Questions and Answers about Tracking and Ability Grouping

[I am not an expert. Maybe you are, in which case I would greatly appreciate a comment pointing out things I’m missing. Much thanks to my anonymous partner in crime, the unlinkable TracingWoodgrains. We’ve been reading this literature together, and while we don’t agree entirely about tracking this piece is a result of our work together. Any good parts of this wouldn’t have been possible without his collaboration.]

What does “tracking” typically mean in American schools?

European-style differentiation into vocational/academic tracks is rare in the US, though it used to be very common earlier in the 20th century.

Now, most elementary classrooms have tables or little within-class groups for reading and math. As kids get older, it’s more and more common for schools to create high/middle/low classes for various subjects, but especially for math. By 8th Grade, most kids are assigned a class based on past performance, and sometimes those classes are “accelerated,” meaning they take Algebra 1 in 8th Grade. By high school, high/middle/low tracking is near universal in math.

(Some of this picture I get from Loveless. Loveless also notes that there’s a lot of terminological confusion between tracking and ability grouping. I’ll use the terms interchangeably here.)

In addition, though it’s not called “tracking,” a lot of school resources are dedicated towards students who aren’t performing highly. This amounts to a kind of ability grouping and is super-common throughout k-12. It’s federally mandated ability grouping, in fact.

Who benefits from conventional tracking? Who loses?

If anyone benefits, it’s almost certainly students in the higher tracks who gain and students in the lower tracks who lose. But the effects aren’t clear, and the impact of tracking isn’t particularly well-understood.

In 1987 Robert Slavin reviewed the existing literature for elementary and secondary students and found practically no benefits for anyone from conventional tracking — but also no real harm done to any group. On this basis he argued against conventional ability grouping, seeing as it helped no one and was morally noxious.

But the studies he reviewed had limits (small size, not nationally representative). Using better data, a number of researchers (Hoffer, Gamoran and Mare, Argys, Rees and Brewer) came to the conclusion that conventional tracking benefits students in the high tracks and hurts those in the low tracks. But it’s really hard to control for the right factors in these definitely non-experimental studies, and Betts and Shkolnik raise questions about the results of these papers (summarized, as is this whole story, in this review by Betts).

(Just to mess with everybody, Figlio and Page argue that by attracting stronger students to the school (because parents seek tracking) students in low-tracks benefit, indirectly, from tracking.)

Recently, there was an experiment in Kenya — one of the very few true experiments! — where they randomly instituted tracking into some schools and measured the impact. It was positive for everybody, but there are a million differences between this context and the one in the US, starting with the number of ability groups (just 2) and class size — over 45 kids are in each classroom! It’s hard to know what to take from this study for the situation in the US.

And there was also a recent big meta-meta-analysis that found no benefits and no harm for between-class grouping, echoing Slavin.

Loveless says the evidence is inconclusive, and that’s echoed by Betts, and the fact that it’s not a clear effect tells you something about how tangly this whole issue is. But if it helps anybody, it’s probably top-track students, and low-track students would be the ones hurt by tracking.

Wait, you said “conventional tracking.” What’s unconventional tracking?

Slavin, Destroyer of Tracking, has a school-turnaround program with good results called Success for All that depends heavily on grouping students by ability. How? By adopting something like the Joplin Plan, which assesses frequently and regroups students based on those assessments. Students across ability groups show benefits in these programs. (Though, not without controversy.)

Another form of grouping that isn’t widely used is acceleration, e.g. placing 1st Graders in 2nd Grade math if they’re ready for it, and continuing down the line. There is research supporting the notion that acceleration benefits the accelerated student in completely straightforward ways — they learn things that they wouldn’t otherwise have access to. See that meta-meta-analysis for instance.

Does race impact where you’re tracked?

Using one of those large, nationally reprsentative samples mentioned above, Lucas and Gamoran (fierce opponents of tracking) found that race wasn’t a factor in track placement. Meaning, controlling for academic performance, race isn’t a further factor in deciding where a student gets placed.

Because of the gap in academic success that Black and whites collectively experience, this still means that Black students disproportionately occupy the lower tracks.

In contrast to Lucas and Gamoran, Dauber et al, found that race was a factor in track placement in Baltimore schools. It’s hard to know for sure how to fit Dauber with Gamoran’s bigger picture results.

What about other non-academic factors? Do they impact track placement?

