Abstract
In this article, we review research literature concerning number sense from several related fields. Whereas other authors have pointed to difficulty defining number sense or to some degree of inconsistency in the literature, we argue instead that this is a case of polysemy: three constructs given the same name. Our purpose is to clarify the research literature concerning number sense by naming and defining these three constructs, identifying similarities and differences among them, categorizing the research traditions associated with each construct, and summarizing the methods used and findings reported. We compare and contrast themes in each body of literature by drawing on a sample of 141 research articles that were focused on number sense. We find evidence that there is confusion of number sense constructs across research traditions. We emphasize the need for clarity in order for research in each of these traditions to progress.
What’s in a name? That which we call a rose, by any other name, would smell as sweet.
Researchers in the social and behavioral sciences have become concerned with impediments to progress resulting from confusion over constructs (Brown, 2015; Gintis, 2007; Larsen et al., 2013; Le et al., 2010; Shaffer et al., 2016). Two particular issues have the potential to plague the research literature: synonymy and polysemy. Synonymy refers to different terms—in this discussion, constructs—having the same meaning. Polysemy refers to the same term being used in different ways. Larsen et al. (2013) argued that these issues result in a proliferation of constructs and meanings, leading to “reverse progress” (p. 1532) as less is known over time about relationships between constructs, relative to the number of constructs that appear in the literature.
There is a history in Review of Educational Research of articles that have sought to clarify the meaning of terms as used in bodies of research literature. Such efforts have concerned terms used to describe methodologies, or categories thereof (Jacob, 1989) or specific research constructs (Nevo, 1983; Tschannen-Moran & Hoy, 2000), as well as work emphasizing the need for rigorous and sometimes multifaceted definitions (Haertel, 1985). While reviews for research, included within empirical articles, may in some cases make contributions to the literature, reviews of research are ideally positioned to do so (Maxwell, 2006). In the vein of such reviews, the present article seeks to clarify the meaning of a term that is used across a variety of fields: number sense. As we demonstrate in this article, the varied uses of the term number sense 1 in the research literature constitute a problematic case of polysemy.
The first known instance of the term number sense comes from Dantzig (1954): Man, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense. This faculty permits him to recognize that something has changed in a small collection when, without his direct knowledge, an object has been removed or added to the collection. (p. 1)
In recent decades, the term has become popular in academic disciplines including mathematics education, special education, and mathematical cognition. There has been increasing interest and research activity centered on the term number sense, as evidenced by a dramatic increase in the number of published articles: Our search revealed that numbers of research articles with “number sense” in the title increased from 13 articles in the 1990s to 40 articles in 2000 to 2009 to 71 articles published in 2010 to 2016. However, the term number sense is used in a variety of ways, and there is evidence of confusion and a need for clear distinctions in number sense research.
Many authors have noted difficulties defining number sense or disparities in the definitions and descriptions found in the literature (e.g., Andrews & Sayers, 2015; Berch, 2005; Dunphy, 2007; Howell & Kemp, 2005, 2009, 2010; Lago & DiPerna, 2010). Before the vast majority of the current body of literature on number sense existed, McIntosh et al. (1992), writing from the perspective of researchers in mathematics education, argued for the need to clarify the meaning of the term: Number sense is a topic of great interest in school mathematics. It is also nebulous and difficult to describe, although it is recognizable in action. Continued productive discussion of number sense (by researchers, teachers, and curriculum developers) must at some stage be based on a definition, characterization, or model which portrays number sense in a clear yet comprehensive manner. The more clearly number sense is understood, the more likely there will be progress made in research, as well as in curriculum development and instruction. (p. 8)
Although McIntosh et al. (1992) recognized this issue more than a quarter century ago, there remains a lack of clarity in the literature, especially as new and different uses of the term number sense have arisen.
In a brief review article published in the Journal of Learning Disabilities, Berch (2005) noted that “cognitive scientists and math educators define the concept of number sense in very different ways” (p. 333, emphasis added). On the basis of literature in mathematical cognition, cognitive development, and mathematics education, he identified two contrasting views: The major disparity here is between what might be considered a “lower order” characterization of number sense as a biologically based “perceptual” sense of quantity and a “higher order” depiction as an acquired “conceptual sense-making” of mathematics. (p. 334)
Although Berch (2005) identified a difference in the use of the term number sense across fields, in his analysis, he assumed that these reflected different conceptualizations of a single number sense construct. This assumption is particularly evident in his list of 30 “alleged components of number sense” (p. 334). His point was to illustrate disparity in the literature and then to argue for practical ways of operationalizing number sense for the purposes of research concerning students with mathematical disabilities.
In a similar vein, Andrews and Sayers (2015) referred to number sense as a “poorly-defined construct” (p. 257). They identified three “distinct, but related, perspectives” (p. 258) concerning number sense, which they termed preverbal, foundational, and applied. Andrews and Sayers (2015) proceeded to focus on foundational number sense and introduced a framework for analyzing it. Similar to Berch (2005), these authors describe number sense as a single construct, which is defined and interpreted in different ways.
In our review, which is the product of a collaborative effort between researchers in mathematics education and special education, our purpose is not to argue for a specific way of defining number sense. Instead, we are concerned with the need for greater coherence in the literature concerning number sense across fields. In this article, we categorize such literature on the basis of researchers’ descriptions of number sense and their associated methodological approaches. Our review of a sample of 141 research articles concerning number sense leads to a clear conclusion that is responsive to the issues identified by McIntosh et al. (1992), Berch (2005), Andrews and Sayers (2015), and others: Instead of disagreement over a single construct, we find three distinct number sense constructs at play in the literature. Thus, we argue that this is a case of polysemy and a microcosm of broader issues of construct confusion in the social and behavioral sciences.
Goals and Focus
It is difficult for research concerning a particular number sense construct to advance so long as authors continue to attempt to draw upon literature concerning different constructs that go by the same name. As our analysis illustrates, these different constructs involve contrasting assumptions about the nature of number sense and are embedded in traditions with distinct orientations and concerns. In particular, in this article, we answer the following research questions:
Thus, we describe the three number sense constructs that we identified through our review of the literature, and we provide an overview of the associated research traditions—the bodies of literature related to each construct, including the populations, research methods, and findings that are typical of that body of literature. We report the numbers of articles in our sample that we identified as belonging to each tradition, and we summarize these traditions in the form of methods, findings, and themes.
By presenting these results, we aim to help organize and clarify a muddled number sense literature. We hope that this review will serve as a guide to researchers who are new to the study of number sense within their respective fields. It should also help more experienced researchers to recognize and appreciate the confusion in the literature, to see the distinctions between constructs more clearly, and to communicate their work more effectively moving forward.
Method
We describe our methods of data collection and analysis. In particular, we describe our search strategy, coding for number sense constructs, coding of articles into research traditions, and coding of article citations across traditions. We report the results of our interrater reliability check. We also describe our conceptual review methods for synthesizing literature belonging to each tradition.
Overview of Methods
We conducted a systematic review of a substantial sample of the number sense literature. Our goal was to synthesize the literature (Noblit & Hare, 1988) with regard to conceptualizations of number sense, as well as methods and findings that characterize each research tradition. We decided to conduct a conceptual review (Petticrew & Roberts, 2006), because such methodology aligns well with our “interest in gaining new insights” into number sense constructs and related research traditions (Kennedy, 2007, p. 139).
