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1 Introduction

It was a great pleasure to read Chap. 13 with its wide-ranging scope concerning different ways in which mathematical structure can play out in primary classrooms in the context of whole numbers and with its wide-ranging suggestions for structuring tasks so as to foster significant encounters with mathematical structure.

As Towers and Davis (2002) observe, the term structure, etymologically linked to ‘strew’ and ‘construe’, has been used in mathematics education in two rather contrasting senses. Its biological use, which underpins Piaget’s genetic epistemology, refers to complex, constantly evolving, co-emergent, contingent and co-implicated forms; its architectural use refers to static interlocked components. Steffe and Kieren (1996) suggest that educational research has been impeded by conflation of these two meanings.

My aim here is to augment Chap. 13 with some examples of tasks which seem to me to promote encounters with mathematical structure and to suggest some directions for future development. The first section offers some observations concerning the recognition of relationships and transition to generalisation through attending to properties being instantiated, drawing on my own experience in supporting the teaching of mathematics. I mention them because they capture something of the growth of my awareness of how mathematical structure can be avoided or circumvented unwittingly through inappropriate pedagogic choices. This segues into a few remarks about attention and structured variation. The final section draws explicitly on Chap. 13 to suggest some potentially worthwhile directions for further study and development.

2 Detecting Mathematical Structure as Recognising Relationships

In the 1970s, as I became more and more involved in the issues and concerns of teaching and learning mathematics, I was inspired by the Midland Mathematics Experiment (1964). There I found sequences of figures made variously out of matchsticks and shapes such as squares and circles. I was excited by them because they seemed to me to offer multiple opportunities for learners to imagine what was not present, but which extended what was present according to some fixed relationships, multiple opportunities to express those relationships and multiple opportunities to develop algebraic expressions for the number of elements required to make a specific but as yet unknown figure in that sequence.

2.1 Expressing Generality

I developed and incorporated dozens of tasks involving sequences of figures in materials designed to support teachers of mathematics at all ages, from early years to secondary school (Mason 1988, 1996). One of the most important principles for me was the necessity that learners formulate a verbal statement of how a pattern continued or how the instances presented fitted into some extendable pattern. Only then is it worth counting the number of objects required. One of my favourites was a figure that we used for an assessment question for would-be and practising teachers upgrading their qualifications (Fig. 14.1).

Fig. 14.1
figure 1

A stick configuration

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The reason for choosing 50 and 32 is to see whether the action of scaling (multiply columns by 10) is either invoked before they have really thought about the situation or used because they want an easy calculation. Then we can talk about parking the first action that becomes available and considering whether other actions might be more appropriate or effective. Thus, the task affords possibilities for work on inner and meta tasks as well as the outer task (Tahta 1981; see also Mason and Johnston-Wilder 2004, 2006).

If I were using the task myself, I would exploit the fact that different people may ‘see’ the configuration differently, and I might even extend it to include asking people to find at least three different-looking expressions for the number of sticks and to indicate how these express different ways of seeing how to build such figures. This is about seeking structure through recognising relationships (one of the forms of attention). The reason is to promote multiplicity in ways of seeing and expressing those ways of seeing symbolically. Not only does this multiplicity lead naturally to the rules of algebra, in order to manipulate different looking expressions that are known to express the same thing, but it is allied to and reinforces parking.

On the original assessment, we found that many of the teachers could cope with generalisation in one direction (number of columns), but not with generalising two things at once. This led me to promote making a copy of the figure for yourself and watching how your body naturally finds an efficient way of doing it (perhaps by doing all the horizontals first or by building column by column). The slogan Watch What You Do (WWYD) emerged as a way to be reminded to do this and applies whenever you are specialising, that is, constructing a simpler specific example of something in order to get a sense of structural relationships. This too was encapsulated in a slogan as Manipulating–Getting-a-sense-of–Articulating (Floyd et al. 1981; see also Mason and Johnston-Wilder 2004, 2006).

