Explicating professional modes of action for teaching preschool mathematics^*

* A previous version of this paper was presented at The Eighth Nordic Conference on Mathematics Education in Stockholm, May 30 – June 3, 2017.

ABSTRACT

Play-based preschool pedagogy usually relies on informal teaching while policy trends and some research call for increased formalisation of the pedagogy. Using Bernstein’s concepts of classification and framing, this article characterises mechanisms that link evaluation of preschool to the push towards the formalisation of teaching in preschool. Moreover, it is suggested how preschool teaching of mathematics can be conceptualised in a way that widens the pedagogical responsibilities of the teachers to include a broader range of social activities than typically expected. These responsibilities concern how teachers are involved in pedagogical situations, if situations are planned and if the mathematics is a pedagogical goal in the situation or instrumental in some other activity. It is argued that a practice built on these principles could both honour well-developed play-based preschool practices and provide a structure for teaching preschool mathematics in which using traditional child–teacher interaction is only one of many options.

KEYWORDS:

• Preschool education
• preschool teachers
• mathematics instruction

1. Introduction

Visibility is not about what you see but about what you teach your eyes to look for. In preschool, the social pedagogy tradition emphasises broad developmental goals and the child’s own interests and learning strategies are at the forefront (Bennett, 2005). In such pedagogy, the learning situations might be complex, but the teaching acts are intentionally kept invisible. The work of the teacher is typically described as tuning in to the child’s interest and through direct interaction challenging the child (Pramling Samuelsson & Asplund Carlsson, 2008). Theories of social pedagogy therefore tend to focus on complex theories of learning, such as those of Piaget, Vygotsky or Bruner, instead of descriptions of teaching. As we will argue, the teaching practices that are used within the social pedagogy tradition remain unarticulated when compared to school pedagogy where explicit forms of pedagogy are often compared and analysed. However, an unexplicated and invisible pedagogy may not be a problem as long as the teaching practice governs itself. It is enough that teachers conform to a common ideology that underlies the pedagogy. Even when the teaching practice is subject to accountability in a larger system, invisibility may not be a problem as long as the effects of the pedagogy, in terms of children’s learning outcomes, can be documented and measured. However, in the social pedagogy tradition, formal assessment of the child is not preferred, and goals are often informally evaluated (Bennett, 2005). This presents a problem – how are teachers to be held accountable for their practice when the outcomes of that practice cannot be measured? What remains to inspect is the teaching itself; thus, what constitutes a teaching act can no longer remain invisible. This is the concern of this article.

I use Sweden as a case and invoke sociological theories of school systems and pedagogy to characterise the Swedish preschool system. By using such a theoretical lens, it will be possible to explain why recent developments in Sweden concerning assessment of teaching risk pushing the preschool system towards more formal teaching and why such a shift is an expected outcome of the system. Instead of taking a political stance and criticising the potentially negative effects of increased assessment of teaching and formalisation of teaching practices, I will take the system and its effects as a point of departure in order to initiate a discussion regarding preschool teachers’ pedagogical choices when teaching. My use of the words “risk” and “negative” in the preceding sentences reveal that I do take a critical stance towards the increasingly formalised teaching of mathematics in preschool. However, I will not spend time arguing this case. Instead I will use somewhat neutral terms to briefly describe different views of young children’s mathematics and different ideas about how to organise pedagogy. Broadly, these different pedagogies are often classified in terms of a social pedagogy tradition and a pre-primary tradition (Bennett, 2005; OECD, 2006). The Swedish preschool historically has a strong social pedagogy tradition but can nowadays be seen as adopting a dual approach that mixes a pre-primary and a social pedagogy (Sheridan, Williams, Sandberg, & Vuorinen, 2011) and has a particular focus on play (Synodi, 2010). I will describe how the outcome of a recent evaluation of Swedish preschool is likely to move Swedish preschool education towards a more teacher-driven, formalised pedagogic approach where teaching is more visible. I will use some elements of Bernstein’s (1971) theory to put the reasoning above in more precise terms.

My hypothesis is that if the complex mathematics pedagogy inherent in a dual approach could be explicated in a structured way, the call for more visibility would not need to push Swedish preschool pedagogy towards a more teacher-driven approach. The purpose of this article is therefore to propose a conceptual framework for teaching mathematics in a play-based preschool practice. I call this framework “conceptual”, but it is not intended as a research framework nor as a basis for evaluation. The framework is not theoretical in the sense of being able to provide predictions. The framework is connected to existing preschool practices but is not analytically derived from such practice. Rather, the framework is conceptual in the sense of providing characterisations of some fundamental aspects of the work that preschool teachers can do when arranging a productive environment for children to learn mathematics.

While most of the reasoning is based on the Swedish preschool context, a framework like this can be relevant in many other preschool practices that are fully or partly defined by a play-based approach. In the remainder of this article, I will first review research on preschool mathematics and its teaching to establish the different perspectives involved and argue that the framework I will propose fills a gap by incorporating a structured way of looking at teaching mathematics in a play-based pedagogical model. I will then review the Swedish system through some elements from Bernstein’s theory of educational codes (Bernstein, 1971, 1975). After this, I will present my framework and discuss it in relation to empirical examples from practice.

