With the November 7 election concluded and new school board members in place, it seems like a good time to share my thoughts on math education in PPS as a parent of two children who have recently completed K-12 in District schools. During 2014-2016, I undertook an investigation—including ten Open Public Records Act (OPRA) requests and numerous letters to public and school officials—that aimed to uncover the demographic details of the accelerated math program. In addition, having lived in Japan and my children being Japanese citizens, I have a deep understanding of the country’s educational system and cultural norms as a matter of comparison. There have been some positive changes to the math program since I was most familiar with it, but to my eyes, the same structural issues and results remain.

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How Math Became an Object of the Culture Wars, by Jay Caspian Kang, 11/15/2022In some ways, the cyclical nature of these arguments makes sense: each generation of parents believes their children are facing problems the world has never seen before. As was true in the nineteen-nineties, today’s fights about math are not entirely about what kids actually learn in their classrooms, or how well those lessons prepare the U.S. for competition against a geopolitical enemy.“

## PART I

Part II: International Perspectives >>

Discussing curriculum reform becomes more productive when we first identify the specific goals in question. Are we prioritizing pedagogy, college admissions, future earnings, or some other objective? For example, if we are talking about the need for students to acquire foundational skills, it is likely that the more time spent on fundamental math topics the better. On the other hand, proceeding too slowly over the years may not prepare students for advanced study, nor give them the challenges needed for their personal and cognitive development. There are many different goals that students and family members have for math study. Below is a list (with clickable details) that attempts to unpack all of these reasons. Which of these reasons for studying mathematics is most important to you?

## Foundational Skills

Math is fundamental to understanding the world and making logical decisions. Many real-world problems, even those not explicitly “mathematical” in nature, require mathematical thinking. Having a foundational understanding of math equips individuals with problem-solving skills and a structured way of thinking.

## Personal Development

Engaging with math helps students discover their strengths, interests, and preferences. It also teaches perseverance and resilience, as students often face challenges and setbacks in their studies.

## Academic Credentialing

Proficiency in mathematics is a standard requirement for graduation in many educational systems and is used as an indicator of general academic competence. Many colleges and universities also consider math performance during admissions, especially for specific programs or majors.

## Preparation for Advanced Study

Exposure to and understanding of mathematics at a young age can better prepare students for higher-level math and science courses in college. It can influence their decision to pursue a STEM major or career, especially if they feel confident in their mathematical abilities.

## Career Advancement

In many professions, especially in the STEM fields, a strong math foundation is essential. Moreover, even outside of explicitly mathematical jobs, the analytical and problem-solving skills honed by studying math can be beneficial. Additionally, individuals with a background in math often command higher salaries in the job market.

## Cognitive Development

Engaging with mathematical problems can enhance memory, attention, and other cognitive functions. The abstract thinking required in math can also promote creativity.

## Life Skills

Basic mathematical literacy is essential for many everyday tasks, from budgeting and cooking to understanding interest rates on loans or mortgages.

## Cultural and Historical Awareness

Mathematics has a rich history that’s intertwined with the progress of civilizations. Studying math can provide insight into human achievements and the evolution of thought.

## Promotion of Logical Thinking

Mathematics is rooted in logic. Regular engagement with math problems can enhance an individual’s logical reasoning skills, which are beneficial in various decision-making scenarios.

## Interdisciplinary Connections

Mathematics is not just an isolated subject. It connects with many other disciplines, from physics and biology to art and music. A solid understanding of math can thus enrich one’s appreciation and understanding of other subjects.

## Test Preparedness

Studying math not only enhances a student’s mathematical proficiency but also equips them with the test-taking skills and strategies needed to excel in standardized exams. Achieving high scores on standardized tests may offer benefits in college admissions, scholarships and financial aid, job applications, self-confidence, and educational placement.

### The Family’s Role in Education

In addition to “regular” math, parents (and other caregivers) have diverse motives for encouraging their children to study advanced or accelerated math. One immigrant family might wish for their child to obtain top-tier educational credentials in this country for reasons of social mobility, while another might be looking more for value in their child’s educational and scholarship opportunities.