Seeking several expressions of the same generality in lots of different situations eventually leads to the question of whether there is a way to move between equivalent expressions without going via the verbal description that they express. We called this multiple expressions and promoted it as a route to algebra, because the ‘rules’ for manipulating letters can be developed and expressed by learners themselves when the desire to do so arises (Mason et al. 1985, 1996).

Some years later, I realised what Mary Boole (Tahta 1972) might have meant when she talked about a particular route to generalisation. I called it tracking arithmetic, and it involves carrying out calculations without actually touching one or more of the initial numbers. In the case of the matchsticks, this means finding a way to count the number of sticks but not touching the 5 (for columns) or the 2 (for rows). It requires expressing everything in terms of these two numbers. Thus, the horizontal sticks are counted by 5 × (2 + 1), the vertical sticks by 2 × (5 + 1) and the diagonal sticks by 2 × 5. Overall, this means that 3 × (2 × 5) + 5 + 2 sticks are needed. The untouched 2s and 5s can now be replaced by r and c, respectively, to give the expression 3rc + r + c for the number of sticks required, perhaps by first going through and marking all the occurrences of 2 as a row count and 5 as a column count. Note also the symmetry between r and c.

Tracking arithmetic (Mason et al. 2005) has proved a powerful route into algebra when working with ‘algebra-refusers’: learners who have decided that algebra is not for them. Instead of letters, I use a cloud to stand for some as-yet-unknown number that someone outside of the room is thinking of. Then I proceed to get them to express some relationships, and to their surprise, they find that what they have done is actually algebra!

A further development (particularly but not exclusively at secondary level) is to ask whether there is a figure corresponding to some specified number S of sticks. Since S = 3rc + r + c, it turns out that 3S + 1 = 9rc + 3r + 3c + 1 = (3r + 1)(3c + 1). Thus, S sticks can be used to make such a figure if and only if 3S + 1 can be expressed as the product of two numbers both of the form one more than a multiple of 3. Furthermore, this structural reasoning can be generalised. It can be used on any expression of the form axy + bx + cy + d with suitable adjustments to take into account b, c and d. Not that this reasoning is accessible to young children, but it is worthwhile at least raising the undoing question so as to immerse learners in the ubiquitous and creative theme of doing and undoing: whenever you find you can perform some action, the doing, ask whether or in what circumstances you can undo it (Mason 2008).

The importance of deciding how all the figures are to be drawn, even those not yet displayed, before embarking on counting, cannot be overstated. In other situations, it is vital that there is some predetermined rule or structure for generating the terms of the sequence. For example, the figure 14.2 is part of a frieze pattern, but without any further information, you cannot be certain how it continues. However, if you are told that it is generated by a repeating block of cells and that the repeating block appears at least twice, you can extend the frieze in only one way (Mason 2014; Fig. 14.2).

Fig. 14.2
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A part of a frieze pattern

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Only when you have decided how it continues does it make sense to ask questions such as the shading of the 100th cell (the nth cell) and the cell number of the 100th occurrence (the nth occurrence) of a lightly shaded cell, which are the doing and the undoing questions. WWYD is still pertinent.

This sort of task has been exploited with pre-school children, where they make up their own intricate patterns and extend their own and those made by others (e.g. Papic and Mulligan 2007; Ferrara and Sinclair 2016). Slightly older children can, with suitable support, proceed to count the numbers of objects required in general and even to extend their thinking to negative numbers (e.g. Moss and Beatty 2006). Zazkis and Liljedahl (2002) report various studies indicating that learners may not find such tasks straightforward, but usually this can be explained by unfamiliarity with expressing generality as part and parcel of thinking mathematically. As Chap. 13 points out, it all has to do with a pedagogy that is in alignment and consistent with a teacher’s ways of thinking about mathematics as a creative endeavour rather than as a process of training behaviour to carry out mindless procedures. Thus, effective teaching is not simply about the mathematical structure and the structural relationships that govern a situation, and it is not simply about the choice of task or how it is structured. It is also about sequence and structure of pedagogical moves made by the teacher in setting up the task and in interacting with the learners. It is all of these, informed by a perspective on and experience of mathematics that values mathematical thinking as much as it does answers.

