NCTM’s Bold and Invigorating Plan for High School Mathematics

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 Gutiérrez’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:

  • Algebra
  • Functions
  • Statistics
  • Probability
  • Geometry

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 – 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!

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

23 Comments

  1. Reply

    I teach at an Early College. We are physically located on a JC campus. We actually do talk/collaborate on articulation with the college professors, for our school. Then, our HS team brought the information to our district math team. These conversations led to our current efforts the improve our math pathways.

    (If you are interested in more info, let me know – we are even within driving distance of you!)

    • Yeah, that’s awesome. Really happy to hear those kinds of conversations are happening. I don’t think we can revise the high school course of study without many more of them.

  2. Reply

    While the Obama administration proposed college and career-readiness as the purpose of schooling, NCTM broadens that purpose here

    Here, yes. Meanwhile, at the NCTM policy shop, this is the take on the purpose of learning math:

    A strong foundation in mathematics for each and every student from pre-K-12 is vital to our national’s economic stability, national security, workforce productivity, and full participation in our democratic society.”

    It would be unfair of me to critique NCTM for pivoting, if this is a pivot. But is it a pivot? Or is NCTM internally divided as to the purposes of schooling? It’s hard for me to see much of a difference between NCTM policy statements and mainstream Obama-style corporate ed reform.

    I haven’t read this document carefully yet. I’m honestly sort of confused how the “publish a document” style of reform leads to changes in the system, but it sounds like they’re saying a lot of good stuff in this doc. Good on them for that.

    As to the 4-year sequence: it reminds me of the original purpose of NCTM. What got the gang together was the fear of dropping enrollment in math courses, and the need to collectively advocate for keeping math courses in the system. In calling for a 4-year sequence for all students, NCTM is fulfilling that original purpose: keeping the demand for math teachers high. It’s only fair to mention that there’s a bit of self-interest mixed up in there, for both us in math edu and NCTM.

    • I’d like to know how NCTM squares both of those statements, one of which chills my vibe significantly. I’m wagering the writing teams for the platform and Catalyzing Change had almost zero overlap, not that that’s any excuse for incoherence.

      As to the 4-year sequence: it reminds me of the original purpose of NCTM. What got the gang together was the fear of dropping enrollment in math courses, and the need to collectively advocate for keeping math courses in the system. In calling for a 4-year sequence for all students, NCTM is fulfilling that original purpose: keeping the demand for math teachers high. It’s only fair to mention that there’s a bit of self-interest mixed up in there, for both us in math edu and NCTM.

      Yeah, fair. I’m ambivalent about a four-year sequence for math. Under the current program of study where second-year algebra is basically inevitable, I’d only argue for only three years. If the world of Catalyzing Change was our reality, I’d be excited about four.

  3. Reply

    I will be anxious to read the new NCTM document. In the early 2000s when they helped with the standards, the result was what I referred to as the lost generation. Arithmetic was taken out of the mix and I resigned my membership in NCTM because I didn’t feel I could have my name associated with the group. I resigned my membership and my participation as a reviewer and referee for The Mathematics Teacher. Each year the result for high school students was that they couldn’t think of numbers in their head. There was no comprehension of arithmetic so Algebra became impossible to comprehend. I am 100% in favor of the equity piece. Bravo to them for the emphasis. The K-8 CC Standards should eventually fix the understanding part for high school but I agree the high school standards were not as tight as the K-8. NCTM owes the lost generation a better effort this time.

  4. Reply

    So I was at a meeting where math educators across the state were brought in to look at our statewide test and determine the “cut scores” (the number of questions students would need to get correct) for what we would define as level 1 “approaching”, level 2 “meeting”, or level 3 “exceeding” the standards.

    Lots of… interesting things from those 2 days, but one thing that struck me was the vast difference between how the math and literacy determined the criteria for each level. The math criteria were organized by content. For example, level 1 had a lot of skills traditionally found in an Algebra 1 class, level 2 corresponds with Geometry, and level 3 with Algebra 2.

    The literacy criteria, however, were organized by theme, or practice – the different levels are differentiated by depth of knowledge. So for instance, “tone” is included in all of the levels but it’s what a student is able to do with tone (identify, analyze, compare, create) that determines whether a student is approaching, meeting, or exceeding the standard.

    Interesting perspective.

    All this to wonder aloud, is a 2.5 year curriculum based around proof and modeling an opportunity to create assessments that are focused on depth of knowledge rather than exposure to specific content standards? I guess I care because I feel like such a content-heavy test has bad implications for teaching and it only exacerbates the inequity in our school systems.
  5. Reply

    Though you question NCTM’s resolve for the future of math education, you are rooting for math educators everywhere to take the gauntlet and become “great” teachers and then there will be a tipping point where scores on the NAEP automatically notch higher to at least where most students are proficient. I wish I can share your optimism. I started teaching HS in 1967. The NAEP scores started in 1971 and there hasn’t been much improvement in those scores since. The only thing that will make a significant difference is if the major textbook companies start to publish books that most kids actually WANNA to read. Right now students who want to understand math deeply usually find their way to the 510+ section of the public library. That’s what I did to survive the poor math instruction I had in college. The books I still treasure today were written by W.W. Sawyer and Harold Jacobs.

    • Hi Ihor, I share some of your cynicism but not all. The document cites the fact that 9-12 NAEP scores have been stagnant but also that K-8 NAEP scores have seen improvement. Obv. that improvement could have lots of parents, but I wonder how much a focused set of content standards helped.