2. Theories of teaching and learning preschool mathematics

There are several reasons why mathematics is a given subject in almost any form of formal schooling and has for many years become an ever more pronounced activity, even in preschool. The failure of schools to properly handle differences between pupils often leads researchers to call for earlier interventions in the mathematical teaching of young children (Gersten, Jordan, & Flojo, 2005), and because it has been shown that young children can learn surprisingly advanced mathematics, researchers have called for such mathematics to be included in preschool (Cross, Woods, & Schweingruber, 2009; Ginsburg, Lee, & Boyd, 2008). Consequently, Nordic preschool models have been updated with more explicit objectives for mathematics (Kunnskapsdepartementet, 2006; Skolverket, 2016). With more and earlier prescribed learning, the question becomes: what types of teacher guidance lead to this learning?

Mathematics as a scholarly practice is typically associated with being abstract and symbolic. Psychologists and educational scholars have repeatedly argued that while preliminary mathematical concepts may well arise from everyday interactions with a mathematically infused culture, formal mathematics must be taught. In Vygotsky’s work, this is almost true by definition in the way he conceptualises spontaneous and scientific concepts (Vygotsky, 1962). From a sociological perspective, Bernstein (1999) introduces a similar dichotomisation in terms of vertical and horizontal discourse, where the vertical component is often associated with taught knowledge, while the horizontal component can supposedly be picked up in everyday situations. When mathematical objectives for an educational practice are explicated, they tend to involve what Vygotsky calls scientific concepts and what Bernstein calls vertical discourse. Some researchers claim that this calls for teaching activities to become more formalised. For example, after reviewing play-based approaches and instruction built on teachable moments, Ginsburg et al. (2008) conclude that

[d]eliberate instruction – teaching – is of course required by curriculum […]. It is the responsibility of educators to do more than let children play or respond to teachable moments. […] Preschool teachers need to engage in deliberate and planned instruction (p. 8).

However, this conclusion is heavily debated. Interestingly, proponents of increased formalisation of preschool teaching often try to tone down the differences between ideas such as: play versus learning, adult-directed versus child-directed activities, adult-initiated versus child-initiated and student-centred versus teacher-centred/directed (Fuson, Clements, & Sarama, 2015). In doing so, they claim that more guided forms of teaching include playful components. By contrast, sceptics of formalised teaching at preschool level tend to promote more guided forms of play as something very different from formal teaching, even when the latter includes playful components (Weisberg, Kittredge, Hirsh-Pasek, Golinkoff, & Klahr, 2015).

Advocates of formal teaching in preschool understand that formal teaching approaches are not feasible for very small children. A central dilemma then becomes: when should formal teaching be introduced? A typical answer to this question is described by Ginsburg et al. (2008). Their approach involves having the child learning mathematics through what Dewey (1976) described as the child’s crude impulse to count, measure and arrange things in order. It is usually assumed that this everyday mathematics can be picked up by the child without formal teaching and that such knowledge can be developed through play-based approaches. When children get older, the play-based approach is gradually merged with a formal instruction approach, and the everyday mathematical concepts are supposed to be developed into, or subsumed by, formal concepts (Ginsburg et al., 2008).

Dijk, van Oers, and Terwel (2004) suggest an alternative approach to that of Ginsburg et al. (2008). Instead of starting with a psychological perspective concerning the mathematics that young children can do and learn, they re-conceptualise mathematics as a form of semiotic activity, that is, an activity of meaning-making. They then connect this to play by noting that “early semiotic activity can be identified in schematising activities in early childhood play” (p. 71). A significant feature of this approach is that the same conceptualisation of mathematical activity can also be developed in a school context, as seen in the realistic mathematics approach envisioned by Freudenthal (1991).

An alternative to focusing on the semiotic aspect of mathematical activity is to consider mathematics in its cultural context by following the anthropological work of Bishop and his suggestions for curricula in terms of mathematical enculturation (Bishop, 1988, 1991). Bishop suggests that while mathematics comes in many forms depending on the cultural context, there are six general activities that tend to unite cultures when it comes to the production and use of mathematics: counting, measuring, locating, designing, playing and explaining. Understanding mathematics in terms of these six activities has proven useful for preschool teachers in their work (Johansson, 2015), and conceptualising young children’s mathematics in terms of these mathematical activities has been promoted through initial teacher education (Helenius, Johansson, Lange, Meaney, & Wernberg, 2016b) and continued teacher professional development (Helenius et al., 2015) in Sweden. It was also a central perspective in the work leading up to the revised national curriculum for preschool in Sweden (Utbildningsdepartementet, 2010).

It should be noted here that in approaches similar to that of Ginsburg et al. (2008) above, the conceptualisation of the teaching of mathematics and mathematics itself are intertwined. Mathematics starts out as an everyday activity, and both the mathematics and the teaching gradually become more formalised as children develop. In an approach in which mathematics is re-conceptualised in other ways, it remains to be explained how to teach such mathematics. As Dijk et al. (2004) put it: “[…]. how can educators organise schematising activities in early childhood education?” (p. 71). A corresponding question can be asked for the approach based on Bishop’s mathematical activities: how can educators organise such activities? According to Johansson (2015), in the Swedish preschool context, teachers tend to appropriate the six common cultural activities rather easily in the sense of establishing a terminology for the types of mathematical situations that frequently occur in the preschool context. But identifying and labelling mathematics in which children count, measure, locate, design, play and explain is, however, not necessarily enough to ensure that preschool teachers also effectively and deliberately arrange such situations in a complex preschool practice.