It was only after many years that I realised how easily learners’ behaviour can be trained and how learners conspire (often unwittingly) to circumvent thinking (active cognition). For example, I used always to present the first three or four figures in a sequence, but eventually I realised that this led to learners paying more attention to how the figures change from one to the next than to the internal structure of each individual figure (Stacey 1989; Stacey and MacGregor 1999, 2000). Sometimes this inductive approach is powerful, and indeed the only way to proceed, but the main purpose of inviting learners to generalise from sequences was to get them to express generality, counting the number of objects needed to make the nth picture without recourse to the previous ones. These tasks were intended to develop students’ powers so that the teaching of every topic intimately involved learners expressing generality for themselves and justifying it. Offering sporadic instances or even a single ‘generic’ instance is one way to avoid falling into the trap of learners becoming dependent on a particular format for such tasks.

There arose for me the question of what students were attending to when they did, and when they did not, detect and express generality in different situations.

2.2 Attention

Having spent a long time trying to get to grips with what it means to attend to something, I eventually discerned five forms or structures of attention, building on ideas of Bennett (1966; see also 1993), only to find that they were in close alignment with the van Hiele levels (van Hiele-Geldof 1957; van Hiele 1986). Where I differ rather significantly is that in my experience the different ways of attending to something are highly mutable. They are not levels to be climbed like some staircase. I describe these forms of attention as holding wholes (gazing at some ‘thing’ which may be visible or imagined), discerning details (some details may become a whole to be gazed at), recognising relationships in a particular situation, perceiving properties as generalities being instantiated in the particular and reasoning on the basis of agreed properties (Mason 2003).

At the core is the movement back and forth between recognising relationships (a sense of structure in the form of structural relationships, but only in the particular) and perceiving properties as general structural relationships being instantiated. I conjecture that, in mathematics, many students rarely if ever explicitly experience properties being instantiated, and consequently the world of mathematics remains closed to them. I popularised this in the UK with the slogan ‘A lesson without the opportunity for learners to generalise mathematically is not a mathematics lesson’ (Mason et al. 2005). In other words, generalisation is the life and soul, the heart of mathematical thinking. So when we promoted figural generalisations, it was only to provide learners with experience of generalisation. Our main proposal is, was and always has been that teaching mathematics means immersing learners in a culture of generalisation, prompting learners to express generalities as conjectures and trying to convince themselves and others that their conjectures (suitably modified) are actually correct. This applies to each and every topic and each and every lesson. It aligns with a Davydov-inspired approach to number which focuses on units before introducing number.

2.3 Structured Variation Grids

The notion of structured variation arose from a situation in the town of Tunja in Colombia, in which I was asked how to teach factoring of quadratic expressions to learners who were unsure about the answer to (−1) × (−1). I came up with what I called Tunja sequences (Mason 2001a). The idea is to call upon learners’ natural powers to extend familiar sequences and then to get them to interpret what they have done, using what Anne Watson (2000) has called with and across the grain. A simplified version for use with young children in a whole number context might be the above (Table 14.1).

Table 14.1 A grid
Full size table

Going with the grain means being able to predict what will be in each cell by detecting and exploiting the familiar sequence of natural numbers, by analogy with splitting wood. Going across the grain is about recognising why it is that the two calculations in each cell always give the same answer, by analogy with seeing the structure of the rings of a tree stump.

Having an applet which enables you to reveal one or other side of the equal sign in any cell makes it easy to show a few parts of a few cells and then to invite learners to conjecture and justify and then check other cells. It is a format in which to provoke generalisation. Learners can then be asked to make up a similar grid for themselves. On a different day, the multiplier 3 can be changed. It doesn’t take long for learners to conjecture and articulate the distributive law of arithmetic and, when expressed as a generality, the distributive law for algebra. Similar grids can be used in upper primary or lower secondary for expanding brackets and factoring (Mason 2015). Note that the effectiveness of structured variation grids lies not in the structure of the grids themselves, though this plays an important role, but in the pedagogic choices that are made, either in preparation or in the moment by moment unfolding of a lesson, informed by a perspective on mathematics conducive to learners taking initiative.