Based on Suri’s (2011) suggestion to adapt Patton’s purposeful sampling strategies to qualitative research synthesis, we enacted purposeful sampling strategies (Patton, 2002) to gather research articles that indicate typical conceptualizations of number sense. To get access to studies representing typical case(s) of number sense conceptualization, we decided to enact criterion sampling (Patton, 2002). Specifically, in our systematic review, we employed the following protocol (Hauari et al., 2014):
The selection criteria to decide which studies should be included in our analysis were composed of (a) date of publication, (b) articles written in English, (c) and articles published in research journals. We excluded the resources that were practitioner articles or book chapters.
In the searching process, we used five databases and searched for “number sense” (with quotation marks) appearing in the titles of the work. Based on our search criteria, our screening was already established as having number sense in the title of the research articles. Thus, it was not necessary to screen abstracts or full texts for possible exclusion. We had only to screen for journal type and to ensure that number sense was, in fact, part of the title of each article.
In the mapping process, we laid out the descriptive characteristics of each study with a particular focus on how number sense was conceptualized and operationalized within the study.
We used methods of conceptual review to identify subcategories of articles, to synthesize methods and findings, and to identify themes.
We provide more details below.
Search Strategy
We searched five databases (Academic Search Complete, Education FullText, ERIC, JSTOR, and PsycINFO) for research articles with number sense in their titles. It was important to control the scope of our review in this way, because we sought to identify how number sense was defined and analyzed and to identify characteristics of number sense research. This being our purpose, including all articles that made any mention of number sense would have muddied our results. In our initial search, the numbers of journal articles that mentioned number sense varied from 168 to 1,587 in the databases listed above, and many of these were not research studies or were studies that did not focus on number sense. For the purposes of our review, the relevant studies were those in which number sense was central to the research. We found the inclusion of number sense in the title of the article to be a reasonable proxy for this centrality.
We searched for all such articles that met the criteria described above and that were published on or before December 31, 2016. (We later updated our search to include articles published in 2017.) We focused on research articles published in peer-reviewed journals. 2 Thus, we filtered out practitioner articles, books and book reviews, and conference abstracts and proceedings. For example, our initial search identified 32 publications in Teaching Children Mathematics, including both journal articles and book reviews. These were all excluded from our study because Teaching Children Mathematics is a practitioner journal, not a research journal. As a specific example, “Developing number sense through real-life situations in school” (Yang, 2006) is oriented toward teachers. It provides in-depth description of a particular lesson and highlights certain aspects of the teachers’ practices. Such publications did not meet our criteria; however, research articles by Yang and colleagues were included in our review. We also filtered out publications not written in English.
A total of 124 articles published by the end of 2016 qualified for inclusion in our sample. (An additional 17 articles published in 2017 were added later.) We recognize that some high-quality articles concerning number sense may not be included in our sample as a result of these requirements. Our purpose was not to provide comprehensive reviews of the literature belonging to each number sense tradition. It was to identify distinct number sense constructs and to describe features of the research literature associated with each construct, focusing on similarities and differences across these research traditions.
In addition to the sample of articles described above, we reviewed seminal works and publications of historical significance that focused on number sense. We identified seminal works based on their being cited frequently in our sample of research articles and/or representing a synthesis of research related to number sense within a particular field. In addition to the seminal works that were research articles and met our initial search criteria, we included The Number Sense by Dehaene (1997/2011), Evolutionary Origins and Early Development of Number Processing (Vol. 1) edited by Geary et al. (2015), Establishing Foundations for Research on Number Sense and Related Topics: Report of a Conference, edited by Sowder and Schappelle (1989), and the handbook chapter “Estimation and Number Sense” by Sowder (1992). These works were not part of our sample of articles. Instead, they were consulted to provide us greater access and insights into the history of number sense research and to enable us to answer questions concerning definitions, assumptions, findings, and themes in cases in which a consensus could not be identified within our sample of articles.
Coding for Number Sense Constructs
Based on our knowledge of the literature, together with the previously mentioned issues raised by other authors (Andrews & Sayers, 2015; Berch, 2005; McIntosh et al., 1992), we expected to identify in the literature more than one distinct number sense construct. We did not know at the outset how many constructs we might find or exactly how we might distinguish them. We began by reading selected articles from our sample with a focus on the authors’ interpretations of number sense and the apparent origins of those interpretations (based on citations and use of key constructs, as well as the populations studied and methods used). We proceeded by using open coding to generate provisional definitions based on salient contrasts in authors’ ways of using the term number sense, including the authors’ evident theoretical assumptions and phenomena of interest. We refined our definitions through constant comparative analysis as we reviewed additional articles (Corbin & Strauss, 2008). After we had reached a saturation point, we settled on three constructs: (a) approximate number sense (ANS), (b) early number sense (ENS), and (c) mature number sense (MNS; see Table 1).
Characteristics of number sense constructs
Note. ENS = early number sense; MNS = mature number sense.
Note that the analysis described above pertained to number sense constructs and was concerned with answering our first research question (“How is the term number sense used in the research literature in the social and behavioral sciences?”). Our subsequent analysis concerned our second research question. This analysis pertained to research traditions related to number sense in the literature. Although we had identified three number sense constructs, these would not necessarily serve to categorize articles as belonging to research traditions. Given the apparent confusion regarding number sense in the literature, some authors might draw upon work related to more than one number sense construct or belonging to more than one research tradition. In fact, when we identified instances in which an article from one tradition cited an article from a different tradition, we coded for the nature of this citation as being consistent or inconsistent. (Further details appear near the end of the Method section.)
Categorizing Articles Into Traditions and Characterizing Traditions
Indeed, as we worked to categorize articles as belonging to research traditions, we became attuned to finer grained similarities and differences between these traditions. Although we could clearly distinguish the three number sense constructs, we identified some overlap among the related traditions. Thus, we coded the articles on the basis of a more refined scheme. Articles belonging to the approximate number sense tradition were coded as 1, those belonging to the early number sense tradition were coded as 2, and those belonging to the mature number sense tradition were coded as 3. Articles that were in the overlap between Categories 1 and 2 were coded as 1.5, and those in the overlap between Categories 2 and 3 were coded as 2.5. Articles coded as 1.5 drew from both Traditions 1 and 2.
Initially, our review focused on 124 articles published through the end of 2016. For the purposes of ensuring interrater reliability, each of the three authors independently coded 62 of those 124 articles using the refined coding scheme {1, 1.5, 2, 2.5, 3}. This coding resulted in half of the articles being coded by two authors. We initially agreed on 51 of these 62 coding decisions (~82%). Levels of interrater agreement above 80% are commonly regarded as very high or excellent (Gwet, 2014). Furthermore, each disagreement was of size 0.5. In other words, the disagreements exclusively involved close decisions between neighboring categories (1 vs. 1.5, 1.5 vs. 2, 2 vs. 2.5, or 2.5 vs. 3). After the interrater reliability check, all three authors discussed the decisions that were under dispute until we reached consensus on each decision. We did likewise for the 17 articles that were published in 2017, coding them independently and then comparing and discussing coding decisions until consensus was reached.
Categories 1.5 and 2.5 are addressed within our findings concerning similarities and differences across traditions and in relation to the crosscutting theme of overlap and confusion between research traditions. Figure 1 shows examples of articles that we coded as belonging to each tradition, including those belonging to the overlap categories. 3
Venn diagram depicting examples of articles in (1) approximate number sense (ANS) research, (2) early number sense (ENS) research, (3) mature number sense (MNS) research, and the overlapping regions between these categories.
Coding of Cross Citations
As mentioned above, a theme that emerged from our analysis was the prevalence of citations across traditions, which we refer to as cross citation (e.g., an article in the MNS tradition might cite an article from the ANS tradition). We identified all instances of cross citations and coded these as consistent or inconsistent. By this, we mean consistent or inconsistent with the assumptions underlying the relevant construct. For example, an article in the ENS tradition that might appropriately cite work from a neighboring tradition in order to clarify the construct under investigation by comparing and contrasting with other literature. It is also possible that a result from a different tradition might be relevant and worthy of citation. Such instances would be coded as consistent.
