This is part three of three.

Part 1: Teaching Social Justice Instead of Math is an Injustice

Part 2: Differing standard deviations give some people the vapors

Study Claims Gifted Math Classes Promote 'Academic Apartheid'

A math education professor is arguing that gifted math classes cause “academic apartheid” among students, claiming that the practice is rooted in “capitalist exploitations and settler colonialism.”

The study, “Understanding Issues Associated With Tracking Students in Mathematics Education,” was published in the new issue of the the Columbia University journal Mathematics Education by Cacey Wells, a professor at the University of Oklahoma.

In his article — which relies heavily upon social justice math theory — Wells takes aim at what teachers call “academic tracking,” which is the practice of placing students in different math classes (such as pre-algebra or gifted classes) depending on test scores.

Under the tracking system, for example, a student who scores in the top 10 percent of his peers may be placed into a precalculus course. On the other hand, a student who scores in the lowest 10 percent may be placed into a remedial math class, or perhaps pre-algebra.

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“This is particularly true in schools. Many times students are pigeonholed into particular academic tracks based purely on socially constructed potentialities,” he added.

Drawing from this, Wells — who teaches aspiring math teachers at Columbia University — argues that separating students by ability is a form of “academic apartheid,” a logical consequence of what he calls “settler colonialism” in math education.

“Settler colonial ideas routinely become evident through colonization of intellect. When there is no longer territory to conquer or people to physically oppress, there exists opportunities to colonize knowledge,” writes Wells.

Separating by ability isn't apartheid or colonialism. It's industrialism.

The whole point of the tracking is so that

**groups**of students... hopefully in large enough batches... can all be taught at the same time. If you throw

**everybody**in the same class, it's even more industrial, but you know you're going to be losing both the top and the bottom students, at the very least. One set because they're bored, and the other because they can never catch up.

Ideally, paths through math would be individualized, because math really requires understanding certain concepts before moving onto other concepts. Some people "get" certain things faster than other things, and this doesn't even have to do with IQ per se. I had some difficulties with a few particular math concepts, and it took me longer to think through those -- and I needed to understand these before subsequent topics could be understood. I took certain topics multiple times - a few by choice (most not by choice) - and in some cases the repetition helped me.

This is why I like the Kumon approach - you can't get farther in their curriculum before you learn earlier topics. And they don't say you've mastered it until you can solve a high enough %age of the problems from one topic quickly enough. There's more flexibility at later levels, but early on, laying that firm foundation is key. You don't get to skip anything.

Now, I get that if one does this tracking, you're very likely to see certain social inequalities play out in the groupings... because it is reflecting real differences. Some kids really know more math than others, and some really don't need as much help to progress, or progress faster, and some are way behind the average levels. This is something that needs to be dealt with. Tracking is not ideal because it only roughly fits with achievement levels, and also, it's often subpar because the same amount of instructional effort is expended on all groups... and that's probably not a good idea, either. I have no problem with more instructional time given to those who need a lot more help in trying to get to average levels. But throwing everybody in the same class, with the same industrialized teaching methods, is not going to work.

Some kids come from backgrounds of higher resources, and from families that are strict on schooling and make sure the students progress. You can try to hold back the highest-achieving ones (and for those, with parents who don't have knowledge or resources, may be effectively held back -- but those like me, with loads of resources at home, cannot be effectively thwarted.) You can try to approach everybody at the lowest-denominator level and end up with average students also not progressing very much. The way to help improve everybody's lot is not to throw the low-level students in a group they can't keep up with, but to provide more intensive help for those who need that, and guidance for those who may not need as much direct teaching.

For what it's worth, while this other teacher talks about dealing with low-level students forgetting a bunch of math stuff taught only weeks before, I had to deal with

**college calculus students**who claimed never to have learned the formula for the area of a circle. Or the equation of a line. These weren't low-level students. (They weren't high-level students, either -- the clue was that many of these were students who had calculus senior year in high school, but were unable to place out of freshman calculus. If you really learned calc, it wasn't difficult to place out of freshman calc, and many did.)

This is the problem with the industrial approach -- mastery is never really tested. If you do "okay enough", then you're moved along with the mass, and many people never really "got" geometry or cartesian coordinates or anything that should be basic if you're taking calculus.

The "benefits" of the industrial model is moving huge lumps of people through at the same time, but the issue is that some get left farther and farther behind, while those who could get ahead, but don't have the resources, are held back at average levels.