Here, the structural relationships which underpin arithmetic are brought to the surface, articulated and then internalised through direct personal experience. Similarly, the multiplication of negative numbers can be addressed by a multiplication grid that extends to the left and down into negative numbers. Going with the grain along rows and columns fills in the cells; going across the grain recognises why multiplying negative numbers works as it does. Recognising that the calculations are correct in each cell and that the left-to-right presentation could be reversed involves attention to specific structural relationships, perceiving the entries in cells as the instantiation of general properties. There are other grids which involve operations on fractions.

2.4 Comment

The reason for presenting some historical developments in my appreciation of obstacles to learning was to provide some specific examples of mathematical structure and to indicate how the shift from recognising relationships in the particular to perceiving properties as being instantiated lies at the heart of school mathematics. Arithmetic is most usefully seen as the study of properties of numbers; getting answers to specific calculations could become a by-product rather than the focus of attention.

3 Possible Directions of Development

To my mind, it would be really helpful if mathematicians and mathematics educators could come to some sort of agreement on how to think about mathematical topics, both as experiences in themselves and in relation to other mathematical topics and to mathematical thinking as a whole. I preface some of my suggestions with a pertinent extract from Chap. 13, marked by an attention point.

3.1 Expressing Generality

There are indications that situations involving spatial awareness can provide useful springboards for WNA working in ways that relatively ‘naturally’ and usefully include attention to structural relations.

As Chap. 13 indicates, there is growing evidence that young children can detect, copy and extend patterns and can create complex patterns for themselves. Teachers can initiate such tasks in the midst of almost any other work (e.g. during theme work on the polar regions, making sequences from polar bears, penguins and seals or whatever is the focus of attention). What matters is the rich way in which pedagogic choices promote the development of children’s natural powers to think mathematically, moving from pattern repetition to counting what is visible to counting what is only imagined and, so, to expressing generality (Mason 1996). It would be helpful to teachers to have more clear descriptions of how teachers have done this with pupils and how pupils have created their own.

3.2 Additive and Multiplicative Reasoning

Distinguishing between additive and multiplicative situations, as well as between different structures within additive and multiplicative situations, appears to be an important avenue into developing understanding of the different underlying structures of these situations. Problem posing in relation to given structures appears to be particularly complex and, therefore, openings for encouraging students to engaging with linking or constructing problems with given structural relations would seem to be an important area for further attention.

Chapter 13 reports research which indicates that structures such as the double number line and the empty number line can be useful for presenting a visual structure which can inform number calculations. To these could be added Numicon, Cuisenaire rods and Exercise Elastics (to manifest multiplication as scaling, of which repeated addition is a special case; see Harvey 2011). What seems to matter most is not the apparatus itself, but how it is used. Mathematics is only embodied in physical objects when someone ‘sees’ it as embodied, so it is all down to pedagogic choices. More work is needed concerning how pedagogic choices influence learners’ seeing mathematics as embodied.

3.3 Mathematical Vision

Ball (1993) points to the importance of teachers’ mathematical vision (mathematical horizon), which includes connections to other topics, relationships to ubiquitous mathematical themes, exploitation of learners’ natural powers to think mathematically and, most specifically, places where a topic has found use or application in the past. In Chap. 13, it is observed that rarely do learners have any sense of where what they are doing fits into a bigger picture, and possibly this is because teachers are similarly unsure about a bigger picture. Artigue (2011) is quoted as echoing this, noting that ‘pupils do not know which needs are met by the mathematical topics introduced’ and, concomitantly, that they therefore have ‘little autonomy in their mathematical work’ (p. 21). Autonomy can be fostered by taking every opportunity to get learners to make significant as well as routine choices.

Connections and vision are enriched through awareness of mathematical themes, such as invariance in the midst of change, doing and undoing, and freedom and constraint (Mason and Johnston-Wilder 2004, 2006). This is part of a framework for preparing to teach any topic. At the Open University, we developed such a framework, called in its later manifestations SoaT (Structure of a Topic). It brings to the surface six aspects of any mathematical topic corresponding to some degree with three aspects of the human psyche as recognised by Western psychology, namely, cognition, affect and enaction.