Educators should appreciate that all parents have gifts of knowing to offer their children. We raise children with the understanding they will bear some resemblance to us, with the desire they should take the best things we have to offer and leave behind the worse. One family might have a history of service in the military, another in professional sports, while yet another might draw inspiration from a distinguished family legacy in academia, and so on. No parent should be dismissed for exercising this right, even as children routinely reject these gifts with brutal indifference and even sometimes outright hostility, for that is our lot.

For instance, imagine a parent attended college at IIT Bombay. If there’s *one thing* that parent knows how to do, it’s how to prepare for an engineering exam. Even though American universities care less about scores on standardized tests than those from most foreign countries, it would be a kind of insult to interfere with such a parent passing on the skill to his child which, in all likelihood, was a major contributor to their career success and reason for living here. Similarly, it would be sensible for school officials to take into account that a parent and Dean of Engineering at a top-ranked university might know something about learning mathematics and providing support for her own children in the process.

Educators and school administrators who make placement and other educational decisions for the students in their care typically know little about these family backgrounds, and even if they do, may not know how to execute educational plans that respect and leverage these aspects of their students’ lives.

In addition to the reasons for studying math above, let’s unpack the underlying motives parents and family members might have for wanting their children to pursue *advanced or accelerated* math courses. For me, two stand out (#4 and #6), but consider your own motives: which of these resonate most with you?

## Aspiration for Social Mobility

**Economic and Social Prosperity**: Parents may hope that excelling in math will lead to high-paying jobs and financial security for their children, as well as facilitate fruitful relationships with well-placed peers.**Status and Prestige**: Success in math is often associated with intelligence and capability, leading parents to aspire for their children to excel in these subjects to attain a certain status or prestige.

## Desire for Educational Opportunities

**Access to Top Colleges**: Parents may believe that excelling in math, particularly in accelerated courses, will enhance their child’s chances of gaining admission to prestigious colleges and universities.**Scholarship Opportunities**: Performing well in math could increase the likelihood of receiving scholarships, alleviating the financial burden of education.

## Concern for Future Readiness

**Job Market Competitiveness**: With the growing importance of STEM fields, parents might see strong math skills as a necessity for staying competitive in the job market.**Adaptability**: Parents may view math as a subject that promotes problem-solving skills and adaptability, preparing their children for an uncertain future.

## Personal Values and Beliefs

**Value of Education**: Parents who highly value education may see math as an essential part of a well-rounded education.**Belief in Hard Work and Discipline**: Parents might view the challenge of accelerated math courses as a way to instill values of hard work and discipline.

## Fears and Anxieties

**Fear of Being Left Behind**: In a highly competitive educational environment, parents might worry that their child will be left behind if they do not excel in key subjects like math.**Anxiety About Future Uncertainties**: Parents might see excelling in math as a way to mitigate future uncertainties and ensure their child has ample opportunities.

## Legacy and Expectations

**Family Tradition**: In some families, there might be a legacy of excelling in math or pursuing careers in STEM fields.**Parental Expectations**: Parents may project their own expectations and aspirations onto their children, hoping they will follow a certain educational or career path.

## Community and Peer Influences

**Social Circles**: The values and expectations of a parent’s social circle can significantly influence their motivations.**Cultural Expectations**: In some cultures, there is a strong emphasis on excelling in subjects like math.

### Tracking at PPS

The practice of “tracking” refers to grouping students based on their academic abilities or achievement levels into different classes or educational programs. Princeton’s math tracking program is relatively rigid, meaning it excludes students and parents from the placement decision process pretty much entirely, and begins a pattern of courses which extends from one year to the next. Parents can lobby on their own to change placement decisions, but there’s no formal process to handle such requests (unless that’s changed in recent years). Many public schools with tracking or accelerated classes allow parents to opt their child in to such classes with a minimum of fuss, and most districts do not actively discourage students from taking advanced classes the way Princeton does.