Gamoran found that, unlike race, socio-economic status does statistically impact track placement (modestly) suggesting that, somehow, high-SES students tend to get tracked above their academic performance. Why? We don’t know for sure. Maybe it’s parents? Maybe it’s the intangibles, like being a good student with homework done and things organized because they have parents at home who can help manage the academic lives since they aren’t coming home at 10 PM from their second job?

But we can’t really know yet how precisely SES helps determine tracking.

What do we know about the quality of these low-track classes, as compared to higher-tracks?

Even defenders of tracking agree that low-track classes are often very poorly taught and that this is a major problem. Here’s Loveless: “Even under the best of conditions, low tracks are difficult classrooms. The low tracks that focus on academics often try to remediate through dull, repetitious seatwork.” Much of Oakes’ contribution is documenting the lousiness of a lot of low track classes.

How does this square with researchers who find no negative impact of tracking on low-track students? All this would mean is that instead of failing these students in low-track classes that schools typically fail these students at similar rates in untracked schools.

Speaking personally, it seems to me that the strongest argument against tracking is the state of low-track classes. Forgetting academic performance, these students need to be placed in safe, respectful, happy classrooms staffed by competent teachers who believe in and care for their students. I think we have plenty of reports showing that this is often not the case in low-track classes, and this is what I saw at the first school I taught at.

So is the tracking status quo bad for racial inequality?

Put it like this: if we immediately removed all US tracking and replaced them with heterogeneous classes, the result would possibly be narrowing of the black-white score gap somewhat — a bit from improving the performance of low-track students, but mostly by limiting the advancement of high-track students.

Those high-tracks don’t just contain white students (schools are also highly segregated remember), and another national priority is increasing the representation of Black and Latino students in the highest ranks of achievement. Some of the tools we have for increasing representation are universal screening for tracked gifted programs, and removing tracking would also remove these programs. Without public access to gifted programs, would wealthier, whiter students just pursue these out of school, exacerbating inequities at the highest levels of achievement?

Still, the net effect would probably be a narrowing of the black-white gap.

OK, so let’s get rid of tracking entirely.

Only if you’re willing to really restrict the amount of learning that some students are capable of — either through deliberate acceleration or by maintaining track differences — for the sake of equity. After all, the flipside of the evidence that tracking exacerbates inequalities is that it really does help some students, usually those in the top tracks. (And, if you doubt that evidence, there are still unconventional tracking methods that we could be using to further accelerate more students, deliberately, from younger years.)

The tough question here is what happens to the learning of students who are ready for more than their heterogeneous class is offering them.

But can’t you teach in a way so that everybody maximizes what they could potentially learn?

This is the golden snitch of teaching, right? You win the game if you can grab it, but it always manages to slip away.

Maybe there are schools that have pulled this off (Boaler, Burris Heubert & Levin), but they seem to be relatively rare. In general, schools that tended towards untracking amidst the heights of the anti-tracking movement inched back towards tracking (Loveless).

Another note: a lot of untracked elementary schools just use ability grouping within classes. Maybe there’s increased mobility between those groups, but teachers need to find ways to deal with heterogeneity. Pedagogies that benefit everybody with no costs are highly vaunted within education, but I’m skeptical, and there isn’t evidence that these schools provide widely replicable models.

But if you don’t remove tracking, is there any way to improve the status quo?

One approach might be to pursue some of the unconventional tracking options, though that would involve pushing against what Larry Cuban calls the “grammar of schooling.”

But there are also many examples of tracked schools that offer a good education to their lowest-tracks. In fact, Rochelle Gutierrez studied eight high school math departmenets, some tracked and some untracked. She came to the conclusion that “tracking is not the pivotal policy on which student advancement in mathematics depends.” What is crucial for her are a whole host of other factors, including strong pedagogy, school culture, and solid, shared curricular resources.

More examples of effective tracking programs that promote mobility come from Catholic schools. (Wait, Catholic schools are closing left and right as they lose students to those charter schools that politicians made such a big deal about? Whoops.) See Camarena and Valli.

Likewise, Adam Gamoran identified examples of schools with successful low-track classes, and identified features of these programs. It’s what you’d expect — high expectation, good pedagogy, making sure good teachers work with the low tracks too.

Which is more promising — expanding hetereogenous instruction or improving low-track classes?

Let a thousand flowers bloom, etc., but I think if you put me in charge of a district or a school I’d focus on improving low-track experiences. It seems to me as if there are more cases of working low-track classes than examples of successful heterogeneous programs. And, as a matter of experience, I am not sure I believe in cooperative learning as a pedagogy that mitigates the risks to high-achieving kids.

