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Abstract
Little research exists on young children’s beliefs about mathematics, and current research perspectives on early mathematical activity may overlook a great deal of young children’s sophisticated mathematical thinking. We argue this is attributable, in part, to a need for a broader view of what mathematics is, including cultural practices that are mathematical. Thus, this study investigated a young child’s mathematical activity and beliefs prior to school using a framework reflecting a view of mathematics as cultural activity. Through the case study of Olivia, aged 3 and 9 months, we found that her early beliefs about mathematics did not reflect her rich mathematical activity. Although Olivia engaged in a diverse range of mathematical activity (counting, measuring, designing, playing, locating, and explaining), what she believed mathematics to be was constrained to writing numbers and letters. To help young children perceive their own activity as mathematical, we argue there is a need: (a) for researchers, parents, teachers, and teacher educators to recognize the range of mathematics in children’s everyday activities; (b) to explicitly tell children when their activity includes mathematics; and (c) for teacher educators to emphasize the cultural nature of early mathematical experiences and how to capitalize on these experiences.
Introduction
In the United States, mathematical proficiency is considered a ‘gatekeeper’ for higher education and future career options, and mathematical reasoning is arguably vital to making informed decisions as consumers and citizens. Many children growing up in the US, however, believe mathematics is a particularly difficult discipline in which only the smartest people can be successful, so it is socially and culturally acceptable to strongly dislike math, be ‘bad at math,’ and generally avoid it (Middleton and Jansen, 2011; Willis, 2010). Moreover, math anxiety, a negative emotional response to mathematics, is prevalent in adults as well as children and impacts mathematics achievement and the motivation to engage in mathematics (Maloney and Beilock, 2012; Ramirez et al., 2013). These beliefs and feelings about mathematics and oneself in relation to mathematics are shaped by numerous complex factors, including community values (e.g. Martin, 2000) and experiences with school mathematics (Cobb, 1986; Franke and Carey, 2007; Klein, 2007; Kloosterman et al., 1996). Also, beliefs about mathematics develop early – research has documented math anxiety, for example, in children as young as first grade (Ramirez et al., 2013) – although we know little about young children’s beliefs about mathematics prior to entering school.
When people view themselves as mathematically capable, they are more likely to be successful learning mathematics and to choose to engage and persist in using mathematics (e.g. Wigfield et al., 2009). If children are able to connect their everyday and school mathematical experiences, they may be more likely to see themselves as capable mathematical thinkers and doers (e.g. Nasir et al., 2008). To facilitate children viewing themselves in this way and promote their engagement in mathematics, we argue it is important to counter popular beliefs about children beginning school and recognize that children are not just ‘ready to learn’ mathematics when they start school, but that children have already had valuable mathematical experiences through participating in family and cultural practices (Bishop, 1988). If we as researchers, teachers, and parents recognize the mathematical experiences young children have and help them recognize their everyday mathematical experiences, it may support more positive beliefs about mathematics and their own mathematical ability.
Unfortunately, little research exists about young children’s conceptions of what mathematics is or about their early experiences with mathematics, particularly beyond experiences with numbers and counting (Fox and Diezmann, 2007; Tudge et al., 2008; Wager and Parks, 2014). What adults recognize in young children’s mathematics depends on our views of what counts as mathematics, which points to a need for studies of young children’s everyday mathematical activity and their mathematical beliefs using a broader view of what mathematics is, including more diverse mathematical cultural practices.
Accordingly, the purpose of this case study was to explore the question: How does a young child’s beliefs about mathematics relate to the child’s everyday mathematical activity? We considered this relationship between the beliefs young children have and the mathematical activity in which children engage before encountering school mathematics by comparing the mathematical experiences and beliefs of a child who had not participated in formal day care or preschool education. To do so, we described the beliefs about mathematics expressed by a young child to provide foundational knowledge about children’s conceptions of mathematics before entering school. We also used a framework with a broad mathematical lens that extended beyond numbers and shapes to provide a comprehensive description of the same child’s mathematical activity. Specifically, we used a mathematical activity framework to capture six types of mathematical activity: counting, designing, measuring, locating, explaining, and playing (Bishop, 1988).
In the case of Olivia, aged 3 and 9 months, we found that she had developed beliefs that mathematics is a narrow subject despite both the broad and complex range of mathematical experiences in which she engaged during her everyday activities and the fact that she was raised in an environment in which those experiences were considered mathematically important. In order to help children like Olivia perceive their own everyday activity as mathematical and thus develop more productive beliefs about mathematics, we argue the findings illustrate the importance of researchers, teachers, parents, and other caregivers using a broad lens of what counts as mathematics to acknowledge preschool children’s early beliefs about and knowledge of mathematics.
Related literature
To frame Olivia’s beliefs and highlight the different ways children’s activities may or may not be viewed as mathematical, we provide a brief overview of the implications of different beliefs about mathematics on mathematics instruction in school. Because we focus on the need to recognize and build on children’s previous mathematical experiences in preschool and elementary school mathematics, next we consider the recommended early grades mathematics curriculum using a cultural framework of mathematical activity. Finally, we discuss research on the mathematical activity of young children prior to school and argue for a wider lens to identify young children’s mathematical experiences.
Beliefs about mathematics
The feelings or beliefs children associate with mathematics may be just as important as the mathematical knowledge they gain (Klein, 2007; Philipp, 2007). We define a belief to be ‘the personal assumptions from which individuals make decisions about the actions they will undertake’ (Kloosterman et al., 1996: 39). Children’s explicit experiences with mathematics help shape their beliefs about mathematics – what it is and who can learn it – which in turn shape children’s dispositions toward and motivation to learn mathematics (e.g. Maloney and Beilock, 2012; Martin, 2000; Wigfield et al., 2009). Although we know school experiences in mathematics shape these beliefs (Cobb, 1986; Franke and Carey, 2007; Klein, 2007: 315; Kloosterman et al., 1996), mathematical beliefs may start to develop earlier through cultural and family experiences. Unfortunately, we know very little about the mathematical beliefs children develop before they enter school or even when these beliefs start to develop, though research suggests that children as young as two years old can express beliefs about abstract concepts (see, for example, Hirschfeld, 2008).
To our knowledge, the youngest children studied in extant literature on mathematical beliefs were in kindergarten or first grade (e.g. Franke and Carey, 2007; Kloosterman et al., 1996; Kutaka, 2013). Research suggests, however, that even in the early school experiences, pedagogy and curriculum influence children’s conceptions of mathematics, beliefs about who is good at mathematics, and the extent to which children see themselves as ‘mathematically competent and confident’ (Anderson and Gold, 2006; Cobb, 1986; Franke and Carey, 2007; Klein, 2007: 315; Kloosterman et al., 1996). For some students, unfortunately, these instructional cues lead to unproductive beliefs about mathematics and oneself as a mathematics learner, such as math anxiety (Maloney and Beilock, 2012). Taken together, these results highlight the importance of students’ sense of competence in early mathematical experiences. To help make sense of the mathematical beliefs children may form in and out of school, we discuss two different views of mathematics that are reflected in the pedagogy and curriculum of school mathematics to varying degrees: mathematics as objective and value-free and mathematics as a cultural practice.
