Breaking Through the Mathematics Education System Despite Itself
By: James Tanton
Tags: Mathematics, Education

There is a psychological tactic employed by some classroom teachers following a strict curriculum: to bond with one’s students by casting the enforced textbook as a common enemy. “We’re in the system and this is the game we have to play,” is the sad and perhaps demoralizing approach to this. But if conducted with grace and care, elements of this psychological stance can be respectful, uplifting, and pedagogically good.

After all, any passage of text provides opportunities to question content, probe inconsistencies, explore missed opportunities, counter seemingly inflexible assertions and definitions with exceptions and alternative approaches, and so on. That is, all texts can serve as an invitation to examine the place and context of content. And in this content-rich, content-at-the-ready world, isn’t it all the more important that 21st-century education should help students learn to be the arbiters and assessors of their own knowledge?

This applies to mathematics education and its availability of content too. Once everyday number facts and facilities are at hand by middle school, the remaining six years of mathematics education could attend to the thinking and meta-thinking of mathematics, and the self-reliance of thought this induces. We could, with intention, help students learn how to learn, to personally assess what they know and how they know it, and to differentiate between familiarity and understanding: the familiarity that comes from repetition and rote doing versus the empowerment of knowledge.

So how does one foster student metacognition, self-confidence, and nuanced understanding when presented with a rigid, upper-school curriculum that is content focussed and content laden? The “system” need not be an enemy, per se, but can we identify opportunities within the system that extend beyond itself perhaps despite itself?

For starters, we might argue that the volume and nature of the mathematics content we are expected to cover in high school mathematics in and of itself holds a message. Factor trinomials, use 2x2 matrices to represent certain geometric transformations, analyse the ambiguous case of the Law of Sines, and so on and so on and so on. The count of disparate topics makes us realise that they can’t all be sacred topics that need to be taught. No topic is important because students will “need to know it later on.” (Or, if I am wrong about that, then the topic could wait until “later on” when it is actually needed!) The notion of detailed “sacred high-school content” that must be taught is a myth. Accepting this and being honest with your students about it is freeing and liberating! It gives the community of your classroom permission recognize pieces of content as beautiful vehicles for learning new potent thinking, and to enjoy the power this can bring each time. Sure, answering loads of “what” questions about content will be on the standardised exam, but all will know that that is not what defines the joy of mathematics. How lovely!

Second, in the push to cover content, many curricula present material briskly and with authority. It gives the impression it is removed from human story and exempt from questioning. And this is gorgeous as this is the very stance that should invite skepticism and question! So then, question it! You yourself, and you and your students together.

This point has struck home for me very recently as I am currently reviewing an international mathematics curriculum with a predilection for authoritative commands to students like “simplify, ” “factorise” (U.S. educators say “factor” rather than “factorise”), “solve via the so-and-so method,” and so on. It’s all very intimidating. My inner-student cowers — I grew up in a mathematics education system of this nature and recognise its effect — but now I have adult confidence and am feisty with the confidence to say: “Hang on! Let’s think about this. And let’s have students think about this too!” I want to empower students with the intellectual might and cleverness to push back when they deem it appropriate themselves to do so.

I’ll present next in this essay three examples on this matter that just arose for me in reviewing this particular curriculum. They are the value of questioning the importance of a “sacred” topic, the value of probing deeply into nuanced understanding, and the flawed use of the word “know.” I have taught each of the specific topics mentioned to high-school students and I have practiced what I preach in each of the examples below.

But I understand it is a challenge to change the culture of what defines success in the high-school mathematics classroom. I have the advantage of a Princeton PhD in mathematics under my belt (and a British-esque accent to boot) to help students — and parents — believe in my vision of long term mathematics meaning and value. But I am also very cognizant that assessment, in particular, standardised assessments, are the first order definors of high-school mathematics value. That is not going to change anytime soon. So the key is to take all I offer here as casual advice and simple first baby steps towards inducing cultural change. But please don’t underestimate the power of such simple first steps!

Factoring

Every algebra curriculum has students spend hours — many hours — factoring mathematical expressions that have been carefully crafted to magically factor. (Most expressions don’t.) So why not explore a question of the following type?

In a class of twenty students, each student can pick three quadratics at random and attempt to factor them. From a sample size of 60, maybe one can garner a sense of what proportion of quadratics are actually factorable? (Or why not use Wolframalpha and examine a larger sample size? Or use basic coding software to run through all 900 quadratics? Why stop at single-digit positive integer coefficients?)

The international curriculum I have been examining asks students to identify the “largest common factor” of three terms such as 12x³ , 6x⁵, and 10x⁴. The desired answer, to be written in the box on the exam page, is 2x³. But pause. Think! This is a question from algebra class where x represents a yet-to-be-specified value. It might well be a whole number quantity, but it could just as well be a fractional quantity or an irrational quantity. We don’t know! This question is akin to asking for the greatest common factor of 4, √3, and ⅚ and so is technically silly since every real number is a multiple of any other non-zero real number. (For instance, 4 is a multiple of 17 since 4 = 17*(4/17).) The notion of a greatest common factor is meaningless in the real number system. What a great conversation to have!

But the meta-conversation to be had then is: Okay, so what are the curriculum authors meaning to ask?

One realises that the curriculum chooses to focus on polynomials with integer coefficients involving non-negative powers of x. (Umm. Why?) The authors probably thus seek as an answer an expression of the form axᵇ, with a and b non-negative integers as large as possible. In which case, when later asked to “factorise fully” 6x⁵ + 10x⁴+12x³, one is probably expected to write 2x³(3x² +5x+6) and not x³(6x² +10x+12) or 10x⁴(0.6x+1+1.2/x) or some other expression that might be more relevant for some later, yet-to-be specified, context.

Question: Simplify 6x⁵ + 10x⁴+12x³.

The Actual Correct Answer: It looks fine as it is. It all depends on what you want to do next with this expression. If there is no “next,” then what more is there to do now?

A rendering of a half-remembered example of student work debated on social media many years ago. (Apologies for losing details of origin of this!)

Solving Equations

Mathematics is a language … literally! Every mathematical statement one writes is a sentence. For example, the statement 3 + 4 = 7 has a noun (the quantity “3+4”), a verb (“equals”), and an object (the quantity “7”). The statement 5>8 is also a sentence.

The first sentence happens to a true sentence about numbers and the second a false sentence about numbers. As mathematics tends to focus on truth, it is interested in sentences that represent true statements about numbers.

The statement w²+w = 2 is a sentence about an unspecified value w and is neither true nor false as it stands: it all depends what specific value one might want to assign to w. If we set w to be 1, then we have a true number sentence. If we set w to be 2, then we have a false one.

Similarly, the sentence x = 3 is currently neither true nor false. If x is 3, then we have a true number sentence. If x is set to be 4, then we have a false one.

When presented with an equation it is natural to seek the value(s) of the unknowns that lead to true number statements. For example, the set of all numbers that make the equation w²+w = 2 a true number sentence — its solution set — is {1,-2}. The solution set of x = 3 is {3}.

The Exit Ticket shown above should give full credit to the student who clearly recognizes that “x=3” is an equation and that the solution set to the equation is the set of numbers {3}, and, perhaps, more subtly recognizes that writing solution sets as “w = 1 or -2” or “x = 3” is technically not correct: these are both still equations that each might or might not be true, but their solution sets are so blatantly clear that folk tend to regard these equations as statements of solution sets. (Indeed subtle! A lot to be discussed here.)

Irrational Numbers

Every curriculum I have seen expects students to “know” that √2 and π are each irrational.

Umm ... How?

There are some texts that do share proofs of the irrationality of the √2 . (Start by assuming that we can write √2 = a/b as a reduced fraction and see what goes wrong. Squaring and some algebra gives a²= 2b² showing that a², and hence a, is even. Writing a = 2k then gives 4k² = 2b² leading to b², and hence b, being even too. But this then contradicts a/b being a reduced fraction in the first place. Our beginning assumption that we can write √2 as a reduced fraction just must be wrong.)

But how is one meant to “know” that π is irrational? One is just told!

Okay then, so let’s actually tell the story, the whole story. Scholars across the entire globe struggled over the question of the rationality or irrationality of π for well over 2000 years. That’s worth sharing!

To human eye, wrapping a string seven times around the lid of a jar seems to match perfectly 22 copies of the width of the jar. (Try it with your students!). This suggests that π might be 22/7. But is it?

Next have your students Google the work of Archimedes of Syracuse from the third-century BCE who showed that π actually has value just shy of this fraction, and of how fifth-century A.D. Chinese scholar Tsu Chung-Chi found an approximation we recognise as correct to seven decimal places, and of the general race over time to the digits of pi, more and more of them, all the while scholars still were still unsure whether or not the number was a fraction. (Could it be a fraction with some extraordinarily large numerator and equally large denominator?) Students will read that it wasn’t until 1760, after some 2000 years of wondering, that Swiss mathematician Johann Heinrich Lambert finally settled the question as to the rationality or irrationality of π once and for all. Using advanced techniques he proved that π is simply not a fraction. (And why not look at his proof on the internet too and see the reason why his argument does not appear in school textbooks?)

This story is good. But, as educators, can we go further and ask about what mathematics we could offer students to hold on to and own for their themselves from this story?

My answer is this. Let’s have budding scholars construct their very own irrational numbers! Try this:

Have students use long division to compute 1/3 as a decimal and then think about why one is trapped in a repeating cycle. (Exploding Dots makes this fun and easy.)

Next have students use long division to compute 4/7 as a decimal (do it too, right now!) and see that one is trapped again in a repeating cycle. (Why? As you try this you will see that there are only seven possible remainders that could appear in the division process and so one must soon repeat some remainder. As soon as you do, you are in a cycle.)

Next imagine — but don’t conduct! — long division to write 13/32 as a decimal. What remainders could appear? Must one eventually repeat a remainder and thus fall into a cycle? Yes!

In a jiffy we learn that every fraction has a decimal expansion that falls into a repeating pattern. (Even ¼ = 0.2500000…. falls into a repeating pattern of zeros.) So any number that has a decimal representation that does not fall into a repeating pattern cannot be a fraction. WHOA! And such numbers are possible to write down. For example, here is James’ Irrational Number

0.101001000100001000001000000100000001000…

This number is a little over a tenth. Its decimal digits have a pattern (for instance, I could tell you what the billionth digit if you really wanted me to) but it is not a repeating pattern. Therefore this number cannot be expressed as a fraction. It is an irrational number.

Exercise: Your turn! Now make up an you own irrational number that you actually KNOW to be irrational.

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Each and every curriculum is chock-full of intended and unintended nuances that invite questions and exploration. As they occur to you, share your questions and wonderings with your students. They too will notice and share their own wonderings, and a lovely snowball effect of genuine intellectual curiosity will evolve.

This is joy in empowered learning.