  6. Stephanie Kolitsch

    October 2, 2017 - 7:29 am -
    Reply

    Tennessee now requires 4 years of math in high school as part of their graduation requirements. We recently revised the state high school math standards. During the revision process, we approached every then-existing standard in the traditional Algebra I-Geometry-Algebra II sequence from the standpoint of “is this something that EVERY student in the state needs to know to be college or career ready” and, if not, we tried to move the standard to a relevant fourth-year course. We then restructured the fourth-year courses to reflect career-based interests (precalculus and calculus for those interested in STEM-related majors, statistics for those interested in careers that need basic statistical understanding, and a new course titled “applied mathematical concepts” that has applications and modeling using mathematics as a primary focus and borrowed heavily from earlier discrete and finite math courses). We also included a “bridge math” course that expands on earlier concepts designed for students who have not shown mastery of those earlier concepts in an attempt to improve their ACT scores and their chances of attending college. The team that redesigned the high school math standards included high school teachers and college/university teachers in order to smooth the transition from high school to college. I am looking forward to comparing NCTM’s vision with ours.

    • Super interesting. It sounds like TN got out of ahead of these changes, Stephanie. I’m curious how working teachers reacted.

    • Stephanie Kolitsch

      October 2, 2017 - 9:28 am -

      Most of the response I have heard has been positive. Our Algebra II was overloaded with standards, and we cut those considerably. This is the first year we have been using the new set of standards, so we will probably know a lot more in Summer 2018. To begin the process of revision, our State Department of Education opened a portal for teachers and other constituents to leave feedback on each individual standard in each grade/course in K-12 mathematics. We used those comments as we evaluated and revised each of the high school standards, so that the results reflect the recommendations of those teachers.

  7. Reply

    “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.”

    While I might agree that none of these deserve the label “essential”, I do think that all of them are indicative of an algebraic fluency that is essential, at least for any of our students headed in the direction of calculus. Does it need to be this particular set of manipulations? Do we need to spend large amounts of time on each one? Perhaps not. However, I find nothing more unhelpful in a set of standards than a wishy-washy “emphasis” on algebraic manipulation and fluency. What do I actually do with that? To that point, what the document actually says strikes me as having value:

    “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.”

    First it highlights a key misunderstanding of many students: “x+4” is not two separate numbers, but a single quantity being expressed in two parts. This misunderstanding rears it’s head frequently as students work with and manipulate algebraic expressions, particularly when students attempt to solve equations. Second, the document offers suggestions for how a better understanding could be useful (in several contexts). It gives you a problem to address and a motivation for doing so. I don’t think it prescribes the repetitive, rote manipulation of unrealistic algebraic expressions.

    To your larger point that “transcribing expressions from one form into another for no greater purpose than passing the class” has turned many students off to algebra, I would imagine anyone reading would have trouble disagreeing with your statement. However, I also think it is an unfair characterization of how these things actually play out in the typical high school math class. I teach my students to complete the square, and while I try not to beat them to death with it, many still find the process tedious. We can (and I do) provide students with justification and context for the skill (I certainly do not believe it serves no purpose other than a score in my gradebook), but I think we are kidding ourselves if we believe we can completely motivate the skills and concepts we are teaching to all of our students.

    Further, if we are going to place emphasis on the beauty of mathematics, I think we should be hesitant to classify any portion of mathematics as being uninteresting or off-putting. Must we only look at the bigger picture? Is there no beauty to be found in the very specific process of completing the square?

  8. Reply

    Further, if we are going to place emphasis on the beauty of mathematics, I think we should be hesitant to classify any portion of mathematics as being uninteresting or off-putting.

    Certainly, which is why I called them “inessential,” not ugly or uninteresting.

    While I might agree that none of these deserve the label “essential”, I do think that all of them are indicative of an algebraic fluency that is essential, at least for any of our students headed in the direction of calculus.

    Definitely agree, which is why I’m glad they aren’t excluded from Pathway B, which leads to calculus.

  9. Reply

    The irony here is that before I was actually teaching quasi-math to high school students (functional programming!) I would have dropped everything and at least skimmed this. Now I’ve got lesson planning to do…

    At some point in the distant future I’ll post an update on ye olde blog.

    • Yeah I really wonder what % of people who read this (and give feedback) are HS teachers. If so, please teach me your ways…

    • High school teacher here. I’m on page 16 so far. Out of 130+.

      My ways? Boredom during child’s soccer practice, no internet, and PDF on my phone. (Can’t grade. Too windy.)

  10. Reply

    Probably the same kind of high school math teachers who find time to read Dan’s takes on the world and even those of the other readers on his blog.

    In fairness, you’re right. I’m less than halfway through it, and probably won’t have time to thoughtfully reflect and respond before NCTM’s deadline. I wonder if more teachers would participate if this was released during the summer or around the New Year.

    • It is a lot to get through, but worth it. I teach high school and will be sure to comment. The final document comes out in April 2018. I hope that it will be much improved by public comment. NCTM’s plan is bold and invigorating!

  11. Trevor Warburton

    October 4, 2017 - 8:56 am -
    Reply

    Yes. I’ve seen similar things now in my position working with preservice teachers from multiple content standards. It seems like it could be very beneficial to emphasize the practice standards across all grade levels and use different content to get to them, but make the practice standards the focus of teaching and the specific content secondary.

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