Mathematics as objective and value-free
School experiences seem to reinforce the commonly held view in the US of mathematics as universal, unchanging, impersonal, and value- and culture-free. Such a view implies there is an absolute mathematical truth in the world separate from human thinking and culture that people either can or cannot obtain. Mathematics in preschool and elementary school tends to reflect an objective view of mathematics by emphasizing routine memorization of facts and procedures and the (mis)implication that problems can always be solved by a unique, correct solution (Burton, 1994; Chazan, 1990; Ernest, 1991, 2010). This may be due, in part, to teacher beliefs about mathematics. For example, interviews with early childhood teachers revealed they held some problematic myths, such as the belief that it is difficult to influence a child’s mathematics achievement if the child is not inherently good at mathematics (Lee and Ginsburg, 2009). Although the relationship between teachers’ beliefs and practice is complex, teachers’ beliefs of mathematics influence their practice (Ernest, 2010; Thompson, 1992). Research on teacher beliefs has demonstrated that beliefs and attitudes filter what teachers notice as well as develop into ‘action agendas’ (Pajares, 1992). Consequently, teachers’ beliefs about mathematics are likely to influence their behaviors and decisions when teaching mathematics, which in turn can sway the beliefs of their students. For instance, research on math anxiety shows that teachers with math anxiety pass on negative attitudes to some of their students (Maloney and Beilock, 2010). In sum, school experiences seem to replicate popular beliefs about mathematics in the United States.
Mathematics as cultural practice
Increasingly, however, content standards and suggestions for instructional practice reflect a different view of mathematics (e.g. Clements et al., 2004; National Governors Association Centre for Best Practices and Council of Chief State School Officers (2010), Common Core State Standards for Mathematics (CCSS-M); National Council of Teachers of Mathematics (NCTM), 1989, 2000). Many mathematicians and mathematics educators recognize that mathematics is inextricably linked with the social practices and history of the humans who developed it (e.g. Bishop, 1988; Burton, 1994; Chazan, 1990; D’Ambrosio, 1985; Ernest, 1991, 2010; Nasir et al., 2008). This perspective that mathematics is linked to social and cultural practices is consistent with the argument that people develop mathematical knowledge and engage in mathematical activity outside of school in homes and communities (e.g. Bishop, 1988; Burton, 1994, 1995; Lakoff and Nunez, 2000; Nasir et al., 2008). That is, mathematics is not only a cultural phenomenon among the elite within each culture, but a pan-cultural phenomenon, meaning it exists among the entire population in every culture, spanning all societies (Bishop, 1988).
A natural implication is that children bring to preschool and elementary school prior experiences that reflect mathematical cultural beliefs and activity (Bishop, 1988; Carraher et al., 1985; Cobb, 1986). Unfortunately, many children, parents, and teachers value traditional school mathematical practices over everyday, cultural mathematical activity. De Abreu and Cline (2007) refer to this as valorization – the placing of social value on a certain practice in a specific context. This differential valorization between children’s mathematical beliefs in school and non-school contexts is common and likely develops early in children’s school experiences (Cobb, 1986). An emphasis on young children’s everyday mathematical activity in preschool and elementary grades may promote more productive dispositions toward mathematics – a characteristic of mathematical proficiency – by highlighting mathematics as being useful and personally relevant and highlighting children as being mathematical thinkers (National Research Council, 2001).
How we define mathematics for the purpose of this study
To frame how we conceptualized mathematics in this study as well as how experiences in and out of school might shape children’s beliefs about mathematics, we discussed two contrasting views: (a) mathematics as objective and value-free and (b) mathematics as cultural practice. We considered mathematics to be a cultural activity in which young children do engage, not just a precursor to mathematics but as a form of mathematics. We use informal mathematics (others have used everyday or street mathematics, e.g. Carraher et al., 1985) to refer to the understandings and activities in which people engage in and out of school without explicit instruction and that do not involve written symbolism. Formal mathematics (or academic mathematics, D’Ambrosio, 1985) refers to understandings or experiences linked with intentional mathematics lessons facilitated by parents, teachers, textbooks, and so on; or involving established written symbolism. The understandings children develop through informal and formal experiences need not be mutually exclusive; in fact, in this paper we emphasize the need to recognize and build on children’s informal mathematical experiences in formal mathematics instruction in order to support more productive beliefs about mathematics.
Although a view of mathematics as cultural practice is related to numeracy or quantitative literacy, generally defined as the critical mathematics skills needed to make sense of everyday experiences (Steen, 1997), we intentionally avoided these terms. These terms might denote or at least connote a limited subset of what counts as mathematics. Instead, we emphasize that although young children’s mathematics often occurs informally, it represents an early development of a broad range of formal mathematical ideas and is an important foundation to build on in school.
Early mathematical activity
Children benefit from early, rich (complex and varied) mathematics instruction that capitalizes on their innate understandings and prior experiences in a developmentally appropriate way (Ertle et al., 2008; Seo and Ginsburg, 2004). Until late in the twentieth century, formal mathematical instruction of preschool children was discouraged in the US (Ertle et al., 2008; Hachey, 2013). Current research, however, has demonstrated that early experiences with mathematics are linked with later academic success and young children are able to learn not only more mathematical topics, but more complex mathematical ideas younger than previously thought (Clements and Sarama, 2007; Ertle et al., 2008; Hachey, 2013; Seo and Ginsburg, 2004). As a result, mathematics educators, mathematics education researchers, developmental psychologists, and policy makers are paying increased attention to preschool mathematics standards and curricula (Hachey, 2013). Teaching mathematics earlier and more formally, however, is only one part of improving preschool mathematics education. Recognizing and building on children’s cultural, everyday mathematical activity and knowledge is also crucial to building both their mathematical knowledge and positive dispositions toward mathematics. Building most effectively on children’s activity and knowledge, however, may require a broader lens through which to view children’s early mathematical experiences than is evident in extant literature.
Broader perspectives needed in practice
When children in the US enter preschool or kindergarten, a popular phrase for their potential success is whether they come to school ‘ready to learn’ (e.g. La Paro and Pianta, 2000). This perspective implies that children’s mathematics learning begins at the point they enter school. Our stance is that when children begin school they are not simply ‘ready to learn’ the preschool mathematics curriculum. Children bring to school knowledge of the actual mathematics in the intended curriculum, which they developed through the practices of the culture in which they live (Bishop, 1988).
