Teaching reasoning in mathematics…but what do we mean by reasoning?

It’s an ongoing issue, one on which I frequently cite Robert Recorde’s sixteenth century observation that students must not only learn by rote but also by reason. Recorde’s The Ground of the Artes is one of the earliest mathematics textbooks in English, it is in the form of a dialogue between teacher and pupil (Master and Scholar) and explains the processes of addition, subtraction, multiplication and division. The following dialogue from the chapter on subtraction reveals Recorde’s thoughts on teaching and learning. The discussion of learning by rote and reason is still relevant today.

Master. So may you if you have marked what I have taught you. But because thys thynge (as all other) must be learned by often practice, I wil propounde here ii examples to you, whiche if you often doo practice, you shall be rype and perfect to subtract any other summe lightly…

Scholar. Sir I thanke you, but I thynke I might the better doo it, if you show me the workinge of it.

M. Yea but you must prove yourself to do som thynges that you were never taught, or els you shall not be able to doo any more than you were taught, and were rather to learne by rote (as they cal it) than by reason [my emphasis] (The Ground of Artes, sig.F, i, v cited in Howson 2008, p. 20).

Yet reason has vexed philosophers, psychologists and teachers alike. What do we mean by reason? And how might it be taught? That is, if it can be taught at all.

Here, I intend to address the former, what do we mean by reason? But in the context of learning mathematics. And to do this, I want to draw on the work of Philip Johnson-Laird, who has researched, extensively, the psychology of reasoning since the 1970s. He presents a comprehensive overview of his work in his book, How We Reason (2006). In spite of Johnson-Laird’s groundbreaking and illuminating work, very few of his ideas have been made use of in mathematics classrooms. I want to present an overview of his central argument and offer a suggestion about how that might be applied in the classroom.

A definition of reasoning

For Johnson-Laird goal-directed[1] thinking can be deterministic or nondeterministic. Deterministic thinking is where each step of the thinking process is based on a current state, like in a computer or when we carry out mental arithmetic. Johnson-Laird considers nondeterministic thinking as processes where we explore worlds of possibility. This leads to Johnson-Laird’s definition of reasoning (which he uses interchangeably with the word inference):

A set of processes that construct and evaluate implications among sets of propositions.

For example, pupils are often asked to compare the magnitude of two or more fractions, like in the example below. The propositions are that they are either the same or one or the other is greater. The reasoning process involves evaluating the possibilities. It involves constructs and representations that permit this evaluation process.

In the following section I explain how Johnson-Laird explains the reasoning process.

Reasoning processes: dual processing theory

Dual processing distinguishes between rapid intuitive inferences and slower deliberative reasoning. This is also connected to subconscious and conscious reasoning, subconsciously we make rapid assessments and judgements and consciously we carefully and logically evaluate. Drawing on this model Johnson-Laird demonstrates how reasoning and inference involve mental processes which are carried out on mental representations. Some of which is subconscious and intuitive and some conscious and rational. The mental representations he refers to as mental models. To illustrate this, consider the following problem:

The cup is to the right of the plate.

The spoon is to the left of the plate.

What’s the relation between the cup and the spoon?

Unconsciously we produce a mental model based on our understanding of the premises and from this we draw a conclusion. In some situations we reason unconsciously and rely on intuition and may not be aware of the premises. In conscious reasoning we become at least aware of the premises as well as the conclusion we draw.

This process can be applied to the fraction example we can make intuitive judgements based on simple insights into the situation, or we can consciously evaluate, using mental models to reason the correct solution that 3/8 is the greater.

In many situations, where we are experts particularly, for example as an experienced teacher, we can quickly form impressions of a situation and act without thinking.

When logic and intuition conflict

One of Johnson-Laird’s most important contributions is the recognition that reasoning based on logic is less frequent than reasoning based on unconscious reasoning and heuristics, where we draw on long term memory and use rules of thumbs to draw conclusions. When we use logic and conscious reasoning it is demanding and can present us with contradictions with our everyday reasoning. Take the following example.

We’re all prejudiced against prejudiced people.

Anne is prejudiced against Beth.

So, does it follow that Chuck is prejudiced against Di?

Intuitively we say no, because nothing has been said about Chuck or Di. However, if we follow through the argument, Anne is prejudiced against Beth, so Anne is a prejudiced person and it follows that we are all prejudiced against her, so Di is prejudiced against her. Because Di is a prejudiced person and we are all prejudiced against Di, Chuck is prejudiced against Di.