However, results regarding number sense from one tradition do not necessarily—in fact, would not in general—be relevant to work in another tradition. Citations of such results might instead be owing to confusion due to polysemy. Authors may cite other work concerning number sense without being careful to ensure that the work being cited actually pertains to the same construct that they are investigating. In addition to results, authors might cite work from another tradition to support broad claims about the nature of number sense (thus applying claims about one number sense construct to another). Again, such instances of cross citation are inappropriate because they ignore important distinctions between number sense constructs. We coded such cross citations as inconsistent. Note that articles that were categorized as 1.5 or 2.5 by their nature tended to involve consistent cross citations: They drew upon neighboring traditions in consistent ways. Therefore, articles in these overlapping categories were not included in our coding for cross citations. 4
Synthesis of Key Concepts, Analytic Approaches, Findings, and Themes
For the purpose of summarizing the three main traditions, we focused on those articles that were cleanly categorized as 1, 2, or 3. Drawing upon these sets of articles, we summarized the features of the three main traditions. We each took primary responsibility for summarizing the literature belonging to one of the traditions. We subcategorized the sets of articles within each tradition through a process of constant comparative analysis of types of studies. We developed summaries iteratively with feedback from one another. We then compared our summaries on the bases of key concepts, analytic approaches, findings, and themes. We did not apply an a priori coding scheme to identify themes in the literature. Instead, these emerged from our analyses within each tradition. Questions that arose concerning similarities and differences between traditions or lack of clarity regarding any terms led us to return to our sample of articles and/or our set of seminal publications for answers. This process, too, was iterative. We refined our summaries of the aspects of each tradition to focus more clearly on the similarities and differences among traditions. 5
Our purpose here is to report what the literature says about number sense in an effort to clarify such literature. Our claims are about the literature itself. We do not purport to offer truths about the underlying phenomena, nor do we seek to settle debates about those phenomena. Contradictions, inconsistencies, and confusions in the literature motivated us to undertake this work.
Results
We first describe the three number sense constructs that we identified in the literature. We define each construct, and we highlight similarities and differences in assumptions that distinguish these constructs. We then describe how number sense is operationalized and analyzed in the research tradition associated with each construct. We summarize findings and themes from each major tradition (our Categories 1, 2, and 3). Finally, we focus on the overlap between these bodies of literature, including issues of confusion across them.
Overview of the Three Distinct Number Sense Constructs
Table 1 provides a side-by-side comparison of characteristics of the three number sense constructs. Approximate number sense is believed to be an inborn set of neurological abilities that is common to humans and some animals. This construct concerns perception and discrimination of magnitudes rather than explicit knowledge of number words or symbols. ANS research involves infants, children, adults, and nonhuman animals (e.g., Halberda & Feigenson, 2008; Libertus & Brannon, 2009; Low et al., 2009). Much of the research with humans involves observing brain activity while participants perform various tasks, such as determining which of two sets consists of more items (e.g., Dehaene, 2001; Libertus & Brannon, 2009; Stoianov & Zorzi, 2012). Dehaene (2001) uses the term the number sense (with the definite article the and emphasis on the word number). This language is indicative of the fact that the ANS construct is regarded as a basic neurological ability, related to visual and auditory perception (i.e., in addition to senses such as sight and hearing, there is also a number sense).
In contrast to ANS, early number sense includes learned skills that involve explicit number knowledge, such as counting items using number words and comparing numbers represented symbolically as numerals. Typically, studies of ENS involve young children and/or students with disabilities. Some researchers claim that ENS skills build upon the more basic ANS abilities (Andrews & Sayers, 2015; Aunio et al., 2005; Geary et al., 2015). Levels of ENS skills vary from person to person (Aunio et al., 2004, Aunio et al., 2006; Cheung & Chang, 2015; Ivrendi, 2011), and ENS is regarded as an important predictor of success in school mathematics (Dyson et al., 2013; Jordan et al., 2009; Locuniak & Jordan, 2008). Accordingly, ENS skills are well aligned with school mathematics, especially in the early childhood years (preschool to Grade 2). ENS research does not belong to a single field. Rather, it is conducted by researchers in fields including mathematics education, special education, and cognitive psychology.
We use the term mature number sense to distinguish the number sense construct that features prominently in the mathematics education research literature. This construct encompasses multidigit and rational-number sense, 6 with studies focused primarily on middle-grades (i.e., upper elementary and middle school) students and preservice teachers. Like ENS, MNS is learned. In contrast to descriptions of ENS, researchers’ descriptions of MNS tend to be articulated in terms of components, which refer to underlying conceptual structures rather than directly observable skills (e.g., McIntosh et al., 1992; Reys & Yang, 1998). It is also presented in more holistic terms such as “general understanding” and described in terms of “inclination” and “expectation” (Reys et al., 1999). Such descriptions are not as easily operationalized as are skills. Furthermore, whereas ENS is well aligned with the mathematical skills that students typically learn in preschool and the primary grades, MNS is often contrasted with the mathematics that students learn in school. In particular, students who rely heavily on the standard algorithms that they were taught in school may not exhibit characteristics of MNS, such as flexibility (Reys et al., 1999; Reys & Yang, 1998).
Key Concepts in Number Sense Research Traditions
Having provided an overview of the three constructs, we delve deeper into related concepts that appear in the three corresponding research traditions.
Key Concepts in ANS Research
According to Dehaene (2001), “Number sense is a short-hand for our ability to quickly understand, approximate, and manipulate numerical quantities” (p. 16). The ANS is considered part of an evolutionary process related to specific neurological abilities. Dehaene (2001) suggested that lower levels of ANS could be attributed to lesions in specific areas of the brain.
Three neurological abilities are associated with ANS: perceptual subitization, magnitude discrimination, and the use of a mental number line. Perceptual subitization can be defined as the ability to rapidly or immediately identify the cardinality of a set consisting of up to three or four items (Clements, 1999; Dehaene, 2001). Any numbers of items beyond four may be approximated with less precision (Clark & Grossman, 2007). Magnitude discrimination is an individual’s ability to readily determine which of two sets consists of more or fewer items (when those items are presented visually or auditorily; Dehaene, 2001). The mental number line refers to “an analogical representation of number” that is sequential in nature (Dehaene, 2001, p. 17). The use of a mental number line is inferred from the ability to quickly make comparisons, such as to “decide that 9 is larger than 5” (Dehaene, 2001, p. 16). 7
Key Concepts in ENS Research
Andrews and Sayers (2015) used the term foundational number sense to label their particular conceptualization of ENS: “Foundational number sense (which builds on children’s preverbal number sense), comprises number-related understandings that require instruction and which typically occur during the first years of schooling” (p. 258). Rather than being innate, ENS is acquired through experiences that are attained in or out of school. In determining the key elements of ENS, Jordan et al. (2006) examined the “assessed skills that have been validated by research and are relevant to the math curriculum in primary school” (p. 154).