Early childhood teachers are positioned to build on the mathematical understandings that develop in children’s everyday activity (van Oers, 2010). In order for educators to build on children’s everyday cultural mathematical knowledge, however, they have to be able to recognize it. A broad conception of mathematics, such as Bishop’s (1988) framework that illustrates different categories of mathematical activity, may support educators in doing so. He proposed that all people in every culture – including children – exhibit the mathematical activities of counting, locating, measuring, designing, playing, and explaining (see Table 1), which we refer to as a mathematical activity framework. The framework focuses attention on how culturally expressed mathematical activities relate to what is typically recognized as formal mathematics.
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Table 1. Mathematical activity framework (Bishop, 1988; Roth, 1902, as cited in Bishop, 1988).
One type of activity in the framework not commonly identified as mathematics is play. Not all of children’s Play in a general sense is mathematical. We use the general capitalized Play to refer to an activity that is ‘incompletely functional in the context in which it appears’ and is ‘spontaneous, pleasurable, rewarding, or voluntary’ (Burghardt, 2011: 17). Particular ways of playing, however, do involve mathematics, and we refer to mathematical play with a lower-case p indicating it is a subset of the broader Play category. Several studies have identified the importance of play in young children’s development of mathematical ideas. For instance, research has linked children’s future mathematics ability and performance with playing a variety of games – including linear number board games, physical-knowledge games such as Jenga and Pick-up Sticks, blocks, and sociodramatic play (Hanline et al., 2008; Kamii et al., 2005; Siegler and Ramani, 2009; Wolfgang et al., 2001). In addition, the sophistication of symbolic substitution exhibited in preschool during imaginative play (e.g. the use of a real or imaginary prop, such as using a washcloth to wash a doll or eating an imaginary cookie) has been positively linked to children’s mathematics skills in early elementary grades (Hanline et al., 2008). Bishop (1988) suggested that the distancing of oneself from reality and the rules generated during play may be at the root of hypothetical thinking, a basis for mathematical reasoning.
All of the main topic areas included in major documents for early childhood mathematics content standards are related to categories of mathematical cultural activity. In Figure 1, we highlight how each of the six types of activity interact with the intended preschool mathematics curriculum in multiple ways. This suggests that teachers should have a rich source of prior knowledge from which to draw. That is, children entering school bring knowledge and experiences relevant to the core topics in the intended early mathematics curriculum that can be valuable resources for teachers. Teachers, however, may not recognize the potential range of young children’s mathematical activity (Anderson and Gold, 2006; Lee and Ginsburg, 2009; Moseley, 2005). Studies of early childhood teachers have indicated they believe preschool mathematics should only include basic number and shape concepts (Lee and Ginsburg, 2009). A study of pre-service early childhood teachers’ reactions to mathematical meaning showed that the future teachers recognized a narrow set of terms, such as those related to addition and subtraction, as mathematical (Moseley, 2005), which may limit their recognition of the possible range of mathematics in young children’s activity (Wager and Parks, 2014).
Figure 1. Illustration of relationships between categories of pan-cultural mathematical activity (Bishop, 1988) and mathematics content in sample early childhood standards documents, such as the Head Start Child Development and Learning Framework (Office of Head Start, 2011) and Common Core State Standards Mathematics for Kindergarten (CCSS-M, 2010).
Broader perspectives needed in research
Both long-standing and recent research provides evidence that young children are capable of learning and understanding relatively complex mathematical ideas (e.g. Baroody et al., 2006; Baroody and Wilkins, 1999; Sarama and Clements, 2009). Yet, there is a dearth of research about how young children without any formal school experience engage in and think about mathematics (Fox and Diezmann, 2007; Parks and Bridges-Rhoads, 2009), particularly beyond number and shapes and in natural settings (Tudge et al., 2008; Wager and Parks, 2014). Few studies have captured young children engaging in mathematical activity in everyday settings, leaving gaps in our understanding about the relationship between children’s early experiences with mathematics and the development of their mathematical knowledge (Tudge et al., 2008).
This lack of evidence may be because beliefs about mathematics and the methods used to capture children’s activity (e.g. parent reports, researcher observations with or without audio and video recording) heavily influence the extent to which research detected children’s everyday mathematics (Tudge et al., 2008). Research has typically reported results based on a narrow perception of what counts as young children’s mathematics activity. One study that investigated 39 three-year-olds’ everyday activity reported that they engaged in few or no explicit mathematical activities (Tudge and Doucet, 2004). Although this important work filled a research gap by attending to young children’s informal mathematical activity, it is important to note that for at least two reasons it was conducted through a narrower view of what counts as early mathematics: (a) the research emphasized mathematical activities that arose with academic materials and during adult-directed lessons in everyday settings, and (b) the research provided little detail about what they identified as mathematics but specifically identified activities in the results that involved shapes and numbers. If we use the mathematical activity framework (Table 1) to interpret the results of this study, it primarily documented just two types of mathematical activity (counting and designing).
Reconsidering other research with this framework, we found that the few studies that used a slightly broader view of mathematics (including aspects of designing and locating) did report children engaging in more mathematical activity (e.g. Anderson, 1997; Seo and Ginsburg, 2004). For example, observations of 90 four- and five-year olds’ free Play revealed that 88% spontaneously engaged in at least one type of mathematical activity, such as counting, comparing, and engaging with shapes and patterns (Seo and Ginsburg, 2004). Interpreting this study in terms of the mathematical activity framework (Table 1), it documented counting, designing, and measuring. Another study indicated that parents engaged their children in a broad range of mathematical activity when interacting with materials such as blocks, storybooks, and drawing supplies (Anderson, 1997), documenting counting, designing, measuring, and locating.
Thus, narrow views of mathematics may lead to an underestimation of young children’s mathematical experiences. As Tudge et al. (2008) stated:
we can say that whether from studies involving parental reports or [researchers’] live observations in the home or preschool setting, children do not seem heavily involved in mathematics, which leaves open to question how it is that by the time they arrive in school many of them have developed rather sophisticated mathematical understandings. (p.199)
This claim points to not only a dearth of literature on early mathematics experiences, but suggests current research perspectives may overlook a great deal of young children’s sophisticated mathematical thinking. What is reported depends on the researchers’ underlying views of mathematics. We argue this is due, in part, to a need for a broader view of what mathematics is, including more diverse mathematical cultural practices.
