NCTM released Catalyzing Change in High School Mathematics last week. I anticipated it would address the gap between the K-8 Common Core State Standards, which feel tightly designed both within and across grades, and the high school standards, which feel loose and shaggy by comparison.
NCTM went about that goal in the second half of Catalyzing Change, enumerating a set of “Essential Concepts” along with two pathways students can take to learn them. I’ll comment on those concepts and pathways in a moment. But it’s worth mentioning first what I didn’t anticipate: a document full of moral ambition, the first half of which is a reimagination of the purpose of a math education along with a high-decibel endorsement of equity in that education.
You should read the latter half of the document if you have any stake in high school math education. But you should read the first half of the document if you have any stake in math education at all, at any level.
While the Obama administration proposed college and career-readiness as the purpose of schooling, NCTM broadens that purpose here to include “Understanding and Critiquing the World,” addressing the question, “When will I ever use this?”, and also “Experiencing Wonder, Joy, and Beauty,” acknowledging the millions and millions of people who love studying math even apart from its immediate application to the world outside the classroom.
NCTM reinvokes its call for equitable math instruction, citing GutiÃ©rrez’s perspective that until it is no longer possible “to predict mathematics achievement and participation based solely on student characteristics such as race, class, ethnicity, sex, beliefs, and proficiency in the dominant language,” we haven’t finished the work. To advance the cause of equity, NCTM pulls precisely zero punches in its condemnation not just of student tracking (which allocates students inequitably to the best classes) but teacher tracking (which allocates teachers inequitably to the most underserved students), also double-year math courses, and other less overt ways in which students are tracked even in elementary school.
This is what I mean by “moral ambition.” NCTM hasn’t merely underlined its existing statements on equity or de-tracking. Rather it lets those statements stand and then opens up several new fronts and runs at them. Catalyzing Change doesn’t arrive pre-compromised.
So again: everyone should read the first half of Catalyzing Change, which addresses much of the “why?” and “who?” of mathematics education. The second half of the document makes several clear and ambitious claims about the “what?”
NCTM proposes that all students take four years of math in high school. 2.5 of those years will comprise “essential concepts,” taken by every student regardless of career or college aspiration. Students may then take one of two paths through their remaining 1.5 years, one towards calculus, the other towards statistics and other electives.
40 essential concepts cluster under five conceptual categories:
If we only examine the number of concepts and not yet their content, this proposal compares very favorably with the Common Core State Standards’ over 100 required standards for high school. Under NCTM’s proposal, students may come to understand a proof of the similarity of circles (Common Core State Standard G-C.1) or a derivation of the equation of a parabola from its directrix and focus (G-GPE.2) but only as an incidental outcome of high school math, not an essential outcome.
Then, as I read the content of the concepts, I asked myself, “Do I really believe every student should spend 2.5 years of their limited childhood learning this?” In nearly every case, I could answer “yes.” In nearly every case, I could see the concept’s applicability to college and career readiness, and even more often, I could see how the concept would help students understand their world and nurture their joy and wonder. (I wouldn’t say that about the derivation of a parabola’s equation, by contrast.)
That’s such an accomplishment. The writing team has created a “Director’s Cut” of high school mathematics â€“Â only the most essential parts, arranged with a coherence that comes from experience.
If I’m concerned about any category, it’s “Algebra” and, particularly, essential concepts like this one:
Multi-term or complex expressions can represent a single quantity and can be substituted for that quantity in another expression, equation, or inequality; doing so can be useful when rewriting expressions and solving equations, inequalities, or systems of equations or inequalities. [emphasis mine]
Without any evidence, I’m going to claim that one of the top three reasons students leave high school hating mathematics is because their algebra courses required weeks and weeks of transcribing expressions from one form into another for no greater purpose than passing the class. I’m talking about conjugating denominators, converting quartic equations into quadratic equations through some clever substitution, factoring very special polynomials, completing the square, and all other manner of cryptic symbology, none of which deserves the label “essential.”
NCTM has done much more work here defining what is “essential” than what is “inessential,” which means their definitions need to be air tight. Some of their definitions in “Algebra” and “Functions” leave room for some very inessential mathematics to slip through.
My other concern with Catalyzing Change is the bet NCTM makes on technology, modeling, and proof, weaving that medium and those habits of mind through every category, and claiming that they have the greatest potential to enable equitable instruction.
I don’t disagree with that selection or NCTM’s rationale. But add up the bill with me here. NCTM proposes a high school course of study premised on:
- modeling, which students most often experience as pseudocontextual word problems,
- proof, which students most often experience by filling in blanks in a two-column template,
- technology, which students most often experience as a medium for mealy, auto-graded exercises,
- to say nothing of joy and wonder, which most students typically experience as boredom and dread.
This is a multi-decade project! One that will require the best of teachers, teacher educators, coaches, administrators, edtech companies, assessment consortia, policymakers, publishers, and parents. It will require new models of curriculum, assessment, and professional development, all supporting modeling and proof and eliciting joy and wonder from students. It will require a constant articulation and re-articulation of values to people who aren’t NCTM members. That is, changes to the K-8 curriculum required articulation to high school teachers. Changes to the high school curriculum will require articulation to college and university educators! Does anybody even know any college or university educators?
I’m not finding fault. I’m identifying challenges, and I find them all energizing. Catalyzing Change is an invigorating document that makes a clear case for NCTM’s existence at a time when NCTM has struggled to articulate its value to members and non-members.
If you haven’t heard that case, let me try to write it out:
Hi. We’re NCTM. We want to restore purpose, joy, and wonder to your high school math classrooms. We know that goal sounds ambitious, and maybe even impossible, but we have a lot of experience, a lot of ideas, a lot of resources, and a lot of ways to help you grow into it. We’re here for you, and we also can’t do any of this without you. Let’s do this!