This illustrates the limited capacity of working memory, it is difficult to hold all the information and make correct inferences. As Johnson-Laird says “Our reasoning is limited in power” (Johnson-Laird, 2009, p. 74).

The self regulation of the management of reasoning is important: knowing when and when not to consciously reason about the situations we meet. Self regulation takes time and experience to develop and this is just one justification for reasoning being part of the mathematics curriculum. It also relates reasoning to emotion and motivation as I explain next.

Emotions and reasoning

Many problems in our lives are concerned with emotions and affective states e.g. pain, stress, fatigue and anxiety. Johnson-Laird suggests that we read our emotions based on mental models of ourselves or self-theories. This has a role in how we reason, we construct mental models of the premises we confront and draw conclusions based on self-theories and our mental models of the world. It is also important to recognise that emotions have an important role in motivating us to act. Johnson-Laird recognises that any theory of reasoning must also account for affect and emotion. There is an interesting and useful link here with the work on self-theories and mindsets by Carol Dweck and on self-efficacy by Albert Bandura. Both of whom use constructs within social psychology that are analogous to Johnson-Laird’s mental models.

The implications for learning mathematics is that it is necessary for teachers to recognise the interconnection of emotion, motivation, confidence and reasoning.

Mental models and reasoning

The cornerstone of Johnson-Laird’s theory of reasoning is a recognition that we don’t simply think about possibilities, we represent possibilities as mental models. The manipulation of mental models leads us to draw conclusions and decide how to act. In learning mathematics students have to become familiar with mathematics as a series of abstract, but interconnected models and processes, and the axiomatic principles that define the rules with which we can manipulate the models. In learning mathematics we have to learn the abstract models that are the basis of mathematics[2] and processes that set the parameters within which these abstractions can be transformed. In the fraction example above, students must know how to mentally represent fractions and the rules that dictate how they can be manipulated in order to derive a conclusion.

The process of teaching and learning of mathematics requires teachers to engage with students’ mental models as articulated through the argumentation. Teachers have to interpret, diagnose and guide the development of more sophisticated mental models of mathematics. This is often referred to as diagnostic teaching in the context of misconceptions (Swan, 2001; Bell, 1993).

To conclude I summarise some implications.

Issues in the teaching of reasoning in mathematics

• Students need to be more aware of what reasoning means — they need to develop more sophisticated understanding of how we reason: dual processing, the impact of emotions and self-theories (see, for example, Dweck’s Mindsets).
• The development of reasoning skills involves having increasingly sophisticated models of mathematics and its concepts. The developmental process involves the diagnosis of existing mental models and supporting the development of new schema.
• Problem solving has an important role in developing reasoning; in testing possibilities and conjectures; in allowing students to explore and develop their own mental mathematical models; in developing argumentation and justification; and building confidence in reasoning.
• Students need to experiencing work on logic problems (like the ones above) to understand logic, but also to recognise the differences between logic and intuition.

Notes

[1] Goal-directed thinking refers specifically to thinking that has a purpose, there is an intention to achieve a result. This contrasts with musing or day dreaming, for example.

[2] It is worth considering here how abstract representations are equivalent to Johnson-Laird’s mental models. This also relates to Lakoff and Núñez’s (2000) notion of embodiment. This relates action and behaviour, mental models and metaphor.

References

Bandura, A. (1997). Self-efficacy: The exercise of control. New York: W.H. Freeman.

Bell, A. W. (1993). Some experiments in diagnostic teaching. Educational Studies in Mathematics, 24(1), 115–137. https://doi.org/10.1007/BF01273297

Dweck, C. S. (1999). Self-theories: their role in motivation, personality, and development. Philadelphia, PA: Psychology Press.

Howson, A. G. (1982). A history of mathematics education in England. Cambridge: Cambridge University Press.

Johnson-Laird, P. N. (1983). Mental models: towards a cognitive science of language, inference, and consciousness. Cambridge: Cambridge University Press.

Johnson-Laird, P. N. (2009). How we reason. Oxford: Oxford University Press.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. New York: Basic Books.

Swan, M. (2001). Dealing with misconceptions in mathematics. In P. Gates (Ed.), Issues in mathematics teaching. London: RoutledgeFalmer.