Purpose
We know little about how young children perceive mathematics before they enter school and how those beliefs are shaped by parents, caregivers, media and other factors such as engagement in everyday activities. Because the US culture is plagued by unproductive beliefs about and dispositions toward mathematics that likely contribute to children and adults avoiding mathematics, it is worthwhile to pursue a better understanding of young children’s mathematical beliefs and activity prior to entering preschool. We explore a young child’s beliefs about mathematics not because we think that holding certain (or any) beliefs is a precursor to mathematical understanding, but to provide foundational knowledge about children’s conceptions of mathematics before entering school. Future research might consider how experiences with preschool and early elementary mathematics can build on and influence children’s beliefs and mathematical activities. Moreover, narrow views of what counts as mathematics may mean that research on informal and prior-to-school mathematics activity overlooks some aspects of young children’s mathematics. Our study intended to broaden the conceptual lens used in early childhood mathematics research and instruction by providing an in-depth exploration of a young child’s beliefs about mathematics compared to her mathematical activity using this broader view of mathematics as pan-cultural activity.
Method
We chose to use a case study in order to provide a deep and comprehensive picture of a single young child’s beliefs and mathematical activity (Stark and Torrance, 2005; Yin, 2009). A single case was appropriate for this initial study intended to demonstrate that children could exhibit rich (in the sense of being complex and varied) mathematical activity and potentially corresponding beliefs about mathematics (Yin, 2009). We suspected that a child would be most be likely to demonstrate broader beliefs about what mathematics is and engage in varied mathematical practices if she: (a) grew up in a home with a parent who believed that mathematics is a pan-cultural activity and (b) had not participated in formal educational experiences such as daycare or preschool, which may replicate popular cultural beliefs about mathematics. It was also important that the child could verbally communicate well. Olivia, the daughter of the first author, fit these criteria.
Furthermore, the methodological choice to use parent research offered several affordances. The parent-researcher has a long history in education research (going back, for example, to Jean Piaget and Erik Erikson), but still plays an important role in education research (e.g. Adler and Adler, 1996; Kabuto and Martens, 2014). For instance, a recently published book includes multiple parent-research studies on children’s early cultural literacy practices (Kabuto and Martens, 2014). The editors argue this provides an ‘insider perspective’ and contributes ‘scholarly discussions about learning in the home: how it is organized, who the participants are, and what children are learning’ (Kabuto and Martens, 2014). Some benefits include diminished pretense (the parent is an expected and familiar presence), immersion, and triangulation, all promoting increased validity (Adler and Adler, 1996). As a parent, Wernet had extensive access to Olivia’s activity without the intrusion of strangers or recording equipment and could understand the contexts of Olivia’s everyday life in order to better interpret her mathematical beliefs. This allowed for the observation of both verbal and non-verbal behavior and provided access without time sampling (e.g. observing for 30 seconds every six minutes, Tudge and Doucet, 2004) – access that is limited with researcher observations of young children’s everyday mathematical experiences (Tudge et al., 2008). As a parent-researcher, Wernet was more likely to capture subtleties in Olivia’s mathematical activity consistent with a broad definition of mathematics than most parent-participants would report (Tudge et al., 2008). Owing to the sensitive nature of research with a child, particularly one’s own child, the first author carefully considered ethical concerns and researcher subjectivities.
Interviews and observations focused on evidence of Olivia’s beliefs about mathematics and unprompted and unstructured events of mathematical activity in her typical daily experiences. Most observation data was collected in the first author’s home during unstructured typical everyday activities that Olivia initiated, including Play activities as previously defined as well as other daily tasks, such as eating meals and helping with household chores. Semi-structured interviews and informal observations focused on these operationalized questions:
What does a young child believe about mathematics? More specifically, how does a child talk about what mathematics is, who does mathematics, where and when it takes place, and why people do mathematics? To what extent do these expressed beliefs reflect beliefs about mathematics in the broader culture, mathematics, and mathematics education?
What is the nature of a child’s mathematical activity according to Bishop’s (1988) mathematical activity framework?
The first author collected the data with the second author providing inter-rater agreement for all data and contributing as an author. Consequently, future references to the implementation of the study will use the term ‘I,’ referring to the first author, and the word ‘we’ will refer to both authors in references to the data analysis.
Data collection
Participant and settings
Olivia was age 3 years and 9 months at the time of the study and had no experience in daycare or preschool, so she had not participated in formal mathematics lessons. It is important to note, however, that she may have had some encounters with school mathematics, because I occasionally tutored students at home. Although her father and I never formally instructed Olivia in mathematics, our experiences and perspectives likely affected our natural interactions, such as through informal mathematical activities (e.g. encouraging Olivia to conjecture and explain why things occur).
During the data collection, Olivia and her younger sister spent one day a week with their father (my husband) and one day in our home with a babysitter, who was a college graduate. During the other five full days I was the sole caregiver and shared parenting each morning and evening with her father. I collected observation data in the course of regular activity, mainly at our home, located in a small urban area near a university. Some observations took place in the car, while running errands, or at the homes of family and friends. One interview took place at home and the other in the car.
Interviews
The mathematical activity framework (see Table 1) guided the development of a list of potential questions to assess Olivia’s beliefs in two semi-structured 15-minute interviews. Both interviews occurred within the two-week observation period. Olivia responded to questions such as, ‘What do you think math is?’ and ‘George [a friend] does not know what math is. How would you explain to him what math is?’ I opted not to use an audio or video recorder because I believed they would be too distracting and affect Olivia’s responses. I typed Olivia’s responses as close to verbatim as possible during the interviews, which she occasionally asked to see. Immediately following each interview, I elaborated the notes and her father confirmed the transcript accuracy.
Observations
For two consecutive weeks, I observed Olivia and recorded instances of her mathematical activity as it occurred in natural settings and situations that she initiated during Play, dining, traveling, and so forth. That is, I did not create opportunities to engage in certain mathematical activities, but rather observed what she did as part of her unstructured regular experiences. The observation period spanned a total of about 156 hours. Although it is possible I missed events during the study period, I was careful to attend as closely as possible to mathematical activity as defined by Bishop (1988). I took fieldnotes as events occurred, writing what happened during the event and the details of our setting. Each evening I typed the notes into more structured field notes, listing all of the day’s events of interest. If instances of an event were repeated throughout the day (e.g. drawing), I counted that event only once. We did not actively analyze any of the data until all the data were collected, although we acknowledge that the selection of events and their description inherently involved some degree of interpretation.
Researcher positionalities and subjectivity
During data collection my role as parent always took precedence over my role as researcher. To ensure that Olivia was comfortable answering questions, Olivia’s father was present for both interviews with the understanding that he would intervene if Olivia expressed any signs of discomfort. Olivia’s father previously worked as a paraprofessional in a mathematics classroom. Both authors are mathematics and teacher educators who consider mathematics a cultural practice.
Data processing and analysis
Interview coding
We first coded the interview responses using the mathematical activity framework. Then, we looked for patterns in the observation and interview data to assess Olivia’s beliefs about mathematics using five questions to frame the analysis (see Table 2): who does mathematics; what is mathematics; and when, where, and why people use mathematics. To determine if an idea Olivia expressed about mathematics should be identified as a belief, we used a framework designed to attribute beliefs to middle school students through interview responses (Jansen, 2006). Jansen (2006) used language cues including modal verbs (e.g. must, should, could, might), expression of affect, and repetition. Olivia only used modal verbs twice (i.e. ‘I could do science or something,’ and ‘Sometimes we need to go to school to do math) and showed positive affect throughout the interviews, so we largely depended on Olivia’s use of repetition of ideas in the interviews and observations to identify her beliefs about mathematics.
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Table 2. Analytical framework part one: Beliefs framework.
Observation coding
Each author individually coded the events using the operationalized definitions of the six categories shown in Table 3 (Bishop, 1988). To determine whether certain events should be identified as mathematical play, we referred to Roth’s framework (1902, as cited in Bishop, 1988), which provided definitions of subcategories of play (see Table 1). To be consistent with research that showed the mathematical significance of Play during games and sociodramatic play (Hanline et al., 2008; Kamii et al., 2005; Siegler and Ramani, 2009; Wolfgang et al., 2001), we focused on discriminative, imaginative, and imitative play. When events fit multiple categories, we noted all relevant categories and identified these events as complex mathematical activity. This procedure was consistent with previous research using Bishop’s framework for mathematical activity, which found that it was difficult and perhaps inappropriate to categorize preschool children’s mathematics into single unique categories (Johansson et al., 2012).
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Table 3. Analytical framework part two: Mathematical activity framework.
Inter-rater agreement
Each author coded the observed events as evidence of, or not of, the six categories of mathematics activity using the framework in Table 3. The initial agreement was 93% of all possible codes (n = 318). After individually reviewing the scoring guidelines and the other’s codes without discussion, each author identified events she had miscoded, which resulted in 97% agreement. Complete consensus of coding was reached through brief discussions of the nine events for which there was still disagreement.
Results and discussion
In our investigation of the relationship between children’s beliefs and the mathematical activity in which they engage before entering school, we found that Olivia’s beliefs about who engages in mathematics and the purposes it serves generally reflected broader cultural beliefs, but not the richness of her own mathematical experiences. Olivia engaged in a diverse range of mathematical activity (counting, measuring, designing, playing, locating, and explaining), including complex activities involving multiple types. What she believed mathematics to be, however, was constrained to writing numbers and letters.
Beliefs about mathematics
Olivia’s interview responses indicated that Olivia believed mathematics is about reading and writing numbers and that mathematics is mainly something adults do in school. Her beliefs generally reflected popular beliefs about mathematics rather than the beliefs of her primary caregiver, her mother (e.g. that mathematics includes cultural activities in which young children engage).
What mathematics is and when we engage
Related questions were posed in multiple ways, yet as the following excerpt shows, Olivia’s interview responses were consistent about what mathematics is – writing numbers and letters.
1 J: Olivia, what do you think math is?
2 O: Writing numbers. Writing letters.
3 J: How do you know that’s what math is?
4 O: Because I’m smart.
5 J: What kinds of things do you do when you are doing math?
6 O: Write numbers and letters.
7 J: George [a friend] does not know what math is. What would you say to help him know what math is?
8 O: Writing numbers.
9 J: Who do you think needs to do math?
10 O: Jamie, Uncle Dave, Kevin.
11 J: Why do we need to do math?
12 O: Because you guys are smart.
13 J: Do you do math?
14 O: Yes [nodding]. Sometimes I get paper and write some letters and numbers.
15 J: When do people do math?
16 O: At schools. I do math on Sundays. Or on Tuesdays.
17 J: Is there anything else you want to say about math?
18 O: Sometimes old people need to do math. Sometimes we need to go to school to do math.
Olivia expressed the belief that doing mathematics means writing numbers and letters each time she was asked about what mathematics is and what kinds of activities are mathematical (lines 2, 6, 8, and 14). Accessing Olivia’s beliefs about the situations in which we use mathematics was difficult, however, because of her specific understanding of the word ‘when’ to mean something temporal rather than activities that might be tied to other events. Line 16 illustrates this understanding and it resurfaced in response to the question, ‘when do we count?’. Olivia answered, ‘On Tuesdays and Wednesdays.’ Yet, Olivia responded quickly and without further prompting when asked what math is and remained consistent in her responses. Thus, in spite of her age, she articulated beliefs about mathematics.
Observation data also supported our assertion that Olivia believed mathematics is something you do in school and involves writing numbers and letters. As previously discussed, beliefs can turn into action agendas (Pajares, 1992). The first example occurred when Olivia described to an imaginary friend what people do at school; she listed math as one of the activities. The second occurred while I was tutoring an algebra student and Olivia said, ‘I want to do math, too.’ When I agreed that she was doing math because I identified what she drew as a triangle, she corrected me ‘see, it’s a one’ and attempted to convince me by coloring in the numeral. This was consistent with her expressed belief that mathematics involves writing numbers.
Who does mathematics where
In interviews Olivia twice identified mathematics as an adult activity (lines 10, 18). When first asked who does mathematics, Olivia immediately identified three adults in her life: her father, her uncle, and her mother (line 10). She reiterated that it is adults who do math when she referred to ‘old people’ (line 18). Olivia also identified school as being the place people do mathematics (lines 16, 18). Her beliefs about who does mathematics and where they do it are likely related, because all three adults she named were graduate students or teachers. Her beliefs that we use mathematics because we are ‘smart’ (line 12) and that she knows what mathematics is because she is smart (line 4) also reflect the common cultural belief that mathematics ability is tied to intelligence (Nasir et al., 2008; Willis, 2010).
Although Olivia indicated she knew what mathematics was because she is smart, Olivia did not spontaneously identify herself as a doer of mathematics. When pressed, however, she did say that she also does mathematics (line 14). Again, this was limited to writing numbers and letters. Because Olivia did not attend school at the time, her beliefs about mathematics as a school activity involving writing numbers and letters may have made it more difficult for her to perceive the mathematics in her own activity.
Why do math?
Finally, our investigation provided some insight into Olivia’s understanding of why we do mathematics. When specifically asked in the interviews why we might need to use mathematics, Olivia could identify situations in which measuring and counting served a clear purpose. For instance, when asked about measuring she remembered her father using a tape measure to measure a turkey and described that as a situation in which measurement was useful. When asked, ‘why do we need to count?’ she responded, ‘Oh, sometimes I do puzzles and I count the pieces.’ Yet, Olivia did not consider counting and measuring to be mathematics. This may suggest that she did not recognize mathematics as personally useful for solving or making sense of problems, as further evidenced by another interview response. When asked how to figure out how many candies were in the bowl, Olivia answered, ‘I could do science.’ Her response likely stemmed from regularly watching (at least weekly) the show Sid the Science Kid (The Jim Henson Company, 2010), which centered on using science to solve problems. So Olivia seemed to associate science with figuring things out and solving problems, whereas mathematics was about writing numbers.
