While the Greeks chose the geometrical concepts of point and line as the basis of their mathematics, it has become the modern guiding principle that all mathematical statements must be reducible ultimately to statements about the natural numbers, 1, 2, 3 . . . .

—Courant and Robbins (1981)

Is it racist to expect black kids to do math for real?

—John McWhorter, Columbia University (2021)

Recent web searches using the words “anti-racist mathematics” yielded an article posted by Ontario’s public affairs channel, TVO. The Province’s Ministry of Education is promoting nostrums in an attempt to solve a chronic problem: underachievement in mathematics performance by black and indigenous Canadian students. In 2021, Ontario’s Grade 9 math curriculum was both de-streamed and revised to include the assertion “Mathematics has been used to normalize racism and marginalization of non-Eurocentric mathematical knowledges.” It also claimed that “mathematics can be subjective,” and asked teachers to “create anti-racist and anti-oppressive teaching and learning opportunities ...” (Okwuosa, 2021). (Later that year, Doug Ford, the Premier of Ontario, had these assertions removed.)

Many educators approve of these assertions. For example, a Toronto District School Board official muddled the distinction between mathematics as a discipline and its applications: “They’re trying to keep this perceived objectivity of mathematics as something that should be sacred and untouched.” An advocate for inquiry-based teaching methods argues that teachers should use “math to understand issues of power and privilege.” An Ojibway language teacher complains that “Eurocentric mathematical knowledge overshadowed ‘The invaluable knowledge that indigenous people have about mathematics ... and allows for the perpetuation of harmful myths about indigenous people’ ” (Okwuosa, 2021).

While these comments imply educators’ ignorance of modern mathematics’ structure, nevertheless they reflect attempts to rectify a chronic problem: under-performance of black and indigenous North American children in this subject. This problem, which includes deficiencies in vocabulary and reading, was described a quarter century ago in a Brookings Institution publication (Jencks and Phillips, 1998). Similar problems exist in Canada (Parekh *et al*, 2018). In this respect, it is important to note that there was then, and is now, considerable overlap in the distributions of abilities, so that there are many talented black children who have become highly educated, an example being the author of the second quote above.

Okwuosa’s piece is part of broader problem: educators’ passion for introducing ethnomathematics into mathematics courses; this has been defined as “studying the relationship between mathematics and culture ... often associated with ‘cultures without written expression’ ” (Wikipedia 1, 2023). This idea has attracted strong criticism; for example, noted U.S. education critic Dianne Ravitch (2005) observes that:

In the early 1990’s the [US] National council of Teachers of Mathematics issued standards that disparaged basic skills like addition, subtraction, multiplication and division since all of these could be easily performed on a calculator.... One of its precepts is using ethnomathematics.... [They] advocate an explicitly political agenda....

To understand problems that make ethnocentric and anti-racist teaching policies counterproductive I make two points. First, following a history extending over millennia and involving contributions from many cultures, it is so-called “Eurocentric” mathematics developed in 19th Century Britain that—among other things—provides the logical structure essential to the operation of both electronic computers and hand calculators. Second, teaching practises widely advocated by mathematics educators are based on a controversial doctrine called *constructivism* (Wikipedia 1, 2023). But evidence cited below shows that sensitively executed direct instruction is clearly superior.

For the first point, following the destruction of the Great Library of Alexandria and the murder of the female mathematician Hypatia in 415 CE by a Christian mob, no significant mathematical developments occurred in Europe for nigh on a millennium. They continued elsewhere, especially in Muslim countries. The debt of the West to Islamic scholars is twofold: they collected and translated the Classical Greek mathematical texts, and they adopted the Indian decimal-positional numeral system, the latter being “transmitted to the West and “eventually to the whole world” (Hollingdale 1991, 92 & 101). The introduction of these ideas to Venice in 1202 CE by Leonardo of Pisa (also known as Fibonacci)—at a time when merchants still used Roman numerals in their documents—began a process by which Europeans developed the mathematics that became the foundational basis of modern science and technology. (For their calculations they used abacuses, which are mechanical analogs of the decimal-positional system.) As Ravitch (2005) puts it: [modern] mathematics is “a universal language that is relevant and meaningful in Tokyo as it is in Paris, Nairobi and Chicago.”