The cognitive axis concerns aspects such as concept images (the associations and images that are usefully associated with the topic; the concept images) together with classic confusions and uncertainties that arise for learners in the topic. The enactive axis includes looking for how terms used technically in the topic are based on or derived from everyday words and the ‘inner incantations’ or ‘patter’ (Wing 2016) that can usefully accompany the carrying out of techniques and procedures, as well as the procedures themselves. The affective axis, being connected with emotions and motivation. And hence with desire and disposition, includes the sorts of problem(s) that the topic resolves, the problems that historically gave rise to the topic and in what contexts the topic has proved to be useful. It also includes questions about how the pedagogic choices are likely to support the development of a positive disposition towards the topic, its language, its concepts and its techniques.

Because different groups of students in different situations are different, it does not seem reasonable to try to find one perfectly effective way to introduce students to algebra. An alternative is to see that there are several routes into algebra (generalising structural relationships and expressing these; tracking arithmetic; multiple expressions for the same thing; axioms of arithmetic expressed generally so as to be the rules of algebra). What is worth dwelling on in any particular lesson depends on the people and the situation, so this is where the art of the teacher is required. Lessons based on textbooks which are in turn based on a single hypothetical learning trajectory (Simon and Tzur 2004) are likely to succeed sometimes, but not always. Successful teaching requires sensitivity both to the mathematics (topic and thinking) and to learners, because teaching mathematics is a caring profession. Balancing care for mathematics and for learners is not at all easy. As is well known, two people co-planning a lesson and then teaching it very often end up doing quite different things because of all the differences. Fundamentally, the issue is what the teacher is aware of (what pedagogical and mathematical actions become available) and what they are currently sensitised to notice. That is what makes the difference between effective (in the long term) and successful (in the short term) teaching.

When teachers are themselves thinking mathematically, whether alone or collectively, there is an ethos and a sensitivity to learners that fades when teachers stop doing mathematics themselves.

3.4 Word Problems

The use and abuse of word problems has been much discussed (Gerofsky 1996; Greer 1997; Verschaffel et al. 2000, Mason 2001a, b). Since word problems seem to be unavoidable, it seems sensible to work with them structurally. Some people have tried to teach learners to analyse verbal statements, to locate keywords and, from these, to work out how to find an answer, while the so-called Singapore method is to depict quantities using a bar diagram and then work with them. Ultimately, what has to happen is that the learner uses their mental imagery to enter into the situation to recognise and express relationships in the situation using whatever support devices and modes of presentation are recommended for this purpose. Word problems cannot be solved effectively at ‘arm’s length’ so as to avoid thinking.

As Bednarz et al. (1996) noted, in arithmetic you proceed from the known to the unknown, whereas in algebra you start with the unknown and proceed towards the known. But, as Mary Boole (Tahta 1972) pointed out, what you do is acknowledge your ignorance and denote what you do not yet know by some symbol, and then you express what you do know using that symbol. This is tantamount to tracking arithmetic, when you start by trying to check whether some guess is actually the answer but track that guess so that it can be replaced by a symbol, in order to reach some equations to solve.

If word problems are treated as a domain of play and exploration, so that learners construct their own, changing the context as well as numerical parameters, then the power to imagine a situation, to locate structural relationships and to express them can be enjoyed rather than feared. For example, take the simple context of sharing marbles:

If Anne gives 3 of her marbles to John, they will then have the same number. How many more marbles did Anne have than John to start with?

Of course, you could also be told how many marbles one or the other has afterwards. But look at all the potential dimensions of possible variation, all the features that can be changed: the number of marbles Anne gives away, the effect of her giving them away (maybe she then has twice as many, or half as many, or 5 more than, or 6 less than John), the number of people involved, the number of actions of giving and receiving involved (perhaps John then gives Anne some marbles or gives some to someone else, etc.), the nature of the actions (perhaps Anne exchanges each of her red marbles for two of John’s blues, etc.) and the things being exchanged (sweets, counters, teddy bears, penguins, etc.). Pleasure can be obtained from making up your own variations and trying to resolve them, not simply in the particular, but in the general. This can be done (in simple instances) with very young children, inducting or enculturating them into the ways of mathematical thinking.

