Keywords

Introduction

Early mathematics education has received increased attention in both politics and research during the last decades. Even so, there are large differences when it comes to perceptions of what preschool mathematics is, how it should be designed and what constitutes appropriate content (Palmér & Björklund, 2016). Despite these differences, there is a consensus that early mathematics matters, and a large number of studies have shown that mathematical competencies acquired in early childhood have positive effects on later school achievement (e.g., Aunio & Niemivirta, 2010; Duncan et al., 2007). Mathematics thus concerns even the youngest learners, and there is thus a need to conceptualize what early mathematics might be. Some notions, such as “mathematizing”, are introduced to the education and learning of young children (e.g., Björklund et al., 2018; Gejard, 2018; Reis, 2011). These notions are important for communicating and developing ideas that are to be implemented in educational practices, but they also call attention to what notions are used, the meaning they mediate and what implications for practice they might have.

As a contribution to the development of the field of early childhood mathematics education, the focus of this paper is on the notion of mathematizing, an expression introduced by the famous Dutch mathematician Hans Freudenthal. He pointed out several challenges in mathematics education, one of which was “How to create suitable contexts in order to teach mathematizing” (Freudenthal, 1981, p.145). Even though Freudenthal’s studies were not conducted in a preschool environment, the notion of mathematizing is often used in relation to preschool mathematics. However, quite often the expression is used as equivalent to “everyday mathematics” or “mathematics in everyday life” without any reference to Freudenthal’s original writings. In this paper, we elaborate on what mathematizing may imply in a context of preschool mathematics, taking Freudenthal’s original writings as the starting point. The aim is not to evaluate or rate how mathematics in preschool is or ought to be taught, but only to elaborate on the meaning of the notion of mathematizing in relation to empirical examples from mathematics education in preschool. The context for these empirical examples is Swedish preschool, thus a pedagogical practice with a play-oriented approach.

Mathematizing

The starting point for Freudenthal’s work on mathematizing was at that time the common teaching in school mathematics. According to Freudenthal, mathematics teaching commonly took its departure from the sophisticated knowledge and strategies of experts resulting in a series of learning objectives that made sense from the perspective of the experts but not necessarily from the perspective of the learners. Based on this, Freudenthal suggested a change in instructional approach, so that instead of decomposing ready-made expert knowledge, students would elaborate, refine and adjust their current ways of knowing (Gravemeijer, 2004). According to Freudenthal (1981), mathematics is always both form and content and therefore it should not be taught as isolated form or as isolated content, but always with regard to the interplay that exists between the two. For example, even though children can learn to perform mathematical procedures (e.g., read the numbers on a ruler) and memorize facts (e.g., a square is a plane figure with four equal straight sides and four right angles), such skills and abilities are in themselves of little value if the child does not understand the purpose of the procedure or the memorized fact, or how and why a procedure works or a statement is true. Based on this, Freudenthal’s starting point was that mathematics should be taught so that the knowledge becomes useful for the learner, which is why all mathematics teaching should be based on the learner’s world and experiences (Freudenthal, 1968).

Mathematizing means, in brief, the process of making use of mathematical thinking and skills in problem solving where there is an actual need for mathematics in order to complete a task:

Sets will not be formed, unless there is some need that they should be. In the laboratory experiment the child is expected to view some hotch-potch as a set, but why should it? What could be the genuine need to form sets? (Freudenthal, 1978, p. 217).

Freudenthal had an idea of mathematics as a human activity where students should be given the opportunity to reinvent mathematics by mathematizing in the sense of “mathematizing subject matter from reality and mathematizing mathematical subject matter” (Gravemeijer, 2004, p. 109). For education, he distinguished between horizontal and vertical mathematization. Horizontal mathematization is the use of mathematics as it appears in the “real world” based on the learner’s life and experiences, while vertical mathematization refers to a process where symbols are shaped, reshaped and manipulated to make a problem solvable by reducing “noise” that the “real world” usually induces. To learn and develop mathematical skills, both are needed. However, in both cases, the subject matter that is to be mathematized should be experientially real for the learners. Freudenthal pointed out that “real world” implies different things to different individuals; for example, mathematical objects are part of the “real world” for mathematicians in a different way than they are for students in school. Thus, in school mathematics, he emphasized first the real world and then mathematizing and clarified that “the real world” in school mathematics implied a context that includes a mathematical problem that is relevant to the learners and where mathematics is needed to solve the problem (Freudenthal, 1981).