Anyway, the paper is here: Understanding Issues Associated with Tracking Students in Mathematics Education. Actually reading through recommendations in the paper, just ignoring the colonialism crap, one finds some actually good proposals that could be made practical.

The specific proposals from the paper are:

Collaborative Environments

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Because of this, it is crucial to begin creating educational environments where

students can share their thoughts, needs, and concerns about what they study.

To do this, schools must rethink top-down practices that quash students’ and

teachers’ voices by providing space for democratic communities of practice to

emerge.

In many schools, top-down practices have been implemented as one-size-fits-all

solutions for controlling others. These practices aim to “produce

desirable student behavior” and “maintain procedures, routines, rules, and

standards” (Casey, Lozenski, & McManimon, 2013). A potential problem with

this is that students’ voices are homogenized. Rarely, if at all, are students’

interests taken into consideration in the current model of mathematics

curriculum.

This won't work. Why? Because math builds on itself. You want to learn the math behind, say, sports? Well, I hope you started learning arithmetic years ago, and understand fractions, and then all sorts of other things so you can understand things like percentages or parabolas. So, do we use a time machine to get to when you're in 10th grade to ask what you'd be interested in, so we can build up from first grade?

Now, if they mean, allow students to choose the math courses to take - maybe harder or less hard than test scores would recommend - I could see allowing that to an extent. But that's not what the author means:

Traditional disciplines, like mathematics, tend to be sequential,

hierarchical, and impermeable. One alternative is for schools to open their

mathematics curricula to become more interdisciplinary, providing access for

more students (Bernstein, 1977).

This author is poorly choosing to critique math for being sequential... BECAUSE IT IS. You can look at it as a web, as Khan Academy has (and it is, to a certain extent). But you need to learn certain more fundamental concepts before moving on to higher concepts. Now, to be sure, I did stuff like read Martin Gardner and Raymond Smullyan in elementary school - there's all sorts of "fun" stuff like that one can do. But I was concurrently learning long division and other stuff I needed to understand once I got to algebra, etc.

Additionally, many students struggle with mathematics at one point or another

in their schooling, and as a result, struggling students are often tracked into

lower-level courses. However, if students begin to take an interest in

mathematics (or any subject for that matter), they remain stuck in the track in

which they were initially placed. In order to move out of lower tracks, students

must take on the monumental task of concurrently enrolling in multiple courses

during their secondary schooling, otherwise they remain trapped.

Taking into consideration the limitations of being tracked into less

rigorous courses, many students are sometimes hesitant to pursue their interests

due to negative stigma associated with vocational, home, and non-scholarly

pursuits (Noddings, 2013a). “By making technical/vocational education a

positive choice, the intake of students could change, as students’ curriculum

choices would be based on interest rather than on failure, resulting in a more

heterogeneous student composition in the different tracks in terms of cognitive

capacity and social background”

I do agree with this. I think tracks should be "escapable". And there are other math things, at the high school level, that can be done. The strictly sequential approach is definitely for elementary education, but the "web" approach can begin once the basic arithmetic is mastered. I think personal finance and/or accounting would be a fabulous high school class. Even the "calculus track" students could get something from it.

Here's another:

Integrating Multiple Perspectives in Mathematics Curriculum

If schools are willing to begin rethinking their current curricular models

of mathematics, there becomes room for multiple perspectives to exist in a

traditionally Eurocentric discipline. Classrooms formed around students’

interests, rather than homogeneous ability groupings, allow for more

heterogeneity in classrooms. The diversity in student’s interests can allow

students opportunities to explore areas of mathematics not traditionally taught

in schools.

.....

Rethinking mathematical labels. Mathematics curricula in the United

States predominantly follow a sequence of courses that builds hierarchically

over time. As students climb the proverbial mathematical ladder, each course

provides a foundation for the next in the sequence. As noted, there exist

problems with hierarchical curricula.

.....

When removing names of mathematics courses, the stigma associated

the class labels becomes less important. In this situation, there is no longer a

necessity for labels such as honors, regular, or advanced. Also keeping in mind

that Math 1 is not the same as Algebra 1, students are able to explore

nontraditional fields in mathematics, which can help cultivate intellectual

curiosity.

Some of this seems to be about "marketing" the classes. I can think of loads of math classes for high school... but it requires to have a huge student population if you want any decent number of students per class. Look, there is a value in having some classes that really are more difficult and faster-paced than others. Some that are practical. Some that are a bit slower, for students who either have more trouble or students who don't want to work as hard at math. Changing the names won't do much, because the students will figure it out.

Historicizing mathematics. Placing mathematics in the context of

history is another possibility for rethinking mathematics curriculum. Rather

than relying on traditional sequences of secondary mathematics that begin with

Algebra 1 and culminate with higher-tracked students enrolling in calculus or

statistics, teaching mathematics through the context of history is one way to

eliminate this hierarchy. While still linear in nature, students are simultaneously

afforded opportunities to learn concepts found in algebra, geometry, statistics,

and calculus. Rather than teaching these subjects as separate courses, students

are able to adopt the role mathematicians play as they uncover mathematical

discoveries found throughout history. For instance, imagine a course where

students work through historical texts like Euclid’s Elements. In a scenario like

this, students learn fundamental axioms associated with high school geometry

curriculum, while also working through interesting dilemmas faced by early

geometers.

I actually think this is a fabulous idea. It need not be Euclid's Elements - I took a class in the history of ancient and medieval math and science in college, and I had a great time (it counted as a history credit for me...because it was more about history than math or science.)

But where it goes off the rails is here:

Ethnomathematics is

essentially mathematics that is “practiced among identifiable cultural groups,

such as national-tribal societies, labor groups, children of a certain age bracket,

professional classes, and so on” (D’Ambrosio, 1985, p. 45). While any form of

mathematics produced can technically be deemed “ethno-mathematical,” a

primary goal of ethnomathematics is to provide alternatives to Eurocentric

mathematical thinking (Borba, 1990).

As fun as it was to learn Babylonian approaches to arithmetic [and thus why there are 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle], it did not much inform me on practical matters as statistics and other useful tools. Ethnomathematics approaches fall into social studies - not math.

There is a reason that "Eurocentric" mathematical thinking - which encompasses much thinking from Arabic, Indian, and Chinese mathematicians historically (are they Europeans? Maybe in current "diversity" metrics they count as Europeans...though all three areas had European colonialism imposed on them more recently) - is privileged. It's because it's relevant to all sorts of useful things, and also "fun" stuff, like keeping sports statistics or looking at popular polling results. You don't need to be European to calculate an average, a rate of change, a standard deviation, etc. These are very useful concepts for figuring about the world. You need not speak a specific language to be able to do any of this stuff.

My main skepticism is that I don't think most K-12 teachers can effectively teach useful "ethnomathematics". And also, because the attitude that, say, trigonometry is "eurocentric" is barring the targeted students from many careers, such as engineering or even accounting. You want to stick the kids into intellectual ghettos? Talk about colonialism.

Speaking of which:

Teaching mathematics for social justice. Just as incorporating diverse

perspectives into mathematics is imperative to rethinking mathematics

curriculum, so is presenting curriculum that is culturally relevant. Building

upon the notion of providing choice in students’ schooling, teaching

mathematics for social justice gives students opportunities to problematize

challenges present in the world today (McGee & Hostetler, 2014).

Implementing culturally relevant pedagogies offers alternatives to traditional

curricular models while also providing opportunities to contextualize learning

and “develop a less mystified view of mathematics” (Brelias, 2015, p. 9).

Teaching mathematics for social justice requires teachers to be up-to-date on

current issues facing their students’ communities. This form of teaching

mathematics is difficult to retrofit into the current hierarchical structure of

mathematical curriculum.

You know what is really difficult? Teaching students relevant math when you barely know math yourself. It's far easier to blather about "social justice".

You want your students to effectively use math to make cogent political points? YOU NEED TO ACTUALLY TEACH THEM THE MATH.

If you have not laid down a good quantitative sense in arithmetic earlier, you're going to have a hell of a time teaching the students about interpreting percentages or percentiles. And that's the simple stuff. Want to explain calculating correlations? That should be fun.

As in my first post in the series, one can address the math that appears in the news media and politicians pontificating. I think that could be useful. But it would be far more useful to teach the students about the math of money at a level relevant to them. It really only requires some basic arithmetic, plus some exponents and logarithms. Some of these high-flown ideas are more likely to end with math-weak teachers asking their students to write essays about how they feel about something only tangent to math, and have very little math content at all.

Giving a non-math class to disadvantaged students, while telling these students it's actually a math class, is doing them a great disservice. Either be honest and explain it's not actually math -- or teach them math instead of the gooey stuff you'd prefer to teach.