Again, it is not the mathematical structure alone (how daunting is a page full of ‘problems’ to be required to ‘do’?), and it is not the pedagogical structure of the task and the interactions, but the two of these together, mediated or held together by the sensitivity of the teacher both to opportunities for mathematical thinking and the particular thinking of her learners.

3.5 Pedagogic Choices

For older children and for teachers, more ‘top-down’ presentations of structure in generalised word sentences or algebraic formats seem to have purchase in drawing attention to the nature of quantitative relations being worked with. This could well be related to, and acknowledging of, extensive prior encounters with additive and multiplicative situations. Parallel approaches for younger children appear to be better supported by the presentation of pictorial models of underlying structure that can be used in similar ways to develop more powerful discourses about the nature of quantitative relations in additive, multiplicative and other patterned situations involving some structural relations.

Chapter 13 covertly acknowledges that terms like ‘direct instruction’ are far from being unambiguous, being used to refer to a wide range of practices. For example, they quote Kirschner et al. (2006 pp. 83–84) to the effect that ‘unguided instruction [is] normally less effective’ than strong instructional guidance. But surely no-one proposes ‘unguided instruction’. Even the much maligned ‘discovery learning’ espoused by Bruner (1966) never meant learners being left on their own to ‘discover’ without any intervention or guidance. The delicacy and importance of an informed awareness, including awareness of awareness (Mason 1998), cannot be overstated.

‘Top-down’ or ‘direct instruction’ is often interpreted as the teacher telling learners what to do, perhaps on a worked example, perhaps as a sequence of instructions. But working in a whole-class plenary mode need not be like this. Rather, the teacher can draw out learners’ ideas and can focus and direct attention while calling upon learners to make use of and develop their own powers. Teachers can shepherd (Towers 1998; Towers and Proulx 2013). Teachers can summon learners’ past experience. Then a little bit of ‘telling’ can indeed be telling, can be effective when it occurs at an appropriate moment (Love and Mason 1992, 1995). Time for learners to work for themselves, to develop a personal narrative or self-explanation (Chi and Bassok 1989), and time for learners to try out their articulations with colleagues and to hear other learners’ narratives is also important. What seems most important is not to be prescriptive as to how a lesson should go. Rather, teachers need to be supported in developing sensitivities to notice, to be aware of, what and how learners are thinking, so that the tasks are used richly. Retaining the complexity of teaching is vital, responding to and making use of the rich complexity of the human psyche, rather than trying to simplify acts of teaching as if on an assembly line. A contribution to structuring teacher-learner interactions can be found in the six modes described in Mason (1979), which outline six modes of interaction based on the systematics of Bennett (1966, 1993). To these can be added the five strands of mathematical proficiency proposed by Kilpatrick et al. (2001), the five dimensions of mathematically powerful classrooms proposed by Schoenfeld (2014) and the habits of mind articulated by Cuoco et al. (1996). There are probably many others. More work is needed on simplifying and coordinating the many different ways of preparing oneself to make effective pedagogic choices when planning and, in the moment, preserving the complexity of the human psyche but not overcomplicating it.

For example, Davis (1996) introduced the notion of hermeneutic listening in which the teacher listens to what learners are saying and watches what learners are doing, rather than listening for what they want to hear or watching for what they want to see. One way to sensitise yourself to listening to is through what Malara and Navarra (2003) called babbling, by analogy with a young child in a cot making the sounds of sentences without yet having the words. The label babbling can alert you to trying to hear what may be behind the words, what learners may be trying to express, even though they may not be using terms correctly. So babbling can serve as a trigger for hermeneutic listening. The didactic tension (Mason and Davis 1989), which arises from the work of Brousseau (1997), suggests that the more clearly and precisely a teacher specifies the behaviour they want learners to display, the easier it is for learners to display that behaviour without actually generating it for themselves. This explains why hermeneutic listening, ‘teaching by listening’, is so important. It is so easy to fall into ‘training learner behaviour’ rather than providing conditions in which learners ‘educate their awareness’ (Gattegno 1970; Mason 1998). As Towers and Davis (2002, p. 338) write:

These attentive and tentative modes of engagement are offered in contrast to those that frame classroom interaction in terms of causal actions and control – which, once again, we might characterise in terms of a shift from architectural to biological senses of structure. An important element in this manner of pedagogy is its embrace of ambiguity and contingency.