Based on Freudenthal’s ideas, an approach to mathematics education called Realistic Mathematics Education (RME) was established. The basic idea of RME was that mathematics is a human activity where students should mathematize real situations in a context that made sense to them. RME implied designed support materials conjectured learning paths along which students, through instructional activities, could reinvent conventional mathematics. Within these learning paths, however, there is a tension between the openness toward the students’ own constructions and the obligation to work toward certain given endpoints (Gravemeijer, 2004).

It is not evident how Freudenthal’s writing can be understood in a preschool context. In preschool, designed support materials conjectured learning paths are not that common. Also, young children may perform actions that we recognize as mathematical even though they may not be mathematical for the child. As expressed by Van Oers (2010, p. 28), ‘we actually cannot maintain that very young children (1–3 years old) perform mathematical actions, even when they may carry out actions that we, as encultured adults, may recognize as mathematical. As long as these actions are not intentionally and reflectively carried out, we cannot say that children perform mathematical actions.” Based on this, in this paper we will elaborate on what “real world” and thus mathematizing may imply in the context of preschool mathematics if taking Freudenthal’s original writings as the starting point.

The Play-Oriented Context of Swedish Preschools

The context of the empirical examples in this paper is mathematics education in Swedish preschools. In Sweden, preschool is available to all children aged 1–5 years, with a national curriculum that clearly states that teaching is to be conducted. Further, play is described as the basis for children’s development, learning and well-being, hence preschool activities should be organized so that children can play and learn together. Based on a play-oriented approach, Swedish preschools should include teaching and opportunities for children to explore different knowledge areas, including mathematics (National Agency for Education, 2019).

There are many ways to describe what play is, but regardless of how play is understood or explained, it is characterized by an openness of the narrative where the direction of the play is not predetermined. Even if play is always about “something”, the direction of the play activity is constantly renegotiated in meta-communication between the players. In play, there is also a constant shift between as is and as if, thus there is movement both towards and away from reality (Pramling et al., 2019). According to Fleer (2011), imagination is the bridge between play and learning. In moving across this bridge in an iterative movement, the child directs his or her attention to the material world – not the physical objects per se but their meaning. Imagination then makes it possible to change meanings depending on the conditions or needs of the activities (e.g., pretending that a banana is a car). Yet the child is perfectly aware of the object’s meaning in “reality” (as is) and in his or her imagination (as if). Thus, reality and imagination are not separated but give meaning to each other and are dialectically related. We see this argument already in Vygotsky’s (1987) writings, where he writes that intervention and creativity build on realistic thinking and imagination working in unison.

In the play-oriented approach of Swedish preschools, there is a tension between the openness of play and the goal orientation of teaching (Björklund & Palmér, 2019). This tension is similar to the tension within RME described above, a tension between the obligation to work toward certain given endpoints and the openness toward students’ own constructions (Gravemeijer, 2004). Previous studies in Swedish preschools have shown, however, that goal-oriented processes can be integrated into play without changing the intentions of play, but depend on the teachers’ awareness and responsiveness to children’s intentions (Björklund & Palmér, 2019).

The Study

The data used in this paper was generated in a combined research-development project conducted in close collaboration between researchers and three preschool teachers for 2 years. The selection of the three preschools was based on the teachers’ interest in participating. The three teachers have a university (bachelor) level preschool teacher exam and have worked in preschool for several years. Twenty-seven toddlers (at the start of the study, the children were between 12 and 27 months old) from three preschools were involved in designed teaching activities and their learning was followed through task-based interviews.Footnote 1 Each activity and task was designed to be engaging for the toddlers, based on experiences familiar to them and aiming to broaden their understanding of mathematical ideas embedded in the play-oriented activities. The play-oriented activities and the task-based interviews were designed based on the following design principles (for thorough description see Palmér & Björklund, 2022):

  • The context of the activities ought to be based on children’s experiences, needs, and interests, being familiar so that they can participate, relate to, and reason about the content based on their previous social and cultural experiences.

