This is my third post on the in-process revision of the California Math Framework. If you haven’t yet, please read the first two: The California Math Framework Revision, and More on the California Framework. I just reread them, and find that I stand by what I wrote: the revision attempts to address an important issue: low-track classes have had low expectations and are essentially dead-end classes. This is the broad professional consensus of math educators, as articulated by the National Council of Teachers of Math. Unfortunately, inequality in school reflects social inequality, and cannot be solved by educational reform alone. Still, it is our responsibility as educators to do the best we can to address it. This is easier said than done, which is why NCTM is at this point hoping mostly to launch a discussion rather than rush into making hasty changes.

Not having read much of the original draft of the Framework revision, I was careful to not endorse it. In both posts, I warned that no reform along these lines can possibly succeed if it is not done well. As I predicted, the resistance to the revision seems to be largely about the issue of tracking and acceleration. The Twitter response to my second post made me realize how polarized this conversation already is, and convinced me I need to spell out my views on these topics. (As for coming out for or against the Framework, that will have to wait until I see the next iteration of it. I just don’t have it in me to read that sort of document more than once.)

First of all, I agree with both the reformers and their critics that (almost) every student can learn significant math, and deserves to be treated accordingly. I also know that students vary widely, bring different backgrounds and skills to the classroom, and learn math at different rates. Denying a reality that every teacher, parent, and student knows does not serve the goal of equity. The question is not whether those differences exist, but what to do about them. The writers of the Open Letter offer no suggestion on this. This is not surprising, as they are among the people who have benefited from the status quo. (I have no animosity towards them: I too have benefited from the status quo! But like many others in NCTM, I have spent much of my career thinking about ways to democratize math education, which I would not have been able to do if I had pursued a career in STEM.)

In the remainder of this post, I will share a summary of my ideas about detracking high school math classes. Those are not ideas that just arose spontaneously in my head. They are based on decades of experience as the chair of an untracked high school math department, where we implemented those ideas. We were able to help students who would have been sunk by low expectations in a tracked system, without harming our strongest students who still managed to get into elite universities. This was in a private school, so you may want to exclude me from this conversation, which is your prerogative. I’ll just say that the range of our students’ math ability (as measured by standardized tests) was huge as they came into the ninth grade, and that the approaches described below were successful with the whole spectrum.

1. *The least obvious and most important ingredient in detracking is the central role of an alliance with our strongest students*. In a student-centered classroom, they are the engine that drives the class. If we do not keep our courses challenging and interesting to them, we lose their respect and their cooperation. Addressing their needs is paramount politically, ethically, philosophically, and pedagogically. Understanding this in no way harms other students — quite the opposite.

2. The sort of *curriculum and pedagogy* that best supports the widest range of students requires some departures from the traditional “I explain, you practice” approaches. It involves a combination of rich problems and activities, engaging classroom routines, teacher-led discussions and instruction, access to learning tools, and student collaboration. This is complicated and cannot be wished into reality by fiat, or merely by adopting a different textbook series, or a different state Framework. Teachers need training and support, and time to grow, not to mention buying into these changes.

3. *We must combine forward motion and extended exposure.* In a heterogeneous class (in other words, in any class) if we wait for every student to “get it” before “moving on”, we are sabotaging everyone’s learning, and undermining necessary “coverage”. But if we only move forward, we are only serving a small fraction of our students, and not doing even that very well. If we want depth of understanding and long-term retention for most, we need to use techniques such as lagged homework and assessments, quiz and test corrections for “points”, separation of related topics, and perhaps other manageable approaches to spiraling. This is what I call differentiation by time, not content.

4. *We should offer outside-of-class math help to students who need it, and math enrichment to students who want it. *

5. *Students in a given course should span two grades.* For example, a Geometry course can include strong 9th graders and not-as-strong 10th graders. This is quite different from tracking, as all students take the same classes — they just take them at different times in their school career.

Any approach to detracking in high school that does not include at least these ingredients is not likely to succeed. Worse, it could lead to a backlash that would set us back dramatically.