Two potential influences may have worked together to reinforce Olivia’s beliefs about mathematics. First, because the beliefs Olivia communicated about who engages in mathematics and why were inconsistent with her mother’s beliefs, we argue that the cultural messages young children receive about mathematics may be at least as influential as parental influences. Second, the study brought to my attention that situations explicitly identified as mathematics around Olivia, specifically the instances in which I said I was doing math, were likely limited to tutoring sessions and working on problems with pencil and paper. Since Olivia likely formed conceptions of mathematics by extrapolating from contextual examples in her experiences (Rosch, 1999), these limited situations likely influenced her beliefs about what mathematics is, highlighting that what we identify as mathematics with children may shape their beliefs.
Mathematical activity
Although Olivia communicated a view of mathematics as a limited activity, during the two-week observation period she engaged in a rich range of mathematical activity on a daily basis. Thus, Olivia’s beliefs about mathematics were inconsistent with her role as a doer of mathematics. To illustrate this inconsistency, we describe some examples of Olivia’s complex mathematical activity, highlighting the range of diverse and spontaneous mathematical behavior she exhibited. We emphasize how the framework helped detect events that might not typically be recognized as mathematical.
Complex mathematical activity
We identified 21 instances in which Olivia engaged in complex activity, that is, spontaneous behavior that fit into more than one category of mathematical activity. Events identified as complex mathematical activity illustrate how the types of activity overlap and how varied and nuanced a young child’s mathematical activity can be. Subsequently, we describe examples from observations that both illustrate the range of mathematical activity in which Olivia engaged and represent themes (e.g. prevalence of imaginative play) in the data. These particular examples also highlight that a view of preschool mathematics as primarily counting, shapes, and measurement could limit how researchers, teachers, and families perceive the mathematical activity of young children.
While our family rode in the car, Olivia drew a map to church that included shapes, such as a heart and an oval. She ‘read’ the map out loud to give directions to church. This event was noteworthy for several reasons. Most importantly, it was an event that might not be perceived as mathematical through a more traditional lens of young children’s mathematics if not for the shapes. Even without considering the geometric figures, drawing the map involved both designing and locating. Olivia explicitly represented objects in her environment with symbols and she used those symbolic representations to direct the way to church. Drawing the map also reflected a mathematical cultural practice; she understood that maps are used to guide navigation during traveling. Such activities are precursors to navigating with coordinates, which is a formal mathematical activity (Lindquist and Clements, 2001).
When Olivia set up a pretend store at her grandma’s house, selling ice cream, books, and crayons for various amounts of money, she demonstrated complex mathematical activity that was more than counting. She had a cash register with some change, and she gave everyone in the room a ‘dollar’ to spend. Due to Olivia’s use of numbers and counting items, adults might identify counting as the only mathematics in this event, but we recognized other types of activity as well. The entire event involved pretending, which is itself mathematical as a form of imaginative play. Moreover, Olivia’s use of monetary units exemplified the mathematical activity of measuring.
Olivia also demonstrated complex mathematical activity when, after seeing Sid the Science Kid make a chore chart (The Jim Henson Company, 2010), she asked me to help her make a similar chart. Presenting data in charts and graphs is typically seen as mathematical. More broadly this event involved explaining because it represented a way to organize information about the relationships between different phenomena (Bishop, 1988), designing because it required written descriptions and pictures, and imitative play because she was modeling behaviors she saw on the television show.
Parents, teachers, children, and researchers who view early mathematics as limited to numbers, shapes, and measurement would likely perceive some, but not all, aspects of children’s mathematics in these examples. In particular, instruction and research may highlight the counting and shape aspects of events at the expense of other layered aspects of mathematical activity such as explaining or locating. Using a broad framework for mathematical activity allowed us to see more activities as mathematical as well as more diverse mathematics (i.e. various types of activity) in the events.
Range and frequency of activity
The previous examples of complex events highlighted the complexity of Olivia’s mathematical activity within and across particular events. The following results illustrate the range and frequency of each type of mathematical cultural activity during the two-week observation period. I identified 54 distinct events of mathematical activity across all six categories (see Table 4 for illustrative examples of each type of event). The events took place in various observation contexts, including home, the car, running errands, and visiting friends and family. Olivia engaged in an average of almost four instances of self-initiated mathematical activity per day, and because some of the events occurred multiple times (e.g. counting down days to swimming lessons, writing and drawing) but were not identified as distinct events, the total count was higher than captured with this coding procedure. Thus, we found a greater frequency of child-generated mathematical activity than most studies of young children’s everyday mathematical experiences (see Tudge et al., 2008 for a review), which reported on average zero to two instances of mathematics per day. The greater frequency of mathematical activity found may be due to two aspects of this study’s methods: (a) the use of a broader framework for capturing mathematical activity, and (b) the greater access we had to Olivia’s everyday activity over several full days and in various contexts. Subsequently, we describe how Olivia demonstrated activity in different categories to illustrate the range of a young child’s everyday mathematics.
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Table 4. Summary of Olivia’s mathematical activity.
Olivia’s counting activity generally included recognizing numbers in her environment, counting objects, counting events (e.g. the number of bites she had eaten of her meal), and writing numbers. Because counting is generally accepted as a mathematical activity, if a child engages in situations that involve counting, an adult might only see the counting and overlook other types of concurrent mathematical activity. Half of Olivia’s counting events integrated other types of mathematical activity, such as counting down the days until her swimming lessons started. Counting down the days involved both counting and measuring, because she explicitly counted days as a measurement of time. In addition to the store example described previously, the following events illustrate counting integrated with other types of mathematical activity:
Playing
Counting while playing hide and seek
Counting game pieces to see who won
Pretending to be a monster family with ‘three toes, 89 ears, 89 eyes…’
Designing
Writing and recognizing numerals
Measuring
Reading numbers on speed limit signs, recognizing this as a speed and asks, ‘are you going too fast?’
Comparing ages of friends and family
Here, again, we found that the mathematics activity framework helped us identify the depth of mathematical behavior in these examples beyond counting.
Although I never observed Olivia using a measurement tool during the observation period, she engaged in various activities in which she referenced measurements. The thirteen measurement events were mainly related to time, age, speed, height, and money. Olivia’s references to measurement were sometimes explicit due to use of units to describe money (dollars) and time (days, weeks, years). In one event she wondered about her sister’s shoe size, asking, ‘what T are they?’ This demonstrated that Olivia generalized ‘Ts’ as the unit of measure for clothing sizes, because young children’s clothing is sized 2T (2 toddler), 3T, and so forth. In other examples, Olivia’s measurement was more implicit, such as when she read her pretend watch and said it was ‘45.’ When she read speed limit signs, she knew it had to do with how fast we should drive, but she did not identify any units. She may have been modeling adult references to speed limits, knowing that most of us say just the number in which adults leave off the units (e.g. ‘it’s 70 here’). She also used words to represent relative size such as ‘miniature’ and ‘teeny.’
Olivia’s comparison of sizes provide some examples of mathematical activity we noticed because of the mathematical activity framework, which may be overlooked if considered through a more traditional lens of what is valued as measurement. Typically, people associate measurement with the use of standard tools with standard units. People developed such technologies, however, to serve the core purpose of measuring – to compare sizes when it was not possible to physically compare things side by side (Ritchhart, 1999; Van de Walle et al., 2010). These examples of Olivia’s measurement thinking highlight how young children may engage in measurement activity by communicating about comparative size and time – the foundation of measurement – without explicitly using numbers or standard units (Van de Walle et al., 2010).
Olivia engaged in locating activity a few times, in varied ways. In addition to the event described earlier, in which she designed a map to help orient her family as they drove to church, she used landmarks while we drove and told me when we were ‘almost there.’ She used directional words and phrases like ‘up the ladder, down the ladder,’ ‘top,’ and ‘bottom.’ These types of direction words are part of describing ‘the spatial world,’ which is a locating activity (Bishop, 1988: 28).
Bishop (1988) described explaining as ‘answering the complex question of ‘why?’ and exploring the logical relationships between events’ (p.48). Explaining may not typically be perceived as mathematical activity in young children, but by using the framework we noticed that this was an unprompted part of Olivia’s everyday experience. Of the six events identified as explaining activity, three were related to conjecturing, which is a primary disciplinary practice of mathematics. In these cases, Olivia made an if-then hypothesis as she wondered aloud about how events were related. For example, she said that when her cousin turned two, he would start to talk; and if the snow melts, then the puddles ‘melt’ too. These examples demonstrate her cause and effect relational thinking related to time and events. Olivia also demonstrated logical relations about spatial play when playing with puzzles; for example, she commented that maybe if she turned a piece, it would fit (also designing, locating, and playing). With growing emphasis on conjecturing, explaining, and forming mathematical arguments as part of the K-12 mathematics curriculum (e.g. CCSS-M, 2010), these examples of Olivia’s activity serve an important role in preparing for formal mathematics.
Although Olivia engaged in discriminative, imitative, and exultative play, about half of her playing activities were imaginative games. In an interview, Olivia emphasized imaginative and imitative types of play, suggesting that these types of play are an important part of Olivia’s daily life.
What’s your favorite thing to play?
Ball…Pretending. Imagining. I like to imagine that I’m a grown-up…I like to pretend that I’m a tree.
Olivia mentioned playing ball, but then gave specific examples of imaginative or imitative play and clearly identified pretending and imagining as her favorite ways to play.
The prevalence of imaginative play does not simply reflect Olivia’s preferences; it also exhibits mathematical activity that may have implications for future mathematics learning (Hanline et al., 2008; Lakoff and Nunez, 2000; Thurston, 1994). Examples of Olivia’s imaginative play suggest abstract thinking, but more specifically, these activities are closely linked to association and metaphor, which are important aspects of mathematical thinking (Lakoff and Nunez, 2000; Thurston, 1994). Association and metaphor are integral parts of imitative play and particularly imaginative play, and as such are an important part of Olivia’s early cultural mathematical experiences. For instance, Olivia declared she was ‘Jungle Girl,’ pretending the couch was a jungle and the curtains were ‘vines.’ Here, as in every instance of imaginative play, Olivia attributed characteristics of something that was not present to some object in her environment (i.e. she exhibited symbolic substitution). Curtains became vines. Swimming goggles became owl eyes that could see in the dark. Her teeth become dragon fangs. Certainly, in these examples Olivia was ‘sensing that some phenomenon or situation or object is like something else (association)’ and ‘holding two things in mind at the same time (metaphor)’ (Thurston, 1994: 165). These instances of sophisticated and organized levels of symbolic substitution in Olivia’s imaginative play are noteworthy, because Hanline et al. (2008) found that the frequency of such symbolic substitution during sociodramatic play was a powerful predictor of mathematics ability among preschoolers, arguing it provides evidence of abstract thinking. Thus, Play is not only valuable for children’s mathematics learning because of the opportunity to engage in counting, spatial reasoning, and so on – Play itself can be mathematical activity.
Olivia engaged in many diverse behaviors recognized by some in the mathematics education community to be mathematical (e.g. Bishop, 1988; Nasir et al., 2008), or at least recognized as preliminary mathematical thinking (Thurston, 1994). Thus, it is clear that even a young child without school experience can engage in rich mathematical activity. Our findings contradict some earlier results that indicated young children engaged in little or no naturally occurring mathematical activity (e.g. Tudge and Doucet, 2004; Tudge et al., 2008). Our study suggests that a broader perception of mathematics allowed us to detect more of the important mathematics activity in children’s regular experiences. The fact that Olivia exhibited a wide range of mathematical activities is promising for entry into school. Olivia’s daily unprompted experiences spanned the core topics in the early grades mathematics curriculum (Ertle et al., 2008; NCTM, 2006; Samara and Clements, 2010), suggesting a rich basis for formal mathematical learning in school. Yet, Olivia’s complex activity was inconsistent with her beliefs that mathematics is a limited activity. As we discuss in the next section, this mismatch indicated the early development of potentially problematic conceptions of mathematics.
General discussion and implications: Relating Olivia’s beliefs and activity
Using a broad mathematical activity framework as an interpretive lens, it became clear through observations and interviews that Olivia’s everyday mathematical activity was plentiful and widely varied. To Olivia, however, mathematics was quite limited in terms of what it was (i.e. writing numbers and letters) as well as where and when people engage in it. On one hand, the difficulty people have in identifying how mathematics is used in their own lives and by others is well-documented even for adults (e.g. Martin, 2000), so it is not particularly surprising that Olivia did not recognize the mathematics in her everyday activity. On the other hand, we suspected that Olivia would be a unique case of a child who would be most likely to demonstrate broader beliefs about what mathematics is and engage in varied mathematical practices. We argue that the incongruence between the richness of young children’s activity and their beliefs could be problematic for later learning of and perseverance in mathematics. It points to implications for parents, teachers, and caregivers, who through formal and informal interactions are in a position to help children perceive their activity as mathematical.