Tracking or untracking: what do you say?

Well, it really depends on the school. I think if you put me in charge of a school I’d want to follow Gutierrez in focusing on things like curriculum, high expectations for every kid, safe classrooms with comptetent teachers for every kid. Tracking or not wouldn’t necessarily be my most important decision.

I don’t think I could stomach a school that tracked strongly along racial lines. That’s not good for school culture or the experience of students in the lower-track, and so I’d probably want to untrack that school as much as possible. That said, I’d still want to see programs for students at either extreme of the achievement spectrum. (And I’d be federally mandated to provide a lot of such resources at the lower end of that spectrum.)

Otherwise, I’d be fine with tracking probably, as I’m fine with the tracking that my current school uses. And I’d be really interested in seeing if I could employ some of the unconventional tracking plans like the Joplin plan or reasonable acceleration, like letting 4th Graders take math with 5th Graders.

Does that include grade-skipping? That’s rarely used in schools, but it’s a form of acceleration.

Grade-skipping seems to generally benefit those who skip (Park Lubinski & Benbow). I’d want to be able to use it, mostly when kids aren’t happy and we think it’s because they’re unchallenged by their current grade.

Why don’t more schools use unconventional tracking?

There are two ways of putting this, I think. The first is to state, as Cuban and Tyack do, that there is a grammar of schooling that resists reform. The typical age-graded classroom is a strong feature of schools, and both of these unconventional tracking methods push against age-grading.

But why should age-grading be such a persistent element of schooling? I find David Labaree helpful in explaining this, because what educational consumers seek are either markers of distinction for their kids, or equitable access to those markers.

Learning is only of secondary value to most parents — they don’t seek learning without distinction — and so something like acceleration is very hard for schools to offer more widely without leading to chaos as parents demand ever-increasing acceleration for their kids.

Age-grading as a strong default is a compromise that helps schools manage the demands of the educational consumer.

But the Joplin Plan, and other plans that assess kids frequently to better determine and match their curricular needs, seem like they deserve more attention than they get.

So does anybody like the status quo?

Maybe not, but that’s by design. Schools aren’t designed optimally for learning or for equity. School as it exists is a sort of uneasy compromise between contradictory principles — fair access and award of distinction — and the competing demands of different groups. Some claim to have revolutionary solutions, but these probably don’t exist. You can reduce inequity, but only if you’re willing to curtail the learning of some. You can improve learning for all, but risk exacerbating inequity. This is an optimization problem with more than one possible solution. Or, as Rochelle Gutierrez says in a different context, the answer to the questions of tracking are usually “neither and both” sort of answers.

That said, in math education circles, tracking is unfairly maligned, in particular by NCTM. In Catalyzing Change they say that the research is “unequivocal” that tracking harms low-track students in permanent and irrevocable ways. Looking more broadly at the literature, it’s hard for me to agree with that take. 

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14 thoughts on “Questions and Answers about Tracking and Ability Grouping

  1. Michael,

    As usual you write about a potentially controversial topic with some nuance and skill. Thank you for doing so.

    One thing occurred to me and I wonder if this is actually true or not. In age-based grades, there is always a cut-off where some students are held back for the next kindergarten class and other students are promoted out of pre-school for kindergarten. In fact, typically this cut-off date exists for pre-schools as well. Usually the cut-off date is arbitrary from the perspective of child development. Has anyone studied the performance of children on either side of this arbitrary cut-off date? Some people, as far a I know from Malcolm Gladwell (who is apparently sometimes wrong on the research), has studied this issue of cut-off date for sports quite closely and found some really weird things, 36% of NHL players drafted are born in the first three months of the cut-off year with only 14% of players drafted born in the last three months of the cut-off year.

    It makes sense that this would be a huge problem in kindergarten and because of the Matthew Effect the problem might carry forward into the following years. Children who are younger than their peers would likely both have lower expectations placed on them their peers but also be, in terms of percentages, well behind their peers in what they know and can do. A four year old born just after the cut-off date would know, on average, 20% less than a five year old born just before the cut-off date.