For the pre-primary tradition, the challenges associated with understanding preschool teachers’ work are less than those associated with social pedagogy or play-based approaches. In a pre-primary tradition, teachers and researchers alike can translate many teaching concepts from school teaching. Approaches such as these have proven successful, with work by Clements and Sarama serving as a particularly distinct example. The professional development designed and researched by Sarama, Clements and colleagues revolves around the concept of a learning trajectory (Sarama, Clements, Wolfe, & Spitler, 2016; Sarama, DiBiase, Clements, & Spitler, 2004). Essentially, the teachers are trained to understand how children learn and to understand progressions of tasks that are designed to provide opportunities for such learning. Furthermore, in the pre-primary tradition, research on preschool teachers’ knowledge and competence tends to follow in the footsteps of similar research for school by, for example, developing constructs and measurements for preschool teachers’ pedagogical content knowledge (McCray & Chen, 2012).

By contrast, a play-based or social pedagogy approach places different demands on the teacher because s/he cannot rely on the possibility of arranging sequences of activities in a structured manner. One widespread idea in play-based pedagogy is to use so-called “teachable moments” (Copley, Jones, Dighe, Bickart, & Heroman, 2007) where the teacher observes the children’s play and other activities to identify and interact with spontaneously emerging situations that can be exploited to promote learning. However, research on play-based pedagogy tends to focus on direct teacher–child interaction (Pramling Samuelsson & Asplund Carlsson, 2008), such as in the form of questioning and affirmation (Sæbbe & Mosvold, 2016). Interactionist approaches such as these are quite common (Sheridan, 2011). What seems to be missing from this research is an analysis of what kind of work outside of child–teacher interactions teachers can do to enable learning in play-based practices.

When teaching is analysed from an interactionist perspective, it is unsurprising that more explicit and planned direct interactions characteristic of a pre-primary approach comes across as more efficient. An explanation for this can be provided by using the famous war theoretician von Clausewitz’s characterisation of strategy and tactics: where strategy refers to pre-planned modes of action and tactics refers to “in the moment” decisions and arrangements (Von Clausewitz & Graham, 1873). In a pre-primary approach with more direct instruction, a teacher has control over time and space for implementing tasks and can use powerful tools in the form of tasks or sequences of tasks to arrange learning situations. The choice of tasks and other pedagogical tools, as well as when and how to implement them, can be planned in advance and are, therefore, more strategic. The teacher need not rely on acting in the moment. In an interactionist play-based pedagogy, the teacher instead must use a tactical approach and work effectively to make decisions and arrangements in the moment, when chances appear. An argument for such an approach is that the teacher can use the interests of the child as a driver of learning in a more appropriate manner. However, a difficulty with the child-initiated approach is maintaining control over what kind of interactions and how many chances of such interaction will occur. Such objections motivate Ginsburg et al. (2008) to conclude that it is not enough to rely on teachable moments. Nevertheless, it is worth noting that in a play-based approach, preschool teachers may do a lot of planning and other work that is not visible in child–teacher interactions. In von Clausewitz’s terminology, this amounts to recognising and describing the strategic work that underlies a play-based teaching approach.

The framework I will propose removes the focus on the interactionist approach by expanding on the line of work previously done with colleagues (Helenius et al., 2015, 2016b). Building on the observation that in the Swedish preschool tradition, a significant amount of children’s time is not guided by teachers (OECD, 2006), I will suggest an alternative approach. There are many situations in which children learn mathematics, without this being the direct pedagogical intent. Moreover, children learn mathematics in situations in which the teacher is not a participant. What happens in this seemingly “unorganized” time is hence of vital importance for the child’s interaction with mathematics. Thus, it is my intention to suggest modes of action for preschool teachers that involve a wider class of practices than direct teacher–child interactions. Before presenting my suggestion, I will look at the Swedish pedagogical preschool practice using Bernstein’s theory to locate my framework within a wider school system perspective.

3. Play-based preschool pedagogy: Practice, framing and classification

Among the plethora of sociological theories that describe the complex co-dependence of institutionalised learning, knowledge and teaching, I find Bernstein’s (1971) typology of educational knowledge useful for my purposes. Bernstein’s theories in the sociology of education, which typically connect theories of language, class, pedagogy and school systems, were developed over 30 years and culminated in his 2000 book (Bernstein, 2000). Bernstein’s work has been extensively used by educational researchers and has proven relevant for analysis of preschool because of how it “provides a conceptual framework with great power in describing the empirical” (Tsatsaroni, Ravanis, & Falaga, 2003, p. 391). From Bernstein’s both very broad and very fine-grained theory, I select the concepts of framing and classification, which are particularly well suited to discuss the visibility of pedagogy discussed in the introduction. Classification refers to the strength of the boundary between what is considered educational knowledge, in our case mathematical knowledge, and what is not. It should be noted that classification does not refer to what type of content is considered but only the firmness of the boundary. To define framing, Bernstein (1971) provides the following definition:

Frame refers to the form of the context in which knowledge is transmitted and received […] Frame refers to the strength of the boundary between what may be transmitted and what may not be transmitted, in the pedagogical relationship […] Frame refers us to the range of options available to teacher and taught in the control over what is transmitted and received in the context of the pedagogical relationship. (p. 50)

In both Bernstein’s later works and its applications, framing has become especially focused on pacing, choice and sequencing of content, as well as how the power over these processes is distributed between teacher and student. For my purposes, however, the two other aspects of framing are more relevant. First, I focus on the issue of the form of the context. As I will show in later examples of play-based preschool mathematics teaching, a relevant aspect of framing concerns which parts of the child–teacher interaction should be considered part of the educational practice. In some sense, this is analysing framing at the roughest possible grain size. In terms of power and control, I am not primarily interested in its distribution between child and teacher. As Bernstein already remarked in 1971, framing and classification relate to power and control at all levels in the school system. My primary interest will be how framing is affected by stakeholders outside of the preschool.

To explain how weak framing and classification can be in a play-based preschool setting compared to standard school settings, let us imagine being a visitor to a grade 2 mathematics lesson that starts with the teacher and children discussing something that happened during recess. Even for an untrained observer, it will be perfectly clear when this part of the lesson ends and the mathematical pedagogical activity begins. The shift will be signalled either by the teaching beginning to discuss the mathematical matters or the pupils beginning to work in their textbooks or on a worksheet. Now, consider a movie clip from a preschool that was gathered during a professional development project.

Example 1, the gingerbread situation. A preschool teacher and two children are baking gingerbread cookies. They are talking about one of the children’s grandmothers while cutting three pieces of dough from a big one and distributing them among themselves. They roll the dough and discuss how thin it should be and so on. An observer can only speculate whether this is an activity with any particular pedagogical intent and, if it is, whether the intent is mathematical. As it happens, this clip was indeed a planned mathematical activity, that is, the students were baking cookies, but the teacher had planned to use the situation to raise the children’s awareness of certain mathematical matters. Different experts, other teachers and certainly the children may have quite different ideas about what was mathematical in this situation as well as different ideas about what activities were pedagogical with respect to this mathematics.

The fact that an observer in example 1 would have hard time distinguishing what are the pedagogical acts and what content the pedagogy is supposed to transmit is an indicator of weak framing at play. By contrast, the content is more obvious in the grade 2 example, which indicates how typical school mathematical activities are more strongly framed and more strongly classified than most preschool activity concerning mathematics. Later in the article, I will provide seven more examples from preschool, that, while showing some variation of framing and classification, all still strongly differ from a school setting.

Bernstein (1971) argued that a combination of weak classification and weak framing “may only work when there is a high level of ideological consensus among the staff” (p. 64). Given the long tradition of social pedagogy in Swedish preschools, this consensus is probably at work in Swedish pedagogical preschool practice. Moreover, Bernstein (1971) discussed an infant school in England which at that time was not subjected to accountability to outside authorities. He argued that one factor that made the school weakly framed and classified was that the school was “relatively free of control by levels higher than itself” (p. 69). This has also been the case in Sweden but has recently changed. The Swedish School Inspectorate has, in recent reports, formulated strong views on what teaching in preschool should entail (Skolinspektionen, 2016). The government has commissioned the National Agency of Education to work on a new national curriculum that, among other things, clarifies the meaning of the term “teaching”, which is a term that is in fact not present in the current national curricula (Regeringen, 2017; Skolverket, 2016). In its reports, the inspectorate starts out by using the definition of teaching from school law: “In this law teaching refers to such goal directed processes that under the guidance of teachers or preschool teachers aims towards development and learning through acquisition and development of knowledge and values” (Skollagen, SFS-2010-800, §3, my translation). An example of how the inspectorate applies this definition is the following.

A grown-up sits with some younger children on the floor when one of the children points at a car and says “there, car”. “Yes, there it is”, the grown-up replies. The child points at another car and says, “one more”. The grown-up nods but does not further engage in the child’s curiosity for the cars. […] Through this, the child invites dialogue and play, but the grown up does not use the situation to teach and cooperate with the child, and thus build on the interest of the child (Skolinspektionen, 2016, p. 19, my translation).

The inspectorate establishes that situations such as these “mean that the conversation does not develop into a goal-oriented process that aims for development and learning” (Ibid, p. 19). Hence, the inspectorate presents their interpretation of teaching in preschool as a process that is initiated by a child showing an interest in something. Preschool teachers are then to interpret this interest in line with some of the goals in the national curriculum and direct the conversation towards those goals. This is what represents a goal-directed process for the inspectorate. This is also illustrated by explicit schematic illustrations in the report.

Further, the inspectorate writes that there is a pervading ambiguity about what teaching means among preschool teachers and preschool leaders. Some explicitly do not want to use the word “teaching”, as the word is associated with school. Instead, teachers may say that “the children learn all the time” (p. 15) or that “we don’t have teaching, we explore together” (p. 15). However, it should be noted that neither the section of the school law that concerns preschool particularly nor the national curriculum uses the word “teaching”. Still, the inspectorate concludes that preschool teachers must think about their role in children’s learning and move forward as teaching preschool teachers (Skolinspektionen, 2016, my translation, italics in original).

