A few months ago, a Canadian math teacher tweeted this picture stating “I find this illustration of how 2+2=5 to be particularly elegant.” His statement is neither elegant nor coherent.
The teacher, whom I am not going to identify, is exemplifying a growing trend in education. In an effort to achieve equity in student math performance across different racial groups, educational activists are lobbying to inject relativism into math instruction.
In this example, the idea is that an individual’s unique vantage point allows there to be multiple interpretations of a math equation. This is similar to anthropologists’ assumption of cultural relativism, which states that it is not appropriate to normatively judge a particular culture through the lens of another culture, when studying a particular cultural group. This does not translate to mathematics. For example, it is considered rude in Chinese culture to make prolonged, direct eye contact. In America, it’s often considered rude to NOT make prolonged direct eye contact. I am perfectly comfortable stating it is not appropriate to normatively assess China’s practice through my American cultural lens. However, adding two to two will always yield four. This is universal across all cultures.
If the teacher’s remark appears clever to you, I suggest you treat yourself to a refresher of basic arithmetic. It’s only possible to arrive at 2+2=5 by circumventing the rules of algebra. There is tremendous pedagogical disutility in sharing this with students. The teacher is using a cute trick employed by many white-collar criminals (“cooking the books” to exaggerate profits or underreport losses) to attempt to refute a mathematical identity. He conflates dubious dimensional analysis with the rules of algebra.
For example, if I, as a child, saved four quarters and then told my dad “I saved four quarters PLUS one dollar,” that would be as coherent a mathematical statement as saying that 2 small boxes + 2 small boxes = 4 small boxes + 1 large box = 5 boxes. The scaling and denomination of our variables matter.
Four quarters, ten dimes, twenty nickels and one hundred pennies are all different ways to refer to one dollar (or one hundred cents). I can say that 2 quarters + 5 dimes = 1 dollar. I cannot say that 2+5=1 or 2+5=2+5+1. The latter variant, which double counts, is what the teacher utilized, but both exploit dimensional confusions.
The correct (and hopefully obvious) way to do this, as is often done in chemistry, is to take the defined ratio of large boxes to small boxes (i.e., the definition of a large box in terms of small boxes) and change the denomination of the quantity.
Businesses who “practice” algebra as the teacher does tend to get accused of fraud and/or misleading the shareholders. To call this mathematical incoherence “elegant,” as the math teacher does, is like calling Bernie Madoff or Enron elegant for “creatively” using fraudulent accounting practices to camouflage businesses in crisis. It is deception in service of a larger goal; in Madoff and Enron’s case it was profit, and in the math teacher’s case it is inclusion and equity. The latter is undoubtedly a more noble end, but both means entail misleading smoke and mirrors
Math is racist?
Activist educators, which constitute a supermajority of all educators, claim that contemporary mathematical education is both a symptom and driver of white supremacy. α Some states, such as California, are removing accelerated math programs because activists claim these programs perpetuate socioeconomic inequities.
One of the primary gripes that these activists have against mathematics is the objective truth of mathematical axioms (2+2 always equals 4); they believe that the objectivity of mathematics (and objectivity anywhere else) is a weapon to preserve inequality and perpetuate White privilege. Hypotheses like these, which point to non-concrete abstractions as concrete catalysts, cannot be tested and therefore cannot be falsified.
Instead of asking “How do we raise math test scores for minority students?,” they ask “How do we dismantle the notion of objective measures of mathematical mastery (which are used to sort minority children into underperforming groups)?” It is the equivalent of banning weight scales to deal with the problem of obesity.
The Wrong solutions to a real and urgent problem
American students do not perform well on standardized math tests. This is particularly true for Black and Latino students. It’s a problem with profound future consequences. Unfortunately, the proposed solutions to this problem are more harmful than doing nothing, and they disproportionately harm the very racial groups they’re supposed to help.
Ibram X. Kendi, Director of the Center for Antiracist Research at Boston University and author of the mega bestseller “How to be an Antiracist” has suggested that, rather than grades based on students’ performance on homework and exams, we should evaluate students based on their “desire to know.” β Kendi does not offer a coherent method for measuring this intangible quality.
In May, the Bill and Melinda Gates Foundation sponsored a toolkit called A Pathway to Equitable Math Instruction, which seeks to “support equitable access to math standards for Black, Latinx, and multilingual students in grades 6 – 8.” γ The toolkit is divided into five different sections called “strides”, which are shown below.
On page 6 of the document, the authors lay out the infamous characteristics of “white supremacy,” which come from a 2001 article by Okun and Jones. δ This list is similar to the museum exhibition of “Whiteness & White Culture” that caused controversy at the Smithsonian National Museum of African American History and Culture last year. ε Slides from the exhibit are shown below.
Both lists are drawn from the same domain of academic literature that has been pejoratively labeled as “grievance studies” by critics. In addition to English, History, and some other humanities fields, this literature is produced by academic departments that have “studies” in their title. Examples include women’s studies, gender studies, African American studies, Whiteness studies (Robin DiAngelo), etc.