First, the incongruence may be an early sign that Olivia valued mathematics as a subject learned in school more than the mathematical activity she engaged in as part of her life experience. This suggests that Olivia (and possibly other young children) valorizes academic mathematical practices over everyday practices based in the home or community, as do many children, parents, and teachers in the US (Civil, 2007; de Abreu and Cline, 2007). We recognized that her mathematical activity (e.g. doing puzzles, keeping track of days until swimming lessons) was plentiful and often served to help her determine how to proceed or make predictions, consistent with suggestions by Head Start and the CCSS-M (CCSS-M, 2010; Office of Head Start, 2011). As we discussed, however, Olivia viewed mathematics as something school-related involving writing symbols rather than ideas to be explored or learned on her own, and mathematics did not serve a clear purpose. This mismatch between Olivia’s mathematical beliefs in school and non-school contexts may interfere with her valuing her own mathematical experiences and her self-generated problem solving processes in favor of teacher-given content and strategies (Cobb, 1986). This, in turn, may hinder the development of positive beliefs about mathematics because beliefs about diligence and one’s self-efficacy are important aspects of productive dispositions toward mathematics (NRC, 2001).
Second, the results suggest a need to be more explicit with children about how their everyday activity is mathematical by labeling what they are doing with the term ‘math.’ This may be especially important for activity that is not traditionally viewed as mathematics such as imaginative play, locating, designing, and explaining. It seemed that Olivia noticed the limited use of the word ‘math’ in the house, and these instances probably contributed to (or created) her conceptualization of mathematics. Unfortunately, her conceptualization of what is involved in doing mathematics was much narrower than her rich range of mathematical activity – activity that I valued as contributing to her mathematical understanding. So although we did not find a direct link between parents’ and children’s beliefs about mathematics, as part of a larger culture, parents and teachers need to be aware of what their actions and words communicate to children about mathematics. Future research should investigate how explicit labeling of their mathematical activity can influence young children’s beliefs about mathematics.
The generalizability of these results is unclear. As discussed, very few studies on children’s beliefs about mathematics include young children without school experience. One important outcome of this study, then, is to provide evidence that young children generalize from their experiences with the word ‘math’ to form early beliefs about mathematics. Increasingly, ‘math’ is used in the US in public television shows for young children, such as Sid the Science Kid (The Jim Henson Company, 2010) and Peg +Cat (The Fred Rogers Company and 9 Story Entertainment, 2014), and popular children’s books, such as the Pinkalicious series (Kann, 2010). Coupled with the growing emphasis on preschool mathematics, it may be increasingly likely that young children will have earlier exposure to broader cultural beliefs about mathematics. Unlike Olivia, some children may not be able to or may not have the opportunity to articulate beliefs about mathematics. Nevertheless, it is possible they are forming such early beliefs shaped by family and cultural influences.
Third, Play is an important context in which to help shape children’s conceptions of mathematics and their perceptions of themselves as doers of mathematics. Play provides opportunities for parents and teachers to explicitly ‘name’ the mathematics with which children spontaneously engage. The richness of Olivia’s mathematical activity observed during unstructured Play provided additional evidence that young children build mathematical knowledge in these contexts (Bishop, 1988; Samara and Clements, 2010; Seo and Ginsburg, 2004). This highlights the importance of Play in children’s everyday lives and early school experiences.
Fourth, this incongruence between beliefs and activity in the case of Olivia raises questions about how to connect children’s everyday and school mathematical experiences. As research has noted and this study confirmed, children might not recognize or value their mathematical activity outside of school as being mathematics (Civil, 2007). As a three-year-old, Olivia already had fairly strict notions that mathematics is an adult, school-related activity. This tension regarding beliefs is something teachers and researchers need to account for in their efforts to bridge the gap between the use of mathematics in and out of school.
Finally, early childhood and elementary teachers themselves may not recognize some forms of playing, designing, locating, and explaining as mathematics (Anderson and Gold, 2006; Lee and Ginsburg, 2009; Moseley, 2005) and thus miss opportunities to scaffold some of the ways children engage with mathematics. Therefore, this study has implications for teacher education, suggesting that professional development for existing and future preschool and early elementary teachers could emphasize the nature of children’s mathematical activity and how to recognize and capitalize on those experiences at home, in childcare, and in the classroom. An assignment to observe children’s mathematical activity using a framework that emphasizes cultural mathematical activity could help new and continuing teachers view children’s mathematical activity with more depth and breadth.
Reflections as a parent
Multiple friends and family have come to me with concerns about their children who are ‘struggling’ in math in kindergarten and first grade, usually because they have difficulty with automaticity of facts. With their children and my own (since the data were collected, Olivia’s sister has passed the stage Olivia was in during the study, and now her brother is two years old), I try to take the advice in the implications discussed and be more careful and explicit with my use of the word ‘math.’ For example, when they conjecture about relationships in their environment, identify a pattern, or try to determine how to build a taller block tower, I might say, ‘that’s good math thinking!’ Or I suggest math activities (and refer to them as such) that do not involve numbers or shapes, such as hiding an object and using clear directions to help each other find it. I also try to be explicit when I am using mathematical reasoning to solve a problem, such as when we need to figure out when we need to leave home to be somewhere at a specific time. In all these examples, my goal is to repeatedly communicate that people do use mathematics outside of school, that a diverse range of activities in our everyday lives involve mathematics, and that they are doers and users of mathematics.
Conclusion
We selected Olivia as a unique case, suspecting her contexts would be optimal for revealing both diverse mathematical activity and broad views of mathematics because both of her parents have been involved to some extent in mathematics education. The results of the study provided evidence that even very young children may hold firm beliefs about mathematics that may be inconsistent with what we (the authors) view as the richness and breadth of their mathematical activity.
This inconsistency has implications for mathematics instruction in preschool and early elementary grades. Most notably, it suggests the importance of helping children detect the mathematics in their everyday activity, including Play. Whether preschool mathematics emphasizes teachable moments in informal Play or more formal and structured instruction through standard-based curricula, the adults in children’s lives need to attend to the potentially rich and diverse prior mathematical knowledge young children bring to the classroom.
The inconsistency also warrants further research with a larger and more diverse sample of children about how young children’s mathematical beliefs develop in relation to their mathematical activity. These findings and future research should also encourage researchers, preschool and early elementary teachers, and parents to consider their own conceptions of mathematics and how these conceptions filter what they notice in and how they respond to young children’s mathematics. Moreover, research and teacher development should attend to how teachers and parents can implicitly and explicitly convey views of mathematics to children that promote effective mathematics learning and productive dispositions.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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Author biographies
Jamie L Wernet (jwernet@lansingchristianschool.
Julie Nurnberger-Haag (nurnber3@msu.