Six main skills are prominent in ENS research: number recognition, counting, recognition of number patterns, number comparison, performing number operations, and estimation (Andrews & Sayers, 2015; Baroody et al., 2012; Jordan et al., 2006; Malofeeva et al., 2004; McGuire et al., 2012). Number recognition requires children to associate the number symbols with the vocabulary and meaning of numbers, such as to associate the symbol 6 with the word six and with six fingers (Andrews & Sayers, 2015; Baroody et al., 2009; Baroody et al., 2012). The skill of counting can be subcategorized into rote counting and systematic counting, including ordinality, cardinality, and counting backward and forward starting with an arbitrary number (Andrews & Sayers, 2015; Jordan et al., 2006; Jordan et al., 2007; Jordan et al., 2012; Jordan, Glutting, & Ramineni, 2010; Jordan, Glutting, Ramineni, & Watkins, 2010). Skill in dealing with number patterns is the ability to copy a given pattern and identify the missing number in a sequence of numbers (Andrews & Sayers, 2015; Jordan et al., 2006; Jordan et al., 2007; Jordan et al., 2012; Jordan, Glutting, & Ramineni, 2010; Jordan, Glutting, Ramineni, & Watkins, 2010). Number comparison refers to awareness of the magnitude of given numbers and the ability to make comparisons between different magnitudes (Andrews & Sayers, 2015; Howell & Kemp, 2005, 2009, 2010). The skill of number operations includes the ability to perform simple arithmetical calculations (i.e., using whole numbers up to 10 or 20) with addition and subtraction (Andrews & Sayers, 2015; Baroody et al., 2009; Baroody et al., 2012). Estimation here refers to magnitude estimation of symbolic and nonsymbolic quantities, including the use of a physical number line to identify the approximate location of a number or by using reference points (Andrews & Sayers, 2015; Ivrendi, 2011; Jordan et al., 2006; Jordan et al., 2012; Jordan, Glutting, & Ramineni, 2010; Jordan, Glutting, Ramineni, & Watkins, 2010).
Key Concepts in MNS Research
McIntosh et al. (1992) provided an influential definition of MNS: Number sense refers to a person’s general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for handling numbers and operations. It reflects an inclination and an ability to use numbers and quantitative methods as a means of communicating, processing, and interpreting information. It results in an expectation that numbers are useful and that mathematics has a certain regularity. (p. 3)
The above description seems to capture the gist of the term number sense, as it is commonly used in the mathematics education community. It focuses on habits of mind and ways of behaving mathematically that are considered desirable, such as flexibly manipulating numbers.
Despite this holistic definition, MNS is typically partitioned into several components: understanding of the meaning and size of numbers (e.g., to compare fractions), understanding and use of equivalent representations of numbers (e.g., to write rational numbers in different ways), understanding the meaning and effect of operations (e.g., to reason about the effect of dividing by a number between 0 and 1), understanding and use of equivalent expressions (e.g., to compare expressions involving different numbers and/or operations), flexible computing and counting strategies for mental computation, written computation, and calculator use (e.g., to select strategies and perform mental computation), and measurement benchmarks (e.g., to estimate the height of an object; Reys et al., 1999). The components view has influenced much of the empirical research concerning MNS. Many researchers have used assessments designed to measure specified components of MNS (e.g., Yang & Lin, 2015).
Similarities and Differences in Key Concepts and Assumptions About Number Sense Across Research Traditions
In summary, ANS is described as an innate set of basic neurological abilities that are common among people with normal brain development. It has also been found in some animals. ENS, by contrast, is regarded as learned. It is unequally distributed among people, depending on their mathematical experiences, and it is not found in animals. ENS is described as consisting of a set of skills developed through practice. MNS is also learned and unequally distributed among people. It is described as consisting of a set of components, which include underlying conceptual understandings and habits of mind. In contrast to the basic ENS skills, MNS is associated with more sophisticated and non-routine mathematical thinking.
Analyses of Number Sense
Each research tradition has its own ways of operationalizing and measuring number sense, which relate to the assumptions and key concepts described above. Here, we summarize the approaches to the analysis of number sense in each tradition (see Table 2).
Typical methods used in number sense research by tradition
Note. ANS = approximate number sense; ENS = early number sense; MNS = mature number sense.
Analyses of ANS
Weber’s law is commonly used to explain the acuity of the ANS (Castronovo & Göbel, 2012). Weber’s fraction indicates the amount of change in the ratio between numbers of items that is needed for participants to discriminate a difference (Sasanguie et al., 2013). As an example, discriminating 40 items from 50 items is about equally as difficult as discriminating 8 items from 10 items, because both pairs of numbers are in a 4:5 ratio. Weber’s law states that accuracy in discrimination decreases as the ratio between the numbers approaches 1 (Castronovo & Göbel, 2012). Researchers believe Weber’s fraction decreases with age, which means that acuity increases (Castronovo & Göbel, 2012; Halberda & Feigenson, 2008).
ANS is evaluated differently depending on the population. In studies of infants, researchers analyze the amount of attention an infant gives to an observed magnitude change (Libertus & Brannon, 2009). This method can be used for auditory magnitude discrimination (Lipton & Spelke, 2003) and visual magnitude discrimination (Libertus & Brannon, 2009; Xu et al., 2005).
By contrast, researchers assessing the ANS of children and adults typically focus on participants’ abilities to discriminate magnitudes between two sets (Halberda & Feigenson, 2008). This approach differs in two ways from how infants are assessed: (a) Infants indicate a change on the basis of focus, or attention given, to a set of objects to discriminate magnitude, whereas children and adults indicate a difference typically through pushing a button on a computer to indicate which stimulus has the greater magnitude (Halberda & Feigenson, 2008; Norris et al., 2015); (b) Infants are assessed for the ability to discriminate when a change in magnitude has occurred, but children and adults are assessed on their abilities to compare two static magnitudes observable at the same time.
Researchers measure the ANS of animals by observing the animals’ preferences. For example, Low et al. (2009) observed the preferences of brood-parasitic cowbirds in deciding whether to lay their eggs in the nest with the largest number of eggs or the nest in which the number of eggs changed from the previous day. The ANS is also evaluated through neurological testing. Scans are taken to observe which areas of the brains are functioning during the tasks. This kind of testing has typically involved infants, children, and adults. Stoianov and Zorzi (2012) expanded neurological testing to also include animals. These neurological tests assess not acuity, but rather the areas of the brain that are active when the subject or participant is performing tasks that are associated with the ANS.
The comparability of these different types of assessments is of concern. In fact, Park and Brannon (2014) 8 found that the various methods used to evaluate the ANS are not well correlated. However, to investigate whether the ANS increases with age, having comparable results between infants and adults is important. Park and Brannon found that adults were less accurate when indicating a change in magnitude than when recognizing a difference between two magnitudes. For example, adults are better able to judge which array of dots has more items than to correctly indicate whether a change occurred when they see one array followed by another. This finding may imply that infants have a higher level of accuracy with the ANS than demonstrated through the current evaluation methods (Park & Brannon, 2014).
Analyses of ENS
Although ENS provides an important foundation for both the development of higher mathematical skills in the elementary years and in diagnosing mathematical disability, exact agreement as to which specific skills comprise ENS has not been reached (Howell & Kemp, 2005; Lago & DiPerna, 2010). On the basis of the findings in their exploratory factor analysis, Lago and DiPerna (2010) proposed that counting aloud, measurement concepts, nonverbal calculation, number identification, quantity discrimination, and rapid naming (colors, numbers, objects) tasks reflect the skills of ENS. Researchers pointed out the design of the tasks used in measurement as a limitation of the study for failing to include estimation and counting. Howell and Kemp (2005) implemented a Delphi procedure to understand which skills reflect ENS. They found agreement on the skills of rote counting, counting on from a number, recognizing numerals, sequencing numerals, using finger patterns, making equivalent groups, distinguishing between quantity and size, and comparing quantities and numbers.