The inspectorate generally concludes that

[t]he concept of teaching and how to carry out teaching need to become manifest. It also needs to become clear how preschool teachers’ responsibility for teaching can manifest itself, so that it makes preschool teachers role as teaching teachers visible (Skolinspektionen, 2016, p. 7).

As such, we can summarise the views of the Inspectorate in three points: 1. Teachers need to be able to explain what teaching is. 2. Teaching should be visible. 3. Teaching concerns direct teacher–child interaction.

In terms of applying Bernstein’s theory here, we can first note that the inspectorate does not describe the goals themselves, their structure, their interpretation or the realisation of concrete goals in practice. Hence, the inspectorate does not deal with the classification of knowledge. When they say that teaching should be visible (point 2), they are asking for stronger framing. Again, I am not referring to finer points, such as sequencing and selection of pedagogical acts, but to framing at the coarser level: what should count as pedagogical acts? In terms of framing, the inspectorate also carries out a fine balancing act, when they firmly insist that the starting point for a pedagogical act should derive from the child’s interest (weak framing) but be clearly directed towards the curricular goals by the teacher, by means of some described pedagogical acts (stronger framing). As noted previously, Bernstein remarks that teaching that relies on weak classification and framing must be based on a closed, explicit ideology (Bernstein, 1971, 1975). My interpretation here is that this starting point of letting the interest of the child lead is an example which draws on such a closed, explicit ideology. In the discussion of preschool teaching, the inspectorate relies on well-known and well-read Swedish literature that emphasise the child-driven interactionist perspective (e.g. Pramling Samuelsson & Asplund Carlsson, 2008). So, at the child–teacher level, the inspectorate insists on weak framing and the type of interactionist view on teaching I discussed earlier. Yet, in presenting a relationship between teachers and “the outside”, the inspectorate tries to strengthen the framing by creating a tighter bond between the pedagogical acts and the curriculum goals. Children are to initiate situations, but teachers are to direct the situations so that they align with the curriculum goals.

4. Theorising preschool teaching

Taking a realist position, I see the pressure for increased visibility of pedagogical acts as an expected outcome of having a School Inspectorate. In a system that has low levels of classification and no evaluation functions, what remains to inspect is the pedagogy itself; thus, a call for stronger framing is unsurprising. I agree with the School Inspectorate in that it is important for preschool teachers to be able to explain their ideas about teaching. On the other hand, I see the exclusive consideration of the child-initiated, interactionist type of teaching as a major limitation. As I will indicate later with examples, many mathematics learning situations that depend on teacher actions in different ways are not of this kind. Therefore, I will suggest a structured way of viewing the complexity of mathematics preschool pedagogy by suggesting three dimensions of teacher action with respect to mathematical pedagogy: pedagogical explication, teacher participation and situational planning. Each of these dimensions in practice represents a continuum of possibilities, but for illustrative purposes I will explain them in terms of dichotomous options. I do not suggest these as normative categories but as examples of how the complex pedagogical practice of teaching preschool mathematics could be explicated. After presenting the three dimensions, I will give eight examples derived from video recordings that were supplied to me by preschool teachers and discuss how the three modes of action are combined in these examples. In the discussion that follows, I will explain how these dimensions can help with the framing dilemma

4.1. A dimension of pedagogic explication.

When considering the teaching and learning of mathematics, two different ways of including mathematics can be identified. The first option is a classical teaching situation. The participants know that the activity concerns the learning of some specific mathematical content. They are experiencing, learning or practising this mathematics as the explicit objective of the situation. Such situations may include activities that are not per se mathematical but can play the role of highlighting some mathematical phenomena or supplying a reality in which some mathematics can be applied. Following Walkerdine (1988), we call such situations pedagogical with respect to mathematics. The second option is that the situation is fundamentally about something else, but some mathematical activity is included as an instrument to achieve the objective of the situation. This type of situation is called instrumental (Walkerdine, 1988). In our research group, we have examined instrumental and pedagogical situations in several studies. In particular, we have observed and analysed the consequences of the fact that one situation can be pedagogical for the teacher but instrumental for the children and at the same time be pedagogical for some children and instrumental for others (Helenius, Johansson, Lange, Meaney, & Wernberg, 2016a). However, for the current theorisation, it suffices to consider if children experience a situation as pedagogical or not, as all situations I am concerned with will on some level have pedagogical intentions or effects from the teacher’s point of view. In example 2 below, the decision to make the robot available to children have pedagogical intentions, but children play with it, they are not thinking of it as a situation where they are supposed to learn some particular mathematics. The central point then becomes that teachers can make the active choice of explicating the mathematical content of a situation or refraining from doing this. This choice can be made beforehand, when a teacher chooses to plan a pedagogical or instrumental situation, or on the fly, when a teacher observes a mathematical possibility in a situation that she chooses to either act on by shifting children’s focus to the learning of some mathematics or not. Similarly, when children act on their own with no teacher involvement in a situation that does contain mathematics, the mathematical learning component of the situation may or may not be visible for the children. It is when the children perceive the situation as being about learning something that can be classified as mathematical that we call the situation “pedagogical”.