— Henri

PS: Given the size limits on a blog post, I once again encourage you to see those ideas spelled out at much greater length in these three articles:Reaching the Full Range, Hyper-Acceleration, and The Assessment Trap. See also some of my other articles About Teaching.

Dear Henri,

Since a friend pointed me to your blog a few days back, I have watched it and I see this post dealing with detracking. You may find it interesting that I largely agree with your ideas, even as I doubt the practicality of some of them. Let me try to explain.

First, I wholeheartedly agree with your first point, that we must not neglect the needs of our strongest students. Education is not a zero-sum game, and challenging advanced students should not detract from our ability to educate less advanced ones.

Second, I mostly agree with you that *in principle* a curriculum that addresses a wide-range of student abilities would be extremely helpful. My problem is that *in reality* have not ever seen such a program applicable at large scale. For example, I have seen problems at the math circles (https://mathcircles.org/ ) that allow multiple level entry points to engage with them, yet they all seem to work only with a smallish group of highly engaged students and an expert teacher/mathematician to lead them. This is clearly unscalable to a typical public school classroom with its largish classes of barely-engaged students and mathematically semi-ignorant teachers, particularly in K-8. Add to that the fact that such intensive teaching as occurs in math circles leaves a lot of basic mathematical skills uncovered, leaving them to parents, regular teachers, or the engaged students themselves. We must remember that school mathematics serves not only to potentially support particular deep interests of students, but also need to support basic mathematical skills to allow progress in other topics that rely on mathematics for skills — the sciences and social sciences.

Third, I fully agree that combining forward motion and extended exposure is important. Yet how to achieve it in a widely heterogeneous classroom is unclear. Sure, we hear the platitudes about differentiated teaching but very few teachers can actually do it, and none can do it if the classroom spans students with 3-5 grade level differences. Your ideas of lagged homework and assessment (I believe your refer to so-called

distributed practice/) and similar are on point, but I must add a word of warning about “spiraling.” Spiraling of *learning* (not of practice and quizzes) such as Everyday Mathematics does has zero basis in research and has been found ineffective. It essentially allows teachers to ignore the failure of their students with the excuse that “they will get it the next time.” Largely they won’t.Fourth, outside-of-class math help and math enrichment are excellent ideas, but they cannot be scaled if the demand is beyond a small fraction of students. Here again more homogeneous classes can help, as they solve the bulk of the problems rather than heterogeneous classes with leave large fraction of students in need of them.

Fifth, your idea of classes spanning two grades makes a lot of sense, but that is effectively already done in HS where students can select the level of the math class based on their achievement rather than their grade. But it has been almost completely eliminated in the middle schools over the recent decades in the name of … DETRACKING.

OK. So far I addressed your five points and, as I’ve said, I largely agree with your principles even as I am somewhat doubtful about the scalability of some. now to some additional comments.

– You ignored the fact that nothing will work if our K-8 teachers are semi (or fully) ignorant of mathematics. Yet that is the current reality in California and much of the country. Teacher colleges and their professors pay little attention to content knowledge and prefer to focus on pedagogy. Check this for an easy read.

– This blog post is in response to the draft California Math Framework. As I have mentioned in my previous comments, there are two issues with the Framework’s proposal:

(a) It proposes to keep *all* students in the same identical rather-basic math course in grades K-10. This essentially goes against all the five of your points above.

(b) The supposed research basis offered by the Framework to this proposal is NONE. One of the three offered studies uses effectively fake data of San Francisco Unified (https://www.independent.org/publications/article.asp?id=13698 ), the second refers non-existent study, and the third study proposes just the opposite of what the Framework suggests https://www.independent.org/news/article.asp?id=13743 ).

So how can you support the Framework?

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I note that the link to what Ed School profs pay attention to didn’t come through in my previous comment (my fault). Here it is:

Cracks in the Ivory Tower? https://fordhaminstitute.org/sites/default/files/publication/pdfs/cracks20in20the20ivory20tower20-20sept2020108.pdf

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[…] I write more about this topic here, and here. […]

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[…] I elaborate on the question of detracking in my next post. […]

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