    David

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    1. I know so little about this, but (based on just a few minutes of trying to start searching) it seems like an actual research possibility. I’m skeptical of the shocking magnitudes that Gladwell cites, just based on experience reading stuff like this in general. Plus the research comes via these big economic papers with enormous datasets and (much like with some of the tracking papers) there are researcher degrees of freedom and it’s hard to quickly get the picture. I don’t know the literature at all, but here are two papers that come to conflicting conclusions about the importance of age:

      The Persistence of Early Childhood Maturity: International Evidence of Long-Run Age Effects

      First in the Class? Age and the Education Production Function

      I’m eager to learn more about this, and while I don’t know if this is the direction you were going for, it raises the possibility that age-grading isn’t “neutral” in its impact on students either.

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  2. A couple thoughts…

    But why should age-grading be such a persistent element of schooling? I find David Labaree helpful in explaining this, because what educational consumers seek are either markers of distinction for their kids, or equitable access to those markers….the Joplin Plan, and other plans that assess kids frequently to better determine and match their curricular needs, seem like they deserve more attention than they get.

    I disagree. Labaree might have part of the answer, but on the whole, I think that response is too cynical. I think it overlooks the positive potential of strong peer relationships. Creating a safe, positive environment–a place where students enjoy learning math together–is difficult. It requires students to trust the teacher, but also each other. I will be the first to admit that I’ve taught many peer-age cohorts where this didn’t happen. But I’ve had a few where it did happen–including one class this year. When it does, the results are remarkable, and are most pronounced for the weakest students. My goal for the next few years is to figure out how to make this happen more consistently (and intentionally).

    From my experience so far, the Joplin plan would have two primary disadvantages:

    1. Groups changing all the time might match a kid’s curricular needs, but it would destroy any sense of community. I recently read the article that everybody on Twitter is posting. My takeaway was that it’s easy to overlook the social-emotional implications of policies that seem logical, but it’s dangerous to do so.

    I also think changing groups frequently could undermine a kid’s sense of agency. When kids believe that learning certain material is expected and normal, so long as there are resources available to help them do so, they can bring quite a bit to the table in terms of effort, seeking out tutoring, etc. On the other hand, if our solution to a kid not learning something is to stick him in another group–that’s something we control, not the kid, and I think it implies that it’s 100% on us to make the learning happen.

    2. There might be ways to mitigate this, and maybe in the long run it would just become the new status quo, but I think this would be pretty stressful for kids initially. It’s no fun being a 5th grader in a class of mostly 3rd graders. But it’s actually also stressful for the 3rd grader to be in a class of mostly older peers, particularly if they were just “early” in developing some skills and they need to move down a level later on. As I side note, I’d be wary of using Success for All as a model of anything. The research supporting its effectiveness is equivocal at best. Beyond that, I have first hand knowledge of what happens in some of those schools, and they are not schools I recommend for any child.

    The tough question here is what happens to the learning of students who are ready for more than their heterogeneous class is offering them.

    Yes. But I don’t see how acceleration is the answer. Suppose you have a kid who moves through math 1-2 years “ahead” of his peers. He completes the K-12 curriculum–maybe even Calculus–in 10th or 11th grade instead of 12th grade. So what? A year or two later, when his age cohort reaches 12th grade, what’s the difference between this kid’s math knowledge and that of his peers? (Of course a lot of the answer depends on what he’s been doing in those two years…so what’s the plan?)

    Here’s another idea for an unconventional approach to tracking:

    There’s a reason we don’t send our gifted junior-high athletes up to the high school to do “advanced” PE. We recognize that the school curriculum is designed to give all students some basic knowledge about healthy living and a few skills that will might them participate in lifelong fitness programs. They aren’t designed to develop giftedness. For that, kids usually play on competitive teams, often traveling to play similarly matched competitors, and receive lots of specialized coaching. The sports situation isn’t perfectly analogous to math, but it’s worth thinking about what it would look like to try something similar in schools.

    One of the more interesting aspects of this analogy: for the most part, our gifted athletes still do PE at school! They really enjoy shooting hoops and playing kickball with their friends, even though they go off and play on Olympic developmental soccer teams on the weekend. Our PE classes are richer for having these kids–they tend to make games more fun. And I actually don’t think they are wasting their time…PE class is developing other skills like sportsmanship (more generally, how to interact appropriately with people who don’t have your gifts) and allows them time to be with friends. Applied to math, I can definitely see some parts of math class (e.g. practice for skills they already know) being a waste of time–but I can also envision ways to provide appropriately differentiated practice (e.g. Art of Problem Solving) during some of that time. At other times–during discussions, activities, games, solving more challenging problems–these kids are a great asset to the class. I’ve also found that even for very gifted kids, communicating about math–verbally or in writing–is often more difficult than solving the problems. Participation in a grade-level class might allow time to focus on these kinds of complementary skills. While a full double schedule might not be feasible, I’d love to see something like a 2 day/3 day split between whole class and tracked (enriched, not accelerated) instruction.