4.2. A dimension of teacher participation.

It is quite obvious that children in preschool spend a lot of time conducting activities with no preschool teacher involved. Such situations can provide children with opportunities to use mathematical knowledge, participate in mathematical activities or encounter mathematical matters in other ways. Therefore, time spent in preschool with no teacher participation can contribute to children’s mathematical development in important ways. The same is true for time spent outside preschool, but the difference is that preschool teachers can influence the mathematical content and children’s experience of mathematics in situations in the preschool environment. One obvious example is simply planning specific activities that they ask the children to do, but there are also more subtle ways, such as planning and modifying the environment in and around the preschool facilities so that it stimulates mathematical activities. In example 2, shown below, children play with a programmable toy robot, which was only available to children for their free play. Telling certain stories or working with specific themes can also, in relatively predictable ways for preschool teachers, influence what children choose to do and what type of mathematics they may encounter.

4.3. A dimension of situational planning.

This last dimension relates exactly to what Ginsburg et al. (2008) term as “teaching moments” which has been discussed as “acting in the moment” versus formally planning learning situations in Swedish preschool mathematics pedagogic discourse. It is of course important for preschool teachers to use the option to plan situations that that they participate in themselves and where the included mathematics is pedagogical or instrumental. Planning has a distinct advantage in that it gives the teacher time to predict what may happen in a situation and prepare for different options. But mathematically interesting situations can occur also outside of planning. Refraining from planning all situations is not only a matter of resources. Situations that are not directed by the teacher also allow for experimentation and play in which children may use mathematics. As already mentioned, there are many ways in which preschool teachers can affect what happens during “free time”. Doing this is also a part of the planning. However, what I discuss here is the planning of distinct situations. My point is that preschool teachers have choices concerning what situations they want to plan and what to deliberately leave unplanned, and the respective professional competence to act in planned and unplanned situations will be very different.

4.4. Eight examples from practice

I will now illustrate and discuss the three dimensions using eight examples.

Example 1: Baking gingerbread (Instrumental, Participatory, Planned). As a first example, we refer to the situation involving baking gingerbread cookies mentioned earlier. The video of this situation was sent to me as a situation that was planned to include mathematical learning possibilities, specifically about measuring. But for the children, the task comes across as clearly being about baking. The mathematical activities intended by the teacher are embedded in the activity. Therefore, this mathematics has an instrumental role rather than a pedagogical one from the children’s perspective. In summary, the situation is Instrumental, Participatory and Planned.

Example 2: The robot situation (Instrumental, Non-participatory, Spontaneous). In this movie clip, a child plays with a small, programmable robot that can move on a map while talking to another child who is watching. No preschool teachers interact with the children. The robot has buttons for 90-degree rotations in either direction as well as for forward and backward movement. You can press those buttons in sequence and then press “go”, and the robot will execute the corresponding programmed sequence of movements. The child tries to make the robot go to a particular spot on the map, following the roads, by executing a number of programmed movement sequences. Sometimes, the child makes errors and must start over.

The movie clip was sent to me as an example of a child-initiated activity that was not directly planned by a teacher but where a teacher retrospectively thought the activity contained some mathematics that may affect the child’s mathematical development. This situation is therefore Instrumental, Non-participatory and Spontaneous. While the existence of the robot is on some level a result of planning, it should be noted that the dimension of situational planning described above only concerns the direct planning of the situation. As in this case, there can be many pedagogical choices made by teachers that enable situations that are not directly planned.

Example 3: Sorting buttons (Pedagogical, Participatory, Spontaneous). This video sequence was provided to me by preschool teachers as an example of a situation that was unplanned (spontaneous). Three children sit at a table, sorting buttons. Initially, they sort the buttons in different ways. The teacher eventually engages in the situation and encourages sorting according to size. Deliberately using words such as big, small, bigger and smaller directs the children’s attention to the meaning of these words in the context of working with round objects.

The involvement of the teacher makes this situation participatory. In this example, it is easy to detect a learning goal, and when viewing the video, it is also relatively obvious when the situation becomes pedagogical. The teacher intervenes and adjusts the situation to be about some particular mathematical concept (hence pedagogical). This situation in itself hence becomes framed in a stronger way but still emanates from a situation initiated by the children. This is the type of situation that falls under the scope of preschool teaching in the report from the School Inspectorate (Skolinspektionen, 2016).

Example 4: Pairs of cups (Pedagogical, Participatory, Planned). This sequence was given to me as an example of a planned activity. The teacher has gathered three children at a table with several plastic cups. The activity is a memory game, and each child has a different number of cups. When the sequence starts, Lina counts her cups while sequentially pointing to them and reciting the counting words in the standard way. She says, “Ten cups, and Alex has two”. The teacher asks Alex how many pairs he has, and Alex responds, “Six.” The teacher says, “The total was six” (referring to numbers written inside the cups as a basis for the number-memory game). The teacher then puts his two cups beside each other while saying, “With pair I mean two together like this. How many pairs do you have Lina?” Lina answers, “Ten.” The teacher responds, “You had ten cups, but I mean like this, this is one pair.” The teacher places two cups beside each other and says, “This is one”. She then takes two more cups, places them next to the first pair and says, “Two”. Lina repeats the word “two”. Lina then continues by grabbing pairs of cups, placing them in line with the previous pairs while counting “three, four, five” and emphasising the last counting word. The teacher asks the last child how many pairs he had, and the child responds with, “Zero”.

This situation is a rather typical planned teaching situation and while it starts out as a game, it ends up being about learning the specific concept of a pair. The teacher carries out direct instruction concerning this concept as an embedded activity of the memory game. The direct instruction and the explicit content means this situation is both classified and framed in a comparatively strong way. It is quite obvious in the video that the attention of the children shifts from the game to the concept of pairs. The situation is thus Pedagogical, Participatory and Planned.

Example 5: Drawing bridges. (Instrumental, Non-participatory, Planned). In this situation, children are drawing images of bridges individually. Earlier, the children had gone on a walk with the teacher during which they crossed bridges; moreover, bridges had been a theme in the preschool for some time. The situation is planned because it was part of a theme, and the teacher had instructed the children to look closely at and think about the bridges during the walk so that the children could draw them when they returned to the preschool. However, the teacher is not participating in the situation. The drawing of bridges contains many elements of the mathematical activity of designing (Bishop, 1988, 1991), but it is largely an instrumental situation for the children. The mathematical elements in the drawings were not in focus until a later follow-up session. Hence, this is an Instrumental, Non-participatory and Planned event.

Example 6: Play with Lego (Instrumental, Participatory, Spontaneous). Here, four children are sitting with the teacher and playing with Lego. At one point, one of the children constructs a symmetrical construction and says she made an airplane. None of the other children had at this point made symmetrical constructions. The teacher talked to the child and directed the other children’s attention to the airplane. While doing this, she used the term symmetry, but allowed the discussion to still be about Lego-building and the airplane rather than about what symmetry means. The situation was thus Instrumental as well as Participatory and Spontaneous.

Example 7: Continuing patterns (Pedagogical, Non-participatory, Spontaneous). In this situation, two children are making bracelets by putting beads on a string. After a while, they instead start to make patterns and then switch bracelets with each other to see if the other child can notice the pattern and continue it in the same way as the first child intended. This is a spontaneous activity initiated by the children. However, the same “game” had earlier been introduced in a systematic way by a teacher, and the pedagogical component of the activity (about following patterns) had been made apparent to the children. Therefore, I see the described spontaneous situation as pedagogical. The situation is thus Pedagogical, Non-participatory and Spontaneous. The situation shows that deliberate teaching of mathematics may have prolonged effects, in the sense that children may pick up on particular mathematical activities on their own.

Example 8: Eighths of apples (Pedagogical, Non-participatory, Planned). This situation stems from an earlier activity in which the teacher introduced some children to cutting apples in halves, fourths, eighths and sixteenths. The children get to choose how much apple they want as well as in what form. For example, they may say, “I want half an apple in sixteenths”. It was discussed how many sixteenths would comprise half an apple. This was a planned situation in which the teacher participated. It was also pedagogical. From the children’s perspective, the situation is clearly not about eating apples but about discussing the concept of fractions.

Example 8 is the situation that followed this one. Here, the teacher is no longer present, but the children continue to share the fruit in quarters and eighths and discuss how many quarters are equivalent to two eighths and so on. This is therefore an example of a planned and pedagogical situation in which the teacher is not participating. I call it planned because the teacher intended for the children to continue sharing the fruit and discussing the fractions after the teacher stopped participating in the activity. The teacher also explicitly encouraged the children to share the fruit and discuss fractions when initiating the second fruit-sharing session. This makes example 8 different from example 7. In example 8, the teacher explicitly sets up the situation and leaves, which is why the situation itself is planned. In example 7, there is an element of planning on a higher level, as teachers have introduced the bracelet switching game earlier. But in the actual situation, it was the children that initiated the game, it was not planned by the teacher. It is interesting to compare with example 2, the robot situation. Both examples 2 and 7 are spontaneous, but only possible because of previous teacher planning (providing the robot and introducing the bracelet game, respectively). In summary, example 8 is a Planned, Non-participatory and Pedagogical situation.

Between the eight examples above, all combinations of the three dimensions are realised, which illustrates that the three dimensions are principally independent. Designing good mathematical learning experiences for children is not only about setting up worthwhile activities in these different situation types but also about choosing which type of situation is appropriate for which purpose. Deliberating which modes of action should be combined for which purpose can be a helpful part of preschool teachers’ work of teaching mathematics.

The eight exemplified preschool situations in themselves show different levels of classification and framing. In some both the mathematical content and the pedagogical act appears as well defined for both children and observers, while in other situations it is hard to identify direct pedagogical acts and the mathematical content is not clearly delineated. An important point is however that when the eight situations together are taken to represent situations in which mathematics can be learned in a play-based preschool environment, the variation in pedagogical approaches as well as focus on particular mathematical content is very large. It is exactly the variation between these learning situations that show that the framing in the preschool practice in general is low. The Swedish School Inspectorate comments that while children can learn when teachers are not actively involved, this is not to be counted as teaching (Skolinspektionen, 2016). I take the opposite stance. The examples indicate that situations in which the teacher does not participate still rely on teachers’ prior preparation or initiation. I see such situations as the result of pedagogical acts, which constitute important aspects of the work of teaching preschool mathematics.

5. Discussion

In Sweden, pre-school mathematics is largely taught through a play-based approach but also through a dual approach (Sheridan et al., 2011) in the sense of including elements of both pre-primary and social pedagogy traditions (Bennett, 2005). Since 1998, there has been a national curriculum that includes goals for preschool mathematics teaching and learning but no evaluation function of children’s achievement and no or very little external evaluation of the pedagogy. Such characteristics have been identified by Bernstein (1971) as important in school systems with low classification and low framing. The introduction of an outside evaluator of the pedagogy (in the form of the Inspectorate) has, arguably, changed the power relations in the system. In a system of weak framing and classification, Bernstein asserts the requirement of a strong ideological function. My interpretation of the work by the School Inspectorate in Sweden is that it has joined forces with a particular aspect of this ideology, namely that pedagogy should emanate from children’s interests. However, in the absence of strong classification (which would make learning situations observable) or evaluation functions (which would make learning observable), the call for visible teacher actions in the form of teacher–child interactions that steer child-initiated activities towards curriculum goals is inevitable or at least predictable. This call from the inspectorate is a form of deadlock, as it tries to simultaneously retain both weak framing (pushing for a child-initiated pedagogy) and strong framing (requesting that teaching become more visible).

The realist position I adopt is to accept that with an outside evaluation function (the School Inspectorate), framing will in some sense need to become stronger. The type of framing that is required is tightly connected to the need for a more visible pedagogy. My answer to how this can be done is, however, to first suggest a much wider class of activities to be counted as pedagogical but then describe these activities more explicitly. It is a general truth that naming and describing phenomena also make them more visible. Visibility is not only about framing and classification but also about what you need to know to see the pedagogy. I do not intend the suggested modes of action to become a norm. Rather, I suggest them as an alternative way of thinking structurally about preschool teaching.

The demands from the School Inspectorate (Skolinspektionen, 2016) have, in this article, been used as leverage for the discussion and theory building. This report has already been very influential and is followed up by a government assignment to the National Agency of Education to rework the preschool curriculum and clarify the meaning of the word “teaching” (Regeringen, 2017). As I have argued, clarifying the term “teaching” might well end up leading to more formalised teaching. It is important to understand that such a change is not an explicit political intention. It is instead stated that “[t]he preschool curriculum is and should continue to be goal directed and should not give any advice on how the pedagogical duties should be carried out in practice (Regeringen, 2017, p. 4, my translation). But at the same time, in the same document, the government states the following:

the concept of teaching, which is central for the accomplishment of the pedagogical tasks of preschool is missing in the curriculum. […] If what happens in preschool is to be counted as teaching, it is required that the work of stimulating and challenging the children has the goals to strive for in the curriculum as an obvious point of departure and direction and aims for a learning process (Regeringen, 2017, pp. 5–6, my translation).

If the National Agency of Education picks up the ball where the School Inspectorate left off and works with a similarly narrow teaching concept, where it is only direct children–teacher interactions that count as teaching, there will be an obvious contradiction in the government’s assertion that it should not advise on how pedagogical duties are carried out in practice. The framework presented here can provide a resolution to this contradiction, as it expands what is understood to be “teaching activities” in the preschool context which simultaneously allows for stronger framing (visibility of teaching) yet much freedom for teachers. In this framework, I characterise this class of activities as spanned by three dimensions of teacher modes of action: dimension of situational planning, pedagogical explication and teacher participation. I have indicated how teaching situations spanned by these dimensions mimic typical preschool activities. Each of these dimensions allow adaptation to particular situations and preferences, and the structure as a whole allows a very wide variety of pedagogical choices for teaching of preschool mathematics to be understood and described.

In Sweden, preschool has for a long time had a solid play-based base and no formal assessment of children. The recent indication of change is important though, and it is possible that a call for more visible teaching may also push teaching to become more formal. The framework I have presented is intended to offer alternative ways of providing a more visible pedagogy and it tries to make explicit many of the choices and actions that underlie a well-functioning play-based pedagogy. In many other countries, preschool teaching already has elements of standardised assessment and formal teaching. As indicated by OECD’s report (2006) such pedagogical practices might well be the practical outcome even when governing documents talk about a play-based or social pedagogy approach. Often it is accountability mechanisms that create such contradictions. I therefore also suggest that in such countries, as well as on the international arena, the framework I have presented can be used by the teaching profession and researchers to indicate what teaching in a play-based approach means. It can also help to characterise some fundamental aspects of the work that preschool teachers can do when arranging a productive environment for children to learn mathematics. Pedagogical situations directed and planned to instruct children about some particular mathematical content are important and effective. But a preschool environment where such situations are the only ones that count would be very austere. There are other dimensions to explore.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Ola Helenius http://orcid.org/0000-0002-0609-7064

Notes

* A previous version of this paper was presented at The Eighth Nordic Conference on Mathematics Education in Stockholm, May 30 – June 3, 2017.