The aspect of tracking most misunderstood by parents is in believing that such curriculum pathways exist for the needs of individual children. In fact, tracking exists to help schools allocate scarce educational resources (such as teachers and classroom space) and make teaching more manageable by allowing teachers to tailor their instruction to a more homogeneous group of students. If, on the other hand, requiring all students to study the same material were viewed as advantageous by teachers and administrators, they would have ended the practice long ago.

If one is an administrator, how can students (and parents) be persuaded to embrace instruction by a single teacher in a classroom where every seat is taken by 28+ pupils, when other classes exist with 20-25 students plus a teacher’s assistant? Call it an *accelerated* section. How does one discourage parents from lobbying for their child to attend classes which have only 8-12 students and multiple teachers? Label it *special education*. This is oversimplifying of course, but it helps to understand how these decisions about scarce resources are made and presented to students and parents.

### Major Money in STEM

This is not to dismiss the value of advanced or specialized instruction. There are indeed big benefits to students who have taken accelerated math classes, one of the most important being the greater likelihood of pursuing a postsecondary STEM major (Bottia 2015, Dalton 2007, Schneider 2013) and ending up in higher-paying jobs with greater financial security. The difference in earnings is stark, as detailed in The Economic Value of College Majors (Georgetown U, 2015).

The report found that median annual wages of college-educated STEM majors were decisively higher than all other groups (p. 9), with Health and Business majors a distant second. Furthermore, the STEM workers’ wages had the largest growth (50%) over the duration of one’s career compared to other types of majors, such as teaching (which grew 28%). The cumulative effect of higher wages along with compounding on wealth over a lifetime is enormous. Without even considering lifetime assets, the report writes:

In some sense, deciding what to major in is more important than deciding whether to attend college. Over a lifetime, the average difference between a high school and college graduate’s wages is $1 million, but the difference between the lowest-and the highest-paying majors is $3.4 million.

The high average STEM income numbers reveal that *if* the primary concern of your child is future earnings, it would be more advantageous to attend a lower-prestige college and major in STEM, than attend a higher-prestige college and major in humanities. Accordingly, we must ask ourselves what makes a young person decide to pursue a particular major area of study. Foundational skills obviously play a role, but in the end, having confidence in one’s ability and a sense of belonging in the field are usually decisive (Zhao 2022, Wang 2012, Good 2012, Zeldin 2008). A useful way to think about math curriculum and advanced math options, then, is whether it helps STEM-curious students build enough interest, skill, and confidence to pursue that type of major in college; not whether the curriculum helps a small cohort of students with little interest in STEM gain access to top-tier schools.

In any case, given these financial effects, it is hard to fault parents who want their children to have full access to math educational opportunities, experience excellence, and gain a sense of confidence before heading off to college.

But let us also consider those students who have been systematically denied these opportunities.

#### The Disparate Impacts of Tracking

Since 1948, when Princeton schools became fully integrated (and earlier), African American students have been consistently lacking from advanced and accelerated courses, and later, Hispanic students (the term used in government and school records). This has been noted in numerous press reports over decades, and according to my calculations based on a 2015 presentation (“JW Enrollment Patterns“, p.5 top), about 96% of the total Black and Hispanic student population at the Middle School were excluded from such classes, resulting in only 5 Blacks and 3 Hispanics having been enrolled in an accelerated section over the prior three academic years. In another report (“Math Achievement – 9th Grade” for 2013-2014), there appear to have been 2 Black and 1 Hispanic students (out of 54 total) enrolled in an accelerated section, none of whom were girls, and even those three did not appear to have come from District schools (presumably Princeton Charter or Cranbury). This is a travesty and was completely whitewashed by the Board at the time.