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    1. Much of my own interest in this topic came as a result of my own experience in age-grouped classes, which were… well, frustrating, to say the least. The only time in twelve years of school that I developed long-term, meaningful friend relationships came in eighth grade, the year after I was allowed to skip. Being with age peers harmed, more than helped, my overall development, since we honestly didn’t have too much in common. Lubinski et. al, studying the cohort most in need of acceleration, found no negative social results and significant positives elsewhere. Miraca Gross found the same in her longitudinal study.

      And those are exceptional cases, to be sure, but in a field built on ambiguous studies and unclear conclusions, it’s actually remarkably hard to find results of negative impact from acceleration for prepared children, even from fervent anti-tracking groups like the NCTM. After acceleration comes early college entrance, or a gap year, or a few other options. My personal favorite solution I heard was one of Gross’s students who accelerated three or four years through elementary and middle school, then repeated his junior and senior year of high school to take a wider variety of courses before going to college around age 16. If you waste less of a child’s time with things they don’t need to learn, you open up their options and chances to learn things they want to learn.

      Ultimately, I do think something like the amazing Art of Problem Solving is a much better answer. My happiest moments with math came when I discovered math competitions, then went away when I realized math classes were nothing like them. I didn’t find out about AoPS until years after finishing schooling, though, since it wasn’t on my teachers’ radar. Acceleration is a simple, stopgap answer for when a more difficult-to-build structure isn’t around, and it’s not as good as a truly enriched curriculum but it’s much more feasible in some environments.

      You make good points about social-emotional impacts of group changing and age peers. I wonder how much of that is built on our assumptions about schooling, though? Children’s most lasting relationships are sibling ones, with all the asymmetrical balances there. In middle and high school, Theater and other hobbies are other good examples. Kids are capable of developing rich friendship networks with people of many ages, but it’s structurally discouraged.

      The frequent changing concerns are real ones. Perhaps giving kids an element of choice there wouldn’t be a terrible idea? In addition to increasing their self-direction over their learning, it would mean that kids who really wanted to advance quickly or grasp the information could, while kids who had healthy friendship groups and were satisfied would not. One of the persistent frustrations in my own experience was that none of the teachers really cared to ask what I wanted in schooling (in my case: as difficult of learning as possible, yesterday) and the structure wasn’t set up to provide it. Elementary school was pretty bleak.

      Your PT analogy is interesting, and informative: In general, advanced athletes love age-level PT. They also have extensive, reliable extra-curricular options. I don’t know that you can say the same for advanced math students and age-level math class. Math class doesn’t have the greatest reputation among any student as being fun, and for a student who loves the patterns and problem-solving at the heart of math, real exposure to those during a typical class is rare. If they’re very lucky, they’ll stumble into the world of math circles and competition math (and increasing the prevalence of these would be great but is a difficult problem with the culture we’ve developed around math). In my own case, I stopped math classes in frustration after tenth grade. Some of that can be solved with curriculum improvement, but not all.

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      1. So interesting to hear others’ experiences and perspectives. Full disclosure: I was not profoundly gifted, and the only reason I seemed exceptional was that I went to very small schools. I skipped going into junior high, coinciding with my family moving across the country. It was a disaster socially, and I think any academic gains were marginal. Fortunately/unfortunately, that was followed by another disaster a couple years later–I got leukemia–which effectively reset everyone’s expectations. I definitely agree that giving kids more of a choice in the decision would be a good thing. Sounds like that would have benefited both of us.