The logical foundation of mathematics that mid-19th Century British mathematicians established starts with five laws for adding and multiplying the *natural* (positive or counting) numbers. With operations in brackets first, they are (Hollingdale 1991, 335-6):

A [commutation]: (1) *a* + *b* = *b* + *a* and (2) *a**b* = *b**a*;

B [association]: (3) *a* + (*b*+*c*) = (*a*+*b*) + *c* and (4), *a*(*b**c*) = (*a**b*)*c*;

C [distribution]: (5) *a*(*b*+*c*) = *a**b* + *a**c*.

Subtraction and division reverse laws (1) and (2), but they do not commute. Negative numbers are governed by the *rule of signs*: the product (−1)(−1) is *defined* as (+1). Rational numbers, *the ratio of two integers*, also obey these rules. The first goal in teaching children mathematics is (or should be!) to help them achieve automaticity in using these rules; otherwise learning algebra, trigonometry and beyond is seriously impaired. This is the equivalent of accomplished pianists correctly using piano keys without conscious thinking.

The development of the idea of negative numbers illustrates the complicated history of mathematics; Rogers (2009) notes that:

Although the first workable set of rules for dealing with [such] numbers was stated in the 7th Century by the Indian mathematician Brahmagupta—who related positive numbers to financial fortunes and negative numbers to debts—it took a long time for mathematicians to become comfortable with the concept.

Thus, for the quadratic equation *a**x*^{2} + *b**x* + *c* = 0, now completely solved in algebra classes, to avoid negative numbers the 9th Century Muslim mathematician Mohammed al-Khwarizmi described five cases, treating them by using numerical examples and geometrical methods (Hollingdale, 1989, 97). The English mathematician John Wallace (1616-1703) gave meaning to negative numbers by inventing the concept of the *number line*: if adding positive numbers corresponds to moving in one direction, adding negatives corresponds to moving in the opposite direction. Nevertheless, as late as 1758, the British mathematician Francis Maseres claimed that they “served only to confuse” (Rogers, 2009).

An example of evidence justifying the need for achieving automaticity in arithmetic is an experiment conducted at Ontario’s Western University. It used 19 male and 14 female grade 12 students who had previously consistently demonstrated a range of mathematical abilities, and had taken the U.S.-based Preliminary Scholastic Aptitude Test (PSAT), which is usually taken at Grade 10 or 11. They were asked simple addition/subtraction questions while subject to *fMRI* (functional magnetic resonance imaging). The images showed that those scoring high on the math part of the PSAT tended to activate that part of the brain “known to be engaged during arithmetic fact retrieval,” while the low scoring students tended to use a region “established to be involved in numerical quantity processing.” The authors conclude that these results highlight “the fundamental role that mental arithmetic fluency plays in the acquisition of higher-level mathematical competence” (Price *et al*., 2013).

On teaching practices, constructivism contends that:

learners do not acquire knowledge and understanding by (direct instruction), rather they

constructnew understandings and knowledge through experience and social discourse, integrating new information with what they already know. (Wikipedia 1, 2023, italics in original.)

Educators widely use this doctrine to sanction student group activities in the classroom known variously as *child-centred*, *problem-based* or *discovery* learning. Teachers—in effect—become *facilitators*, who “help the student to get his or her own understanding of the content” (Wikipeadia 1, 2023).

Constructivism and the associated teaching techniques have been strongly criticised. For example, in 2003 one US mathematician observed that:

[T]hroughout the 20th century the “professional students of education” have militated for child centered discovery learning and against systematic practice and teacher directed instruction.... Because of the hierarchical nature of mathematics and its heavy dependence at any level on prerequisites, high school and even college mathematics courses have at times been strongly affected by [such ideas].... (Klein, 2003.)

A related controversy is the context in which the student learns:

The social constructivist paradigm views the context in which the learning occurs as central to the learning itself.... Decontextualized knowledge does not give us the skills to apply our understandings to authentic tasks because we are not working with the concept in the complex environment.... (Wikipedia 1, 2023.)

Cognitive scientist and Nobel Laureate H. A. Simon and his colleagues excoriate this assertion:

This false rejection of decomposition and decontextualization runs deep in modern mathematics education.... [We] find frightening the prospect of mathematics education based on such a misconceived rejection of componential analysis…. When students cannot construct knowledge for themselves, they need some instruction. There is very little positive evidence for discovery learning and it is often inferior.... [I]t may be costly in time, and when the search is lengthy or unsuccessful, motivation commonly flags. Real competence only comes with extensive practice.... The instructional task is not to “kill” motivation by demanding drill, but to find tasks that provide practice while at the same time sustaining interest. (Anderson et al, 2000.)