It is disheartening to hear members of the school board, administration, and voices in the community essentially calling to maintain the status quo with respect to our district’s tracking policy and math curriculum. They mean well, but these calls are inevitably accompanied by superficial proposals to improve the placement process, as if a little tweaking could solve a systemic problem which has existed for 75+ years. To effectively *de*track, or at least modify our curriculum for the betterment of all students, we must first address the needs of teachers and administrators, since it is primarily for their needs that tracking exists. Instead of asking them to handle ever more students and greater workloads, we should ask them what it would take to give up tracking, and then improve their situations so they can provide more personalized instruction and undermine tracking’s most pernicious effects.

By my reckoning not only should we eliminate rigid tracking, we should strive to make reparations to our African American alumni and community at large. Assuming the population of Princeton averaged 24,000 inhabitants with a 10% African American component that created 20-30 new students annually, one might estimate 1500 graduates over the last 75 years were denied opportunities for advanced study and career advancement in more lucrative fields. These numbers are obviously approximate, but grounded in historical documents and research, and readers are very much encouraged to make their own estimates.

Based on the lifetime earnings gap mentioned earlier, between the highest and lowest paying college majors ($3.4 million) as well as between high school and college graduates ($1 million), a figure of $1 million would be a conservative estimate for the potential lifetime earnings deficit of each affected individual. We can use this figure to come up with a cumulative estimate of the damage to our African American community that occurred as a result of tracking and similar educational policies that kept them out of college and more lucrative fields. Naturally, not all students would have pursued these avenues, but if there was a negative impact on only *one third* of African American students, or 500 affected individuals, we are still talking about a cumulative sum equal to $500 million. If we consider treatment of Princeton’s more recent Hispanic community, as well as the period before 1948, the amount could be well over $1 billion in our town alone. When we talk about the high costs of improving our schools and education, it would be helpful to put those costs into this context.

#### The Positive Impact of Small Classes

In a comparison of private high schools in the area with ours, I identified three concrete advantages that made them objectively better in my opinion, in addition to softer attributes like reputation and quality. Besides mandatory sports participation and all-day programming, they offered seminar-style teaching, which basically meant class sizes were between 10-15 students and conducted while sitting around a table (the “Harkness” method), on a campus with plenty of teaching space. With such conditions in place, there is much less need to maintain homogeneity in instruction, and plenty of room in the system to move classes and students around as needed. While private schools still provide differentiated instruction, they do not practice tracking as we know it. Our public district may not be able to get class sizes that small, but 15-20 students in core classes does not seem out of the question (if classroom space for it existed). This change would allow teaching to be personalized and tracking eliminated, in addition to numerous other benefits. This idea is examined more closely in conclusion, but first we must discuss the unique characteristics of the American educational system and its workplaces…

Quite interesting, like the research

Have you considered advocating for high-dosage tutoring? This is a research-based method that has proven effective and disproportionately benefits historically underserved groups. It is expensive, but much less expensive than reducing class sizes. And much easier for teachers to implement than differentiation in a heterogenous classroom.

In contrast to high-dosage tutoring, the research surrounding math detracking is, at best, mixed and not very reproducible. For example, some of the most high-profile success stories (e.g., SFUSD) were later debunked when the data was made public. Despite the best of intentions, the detracking of the 6th grade math at PPS has led to *worse* learning outcomes (e.g., Algebra I proficiency rates) and inequity for Black students, Hispanic students, and students from lower socioeconomic backgrounds.

Depends on goals. I suspect that extra tutoring as part of a school program would be counterproductive in terms of encouraging students to pursue STEM in college (that it would be viewed as a chore, and proof they’re not good at STEM). What

wouldhelp is clear open communication to parents that many students in elementary school (and upwards) receive private tutoring and if they want their child to keep up, they should provide that as well. And help pay for it. Or ban private tutoring, as is done at Princeton University.Since ‘detracking’ is not a single coherent prescription, and success can be defined in multiple ways, it is hard to generalize about the success of such initiatives. I agree with Molly Kurtz, “Devil Is in the Details When It Comes to Tracking, Detracking“. In any case, I don’t think any conclusions can be reached on instruction and outcomes that occurred during pandemic lockdowns.