        The studies you cited seemed to refer to the profoundly gifted–the one in 10,000 kids. For those very unique cases, I think acceleration may sometimes be the answer–and the early college option makes sense for those kids as well, particularly if they are strong across the board and not just in math. But at the schools I went to, that would be about one kid every 12 years or so. While I certainly wouldn’t want the option to be off the table, I get nervous when schools start organizing classes and schedules, setting the expectation that a significant percentage of each class should accelerate. I teach about 30 kids per grade, and we have to offer an accelerated track because otherwise our kids will be “behind” once they get to high school. Once we start offering the class, we need a certain number of kids to enroll in it. We also have a certain number of parents who demand their kids take it, whether the kid wants to/is ready or not. I prefer the enrichment model, open to all kids: various ratios of passion/discipline/aptitude allow many kids to be successful with more challenging math (i.e. AoPS)…these experiences also tend to be more enjoyable and less stressful for most kids, because they know it’s supposed to be hard, and they aren’t “falling behind” if they don’t get something right away. For kids who aren’t profoundly gifted, but are just ready for more challenge than most of their classmates, this seems a good option.

        On the other side of the spectrum, offering an accelerated track seems to make those not accelerating anxious that they are being left behind. It also makes it more difficult to create community and cultivate rich experiences in the “bottom track” classes. For the most part, kids have already decided they aren’t good at math, which definitely affects their willingness to participate. Good curriculum helps–I’ve definitely had a much better year with Math 8 since we switched to IM this year–but it’s still a much tougher class to teach. Better teachers might be another part of the answer, but I think some of the issues would be difficult for even the very best teachers to overcome.

        I don’t see the accelerated option going away any time soon, so we’ve done the best we can to make the best of it. I don’t think the status quo is the worst possible outcome–acceleration works out ok for lots of kids–but it’s interesting think about more unconventional approaches to tough questions about equity and optimal learning. I think a lot of our answers reflect our vision of what our classrooms should be like. I do think it’s important to let schools grapple with these locally, and to accept that schools might come to different conclusions: I enjoy teaching heterogenous classes more, and they work well at my school, but I can envision other environments, particularly at large schools, whether other solutions might be better, or where more emphasis on improving the low track experience might have greater yield. I also see community/democratic participation as a very important goal; various schools may prioritize that differently. The problem is that schools don’t make these decisions in a vacuum–so much of what we do is influenced, directly and indirectly, by what the schools around us are doing. I’m curious as to (a) how the NCTM recommendations will be interpreted and (b) whether they’ll have much effect on secondary schools in the next few years. I’m also very aware that few “reforms” are successful. The Cuban article/book chapter was interesting and informative–we’re thinking about blowing up the schedule and trying some new stuff next year, and it’s always helpful to have some perspective on how that’s turned out in the past.

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      2. You raise a lot of great points. I wanted to touch on your comment regarding the very unique cases:

        The trouble with a situation that rare is, when it comes along, if people haven’t been properly trained to know the best ways of handling it, they sometimes make strongly harmful decisions. That danger is reflected with Gross’s work: 33 out of 60 in the most extreme category that could be identified were left, lockstep, with age peers. This danger is compounded if a child is from a disadvantaged background, with poorer or less educated parents: their parents may not be as immersed in the culture of education, as prepared to fight for their children, or as knowledgeable as to the needs of unusual children. Whereas richer parents may just withdraw from systems that don’t suit them and find top-quality options elsewhere, disadvantaged ones may not have that luxury.

        The more general the knowledge about appropriate acceleration and willingness to open it up as an option for children to choose, the more a chance that both regular hardworking, smart children will be able to find challenging academics and that the small groups at the very edge of the curve will find the understanding and support they need.

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  3. You’ve correctly identified the tension, as it is experienced in schools: is it more important that our Mathematics classrooms are designed to (1) ensure top students (i.e. those raised in households with higher SES and parental education) score as high as possible on standard measures of mathematical attainment (ignoring mathematical learning or joy, mind you); or (2) produce a future society that intentionally works to reduce classism, racism, and division among peoples based on a “better-than” mindset.

    Tracking in school mathematics is not a question of ensuring all kids learn as much mathematics as possible, it’s about the purpose of schools and the society we aim to create.

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    1. The tension is messier than you state it, I think. Mostly because most people who want (2) also want schools to be a place like (1), just one where you can’t use gender, race, class, etc. to predict who the top students will be. The list of people or organizations is small that are willing to bite the bullet and say, yes, sign me up for reducing the amount of math that our strongest third of students learn for the sake of reducing inequality.

      So from my point of view, the central difficulty is that most people want both (1) and (2) in some form or another, and they’re correct to want both, but this means that school is always an uneasy compromise somewhere along this lived tension.

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      1. We have a false sense of what is mathematics, and what is an indicator that mathematics was learned. Both of those false notions serve to perpetuate a hierarchy-based society. The people who want both (2) and (1) don’t yet understand that they are irreconcilable.