In parallel to the different skills associated with ENS are different measures of ENS. For example, Malofeeva et al. (2004) designed a Number Sense Test focused on six scales of ENS: counting, number identification, number-object correspondence, ordinality, comparison, and addition/subtraction. Jordan et al. (2006) emphasized a somewhat different set of skills in designing their Number Sense Battery to measure counting, number knowledge, number transformation, estimation, and number patterns, which they regard as key elements of ENS. Thus, different measurement tasks are used depending on the skills that researchers treat as proxies for ENS. These measurement tools are developed either for preschool or for the early elementary school grades. The tasks have included open-ended items (Jordan et al., 2006) as well as short-answer and comparison items, including symbolic and nonsymbolic representations (Chard et al., 2005; Malofeeva et al., 2004).
Analyses of MNS
Researchers have used written test instruments and interviews to assess individuals’ MNS, and these have included both multiple-choice and open-ended items (e.g., Markovits & Sowder, 1994; Reys et al., 1999; Reys & Yang, 1998; Yang, 2005, 2007; Yang & Huang, 2004; Yang, Li, & Lin, 2008; Yang, Reys, & Reys, 2009). These have largely been single-snapshot studies of particular populations. In relatively few studies have researchers investigated change in students’ MNS, and those that have done so have focused on comparisons of snapshots (e.g., Markovits & Sowder, 1994; Whitacre & Nickerson, 2016; Yang, 2002). We found no qualitative studies of MNS development that were published by the end of 2017. 9 Measures of MNS tend to be based on a components view. For example, Li and Yang (2010) used a five-factor model based on five components of MNS in their instrument. These instruments have proven useful. At the same time, they typically are not designed to measure all supposed characteristics of MNS, such as “an expectation that numbers are useful and that mathematics has a certain regularity” (Reys & Yang, 1998, p. 226).
Similarities and Differences in Analyses of Number Sense Across Research Traditions
In summary, ANS research involves humans and some animals and focuses on measuring the basic neurological abilities that are associated with the ANS. ANS research is concerned with understanding brain development and factors that might affect the functioning of those neurological abilities that comprise the ANS. ENS research involves young children and students with disabilities. Researchers either measure specific skills associated with ENS or use more general tests of early numeracy. Much ENS research is concerned with predicting success in school and with improving the ENS of particular student populations. MNS research typically involves middle-grades students or preservice teachers. A variety of tests have been devised based on the components view of MNS. Researchers have also used survey and interview methods and sought to measure or describe aspects of MNS based on participants’ strategies for solving particular types of mathematical tasks. For the most part, MNS research does not focus on predicting future academic success.
Empirical Findings in Number Sense Research
Each tradition is characterized by a different set of noteworthy findings. In some cases, there is a clear consensus about major findings; in others, there may be disagreement. Below, we summarize the major findings that we identified in each tradition on the basis of our sample of articles.
Findings in ANS Research
The majority of the research in the ANS tradition is about acuity levels or changes in acuity levels as individuals age. This is true for studies of infants, children, adults, and animals. Few studies in our sample of articles involved interventions. 10 Findings related to the ANS of infants indicate that infants follow Weber’s law in the same way that adults do but with less acuity (Libertus & Brannon, 2009). Differences in types of evaluations may have underestimated the acuity of infants; however, growth is still seen when using the same evaluation methods as participants age. Infants improve their acuity with age by increasing their acuity levels from a 2:1 ratio to 1.5:1 ratio between the ages of 6 and 9 months of age (Lipton & Spelke, 2003). Xu et al. (2005) investigated infants’ ability to identify differences between large amounts (16–32) and between small amounts (1–16). They found that infants are more successful at distinguishing differences between large amounts. This finding seems to be contrary to Weber’s law, which would indicate that the number of items should not matter as long as the ratio between the amounts remains consistent.
Some of the ANS research done with children focuses on possible reasons why some children might be less accurate in their numerical judgments. Friso-van den Bos et al. (2014) found that visuospatial skills and working memory were predictive of ANS ability levels. A similar result was found by Davidse et al. (2014) and indicated that interventions for the ANS would not be effective until these skills were normalized. This finding points to possible ways to improve ANS abilities.
ANS researchers have also studied children with disabilities. Davidse et al. (2014) studied twin girls who were born with a very low birth weight and had significant math delays. These girls had very low acuity in magnitude discrimination and appeared to only be guessing at which quantity was greater. Additionally, Hiniker et al. (2016) found that students with autism spectrum disorder had lower acuity levels than their typically developing peers. However, the ANS was not found to be as predictive of their mathematical abilities as symbolic number understanding (Hiniker et al., 2016). Symbolic number understanding was also found to be a higher predictor of mathematical skill than the ANS for typically developing children (Sasanguie et al., 2013).
There are only a few studies in our sample concerning how to improve the ANS. Wilson et al. (2009) and Davidse et al. (2014) studied the effects of interventions involving a computer program called “The Number Race” to improve the ANS, but they found that the intervention increased only “symbolic number sense” and did not affect “nonsymbolic number sense.” In particular, students improved their abilities to discriminate between numerals but not their magnitude discrimination. The researchers concluded that these results did not indicate improvement in the ANS but rather in strategies related to symbolic information (i.e., ENS skills). Boonen, Kolkman, and Kroesbergen (2011) found that teacher math talk involving number labels for everyday items and numbers of items was related to higher ANS scores in kindergarten students. However, if the mathematical language focused on too many concepts or calculations, it created a negative relationship with the ANS of kindergarten students.
In studies of adults, researchers have been able to distinguish performance on measures of ANS abilities. Lyons and Beilock (2011) found a strong correlation between greater ANS abilities and participation in higher-level undergraduate mathematics courses. According to Tosto et al. (2014), the difference in the ANS is more often related to environmental factors than to biological factors. Norris et al. (2015) addressed the ANS of aging adults and found that there were no detrimental effects of aging on the ANS. Although age does not appear to affect the ANS, it has been found that the ANS is affected by chronic pain (Wolrich et al., 2014). Wolrich et al. (2014) found that one third of the patients with chronic pain scored more than two standard deviations lower than the control group. Researchers have also evaluated whether or not magnitude discrimination is voluntary. Ruusuvirta et al. (2007) found that the ANS is active and enables magnitude discrimination without the need for conscious thought. However, it has been found that if participants are informed to slow down when evaluating magnitudes and are provided feedback on their responses, their acuity improves (Smets et al., 2014).
Findings in ENS Research
ENS research articles can be grouped into the following four categories: defining and measuring ENS, correlational and predictive studies, intervention studies, and other studies.
Defining and measuring ENS
The articles in this category include those aimed to understand the ENS construct by developing a framework (Andrews & Sayers, 2015), defining ENS by using a Delphi procedure (Howell & Kemp, 2005), or studying the construct by using factor analysis (Lago & DiPerna, 2010), and to develop or validate ENS measurement tools (Chard et al., 2005; Jordan, Glutting, & Ramineni, 2010; Jordan, Glutting, Ramineni, & Watkins, 2010; Malofeeva et al., 2004). In parallel with the need to clarify the ENS skills, Andrews and Sayers (2015) identified the need for instructional instruments that provide opportunities for children to practice these skills, as well as tools for evaluating learning opportunities. They use a framework to analyze the learning opportunities provided in different cultural contexts, including English, Hungarian, and Swedish.
Lago and DiPerna (2010) also emphasized the need for clarifying ENS skills and providing a consensus for the properties of measures. Their exploratory factor analysis indicated that number-related skills (counting aloud, measurement concepts, nonverbal calculation, number identification, and quantity discrimination) and rapid naming skills (identifying objects, colors, and numerals) are two factors that explain the ENS construct.