The given characteristics of white supremacy are: Perfectionism, Sense of Urgency, Defensiveness, Quantity Over Quality, Worship of the Written Word, Paternalism, Either/Or Thinking, Power Hoarding, Fear of Open Conflict, Individualism, Only One Right Way, Progress is Bigger/More, Objectivity, and Right to Comfort.
Evidently, objectivity is a tool of white supremacy
There is SO much to object to in Jones and Okun’s list, but I’ll only critique the claim that objectivity is a feature of white supremacy in mathematics. A Pathway to Equitable Math Instruction states:
“The concept of mathematics being purely objective is unequivocally false, and teaching it is even much less so. Upholding the idea that there are always right and wrong answers perpetuate objectivity as well as fear of open conflict.”
After condemning objectivity, the authors of A Pathway to Equitable Math Instruction state that “some math problems may have more than one right answer and some may not have a solution at all…”
I’m a doctoral student in finance. In all my mathematical, statistical, and econometric training, I have never encountered a math problem that had more than one right answer. There are certainly problems that don’t have a solution, but that does NOT mean there is more than one right answer.
For example, in linear algebra, some matrices are singular, which means that they cannot be inverted. If a problem asks a student to derive the inverse of a singular matrix, they will not be able to do so. However, the right answer is for the student to state that there is no solution because the matrix is non-invertible. To do this, they must understand the concept of matrix singularity and the procedure of matrix inversion.
Obviously, in academic research, there are disagreements about whether a population model is correctly specified or whether a particular empirical methodology is the correct one to a answer a research question given the circumstances. For example, one social scientist might object to the conclusions drawn by one of her peers on the grounds that the peer’s analysis did not take into account certain relevant factors. However, these instances are not akin to the notion that there is more than one correct answer to – “2+2=?.” In this example, the social scientist is objecting to the research methodology of the analysis itself, not the inferences from statistical software results (for example, there are not multiple correct interpretations to estimated coefficients in linear regression output).
Often in mathematics, there are multiple ways to derive the correct answer. But that is not the same as saying that there are multiple correct answers. Math teachers do not care which method a student employs to derive the correct solution; their concern is whether the student CAN derive the correct answer.
The claims that the toolkit makes regarding how racism infiltrates mathematics do not stand up to elementary scrutiny. Unsurprisingly, they’re grounded in Critical Theory and postmodern philosophical traditions (constructivism, etc.). After reviewing the entire proposal, it seems the authors assume that it is racist to expect minority children to master mathematics (solve problems on tests under time pressure) in the same manner as White children and Asian children. Ironically, the claim that Black or Brown children are less able to display mathematical proficiency is a White supremacist claim. I’m White and my wife is Haitian. If a teacher used this toolkit to suggest that my child was somehow intrinsically mathematically handicapped because of their skin color, I would have strong words with that teacher.
Minority students’ suboptimal performance on math tests has NOTHING to do with their intrinsic math abilities. They are educated in poor public schools. They lack access to basic nutrition and healthcare services. They are victimized by harmful, unjust pieces of legislation, such as those that led to the War on Drugs, mandatory minimums, and the institution of mass incarceration. These deleterious laws and systems have led to broken homes, increased levels of crime, and unsafe neighborhoods for minority children. Furthermore, many programs that are intended to serve as social safety nets incentivize counterproductive social behavior, which further exacerbate the situations of vulnerable children. They’re simply not given the chance to succeed. These and other factors are what drives minority students’ poor math performance, not objectivity in math class. Expecting less from minority students will only harm them in the long run.
There is nothing Western, colonial, racist, or “problematic” about mathematics – ancient Chinese mathematicians were practicing Gaussian linear algebra two thousand years before Gauss got the credit for Gaussian elimination. Math transcends cultural tribalism. 2+2 will always equal 4, for a right triangle – a²+b² will always equal c², and the derivative of f(x)=4x+92 will always equal 4. Math is difficult enough to learn for children; we don’t need ideologues injecting relativism into a space of total theoretical objectivity by vilifying the concept of objectivity itself.
Telling students that there is more than one right answer to a mathematical equation is about the most damaging thing a math teacher can do to a student.
α Bond, Paul. (2021). “Math Suffers From White Supremacy, According to a Bill Gates-Funded Course.” Newsweek. https://www.newsweek.com/math-suffers-white-supremacy-according-bill-gates-funded-course-1571511
β McWhorter, John. (2021). “More on what modern “antiracism” does to schools, or could not — and some insight on the Kendi thing.” John McWhorter Substack. https://johnmcwhorter.substack.com/p/more-on-what-modern-antiracism-does
γ (2021). “A Pathway to Equitable Math Instruction.” https://equitablemath.org/
δ Cintron, Sonia et. al. (2021). “Stride 1.” A Pathway to Equitable Math Instruction: Dismantling Racism in Mathematics Instruction. https://equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf
ε Koop, Chacour. (2020). “Smithsonian museum apologizes for saying hard work, rational thought is ‘white culture’.” Miami Herald. https://www.miamiherald.com/news/nation-world/national/article244309587.html