        It’s not a question of “amount” of mathematics; it is a question of a system designed to ensure some have and some do not.

        We are all mathematical beings. We all perform mathematics, we all can learn to interact with other’s mathematics, we all (re-)invent new mathematics. What if the focus of schooling were to create opportunities for these experiences. And the end of the day was not marked with who did the most (from the omniscient(?) eyes of the teacher) but with a celebration of everyone’s new insights and an appreciation for one another.

        ***or, if you don’t like this idealism–which demands a radical change to how our schools operate and would disrupt our present society, then I am 100% in on focusing the mathematics classroom to be a completely heterogeneous environment in which the primary learning goal was respect for other’s ideas and democratic participation. Any mathematics learned along the way is nice.

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      2. Your non-idealistic solution is pretty idealistic and radical, though! (I really admire, as usual, your clarity and willingness to take a strong and consistent position.)

        What makes this radical is that many people want a version of (1) out of school. They want every student (including but not restricted to top students) to have the chance to achieve as much as possible. This might be an oppressive belief (it certainly makes equity possible) but it’s one that I believe to be widely held.

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    2. Can I suggest an option 3, in terms of goals?

      Respect the authority of students to choose whether they want to dive deeper into mathematics and explore the subject as thoroughly as possible, providing necessary acceleration and enrichment for students who are excited about the subject and ready to learn more. Respect for others’ ideas and democratic participation are excellent as ideals, but if I go into a mathematics classroom, I’m there first and foremost to learn math.

      Regarding standardized testing: Many kids who score high on standardized tests are uncomfortable with it because it means they’re different to their peers. Many of those same kids are the ones who will sit down and work long, complex math problems for fun, who pause in the middle of whatever they’re doing when there’s an interesting puzzle to be solved. But they go into math class, and they learn that it’s a place where their intellect will not be challenged, where their desire to stretch won’t be encouraged, and they come out tired & cynical as “beneficiaries” of a system that decided its job wasn’t to teach them math, to push and stretch them and help them grow intellectually. They’re not upset because they care about who’s learning “most” as much as they’re upset about whether they’re learning what they want at all. I don’t see a way for your option (2) to serve these students.

      An equitable society means reducing prejudice, yes. It means reducing divisions based on “better than.” It also, though, means equal opportunity for all to progress and stretch themselves in their passions, even the ones who are from privileged backgrounds and who happen to love math.

      Liked by 1 person

  4. Now that I have finally read both Michael’s post and everyone’s comments, I feel like I have even less clarity. (…not that I felt like I had clarity to begin with!) My head is swirling with all different ideas.

    First of all, I am dictating my response while my lovely kids run around, so apologies for typos and errors.

    For background: I teach in a high SES district, a wealthy semi-urban suburb on the edge of Boston. I used to teach in a working class district 20 miles north of the city. The typical family in my current district has money and lots and lots of fancy degrees. Only 1/3 of the parents in my former district had earned bachelor’s degrees, and I think this figure was even lower at my particular school.

    Theoretically, I’m all for keeping kids with their age-based peer group, and promoting depth. There would be lots of opportunities for students to engage, and collaborate, and persevere, and do all the hot buzz words.

    Then I look at some of the kids in front of me. There are students with a profound understanding of elementary mathematics. There are kindergarteners who understand mathematical ideas not only deeper than their kindergarten teachers, but deeper than most of the elementary teachers. (…and I think our faculty is pretty solid.) I’m sure most people have that outlier case study. I could describe a few. But we have also dealt with cohorts that have unusually large numbers of students wrestling with mathematical ideas multiple years beyond their grade level peers.

    Traditionally, my upper SES school don’t with Leslie: parents or teachers requested that the students received testing for subject acceleration. Students were assessed rigorously, and students that demonstrated a well-rounded mathematical proficiency–at least two years above their age-based grade, accelerated two years. (These are not just 4th graders that know how to find percentages.) They would spend the majority of their day their traditional classroom, and take math in another, e.g. a third grader taking math in a fifth grade classroom. Clearly, this is not a perfect solution. Major problems include the equity of the screening process. I am thrilled with the ideas it communicates to the other kids, either, but the screening process was intensely problematic. Also: the classroom two years above was not necessarily any more “enriching.”

