What are we reducing mathematics teaching to? What are we reducing the experience of learning mathematics to?

The overemphasis on testing and the impact of ‘knowledge’ evangelism is reducing mathematics teaching to a shallow empty routine of teach, test, identifying the missing ‘knowledge’ and then giving pupils more practice.

Simply put, it is *finding and filling gaps*.

To someone not familiar with learning theory or with experience of teaching mathematics, this can sound logical and even sensible. A rigorous programme which efficiently leads to progress toward mathematical proficiency. And it is as if what went previously, in mathematics teaching and learning, was shambolic and ad hoc.

But gap filling is a superficial characterisation of mathematics learning. It assumes that mathematics is learnt as a set of knowledges and skills that are, themselves, discrete elements. Teach, test, analyse results and re-teach. Some refer to it as formative assessment. But to paraphrase Dylan Wiliam, if your assessment-for-learning (AfL, or formative assessment) practices involve conditionally-formatted spreadsheets, then it isn’t AfL.

AfL involves the diagnostic assessment of learners’ understanding in order teach responsively.

Mathematics is built upon an interconnected network of concepts and processes. These underpinnings also have cultural, political and historical origins. Mathematics cannot be defined as a set of abstracted and freestanding ideas and methods.

In learning mathematics, pupils need experiences and direct teaching that allows them to understand what mathematics is, its significance in their lives and to develop the confidence to use it in novel situations. Focussing on methods and factual elements does not achieve this.

Let me offer an example.

The English National Curriculum has included, since its inception in 1988, a requirement that students need to learn how to find the terms in a sequence by using a term-to-term rule and then a position-to-term rule.

The following linear sequence includes term values and position values. The general position is ‘*n’* and an expression for the *n ^{th}* term can be found.

position | 1 | 2 | 3 | 4 | n |

term | 5 | 9 | 13 | 17 |

In last Thursday’s Faculty session, I was discussing this with the secondary mathematics PGCE^{1}PGCE = Postgraduate Certificate of Education trainees. The question I posed was: *what is the purpose, in the curriculum, of position-to-position rules and term-to-position rules? *The trainees suggested that it is important so that students understand sequences and patterns and as an introduction to algebra. These answers represent, respectively, a conceptual and procedural view of learning. *Understanding* and *knowing how to*.

But even deeper than this, the underlying purpose is concerned with a fundamental human capacity to identify patterns and use these to make predictions. Mathematics is often considered as the science of patterns. It is through the identification of patterns that we are able to apply inductive reasoning. This is a key part of human survival, as a social being. Through patterns we can make predictions about season, weather and the planning and production of resources.

This capacity can put us on the edge of madness. Some psychologists suggest that conditions such as *patternicity* and *apophenia * are cognitive malfunctions where an individual identifies patterns in what is effectively noise . This has been related to conditions such as schizophrenia. In my view, patternicity is a natural human condition, the difficulty comes in knowing how to examine the validity of a proposed pattern. This is where the investigation of sequences in the mathematics classroom becomes important in learning essential aspects of being human.

The curriculum example, when you really start to dig, is so much more than learning a procedure i.e. learning how to find the next term and how to find an algorithm that links the position value to the term. There are profound philosophical and psychological aspects within this simple concept.

I conjectured in the PGCE session that the original intention and value of this in the curriculum had this in mind. However, its presence has also been shaped by classroom practice. The investigation of linear sequences is an enjoyable activity. One of my favourite resources *New York Cop and Other Investigations * is now sadly out of print, but there are a number of other resources available from my colleagues at NRICH. There are some examples of sequence investigations here.

These kinds of investigations provide students with the experience of considering patterns in concrete form, allowing them the chance to make conjectures and predictions, as position-to-position rules and to position-to-term rules. The danger is that this investigative experience is reduced to a process and the exploration of patterns is reduced to a simple and largely meaningless procedure .

Bruner’s conceptualisation of learning in terms of:

- Enactive representation (action-based)
- Iconic representation (image-based)
- Symbollic representation (language-based)

In this conceptualisation, learners move back and forth between concrete and enactive representations, which may be physical or computer-based manipulatives, moving toward image representations and on to symbollic, language-based and abstract manipulation. Bruner presents a mechanism through which learners move from the real world to the cognitive world.

Returning to how sequences are taught in the classroom.

position | 1 | 2 | 3 | 4 | n |

term | 5 | 9 | 13 | 17 |

At primary school pupils might start finding the next terms in sequences. Linear sequences naturally relate to multiplication tables. In the above example, the position-to-position rule is *add 4*. Primary pupils will be able to identify this and predict further terms.

In secondary school, in year seven, eight and nine, position-to-rules are introduced or what is often referred to as the *n ^{th}* term. It also represents an advance in mathematical reasoning from identifying next terms to understanding how an algorithm can be used to generate the term in a sequence from its position.

In this example the *n ^{th}* term can be found using the expression $latex 4n+1$. Understanding how to find and use this expression is more of a challenge than the position-to-position rule, since it requires multistage reasoning in the construction of an algorithm. Pupils need to have secure conceptual understanding of multiplicative reasoning and the properties of operations in mathematics i.e multiplication, division, addition and subtraction. If they have difficulty in finding an expression for the

*n*term, it is highly likely that they are not secure in the underlying concepts. Furthermore, if it is an algebraic expression that is sought, difficulties and misconceptions in algebra can cause further problems in undertaking what appears to be a fairly simple method. In high quality teaching, we would be looking at pupils’ work in order to diagnose the problems and identify tasks to address these issues.

^{th}The gap filling approach relies on teaching more procedurally. I had a well-refined ‘system’ through which I could demonstrate an efficient routine for generating an expression for finding the *n ^{th}* term. Most maths teachers have something similar. Find the difference between terms, this tells you the coefficient of

*n*. In the example I have used it is 4. But $latex 4n$ would give 4, 4, 8, 12, 16 … We have to add 1 to get 5, 9, 13, 17… . So the expression is $latex 4n+1$.

You would think with efficient teaching and practice this is easy. It isn’t, for many pupils this is not straightforward. They will, with support, produce correct answers, but after time or in another context, they struggle. It is as if they have never learnt how to do it. And not just for the reasons I have set out above. Self-theories and mindsets also have an impact on the extent to which pupils retain procedural learning. I am not going to go into this here but Mark Dawes and I have written about in this paper.

The gap-filling approach does reveal that pupils have not retained procedural knowledge. But all too often the result is to treat the symptom and not diagnose the underlying source of the problem. Research shows us that long-term procedural retention in mathematics is a result of conceptual difficulties and not an issue of retention. Yet, too often the gap-filling approach encourages teachers to repeat the same teaching procedures rather than analyse and diagnose the underlying reasons for non-retention. This means that we need to include more investigative work into mathematics learning. Not because we want learners to ‘discover’ expressions and algebra but in order that they can make deeper conceptual connections.

#### References

*Self-efficacy: The exercise of control*. W.H. Freeman.

*New York Cop and Other Investigations: A Resource Book for Mathematics Teachers*. CUP.

*Toward a theory of instruction*. Beknap Press of Harvard University.

*Self-theories: their role in motivation, personality, and development*. Psychology Press.

*Mathematics Teaching*,

*140*. http://nrich.maths.org/content/id/7768/Train%20Spotters.pdf

*Scientific American*,

*299*(6), 48–48. https://doi.org/10.1038/scientificamerican1208-48