  • The activities ought to make it possible for the children to discern essential aspects of numbers (representations, cardinality, ordinality, and part-whole relations).

  • The activities ought to allow the children to express different ways of understanding and allow a variety of experiences and expressions both between the children and for the same child over a prolonged period.

During the project, a large body of video documentations of play-oriented activities as well as task-based interviews was collected. These have been analysed in detail regarding different issues (e.g., critical conditions for learning numbers’ meaning, the toddlers’ development of number knowledge and methodological issues in the designed activities and task-based interviews). In this particular study, we take a holistic view on the whole data set, now focusing on the practice that was developed in collaboration between teachers and researchers. The aim of this paper is to elaborate on Freudenthal’s notions of “real world” and mathematizing in relation to preschool mathematics. That is, the toddlers’ numerical development is not the primary focus. Instead, we direct our attention to their ways of experiencing numbers in situations they are involved in. We take the toddlers’ perspective as an outset and focus our analysis on how the mathematical content embedded in the activities appears to the toddlers. Their expressed intentions (in words and actions) are the unit of analysis and are described in terms of different ways of experiencing mathematical content and form. This analysis is in line with a phenomenographical approach to interpreting the meaning of lived experiences among certain groups of people (Marton & Booth, 1997) – in this case toddlers. The examples presented below are chosen to illustrate how mathematizing is realized, or not realized, based on our reading and interpretation of Freudenthal and analysis of toddlers’ expressions. Of particular interest is how “real world” can be conceptualized in play-oriented activities. The consolidation of content and form, which Freudenthal held essential for mathematizing, works as a guideline in the elaboration of what “real world” entails. How the toddlers experience the mathematics in different ways thereby becomes the key to interpreting what “real world” entails and how mathematizing may be realized in the preschool activities.

“Real World” and Mathematizing in Preschool

Below we present three examples that illustrate how mathematizing is realized or not. The first two examples are from pre-planned singing activities where materials are used to illustrate the numerical elements of the song. The third example is from one of the interview situations. Thus, the first two examples are teaching-learning situations while the third is a research situation. However, from the perspective of the involved children, the different purposes of the activities are likely not evident.

Mathematics But Not Mathematizing

In the project, the teachers together with researchers designed play-oriented activities where numbers were to be possible to elaborate on with the toddlers. However, the content “numbers” were not always connected to a “problem” that, from the children’s perspective, needed to be solved.

In the following example, one teacher and three children (2 and 3 years old) are involved in a singing activity. The children sit at a table, facing each other. The teacher also sits at the table holding a box containing five plastic elephants:

Teacher::

We’re going to sing about the elephants. Puts one elephant on the table. How many elephants is this?

Holger::

One. Takes the elephant.

Teacher::

One elephant. Holds up one finger. Can you show one elephant with your fingers? How many fingers is one?

Holger starts to play with the elephant. One of the other children starts to sing another song about spiders, with movements. The third child starts to do the movements to the spider song. Then, the teacher turns to Holger who is still playing with the elephant.

Teacher::

Can you show one with your fingers?

Holger::

One. Holds up the elephant. Takes it down and points at the elephant’s trunk.

Teacher::

One trunk. The children start to talk about the elephant’s legs. The teacher asks them to count the legs. One child counts three legs, another five legs. Five feet. Then the teacher starts to sing the elephant song.

In this example, the teacher is trying to teach mathematics within the play-oriented approach. The teaching starts in the “real world” in the sense of singing a familiar song and using props that the children are used to playing with. Before they are about to sing, the teacher asks the children to express the same numerical meaning of one through different representations (words, fingers). In this way, the teacher tries to bridge the “real” lived experiences with the symbolic representations, connecting content and form. However, the children apparently do not experience this as necessary for singing the song. To the children, this activity does not seem to contain any problem that would need to be solved by showing one with their fingers – showing one with one’s fingers serves no purpose. Thus, even though the question of showing “one with your fingers” is a mathematical question based on an activity in the “real world”, this example cannot be seen as an example of mathematizing as the mathematical problem given by the teacher is not relevant for the children in this context.

Mathematics in the Sense of Mathematizing

As shown in the example above, the toddlers’ perspective and directed attention in a situation has great influence on how the mathematical content can be consolidated with form. Even when the starting point is within the toddlers’ “real world”, it becomes a challenge to accomplish mathematizing where symbolic representations, such as counting words and the counting sequence, become necessary tools for solving a “problem” that, from the children’s perspective, needs to be solved.

The following example illustrates the same elephant song as in the previous example, to be sung by one teacher and one child (2 years old) with the exception that they use small toy bears instead of elephants. The teacher has taped a line on a table – the spiderweb – where the bears are to balance. She then takes out a box containing several bears (Fig. 1):

Fig. 1
2 close-up photographs. One photo has hands holding a round container with another hand taking something out of it. Another photo has one set of hands with 3 fingers outstretched and another set of tiny hands pointing at the 3 objects on the table.

Teacher asking child to take three bears out of the box

Full size image
Teacher::

Can you take three bears? Holding up three fingers. Three. 

The child puts one bear on the line on the table.

Teacher::

How many have you taken now?

Sander::

One.

Teacher::

One. And you were to take three.

The child nods and puts one more bear on the line on the table.

Teacher::

How many do you have now?

Sander::

The child takes another bear from the box while the teacher asks the question, and says “three” while pointing at the third bear.

Teacher::

Are there three now?

Sander::

Yes.

In this example, the teacher asks the child to put forth the number of bears they are to sing about. Based on the child’s actions, he is apparently experiencing it as necessary for singing the song. The episode takes place before the playing and singing is about to start, thus the “real world” can be understood as the setting in which preparations are made for the play. This can be seen as an example of mathematizing as the mathematical task introduced by the teacher starts in the “real world” and connects content and form. Thus, the child appears to consider the problem of enumerating as relevant to solve, and he has knowledge of both form (enumerating) and relevant content (items to sing about).

Mathematizing in Play

As the focus of this paper is early childhood mathematics education, the essential play-orientation in this context cannot be overlooked. Therefore, the following example further elaborates, based on the insights described above, how play may affect how “real world” and “mathematizing” are interpreted.

In this third example, one teacher and one child (soon to be 3 years old) are in an interview situation where the child is asked to set the table for a toy cat having a birthday party.

Teacher::

Look, now the kitty was to have a birthday party. Because it was the kitty’s birthday. Now you are to help the kitty to set the plates.

Gustav::

Then we first must bring the cake.

Teacher::

Yes, also cake.

Gustav::

And some muffins.

Teacher::

And some muffins.

Gustav::

All his friend should be.

Teacher::

All his friends are coming.

While talking, the teacher puts forward one toy kitty, two plastic plates and twelve cookies (small circular pieces of wood). Then, she knocks with her hand under the table and says “here comes a friend” as she puts forth another toy kitty. She places the kitties next to one plate each.

Gustav::

I will set the table for them.

Teacher::

Yes please, serve them the cookies.

Gustav::

One for you. Puts one cookie on one plate. And one for you. Puts one cookie on the other plate. This is not fair! (Fig. 2)

Teacher::

Is it not?

Gustav::

No. They must have only, they must have four cookies. Holds up four fingers at the same time as he says four. One for you. Puts another cookie on the first plate. And one for you. Puts another cookie on the second plate. How many is it? One, two, three, four. Points at one cookie at a time as he says each number word. But they must have four cookies! (Fig. 3)

Fig. 2
A close-up photograph of a child sitting on a chair with a table with 2 toys in front. The table also has 2 plates with the child placing round objects on them.

The child starting to divide the 12 cookies putting one cookie on each plate

Full size image
Fig. 3
A close-up photograph of a table with 2 toys and 2 plates with round objects on them, and a child's hand with 4 fingers outstretched.

Holds up four fingers and says that the kitties must have four cookies

Full size image

Gustav continues handing out cookies, one at a time on each plate. Then he says, “How many are there?” and counts the cookies on one plate at a time. When there are four cookies on each plate he says:

Gustav::

I think it was both their birthdays.

Teacher::

Did both have a birthday?

Gustav::

Yes. Now they are to chew on their cookies.

Gustav holds the cookies in front of the kitties making a chewing sound. He says the kitties say that the cookies taste like strawberry cake. Then Gustav says, “What do you do when the cookie is finished?” He puts one cookie behind the kitties saying, “I put the cookie, I mean the stick here.” When all the cookies are eaten, a third kitty joins the birthday party. The teacher puts forth a third plate and collects the twelve cookies in the middle of the table. Gustav says that they have forgotten to sing the birthday song. The teacher says that they can do this after he has handed out the cookies once again. Like the first time, he hands out the cookies one at a time and counts the cookies on the plates. However, he has now decided that the kitty is turning 5 years old and therefore he wants the kitties to have five cookies each. He asks for more cookies but the teacher says that there are no more cookies. He looks under the table for possibly dropped cookies. Then the teacher asks if they now are to sing the birthday song. The boy starts to sing and the teacher claps her hands.

Gustav::

Now we must go get the balloons.

The child collects small gadgets from a box in the room. He divides these between the kitties.

Gustav::

They only got three. They were to have four.

This is an example of an activity with a problem that, from the perspective of the child, needs to be solved with mathematics. It is also an example of an activity with clear indicators of play: there is a narrative (birthday party), meta conversation (e.g., what to do with eaten cookies, the forgotten birthday song) and a continuous shift between as is and as if. Both the child and the teacher act as is and as if, thus the “real world” is sometimes as is and sometimes as if. “Real world” may therefore imply the world of fantasy where the problem to be solved is an imagined problem within the narrative of the play. While the child is free to take the play activity in new directions (e.g., collect balloons), the teacher stays with the intended mathematical content (partitioning of twelve cookies). When all the cookies are distributed and the child wants more, it is an example of the tension between openness of play and the goal orientation of teaching in preschool (Björklund & Palmér, 2019) as well as between the openness toward students’ own constructions and the obligation to work toward certain given endpoints that, for example, is highlighted within RME (Gravemeijer, 2004).

Discussion

The focus of this paper is on what “real world” and thus mathematizing may imply in the context of preschool mathematics if taking Freudenthal’s original writings as the starting point. The three empirical examples in this paper illustrate that not all activities in preschool that involve mathematics can be labelled as mathematizing. The first example illustrates that even if an activity can be seen as taking children’s “real world” as an outset, there may not be a problem involved that, from the children’s perspective, needs to be solved by using mathematics – content and form are not consolidated. On the other hand, the second example illustrates a problem that, from the child’s perspective, needs to be solved by using mathematics, thus it is an example of mathematizing. In both these examples, “real world” implies a situation where the children are to sing a song. The third example illustrates that “real world” can also include the world of play and fantasy. This means that “real world” may imply the world of fantasy where the problem to be solved is an imagined problem within the narrative of the play. “Real world” may in this respect, in the context of early childhood education, imply different kinds of activities, but it only becomes mathematization if there is a problem that, from the children’s perspective, needs to be solved.

According to Van Oers (2010, p. 28), we cannot maintain that very young children perform mathematical actions as long as these actions are not intentionally and reflectively carried out. However, when children mathematize in the sense of using mathematics to solve a problem that according them needs to be solved, this is no longer an issue. Thus, when children mathematize, even when the “real world” is the world of play and fantasy, we can maintain that they are performing mathematical actions. As illustrated in the third example, the problem that needs to be solved can emerge from both teachers and children. The key is to find and consolidate form and content that constitute a relevant asset for the child’s problem solving.