One domain of pedagogic choices that seems not to be mentioned very often has to do with learner involvement in making choices. By getting learners to make significant mathematical choices, and by getting them to construct mathematical objects, exercises and examples, they can push themselves just as much as they feel capable of, rather than depending on the teacher to provide a range of examples suitable for different learners (Watson and Mason 2005). These and other pedagogic strategies could be brought to teachers’ attention more widely, through engaging them in effective personal experiences.

3.6 Reasoning, Justification and Proof

‘Proof’ is another aspect of mathematics that is experiencing a revival in mathematics education. But proving things, justifying conjectures by means of mathematical reasoning, is probably not so easily ‘taught’ as enculturated into. When learners discover that they can ‘know things for certain’ in mathematics, not because someone told them so or because they have seen a convincing number of instances to believe it is always true, but because they can reason it out for themselves, their interest and engagement and their disposition towards mathematical thinking can be enriched. Probing learners’ recognition of relationships, in particular, and perception of properties, in general, such as in Molina et al. (2008), Molina and Mason (2009) and Mason et al. (2009), among many others, can enhance sensitivity to which experiences might be useful for learners and hence what pedagogic choices might be effective, concerning the development of their contact with mathematical reasoning. Alerting teachers to pedagogic possibilities for promoting reasoning and for learners becoming aware of their reasoning in the midst of teaching is an ongoing process.

To reason successfully requires awareness of generality, of properties being instantiated rather than simply some relationships holding in some particular situation(s). Only then is it possible to reason making use of previously agreed properties to reach fresh conclusions. But not all reasoning has to be original to the learner: at first, learners can be shepherded towards such reasoning through participation. They can be immersed in such reasoning and invited to engage in such reasoning for themselves. They can also be shown examples of reasoning that is more complex than they might be expected to construct for themselves, so they are immersed in extending and enriching their experience of reasoning. Examples of teachers doing this are always welcome.

4 Beyond Whole Numbers

As Bob Davis pointed out (Davis 1984), if children experience the operations of addition over a long period of time, followed only then by subtraction, followed then by multiplication and finally by division, and only then encounter ‘numbers’ that are not whole numbers, it is not surprising that they revert to addition whenever they are faced with a situation in which they do not know what to do. Naturally, they enact the first action that becomes available. If they have learned to park the first action, then they have a chance of probing beneath the surface to find out what is really involved; otherwise, they are likely to disappoint their teachers.

Treating number as a complex whole, incorporating all four operations as early as possible, and drawing on Davydovian ideas by introducing number in the context of units, of some feature being measured, is more likely to lead to an appreciation of arithmetic as the study of properties of numbers rather than as the calculating of answers (Thompson et al. 2014). If they are exposed to scaling as well as repetition, so that multiplication is not identified with repetition, then they have a chance of appreciating and comprehending, if not understanding, the basics of mathematics. Complexity is not best taught through oversimplification, through isolating components and then expecting learners to recompose them into a complex appreciation.

Teachers’ mathematical ‘being’ is manifested moment by moment in the classroom and is picked up subliminally by learners. By participating in mathematical thinking themselves, by enriching and complexifying their sense of mathematical structure, by exhibiting mathematical ‘habits of mind’ (Cuoco et al. 1996), by getting to grips with underlying structures in mathematics such as covariation (Thompson and Carlson 2017) and by enriching the range of pedagogical actions to which they have access, teachers can keep themselves fresh and so provide learners with an immediate and enriching experience from which they can learn. What is needed in the future is evidence for and examples of a truly humane way for humans to teach each other.