One of the most powerful recent demonstrations of the superior value of direct instruction is given by University of Melbourne education professor John Hattie’s meta-analysis of the results of over 52,000 studies involving millions of students. Hattie observes that, when he tells teacher trainees about this, they are shocked:

Every year I present lectures to teacher education students and find that they are already indoctrinated with the mantra “constructivism good, direct instruction bad.” When I show them the results of these meta-analyses, they are stunned, and they often become angry at having been given an agreed set of truths and commandments against direct instruction. (Hattie, in Christodoulou, 2014.)

A frequently cited paper by two University of Saskatchewan educators illustrates their tribe’s characteristic mind-set (Brandt and Chernoff, 2015):

[M]athematics and mathematics education is being further divided along ideological lines of those who perpetuate the idea of a single dominant world view and those who support and cherish diversity. The former group believes mathematics is independent of culture and therefore should be taught in a homogeneous curricula and pedagogy.... Conversely, the latter group views mathematics as a human activity very much entrenched in culture and ... can be greatly enriched by intellectual diversity inn curricula and pedagogy.... Mathematics education ... should reflect/embrace the cultural diversity of our classrooms.... Mathematics educators should ... adopt ethnomathematics into their lesson plans.

Falsely asserting that Eurocentric mathematics “is based mainly on Greek texts,” they claim that this reflects a “single dominant world view” that does not “cherish diversity.” As Ravitch (2005) makes clear, this ignores the ongoing universal acceptance of modern mathematical ideas by cultures that understand them; they may be said to acquire a Platonic character. A classic example is the Pythagorean theorem relating the areas of the squares on the sides of a right triangle. The Babylonians and ancient Chinese cultures—among others—accepted it while offering different proofs or demonstrations.

Unfortunately, Ontario’s Ministry of Education reports show that the ministry’s officials and curriculum designers enthusiastically support constructivism and all its counterproductive consequences (MOE 2004 and 2012; Sullivan, 2018).

Another challenge to teaching mathematics is the current fashion to attribute the poor performance of black males to an alleged *whiteness* of the educational system: privileges said to be unconsciously earned by light-skinned people by virtue of unidentified racism in North American culture. This dogma is motivated by a widely cited list of fourteen alleged negative attributes assembled by activists Kenneth Jones and Tema Okun (Jones and Okun, 2001). These include: *perfectionism, valuing quantity over quality, worshipping the written word, paternalism, believing in objectivity*.... It implies that “white culture” people lack a compassionate sense of what it means to be fully human(!).

A widely cited pamphlet prepared by a Californian advocacy group aimed at removing alleged racism in math teaching parrots this list (The Education Trust-West, 2020). Opening with staged photos of a downcast boy and an apprehensive girl, it includes the full list of supposed whiteness characteristics before using it to belittle instruction techniques known to be effective. Three examples are: (1) “Independent practise is valued over teamwork or collaboration”; (2) “Control of classroom is valued over student’s agency over their own learning”; and (3) “Math is taught in a linear fashion and skills are taught sequentially....” The third observation suggests profound ignorance of both the subject of mathematics itself and the challenges of teaching it. As linguist John McWhorter’s (2011) pointed critique observes, the document is itself racist.

To conclude on a hopeful note, I draw attention to an extensive analysis by the black scholar Ronald Ferguson, of Harvard University’s Kennedy School of Government (Ferguson, 2016). Ferguson identifies a “*systematic* predicament” (italics in original) faced by “boys and young men of color (BYMOC)”; it is a “complex web of circumstances for which no individual is to blame and that no one person can unravel.” I cite a few of the factors he identifies:

Efforts to dismantle the predicament should begin at birth.... There is evidence that BYMOC desire to succeed academically as much as any other group. However they tend to start kindergarten as the lowest achieving group in the school.... Even when they misbehave, BYMOC are often responding to peer pressures they would prefer to resist but feel compelled to comply with.... [S]kill gaps measured at the beginning of kindergarten predict all of the racial differences in special education placements by fifth grade.... On student surveys, BYMOC rated their classroom teachers the same, on average, as their white male classmates, but there are clear racial tensions in the hallways.... On average BYMOC have less access to effective teaching than whites do....

It seems that, in the USA, overt racism is—at best—a minor factor. One might expect that aspects of this analysis apply to Canadian black children and youth.

It is clear that, whatever the challenges facing Ontario’s black children and youth may be, they are not going to be solved by methods promoted by Ontario’s educrats. Better would be to find ways of providing competent direct instruction, together with sensitive and appropriate guidance.

## References

Anderson, J. R.. L.M. Ryder, and H.A. Simon. 2000. Applications and Misapplications of Cognitive Psychology to Mathematics Education. *Texas Educational Review* (Summer).

http://www.act-r.psy.cmu.edu/papers/misapplied.html

Brandt, A., and E. J. Chernoff. 2015. The Importance of Ethnomathematics in the Math Class.

*Ohio Journal of School Mathematics*. **71** (spring). 31-36.

https://kb.osu.edu/bitstream/handle/1811/78917/OJSM_71_Spring2015_31.pdf

Christodoulou, D. 2014. *Seven Myths About Education*. London: Routledge. See p. 40.

Courant, R., and H. Robbins. 1981. *What Is Mathematics?: an Elementary Approach to Ideas and Methods*. Oxford: Oxford University Press. 1, 52-55.

Ferguson, R.F. 2016. Aiming Higher Together: Strategizing Better Educational outcomes for Boys and Young Men of Colour. Cambridge Mass.: Urban Institute.

Hollingdale, S. 1991. *Makers of Mathematics*. London: Penguin Books.

Jencks, C. and M. Philips. 1998. The Black-white Test Score Gap: Why it persists and What Can Be done.

Jones, K., and T. Okun. 2001. Dismantling Racism: A Workbook for Social Change Groups. Excerpt in

https://www.thc.texas.gov/public/upload/preserve/museums/files/White_Supremacy_Culture.pdf

Klein, D. 2003. A Brief History of American K-12 Mathematics Education in the 20th Century.

http://www.csun.edu/~vcmth00m/AHistory.html.

McWhorter,. J. 2021. Is it racist to expect black kids to do math for real?

https://johnmcwhorter.substack.com/p/is-it-racist-to-expect-black-kids

Mer, B. 2022. Harvard organization lists 15 traits of “white supremacy culture.” *Why Evolution is True*, July 12.

MOE (Ontario Ministry of Education). 2004. Teaching and Learning in Mathematics. The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario.

https://www.edu.gov.on.ca/eng/document/reports/numeracy/panel/numeracy.pdf

MOE. 2012. Paying Attention to Proportional Reasoning, K-12.

https://www.edu.gov.on.ca/eng/teachers/studentsuccess/ProportionReason.pdf

Okwuosa. 2021. What does an anti-racist math class look like? TVO today. Accessed 9 February 2023.

https://www.tvo.org/article/what-does-an-anti-racist-math-class-look-like

Parekh, G., R. S. Brown and S Zheng. Learning Skills, System Equity and Implicit Bias Within Ontario, Canada. *Educational Policy*. (2018). 1-27.

https://peopleforeducation.ca/wp-content/uploads/2020/06/Parekh-Brown-Zheng-LS-Pub-2018.pdf

Price, G.R, M.M. Mazzocco and D. Ansari. 2013. Why Mental Arithmetic Counts: Brain Activation during Single Digit Arithmetic Predicts High School Math Scores. *The Journal of Neuroscience*. 33(1): 156-163.

http://www.jneurosci.org/content/33/1/156.full

Ravitch, D. 2005. *Ethnomathematics*. https://www.brookings.edu/opinions/ethnomathematics/

Rogers, L. 2009. The History of Negative Numbers. NRICH. https://nrich.maths.org/5961

Sullivan, P. A. 2018. *Mathematics Teaching in Ontario’s Public Schools*. July 19. Unpublished report, available on request (Email: psullivan@utias.utoronto.ca)

The Education Trust-West. 2020. *A Pathway to Equitable Math Instruction*

https://equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf

Wikipedia 1, 2023. Ethnomathematics. Accessed March 15, 2023.

https://en.wikipedia.org/wiki/Ethnomathematics

Wikipedia 2, 2023. Constructivism (philosophy of education). Accessed March 7, 2023.

https://en.wikipedia.org/wiki/Constructivism_%28philosophy_of_education%29