    In this process, seventh graders attend math at the high school. The math department developed a course called “advance math topics” for them, which addresses content that is very rarely taught in a public school. It’s pretty awesome, and unfortunate, only available to those accelerated early. Then, as eighth graders, the accelerated kids take ninth grade advanced geometry. This means that most end up one year beyond their age group.

    The district has been working change the selection process. Now, we have something called a “challenge framework,” which outlines a process for all stakeholders (teachers, parents, students) to ensure that a child is being challenged. It’s great, but, honestly, it’s a long process that requires a lot of data to move to what a lot of parents want: acceleration. Teachers seem put off by all of the work involved along the way. I’m not certain how to fix it. (I’m really glad that particular task is not my job. 🙂 )

    Oh, and a major compounding issue? More than half of our students take outside math classes, e.g. at the Russian Math School. (I’m really glad we don’t even often the option to accelerate only one grade level, because it would be madness.) Not all of the students who have been accelerated take an outside math course, though. Some of them are just naturally propelled by their own curiosity and passion for mathematics. But the ones that do take the classes are generally rich, and white or Asian. The gap widens.

    Oh, and because so many of our kids take outside math, I think it allows some teachers to get lazy with instruction. If half the class comes in already knowing how to multiply,…

    Anyway, a few years ago, we realized that we had an unusual number of kids that may fit the profile of an accelerated student all in one grade level: like 6 or more, in a grade level with 70ish kids. We did clinical interviews with them. Classroom teacher felt like she was at wits end. “today, Ian and Kevin we’re talking about a pattern they noticed with fractions, where they basically converge to 1. They’re six years old. What do I do with them!” We gave a universal screener. Eight kids demonstrated conceptual understanding and procedural fluency three years about their grade level.

    A second grade classroom teacher met with a large team that included district personnel, and it was determined that she would teach her class workshop model, with most of her students going deep with 2nd grade concepts — she still had plenty of kids performing above grade level in there — and all 8 kids would be in her classroom to do 4th grade math, while in 2nd grade. It was like ‘acceleration in place.’

    …and it was a logistical nightmare for the teacher. It’s hard enough to teach one grade level well!

    Thus, when they were in third grade, it was determined by the administration that a math specialist (me) would teach the group as a pull-out acceleration. I didn’t do a standard 5th grade curriculum. It became more like an enrichment class. But it wasn’t without its problematic elements, and it became very controversial. My colleagues hate it. I don’t know if it was the right move, but it has made all the difference for those 8 kids that I think we were at risk of “losing.” Do we really want a bunch of disaffected learners who once loved math?

    Wow, I feel like I could say so much more… but we need to do other things. This is all to say that I don’t know how I feel about tracking in reality. (I’m opposed in theory, generally, but, again, then I see the kids…) I worry a lot about the kids in the “low” tracks. I worry a lot about the kids in the second highest track that felt slighted and/or have parents who tell them that they should feel slighted or that they weren’t good enough or whatever. I worry a lot about ways to screen and assess equitably, and about unequal representation of race and class in the various tiers. I don’t know how to reconcile with any of these worries right now, either.

    Maybe more later?

    Liked by 2 people

    1. Very well dictated! I think our paradigm of what is mathematics education, what counts for knowing, what’s important to learn, etc. are what create our inability to imagine all the same students engaging in, doing, learning, creating mathematics together in the same classroom. We’ve been over-standardized in mathematics since forever–the fifties maybe.

      Liked by 1 person

  5. Jenna,

    I realize this was a school or district solution, not your own, but while I’m in favor of enrichment, I’m not anymore comfortable with a class given to six or seven kids. Moreover, there are a lot of kids who do well with advanced conceptual thinking in arithmetic but don’t necessarily do well with mathematic concepts later (algebra, geometry).

    I think enrichment is fine, but in class only.

    “Do we really want a bunch of disaffected learners who once loved math?”

    Look, the kids were advanced because their parents footed the bill for tutoring. Tell them they don’t have to take math at all in school, or have them do book reports on math books or concepts. Give them an A in math. But fortunately, at this point in time, we aren’t required to support gifted kids with the same guarantees that we are special ed kids (who, frankly, should not be guaranteed anything either, but that’s a different story).

    Or take all the advanced kids and teach them enriched math, and really push them, making math hard and challenging and mess with their sense of “everything’s easy”. But do it for 30 kids, not 6.

    I’m not opposed to tracking, but I’m very unsympathetic to kids who are far advanced because their parents paid for tutoring. It’s not on the school to give up resources to continue their private instruction.

    Like

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