Correlational and predictive studies
These studies consistently address the importance of the experiences children have before they attend school and the experiences they have during kindergarten. ENS was found to be correlated with before-schooling experiences in children. It is also correlated with mathematics performance in the primary grades.
Early exposure to quantities and numbers, called home numeracy experiences, show a positive correlation with children’s ENS development (Jordan et al., 2006). In addition to the experiences that children have, the socioeconomic status of the family, education level of parents (Ivrendi, 2011), and home culture may be predictors for ENS development (Aunio et al., 2004; Aunio et al., 2006). Half of the articles in this category involved examinations of the influence of age and gender on ENS skill performance. The studies addressed a significant age-dependent gain, whereas the results of gender-focused studies are mixed. Aunio et al. (2004) and Aunio et al. (2006) found no gender difference, while Howell and Kemp (2010) stated that girls and boys may outperform one another in particular ENS skills, and Ivrendi (2011) and Jordan et al. (2006) found gender as a significant predictor of overall ENS performance, favoring boys. Aunio et al. (2004) and Aunio et al. (2006) investigated nationality as an additional factor. In these studies, significant differences in counting, organizing, and comparing quantities were found among children in China, Finland, Hong Kong, and Singapore. The authors pointed to language, instructional approaches, and cultural character as possible explanations for these differences.
In addition to examining the factors that may influence ENS development of children, predicting math achievement and math difficulty in the following years of schooling in terms of ENS development of children has been an important motivation for the studies. ENS was found to be a meaningful indicator for the variance in children’s math achievement in the first and third grades (Jordan et al., 2007; Jordan, Glutting, & Ramineni, 2010). In addition, Locuniak and Jordan (2008) found ENS is a good predictor of calculation fluency in second grade.
Intervention studies
Intervention studies constitute an important subset of the ENS literature. The main underlying motivations for developing intervention programs can be summarized as supporting children’s development of ENS skills, creating opportunities to bridge the gap in ENS development of children related to their experiences prior to schooling, and developing effective intervention programs. Many interventions have been designed to provide the early experiences that are critical for children’s future success in schooling, especially for children who start school behind their peers due to socioeconomic disadvantages (Cheung & Chang, 2015; Dyson et al., 2015; Jordan et al., 2012; Schacter et al., 2016). These studies involve various intervention programs designed to bridge the gap between the experiences children have before schooling. Jordan et al. (2012) examined the effectiveness of an intervention program aligning with the Common Core State Standards for Mathematics and focusing on counting skills, number recognition, number comparison, nonverbal calculations, story problems, and number combinations. The comparison of a control group and an experimental group of kindergartners at high risk revealed a meaningful improvement in scores on the Number Sense Brief Test (Jordan, Glutting, Ramineni, & Watkins, 2010) in favor of the children in the experimental group.
In addition to the intervention programs that directly aim to support the development of ENS skills, van Nes and Eerde (2010) targeted early spatial sense, which is expected to support the development of ENS. The construction of three-dimensional blocks and (de)composition of concrete structures in the program connected with mental decomposition of quantities and counting procedures. In this exploratory study, kindergarteners were found to correlate spatial structuring to their preferred counting procedures and abbreviated counting strategies.
In parallel with developing intervention programs to improve the ENS of children in kindergarten and additional elementary grades, there are some supportive instructional materials developed. McGuire et al. (2012) proposed five-frames as an effective instructional tool for supporting the development of the stable-order principle, one-to-one correspondence, and cardinality. Baroody et al. (2012) examined the effectiveness of a computer-assisted discovery learning experience on arithmetic for supporting students’ fluency in adding 0 and 1, and developing mental calculations. The findings indicated that the learning experience through computers provided promising improvement in children’s fluency in practiced and notpracticed arithmetic calculations.
While many intervention programs target prekindergarten or kindergarten children directly, Cheung and Chang (2015) designed a training program for parents that involved playing mathematical card games with children. Playing these games fostered participating children’s performance on six tasks—numerical identification, object counting, rote counting, missing number, numerical magnitude comparison, and addition—and provided critical experiences for improving basic number knowledge of children.
Findings in MNS Research
The MNS articles in our sample were grouped into four categories: snapshot studies, intervention studies, correlational and measurement studies, and other studies.
Snapshot studies indicating the need for improvement in MNS
The most common type of study in our set of articles on MNS was the “snapshot study.” We borrow this term from Mewborn (2001) but apply it to MNS research. For our purposes, snapshot studies of MNS involve investigations into the MNS of a population at a single point in time, typically by means of a test or interview. Paradigmatic of this type of study is a collection of articles by Yang and colleagues. In these studies, the authors test and/or interview students from a population (typically middle-grades students or preservice teachers) and then report on the participants’ performance. Yang and colleagues have repeatedly reported that Taiwanese middle-grades students generally exhibit low levels of MNS. These students are able to perform familiar computations using school-learned procedures, but they perform poorly on nonroutine tasks designed to tap their MNS, unless they are able to solve the tasks by employing procedural methods (e.g., Yang, 2005; Yang et al., 2008; Yang & Huang, 2004).
Yang and colleagues have also found that preservice teachers in Taiwan exhibit “poor” MNS, relying heavily on standard written algorithms (Yang, 2007; Yang et al., 2009). Although the preservice teachers were more capable of correctly answering MNS test items than their middle-grades counterparts, the majority of their answers were also obtained by written computation using standard algorithms. Both the middle-grades students and the preservice teachers in these studies have tended to rely on standard written algorithms, despite the fact that the instructions for these assessments explicitly discouraged such approaches.
Snapshot studies of MNS consistently report “low” or “poor” MNS. They may also report specific performance trends, such as differences in performance by component or differential performance by student population. None of the snapshot studies in our set of articles reported encouraging findings. Instead, a clear consensus emerges from this literature: The researchers are disappointed with participants’ performance on measures of MNS. There is no clear standard of performance against which judgments such as “low” or “poor” are made; rather, it is taken for granted that the scores or tendencies evident in participants’ responses are unsatisfactory. For example, Yang (2007) coded preservice teachers’ responses to a set of tasks involving rational numbers. He reported, “About two-fifths, three-fourths, two-thirds, and three-fourths of participants respectively relied on the rule-based method to answer the four number sense components. . . . This problematic situation was consistent with the results of earlier studies . . . ” (p. 299). In many cases, the preservice teachers gave correct answers to the tasks, but they “heavily relied on the written method, such as finding a common denominator” (p. 299), which is not regarded as being indicative of MNS. Thus, certain kinds of strategies are valued in this literature, and it is unclear what frequency of use of such strategies would be considered appropriate or satisfactory.
Intervention studies aimed at improving MNS
Our set of articles also included those that reported the results of interventions intended to improve MNS. Evidence of improvement comes from repeated measures involving tests and/or interview methods, as described above. The instructional interventions described in these articles involve (a) encouraging students to invent their own strategies and to try strategies used by other students, (b) discussions in which students were expected to justify the validity of their strategies, and (c) and use of technology such as calculators or specially designed software.
As an example of an intervention study, Markovits and Sowder (1994) reported the results of an intervention designed to improve MNS in a seventh-grade mathematics class. The intervention involved engaging students in mental computation, encouraging them to use invented strategies, and discussions in which students were asked to provide (at least informal) mathematical justifications for the strategies that they used. Students participated in pre, post, and retention written tests and interviews. The written tests emphasized tasks concerning number size, while the interviews focused on mental computation. Students’ written test scores improved substantially, and these improvements were sustained on the retention test. The researchers found that the frequency of less standard strategies increased from the pre- to post-interview, and this change was sustained in the retention interview. Markovits and Sowder (1994) argued that these results demonstrated improvement in the participants’ MNS. This article constitutes an exemplar of intervention studies to improve MNS.