Kurtz appears to be a proponent of supplemental instruction provided at the school (such as high-dosage/impact tutoring, double-dose track, or “plus” classes). It’s the same basic principle as private tutoring, but provided in a more equitable and transparent manner. The research suggests that these interventions lead to higher Algebra I proficiency rates and higher participation rates in advanced math courses. Are you aware of any studies that indicate that such interventions are counterproductive toward students pursuing STEM in college?

You and Leo inspired me to look for research that studies any link between tracking and college STEM majors, but I keep coming up short. I bet extra tutoring doesn’t hurt, especially if it’s done in the right way, but mostly I think teachers are overworked.

Keith – thanks for your detailed, thorough, and thoughtful writeup. I had a few quick comments and questions:

1. As I understand it, to a first approximation there is a single math detracking proposal in US K-12 education circles, which can be summarized as “algebra I for everyone in 9th grade” (8th grade in some places). Course pathways that bring some students to algebra I one or two years earlier are delegitimized and removed; there is some funding for teacher professional development to support in-class differentiation but the basic economics of class size and teacher pay are left untouched.

Is it correct to say that the thesis here is that tracking causes vast harm (it creates an “underclass”) and that this harm can be completely removed at a reasonable cost by shifting the “higher-track” students to private schools, other districts, and “regular-track” classes? To be concrete, historically something like 35-40% of PPS students have been put on a track to achieve strong proficiency in algebra I by 7th grade; if this population was whisked to another school district, would the remaining 60-65% of students then perform significantly better in school and achieve significantly better life outcomes, with the exact same K-12 course pathways they take today? Could you provide a causal analysis of why this might be the case, as well as evidence of a school district that has achieved the desired outcome through detracking?

2. There seem to be 3 key facts: (a) tracking occurs in middle school (b) K-12 math ed aspires to culminate in calculus (c) there are troubling racial disparities in the outcomes of racial groups. But treating (a) and (b) as causes of (c) appears to be a matter of personal faith, not any sort of analysis. Why would emphasizing statistics, computing, and linear algebra instead of calculus lead to better racial outcomes? The intuition seems to be that calculus feels like an arbitrary and irrelevant gatekeeper course from the Cold War, so it’s plausible that it’s used mainly as a way to keep undesirable minorities out. But this view is rooted in ignorance, not insight: to take just a single cutting-edge example, calculus plays a critical role in the development of the AI algorithms behind ChatGPT (and in data science and machine learning more broadly). It makes me angry that JustEquations says that the rise of data science as a profession is a reason to de-emphasize calculus, when the exact opposite is true; I guess that shunting minority and low-income students into dead-end math tracks that put data science careers further out of reach is a small price to pay to achieve their grand vision of equity in education.

3. You write: “There is a persistent belief among many that differences in math ability are genetic and should therefore be managed through tracking or independent pathways.” This is uncharitable: math placement is important because preparation and ability are important for success in math courses, and there is considerable variation between students. The fact that (for example) some students are able to achieve algebra I proficiency in 7th grade while others struggle to do it by 9th grade remains a fact independent of one’s belief in the importance of genetics. And this lack of charity can be turned around: as a society we have a mechanism for producing scientists, engineers, mathematicians; if this pipeline isn’t working for some groups, surely the answer isn’t to reduce the amount of math being taught in schools (while banning private tutoring)… unless we’ve given up on the problem out of fatalism, perhaps because we mistakenly think some differences are genetic and unchangeable, and would prefer to focus on the easier problem of making differences harder to detect and measure.

Thanks for reading it through, Leo. I think you appreciate the difficulty of limiting oneself to only a few paragraphs on this topic.

1. I am not so concerned with what other school districts are doing, or whether Princeton is meeting Common Core / PARCC standards. We have the advantage of an academic reputation and culture that should enable us to focus on meeting our own standards, which, in my view, certainly includes offering Algebra I before high school. That said, Algebra I in 9th grade was pretty standard in the U.S. before 1990, when only 16% of students took it in 8th grade, yet many Americans still managed to have pretty distinguished careers in math and engineering.

