Dr Hugh Morrison (The Queen’s University of Belfast [retired])
Professor Jo Boaler’s case for a new approach to teaching and learning in mathematics is an incoherent mix of dubious mathematical reasoning and neuroscience. Boaler’s (2016, p. 1) claim that her “visual mathematics” approach satisfies “an urgent need for change in the way mathematics is offered to learners” is outlined in her TEDx Stanford presentation entitled How you can be good at math, and other surprising facts about learning. Her recent visit to Scotland confirms that her visual approach is now being urged upon that country’s teachers. This short essay is designed to alert teachers everywhere to the dangers of replacing traditional approaches to pedagogy with Professor Boaler’s confused reasoning.
Boaler’s case for her visual mathematics is illustrated using a sequence of patterns each comprising a number of squares (see her TedxStanford talk for a very engaging outline of her analysis): the first pattern (n = 1) has four squares, the second (n = 2) has nine squares, the third (n = 3) has sixteen squares, and so on. (The reader will, no doubt, recognise the three numbers 4, 9 and 16 as “square numbers” because 22 = 4, 32 = 9 and 42 = 16.) The pupil is asked to continue the patterns in the same way and find the general rule of which these three patterns are instances. According to Boaler’s TedxStanford presentation, the general rule which generates the number of squares (4, 9, 16, and so on) in the sequence of patterns is, needless to say: number of squares = (n + 1)2. This isn’t difficult to verify. Substituting
n = 1 in this rule gives 4, substituting n = 2 gives 9, and substituting n = 3 gives 16, and so on.
Every mathematics teacher in England, Wales and Northern Ireland with experience of GCSE mathematics coursework – now abandoned in the UK after decades of effort to promote and assess “discovery learning” – will recognise Professor Boaler’s illustrative example of visual mathematics as one of the GCSE “growing squares” tasks. Indeed, one could be forgiven for thinking that Boaler’s visual mathematics is little more than her UK experience of discovery learning, with a pinch of neuroscience. Once identified, the (n + 1)2 rule can then be used to continue the sequence of patterns onwards, yielding:
4, 9, 16, 25, 36, 49, …
There can be little doubt that mathematical activities such as the “growing squares” task serve to enrich the mathematical experience of children by teaching the principles of problem-solving, and facilitating collaborative learning. However, the case I want to advance in this essay is that it is nonsensical to argue, as Boaler does, that such activities can ever challenge established, “traditional” approaches to the teaching and learning of mathematics. Traditional learning is always prior to discovery learning; without the framework laid down by the traditional teacher (the so-called “fiduciary” framework), discovery learning is impossible. Boaler’s error is to have ignored Polanyi’s (1958, p. 266) warning: “No intelligence, however critical or original, can operate outside such a fiduciary framework.” Boaler’s visual mathematics can never replace the traditional approach to teaching and learning.
Professor Boaler seems unaware of a problem first identified by the great mathematician Leibniz, namely, that a finite number of examples always underdetermines the rule which generates these examples. Boaler focuses on the rule (n + 1)2 as the answer, but there is an infinity of such answers. Anscombe (1985, pp. 342 – 343) presents the Leibniz argument using the even numbers: “[A]lthough an intelligence tester may suppose that there is only one possible continuation to the sequence 2, 4, 6, 8, …, mathematical and philosophical sophisticates know that an indefinite number of rules (even rules stated in terms of mathematical functions as conventional as polynomials) are compatible with any such finite initial segment. So if the tester urges me to respond, after 2, 4, 6, 8, with the unique appropriate next number, the proper response is that no such unique number exists. … The intelligence tester has arbitrarily fixed on one answer as the correct one.”
In her TEDx Stanford presentation, Boaler presents her pupils with three instances of a rule (the first pattern has 4 squares, the second has 9 squares, and the third has 16 squares) and implies that the brain (for Professor Boaler the brain is pivotal to appreciating how children learn mathematics) should, after a process of understanding, arrive at the conclusion that the rule is:
number of squares = (n + 1)2
However, there is an infinite number of alternative rules which begin with the numbers
4, 9, 16, but diverge thereafter. These can be characterised as follows:
number of squares = (n + 1)2 + a (n – 1)(n – 2)(n – 3)
where a can take an infinite number of values.
For example, a = 0.5 generates the sequence: 4, 9, 16, 28, …
and a = 5 generates the sequence: 4, 9, 16, 55, …
and a = 12.5 generates the sequence: 4, 9, 16, 100, …
and a = 100 generates the sequence: 4, 9, 16, 625, …
and a = 2000 generates the sequence: 4, 9, 16, 12025, …
In all these cases, the pupil can protest that he or she went on in the same way. It is tempting to suggest that the pupil’s way diverges from Professor Boaler’s because he or she had a different rule in mind, or should I say, in brain. Indeed, what of the pupil who simply repeats the pattern, arriving at the sequence: 4, 9, 16, 4, 9, 16, 4, 9, 16, … ? Hasn’t this pupil gone on in the same way as indicated by the three initial examples?
Clearly, one rarely, if ever, comes across a pupil who would propose one of these alternatives to the (n + 1)2 rule in real classrooms, but Boaler’s thesis is that understanding is a rational process in the brain. By what neural mechanism would the rational brain select one rule – Boaler’s (n + 1)2 rule – from an indefinite number of alternatives? How, in a finite time period, does the brain brand all of these alternative rules (indefinite in number) as somehow incorrect, and settle on the (n + 1)2 rule as the correct rule? What possible criterion does the brain use to distinguish correct from incorrect?
The role of the brain in mathematics is central to Boaler’s research. It is tempting therefore to think that the “visual” learner, in understanding the problem, attaches an interpretation (something represented in the brain) to the three examples which constitute the statement of the problem, namely, a pattern of four squares, followed by a pattern of nine squares, followed by a pattern of sixteen squares. If a pupil responds by suggesting that the next pattern is made up of 55 squares, for example, (the a = 5 sequence has 55 for its fourth term), Professor Boaler will treat this as a mistake (after all, the “correct” answer is 25).
But nothing in the statement of the problem rules out the answer “55” because it could be argued that the pupil has merely interpreted the statement of the problem in a way which is at odds with Professor Boaler. Of course, the pupil’s interpretation accords perfectly with the three examples used in the statement of the problem. The pupil has responded correctly to the instruction to continue the sequence of shapes in the same way. What makes Professor Boaler’s interpretation correct and the pupil’s incorrect? Indeed, any answer whatsoever to the question “how many squares are in the fourth pattern?” will be correct on some interpretation. Wright (2001, p. 98) captures this intractable situation in the words: “Finite behaviour cannot constrain its interpretation to within uniqueness.” This is at the core of the case made by Leibniz. It would seem that if understanding in mathematics is construed as an activity of the mind or brain, then the notions of a “correct” and an “incorrect” answer are rendered meaningless!
Did anyone at Professor Boaler’s TEDx Stanford talk, or at her Scotland talk, spot this profound error in her reasoning? Thousands of papers and many hundreds of books have been written about Ludwig Wittgenstein’s resolution of what has been called the “rule-following paradox.” Furthermore, Wittgenstein’s resolution is unlikely to be to Professor Boaler’s liking, for the solution emphasises the traditional classroom in which children are trained to adhere to established mathematical practices, and Wittgenstein makes no mention of the brain. According to Wittgenstein, we are forced to conclude that children go to school to acquire a “framework” or “background,” which they grow to accept without question and within which they can be creative. This framework constrains the pupil’s creativity in order that he or she can be understood by peers and teachers, but, it never determines the pupil’s subsequent response to any particular problem.
If the object of analysis is the pupil treated as a separate individual with a brain/mind, divorced from the framework of mathematical practices that the pupil takes on trust from authority figures at school and beyond, one gets incoherent nonsense. Understanding is not an activity, state or process of the brain or mind; understanding is a capacity. This is the error at the heart of Boaler’s analysis: her model omits the framework of mathematical customs and practices which the pupil has come to accept (as common sense, one might say) through his or her training at school. According to Scruton (1981, p. 291), “All attempts to understand the human mind in isolation from the social practices through which it finds expression” are doomed to fail.
In Zettel (§419) Wittgenstein cautions: “Any explanation has its foundations in training. (Educators ought to remember this).” Because she omits the all-important, long-established mathematical customs and practices in which the pupil participates in the traditional classroom, and treats the pupil as separately analysable, Boaler is forced to accept an indefinite number of different answers as the correct answer for the number of squares in the 4th pattern, for example, and she must conclude that it is meaningless for pupils to seek the correct rule.
It is instructive to identify the source of the incoherence in Boaler’s “visual mathematics.” She has a confused grasp of the notion of understanding in general and in mathematics in particular. Her 2016 paper with Chen, Williams and Montserrat has the title: Seeing as understanding: The importance of visual mathematics for our brain and learning. For Boaler, understanding consists in an inner state or inner process in the head, that state or process being the source of the pupil’s subsequent behaviour. Nothing could be further from the truth: Rowlands (2003, p. 5) writes:
Thus, according to Wittgenstein, to … understand something by a sign is not to be the subject of an inner state or process. Rather, it is to possess a capacity: the capacity to adjust one’s use of the sign to bring it into line with custom or practice. And this connects … understanding with structures that are external to the subject.
Note the absence of any mention of the brain in Wittgenstein’s resolution of the rule-following paradox.
This is why the vast majority of children respond to the “growing squares” problem with the answer: number of squares = (n + 1)2. It is custom and practice in mathematics to respond in this way. The conundrum identified by Leibniz (see above) is also resolved. If understanding the even numbers is a matter of adjusting one’s behaviour to accord with the established mathematical practice in respect of the these numbers, then there is only one unique answer we ought to give, namely, “10.” In §6.21 of his Remarks on the Foundations of Mathematics, Wittgenstein writes: “The application of the concept ‘following a rule’ presupposes a custom,” and McGinn (1984, p. 39) defines custom as follows: “A custom, like a habit, is something that gets established, not through the deliverances of reason, but on the basis of what we might call a tradition.” Boaler et al. (2016, p. 5) must appreciate that if students were not “made to memorise math facts, and plough through worksheets of numbers,” in the traditional classroom, mathematical rules would not so much as exist.
 Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied & Computing Mathematics, 5(5), 1-6.
 Polanyi, M. (1958). Personal knowledge. Chicago: University of Chicago Press.
 Anscombe, G.E.M. (1958). Wittgenstein on rules and private language. Ethics, 95, 342-352.
 Wright, C. (2001). Rails to infinity. Cambridge, MA: Harvard University Press.
 Scruton, R. (1981). A short history of modern philosophy. London: Taylor and Francis.
 Wittgenstein, L. (1967). Zettel. Oxford: Blackwell
 Rowlands, M. (2003). Externalism. Ithaca: McGill-Queen’s University Press.
 Wittgenstein, L. (1956). Remarks on the foundations of mathematics. Cambridge MA: MIT Press.
 McGinn, C. (1984). Wittgenstein on meaning. Oxford: Blackwell.