Correlational and measurement studies
Measurement studies focus on the development of assessment instruments, as described earlier. Correlational studies concern relationships between MNS and other constructs. It seems that these relationships are not simple. For example, Yang et al. (2008) found that MNS test scores were significantly related to mathematics achievement (grades in Year 5). In particular, two components—recognizing relative number size and using multiple representations of numbers and operations—were moderately correlated with achievement. On the other hand, Yang and Huang (2004) found that the high written computation scores of Taiwanese sixth-graders differed significantly from their MNS test scores, and skill in written computation did not imply high scores in MNS.
Similarities and Differences in Findings Across Number Sense Research Traditions
The three number sense research traditions are characterized by different foci and relatedly different findings. Table 3 summarizes these. The differences in what researchers are learning about number sense follow from their focus on distinct constructs, populations, and methods of measurement. Findings in the ANS literature are informative with regard to the details of apparently innate neurological abilities, such as the acuity of the human capacity for magnitude discrimination, and how these abilities differ between humans of different ages. Findings in the ENS literature are informative with regard to differences in ENS skills among student populations and how various experiences or interventions may influence ENS skills and achievement in school. Findings in the MNS literature are informative with regard to the level of performance of middle-grades students and preservice teachers on measures of MNS, the kinds of instructional approaches that improve MNS, and the relationships between MNS and other constructs.
Foci and findings of number sense studies by research tradition
Note. ANS = approximate number sense; ENS = early number sense; MNS = mature number sense.
Themes in Number Sense Research
We present themes belonging to each number sense research tradition. We then address similarities and differences in themes across the traditions.
Themes in ANS Research
A prominent theme in the ANS literature is to investigate whether and how basic ANS abilities may be malleable and relevant to formal mathematical learning. The ANS is itself an interesting phenomenon for investigation, such as in studies of infants’ innate abilities and in experiments involving birds. As such, many studies are primarily concerned with measuring ANS abilities. At the same time, some researchers believe that ANS interventions are necessary and important for success in school and that ANS measures should be used to identify children at a young age for math disabilities for early intervention (Smets et al., 2014). Work that focuses on planning and enacting interventions follows from the belief that the ANS is foundational to later mathematical competencies (Halberda & Feigenson, 2008). There is some evidence that ANS is a predictor of mathematical performance in the early grades, although more research is needed (Chen & Li, 2014).
Themes in ENS Research
A prominent theme in the ENS literature is the importance of children’s early mathematical experiences. ENS skills are highly relevant to success in school mathematics. Indeed, ENS scores are an important predictor of mathematics proficiency in the primary grades (Jordan et al., 2007; Jordan et al., 2009; Locuniak & Jordan, 2008). Awareness of the importance of ENS skills motivates educators and researchers to improve instruction and to develop early intervention programs for children, especially those who are at risk. In support of these efforts, clear definitions and valid measures of ENS are regarded as essential (Andrews & Sayers, 2015; Howell & Kemp, 2005, 2009, 2010).
Themes in MNS Research
A prominent theme in the MNS literature is the mismatch between MNS and traditional school mathematics, which has tended to emphasize algorithms and memorization. Authors lament this situation and recommend reforms to mathematics instruction in order to better support students’ development of MNS. In keeping with this theme, prominent arguments for the importance of MNS are less often based on claims of its relevance to success in school. Instead, authors argue for importance of MNS based on recommendations from the National Council of Teachers of Mathematics, the National Research Council, and other policy documents and research articles (e.g., Șengül, 2013; Yang, 2002, 2005). A second theme is the claim that MNS cannot be taught directly; instead, it develops gradually, given experiences that invite students to explore numbers and discuss ideas (Greeno, 1991; Markovits & Sowder, 1994; Yang, 2002). This claim is also based on recommendations in policy documents, as well as arguments made in research articles, practitioner articles, and other publications.
Cross Citations Between Number Sense Research Traditions
A common theme across the number sense literature relates to our motivation for conducting this literature review—that there are different meanings for the term number sense. Many authors note this point, referring to varying definitions or to “disagreement” or “difficulties” regarding how to define number sense (e.g., Akkaya, 2016; Șengül, 2013; Yang & Li, 2013). A related theme is the prevalence of cross citations, in which researchers working in one tradition cite literature from another tradition. In some cases, this is done appropriately, such as for the purpose of distinguishing the authors’ definition of number sense from that of other authors (e.g., Howell & Kemp, 2009). We refer to such cross citations as consistent. 11
In other cases, we found literature from more than one number sense tradition being cited indiscriminately, as if all publications using the term number sense necessarily concerned the same construct. For example, some articles in the ENS tradition point to the work of Dehaene for a definition of number sense and yet it is clear from their study design that they are conceptualizing and operationalizing number sense in terms of ENS skills and are conducting interventions to improve those skills. We refer to instances as inconsistent cross citations if authors cite work that involves clashing or mismatched assumptions regarding the nature of number sense, and we see no evidence that the authors recognize any mismatch.
Table 4 quantifies the phenomenon of cross citations across each tradition. 12 As shown in the table, cross citations were prevalent across all three traditions. Many authors acknowledge the existence of different definitions (or “views” or “perspectives”) concerning number sense but then proceed to cite literature from different traditions as convenient to support their arguments. Given the clear differences between the ANS and MNS traditions, cross citations between them were especially conspicuous. As an example, an article in the MNS tradition cited Dehaene (1997/2011) to bolster its claims about the importance of number sense in everyday life, despite the fact that Dehaene’s (1997/2011) work is concerned with ANS—a fundamentally different construct. Clearly, it does not follow from the point that basic cognitive abilities such as magnitude discrimination are important in everyday life that flexible reasoning with fractions and decimals would also be important in everyday life. Thus, indiscriminate use of cross citations leads to unsubstantiated claims.
Cross citations by “number sense” research tradition
Note. ANS = approximate number sense; ENS = early number sense; MNS = mature number sense.
As another case in point, many studies cite the work of Jordan et al. (2007) to support the claim that number sense is predictive of achievement in school mathematics. However, this evidence comes from the ENS tradition. In the study by Jordan et al. (2007), ENS was measured in kindergarten and used to predict achievement in first grade, based on a standardized test. As previously noted, the ENS skills are well aligned with school mathematics in the primary grades, so it is unsurprising to find a strong predictive relationship. We note, however, that this finding is cited in articles concerning students at different grade levels and even articles belonging to different number sense traditions. It seems especially misleading when MNS articles cite this study to support claims that number sense is predictive of mathematics achievement, being that those articles involve older students for which the relevant prediction would concern mathematics achievement in middle or high school.
Inconsistent cross citations result in research literature built upon shaky foundations. There is no reason for authors to cite Dehaene and state his definition of number sense (i.e., ANS) if their conceptualization of number sense is fundamentally different than his (as in ENS or MNS). There is no apparent justification for researchers who are studying the MNS of secondary students or preservice teachers to assume that it is predictive of success in school mathematics based on studies of the ENS of children in the primary grades. There might a lot of potential for the ANS, ENS, and MNS research traditions to communicate and learn from one another; however, we see evidence that confusion resulting from polysemy has negatively affected the quality of research in these traditions and limited progress.
