) should be spent puzzling over what in the world could possibly be done. Finally, an elegant solution should be shown that reduces the problem to multiplication, and the use of variables highlighted (let S be the sum of these numbers, even though we don’t know what it is…). And then the problem can be extended to a general sum of the first few integers, sums of squares, and so on.

2. They should be exposed to problems that cannot be solved with any technique they know but will foreshadow their education in future classes.

When students are learning about the slope of a linear function, they must be encouraged to wonder how one could reason about the steepness of nonlinear things. For it’s obvious that some nonlinear functions are steeper than others at different places, but how can we use a single number to compare them like we do for slopes of lines? The answer is that we cannot! The “correct” way is to invent calculus, but of course the calculus way of doing it involves extending the usual notion of slopes of lines by taking limits. The students will not know this, nor will they find it out by the end of their algebra class, but it should linger in their minds as a motivating question: there are always more unanswered questions! What about the “steepness” of surfaces? Can we talk about the “steepness” of time? Students should readily ask and be asked such intriguing questions (again, this is generalization at work). The fascinating anecdote is that educators are forced to do this, but rather than have their students wonder about steepness of curves, they make them memorize the form of a “difference quotient” with little indication of why, other than that “it will be useful later.” Seeing the solution before they even hear the problem? It’s completely backwards!

3. They should be asked obvious technical questions that appear not to have any technique at all.

For example, they might be asked the difficult question: is a rational number? Indeed, this is an extremely natural question to ask, since is defined to be a ratio of two numbers: the circumference and diameter of a circle. But despite the fact that there are many proofs using a variety of techniques, almost all proofs that is irrational are beyond the abilities of high school students to follow and not even familiar to the average college math major. I certainly couldn’t prove it off the top of my head. This is quite different than the previous kind of problem, because there it was obvious that you can reason about the steepness of nonlinear functions, the students just don’t know how to formulate it rigorously. But here, the rigorous question is understood (can be represented as a quotient of integers) but it’s unclear whether the problem is easy or hard to solve, and it turns out to be hard.

4. They should be exposed to problems that NOBODY knows how to solve.

When students learn about rational numbers (and if they know about , which I doubt they should before calculus but I’ll use it in my example anyway), they could be asked whether is rational. This is an open problem in mathematics. If they’re learning about prime numbers, they should be asked whether every positive integer can be written as a sum of two primes. Every even positive integer? Every even integer greater than 2? And so they go through the process of refining “stupid” questions (with obvious “no” answers) into deep open conjectures. And then it can be connected to other ideas: can an algorithm answer this question? How long might it take? Can we try to correspond integers to something else? Can we give an approximation argument? There are so many simple open problems in number theory that it baffles me that many students are never exposed to them.

The point of all this is that mathematics, and mathematical problem solving skills, are not just about picking the right tool from the set of tools you’ve been taught. It’s about recognizing when any tool you’ve ever heard of even applies! More deeply, it’s about debating with your colleagues that problems can and cannot be solved using certain methods, and giving principled reasons why you think so. This sounds like “modeling,” and indeed the strategies used for modeling also apply to pure mathematics, a fact that most people don’t realize. Critical thinking and mathematical problem solving is more akin to art and debate than to mechanical computation. And the most interesting problems are the most natural questions one could ask, not the contrived “compute the volume of an oddly shaped wine glass” questions. Those are busy work questions, not open for discussion and interpretation. And the students know it.

Why This Matters

A reasonable objection to my rant goes as follows: why does it matter that the Common Core isn’t super clear about these “big ideas” I’m claiming are so central? If the teachers are knowledgeable they’ll know what is important and what isn’t important, and how to teach the material in the manner that best promotes learning.

The problem is one of intention and misdirection. If the teachers are not rock-solid in their own understanding, then this Common Core, promoted by The World’s Leading Education Experts, can easily narrow their teaching to just what’s in the Core. More disturbing are the people who don’t know mathematics well, the principals, policy makers, and standardized test writers who really take these guidelines at face level. Even if a teacher has a good reason to favor one area like modeling over memorizing facts about hyperbolas, they will be met with the same kind of obtuse opposition by administrators seeking short term business goals. The standards exist, one might argue, to explain to these people (the people who wouldn’t know mathematical reasoning if it hit them in the face) that the teachers are teaching ideas much deeper than the rules of matrix multiplication. The Common Core represents this adequately with regards to modeling, but little else.

The Common Core also claims that the standards should be separated from specific curriculum and pedagogy, and one would reasonably argue that what I’m presenting here is pedagogy, not standards. Regardless of whether you agree or disagree with this, it still remains that the Common Core is designed to influence curriculum and pedagogy. And so even if the Common Core must have “facts” as standards, if it fails to emphasize the deep ideas underlying the factual obligations then it fails to influence pedagogy in the right way. In doing so, it reinforces obviously bad practices like teaching to the test.

The important thing to realize is that the correct pedagogy is already basically known: from a young age students should explore and reason and puzzle without horse blinders. Sometimes there are some dry factual things they cannot escape, but such is true of everything. So the separation of mathematical church and state (pedagogy and standards) claimed by the Common Core seems to be entirely a political one. It would infringe on the freedom of the teachers to impose pedagogical constraints, especially ones that only work for some environments. If this necessarily causes deficiencies of a global set of standards, then it is simply the wrong approach.

Again, I cannot say for sure whether the writers of the standard don’t understand the mathematics well enough, and it would be pointlessly arrogant to imply my own superiority. I hate to think it’s a bureaucracy issue, and that the designers felt the only progress they could make was to emphasize modeling as well as they did. If this is the truth then it is a sad one, because where I and many of the teachers sit, our country is stuck with the results.

We don’t need more compartmentalization by subject and grade. We do need a recognition of the deep critical thinking skills we want to teach. “Abstract reasoning” is not a specific enough goal to warrant policy. We need to admit to our teachers and our students exactly what we’re trying to get them to learn. And then we can organize education based on increasingly sophisticated applications of those ideas, to thinking about shapes, numbers, modeling, to whatever you want. Then students won’t forget about counting as “matching” after kindergarten ends, or only consider approximations related to irrational numbers. They will instead see these ideas blossom over time into the mental Swiss Army knives that they are. And they will use these ideas as a foundation to acquire whatever factual knowledge they might need to succeed in their careers.