The key to detracking is to eliminate it from the beginning of a student’s trajectory. The scenario you’ve described has them being tracked first, then separated. But, if we follow this thought experiment, I suppose in our district the remaining 65% would be yet again subjected to tracking in their further studies, and experience the same problems as before!

It may well exist, but I cannot find research that looks at whether detracked kids were more/less likely to major in STEM, which in my opinion is a more meaningful objective than simply doing well on math tests or taking AP courses. When I asked ChatGPT to name some citations along these lines, it made up false studies, LOL.

2. There can be no doubt that tracking at PPS has a disparate impact since the numbers have been overwhelmingly one-sided from the moment it starts. Regarding life outcomes, there’s considerable research (some of which I cited above in ‘Major Money in STEM’) indicating a strong positive correlation between taking accelerated math and pursuing a STEM major. I suspect that allowing students to align their math study with their interests in senior year will lead to greater confidence in pursuing STEM in college, albeit not necessarily the ‘hard’ engineering fields, and make them appear less generic in the college admissions process. They will probably end up taking Calculus at that time, anyway.

3. My public school ideal would be for everyone to take the same base course in a grade, say Algebra I, but in small, differentiated class groups where even the students aren’t sure where they are on the ability continuum. And then each year, or semester, the students would start fresh, and not be locked out of any courses that their more capable classmates would have access to. This would be based on the premise that teachers could provide additional learning challenges and enrichment to their best-performing students while staying on the same base syllabus as everyone else. As students get up into high school, I could see there being more differentiation depending on their (family’s) college plans.

Can you elaborate on your proposal for detracking the middle school math curriculum? Approximately 40% of PMS student take Algebra I in 7th grade, while approximately 55% take Algebra I in 8th grade. Is the proposal to have (essentially) all PMS students take Algebra I in 8th grade? Or in 7th grade? Or something else?

My dream math curriculum will never happen, so my hope is that the teachers themselves would play a major role in deciding the sequence, in return for agreeing to teach heterogenous (and hopefully smaller) classes. Personally, I think Algebra I in 8th grade (for all) would be preferable and leave enough space to take Geometry, Algebra II, Pre-Calc, and Calculus/etc.

You seem particularly interested in how detracking affects student motivation (rather than achievement). There is research on this as well, but, again, it’s mixed. There are two competing hypotheses. The “labeling hypothesis” argues that mixed-ability classrooms foster the academic self-concept of students with low academic achievement because they lose their negative track branding. The “contrast hypothesis” presents the opposing view that mixed-ability classrooms expose students with low academic achievement to higher achieving peers, thus harming their self-concept due to social comparisons.

https://www.sciencedirect.com/science/article/pii/S0959475223000221

That is quite an interesting set of studies. I have a couple issues with the conclusions. The data looks very comprehensive but they only looked at students in Austria and Germany, which is very far indeed from the American experience, particularly with respect to African Americans and Central American immigrants. Broadly, I don’t think there’s enough consistency to detracking implementations that one could draw conclusions even based on different communities in the U.S. Also, the study does not seem to consider the possibility that the quality of teaching by the German and Austrian teachers may have diminished after they were forced to manage heterogenous classrooms, thus impacting students’ notions of self-efficacy. My hunch is that 85% of teaching success is due to the engagement and empowerment of the teachers themselves, rather than the use of any particular teaching paradigm. It would also not surprise me if one of the results of more heterogenous classes was that some students, particularly in disadvantaged groups, have the realization that they’re not actually very good at the subject. That’s ok, part of the journey. But if they didn’t have that shared experience, they might never be sure if it was their individual qualities, or the system, which had made them less successful. It would have been interesting to see the self-concept scores for the language classes that they took. Maybe the math-challenged immigrants had higher self-concept scores in that area. In the end, though, I think most people would agree that the more personalized the instruction is, the better the result is likely to be, which is why I would love to see small class sizes. Then the whole tracking/detracking debate becomes less relevant.