Discussion
Our review owes a debt of gratitude to previous publications that have raised issues concerning inconsistent definitions of number sense. Berch (2005) and Andrews and Sayers (2015) explicitly identified different ways that the term number sense was used across fields. Our work goes a step further by rejecting the assumption that the literature is plagued by varying interpretations of a single construct, as opposed to fundamentally different constructs with the same name. In particular, Berch (2005) referred to “the concept of number sense” (p. 333, emphasis added), and Andrews and Sayers (2015) described number sense as “a poorly-defined construct” (p. 257, emphasis added). Such language is indicative of the point that both articles assumed a single number sense construct with varying interpretations.
Thus, we find it unsurprising that Berch’s (2005) suggestion of two “views” of number sense did not seem to lead to any dramatic clarification of the literature. He was writing to researchers in special education for the purposes of operationalizing number sense—specifically, ENS—within that particular field. Andrews and Sayers’ (2015) more recent assertion that there are three “perspectives” on number sense is analogously situated within early childhood education and concerned with advancing research on number sense (i.e., ENS) within that field. Unlike our review, these previous observations about the number sense literature were written from the perspective of researchers in a single field and were primarily concerned with the priorities of researchers within that field, rather than with clarifying the broad number sense literature.
Our review led to the identification of three number sense constructs, which enabled us to categorize the associated literature (acknowledging overlapping categories). As there are researchers from more than three fields actively investigating number sense, we have chosen to characterize the research traditions by construct rather than by field. The three constructs that we identified align well with the “perspectives” suggested by Andrews and Sayers (2015). However, we make a crucial point in this article of identifying these as distinct constructs. Furthermore, we identified these constructs through rigorous, constant comparative analysis and were able to use our coding scheme to reliably code the 141 articles in our sample.
To the best of our knowledge, our review of the number sense literature is the first of its kind. Ours is a review of research (concerning number sense) rather than a review for research (Maxwell, 2006) situated within a single research tradition. To achieve our purpose, it was not necessary to review all research articles that mention number sense. It was, however, necessary to review seminal works and to rigorously review and code a substantial sample of research articles that focused on number sense.
Whereas other authors have noted discrepancies in definitions or difficulties defining the term number sense, we identified three distinct number sense constructs. Our review then went beyond distinguishing number sense constructs to describe the research tradition related to each construct. These are not fields of research per se; instead, they specifically characterize the research literature related to each construct. In each tradition—but especially in the ENS tradition—this body of literature involves contributions made by researchers from a variety of fields. We summarized these research traditions, focusing on similarities and differences between them. The contrasting features especially underscore the need to distinguish number sense research by construct and tradition.
We believe it is no coincidence that the distinctions suggested by Berch (2005) and Andrews and Sayers (2015) both arise from research in the ENS tradition. There are clear differences between the two extremes—ANS and MNS. Researchers in the ANS tradition tend to cite the work of Dehaene and to conduct the kinds of experiments with animals and infants that were described previously. Meanwhile, researchers in the MNS tradition tend to cite the work of McIntosh, Reys, Yang, and colleagues. Their work typically involves measuring the MNS of middle-grades students or preservice teachers, based on the components view of MNS. There should be little room for confusion between these two disparate number sense research traditions (although, as we noted, there are cross citations appearing in the MNS literature).
As we illustrated in Figure 1, the ENS tradition lies between those two extremes. Some studies within the ENS tradition lean in the direction of ANS research, while other studies in the ENS tradition lean toward MNS research. Thus, it is ENS researchers who face the most potential for confusion in attempting to navigate the number sense literature. It is no surprise, then, that ENS researchers have taken the lead in recent efforts to clarify the meaning of number sense within their fields in order to facilitate progress.
We also note that our intention is not to blame authors for inconsistent cross citations. After all, there is a great deal of confusion in the number sense literature, and sorting it out has been a major undertaking. It is also unsurprising that more inconsistent cross citations occur in the MNS literature than in the other traditions. Because ANS researchers tend to focus on young children or older adults and to investigate brain activity, it is unlikely that they would cite the MNS literature. By contrast, MNS researchers who study school-aged students or preservice teachers may be interested in the origins of the issues that they see with number sense in their participants and so look to research concerning young children for that reason.
Recommendations
There are two main recommendations that follow from the results presented here. First, authors should take care to ensure that the number sense literature that they draw upon and cite relates to the same construct they are investigating, or is cited for specific and appropriate purposes, recognizing distinctions between constructs. It is absolutely not safe to assume that all literature that refers to number sense concerns the same construct. As we have demonstrated, this is not the case. This does not merely mean that different authors think about number sense in different ways. Across the three traditions, researchers are actually studying different phenomena, and their studies do not necessarily have any relevance to one another. The most important point we wish to communicate is the need for authors to recognize and accept this fact.
Second, authors should clearly identify the number sense construct that is the subject of their articles in order to help their readers to make distinctions and to successfully navigate the literature. In other words, the first point is to make sense of the existing literature by distinguishing number sense constructs, and the second point is to more clearly communicate with readers in new contributions to the literature. We believe such communication would be greatly aided by the use of more specific terms, such as those used in this article. Readers will be less likely to confuse number sense constructs if authors specifically identify their work as concerning approximate number sense, early number sense, or mature number sense while citing appropriate sources from the relevant research tradition(s).
Conclusion
The results presented above contribute to the literature by clarifying distinctions between three constructs that have gone by the same name. The widespread use of the term number sense to refer to three distinct constructs belonging to different traditions has led to confusion, and such confusion has long been noted in the literature. However, these issues have persisted. Previous observations of “confusion” or disparity in definitions have fallen short of clarifying the number sense literature because they have assumed that all mentions of number sense refer to the same construct. By systematically coding the articles in our data set according to the authors’ definitions and assumptions about number sense, we were able to clarify the nature of the construct within each tradition.
We view the confusion over number sense across traditions as a microcosm of the broader phenomena of polysemy and reverse progress. The use of the same term can be problematic, because it may blind some researchers to important distinctions. It is easy to read an article about number sense and filter it through one’s existing conceptualization of number sense instead of recognizing the fact that claims from a different tradition may actually concern a different construct and may be irrelevant to number sense research in another tradition.
An important contribution of this review is to point out the phenomenon of cross citations. To be clear, authors working in one number sense tradition should not be citing authors working in another tradition, unless there is explicit acknowledgment of the differences between traditions and (given those differences) there is an appropriate reason for the cross citation. We suggest that researchers use specific terms to distinguish the number sense construct that they are investigating and take care to ensure that they draw on relevant literature. Although number sense is a catchy term that rolls off the tongue, its loose usage across related research traditions has led to confusion and impediments to progress.
An important question for investigation across research fields is how ANS, ENS, and MNS might be related in terms of ontogenesis. In particular, how might the ANS support the learning of ENS skills? Also, how might ENS skills provide a foundation for the development of MNS? (Note that this question is specific to these constructs and thus distinct from other ontogenetic investigations of mathematical learning.) As research in each field progresses, it is crucial to maintain clear distinctions. By doing so, the number sense research done across fields can move forward both to achieve greater understanding of each construct separately and to investigate relationships between the constructs. Clearly, researchers must first distinguish between number sense constructs before they can investigate how those constructs relate to one another.
Footnotes
Notes
Authors
IAN WHITACRE is an associate professor of mathematics education in the School of Teacher Education at Florida State University, G123 Stone Building, 1114 West Call Street, Tallahassee, FL 32306-4459, USA; email:
BONNIE HENNING is an assistant professor of early childhood special education in the School of Education at the University of St. Thomas, 1000 LaSalle Ave, Minneapolis, MN 55403, USA: email:
ȘEBNEM ATABAȘ is a graduate student in mathematics education in the School of Teacher Education at Florida State University, G123 Stone Building, 1114 West Call Street, Tallahassee, FL 32306-4459, USA; email: