Challenges in Mathematical Cognition

A. Elucidating the Nature of Mathematical Thinking

These questions reflect the ambitious long-term goal of understanding mathematical cognition from infancy to mature expertise. Collectively they address the contribution of nativist theories in proposing foundational “number sense” capacities seen in nearly all human infants (e.g., Feigenson, Dehaene, & Spelke, 2004; Geary, 2007), the mature state of refined formal cognitive capabilities associated with highly successful mathematicians (Weber, Inglis, & Mejía-Ramos, 2014), and the as-yet unexplained mechanisms that link the two. Although individual research studies necessarily focus on a relatively narrow part of this spectrum, participants in the exercise recognised the importance of working towards an empirically evidenced understanding of mathematical cognition that accurately captures effective mathematical reasoning, that provides reliable ways of diagnosing atypical development, and that is specific enough to be leveraged for the design of interventions. In the Critical Review section to follow, we discuss the methodological challenges associated with studying expertise in particular.

B. Mapping Predictors and Processes of Competence Development

These questions are motivated by concerns about the large proportion of individuals of all ages whose achievement in mathematics is considered inadequate relative to standards or criteria specified by governments or related agencies. Compared with the questions in Set A, they are more specific in several ways. First, they explicitly acknowledge that both typical and atypical mathematical development might be influenced by a wide range of environmental and affective factors, as well as by domain-specific and domain-general cognitive competences. As such, it is necessary to consider the full range of influences (and the ways in which we measure these) because such factors might interact in complex ways. Moreover, the nature of these interactions might vary across periods of development. This issue becomes more salient as apparent inconsistencies across research studies come to light, as illustrated by studies of the Approximate Number System (ANS) and formal numerical skills (De Smedt, Noël, Gilmore, & Ansari, 2013; Gilmore et al., 2013). The time is ripe for studies designed to disentangle the effects of different predictors and to map patterns of development, and such studies are beginning to be reported (Fuchs et al., 2010; Göbel, Watson, Lervåg, & Hulme, 2014; Klibanoff, Levine, Huttenlocher, Vasilyeva, & Hedges, 2006; Mazzocco, Feigenson, & Halberda, 2011).

C. Charting Developmental Trajectories and Their Interactions

The questions in Set B acknowledge the continuum of typical and atypical development, but there is no reason to presume that pathways to mathematical understanding are homogeneous among all children whose path is considered either “typical” or “atypical”; in both scenarios, individual differences are likely (Geary, 2011; Kaufmann et al., 2013, respectively). The questions in Set C thus consider how research might capture variations in mathematical learning. They are particularly pertinent for researchers considering cognition about mathematical concepts beyond basic arithmetic because, earlier points notwithstanding, early number and basic arithmetic have been comparatively well studied. This is partly because arithmetic concepts provide a clean context for setting and evaluating performance on a restricted set of problems with definite right and wrong answers, meaning that these concepts provide a sensible setting for investigating more general cognitive processes (e.g., strategy use, for which see Siegler, 1988; Wu et al., 2008; gesture, for which see Goldin-Meadow, Wagner Cook, & Mitchell, 2009). They are also amenable to investigation in the sense that research has identified a fairly restricted number of strategies that a student might use to solve, for instance, a single-digit addition problem. But mathematics as a whole has a hierarchical structure that builds rapidly through many levels of abstraction. At every level, it might be possible to explain learning not only in terms of what students lack – mature working memory systems, proficiency with basic arithmetic facts – but also in terms of what they already know: previous knowledge can interfere with the learning of higher-level ideas (e.g., McNeil, 2014). Researchers have developed pockets of knowledge about how this might play out (e.g., the natural number bias as discussed by Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). But knowledge development is currently understood only in single steps in which a particular existing conception is known to influence interpretations of a new concept – we know very little about longitudinal development in mathematical cognition, and the questions in this section reflect this.

D. Fostering Conceptual Understanding and Procedural Skill

Mathematical thinking does not emerge in a vacuum. By definition, emergence of formal mathematics requires instructional input, but what form should this instruction take? The questions in Set D sit at the heart of a distinction that causes consternation and controversy. Students at all levels must develop procedural fluency so that they can efficiently execute standard manipulations, and conceptual understanding so that they can identify and work effectively with mathematical structures that arise in a variety of situations (Kilpatrick, Swafford, & Findell, 2001). The fundamental distinction between the two is conceptualised in various guises and terminologies in both education (Skemp, 1976; Star, 2005) and psychology (Rittle-Johnson & Schneider, 2015), and the apparent tension manifests itself most dramatically in the so-called “math wars” (Schoenfeld, 2004), which contrast “back to basics” movements advocating drill and practice – intended to develop factual knowledge and procedural fluency – with programmes designed to foster conceptual understanding by providing mathematical learning situations that are personally meaningful to the student (Gravemeijer, 1994; Van den Heuvel-Panhuizen, 2003). A similar tension also appears in a more moderate guise in programmes designed to test the relative efficacy of learning via concrete or abstract representations (De Bock, Deprez, Van Dooren, Roelens, & Verschaffel, 2011; Kaminski, Sloutsky, & Heckler, 2008). If one of these approaches were truly superior in a straightforward way then we would surely by now have conclusive evidence of this, so it seems likely that progress will be made by asking not about the extremes of the dichotomy but about the specific implementation and combination of different types of learning experience. As indicated by the questions in this section, there is much potential for developing and testing specific instructional sequences.

E. Designing Effective Interventions

These questions address issues already raised, but in a more practical way. They focus on the nuances of applied settings, recognising that the effectiveness of any given intervention might depend upon characteristics of the individuals for whom it is intended. The questions also highlight the two-way relationship between theoretical development and experimental testing: by interceding with an intervention based on theoretical propositions about the causes of mathematical errors and difficulties, researchers test both the intervention and the underlying theoretical propositions. It is therefore encouraging that effective interventions do exist and have been tested at scale. Some of these have been designed to allow for individual differences among students who are falling seriously behind, by tailoring research-based interventions to individuals after detailed diagnoses (Holmes & Dowker, 2013; Torgerson et al., 2013). Others have been designed to change instruction in ordinary classrooms by reducing unhelpful regularities in teaching materials and by deliberately designing activities that have demonstrable positive effects (McNeil, Fyfe, & Dunwiddie, 2015). Such interventions so far have been primarily directed at comparatively young children, which is natural due to levels of research-based knowledge and to serious concerns about the cultural and economic implications of children falling behind at early educational stages. But the forms of such interventions, when combined with localised knowledge about typical understandings of a broader range of concepts, have potential for improving instruction at higher levels too.

F. Developing Valid and Reliable Measures

These questions capture issues that pervade the entire list: to obtain convincing evidence within and across these questions, we must give appropriate attention to our measures of key predictors and outcomes. In some parts of the literature, debates about these issues are both obvious and detailed. For example, in the substantial body of work on the ANS, methods have become an overt subject of debate as comparisons across research studies reveal serious inconsistencies (Clayton, Gilmore, & Inglis, 2015; De Smedt et al., 2013). In other areas, the discussion is less specific: for instance, methods have been advanced for measuring constructs such as understanding in relation to certain concepts, but there is a lack of agreement over what constitutes evidence of conceptual understanding (Crooks & Alibali, 2014). At higher levels, with exceptions such as the Calculus Concept Inventory (Epstein, 2013), there are few posited standard measures of mathematical knowledge and understanding. The listed questions reflect the need for valid and reliable measures of understanding and performance across all levels of mathematics. Attention to multiple methods, to their details and differences, is certainly a positive step.

Elucidating the Nature of Mathematical Thinking

Recall that the questions in this section are:

By their nature these questions present serious methodological challenges. A wealth of data suggests that infants have an innate ability to process quantitative information (Izard, Sann, Spelke, & Streri, 2009); habituation and preferential-looking studies indicate that infants can discriminate between sets of small (Starkey & Cooper, 1980) or large quantities (Xu & Spelke, 2000; but see Clearfield & Mix, 2001, for an alternative perspective). It has been argued that these numerical discrimination abilities are guided by two separate systems, Parallel Individuation for small quantities and the ANS for large quantities (Feigenson et al., 2004); both systems are believed to be multi-modal in the sense that they aid the discrimination of visual, auditory and action-based quantitative stimuli (Carey, 2009; Lipton & Spelke, 2004). But a variety of additional characteristics can vary with quantity: for visual stimuli, for instance, increases in quantity generally co-occur with increases in surface area or contour length (Mix & Cheng, 2012). Although researchers may attempt to control for these extraneous perceptual variables, they cannot all be controlled simultaneously (see Cantrell & Smith, 2013 for an extensive review). Therefore, previous findings may be interpreted as indicating that infants are sensitive to changes in a variety of general magnitude dimensions, such as duration, length, convex hull, and area, in addition to quantity (Gebuis & Reynvoet, 2012; Newcombe, Levine, & Mix, 2015). The question of which is the most accurate interpretation has yet to be resolved.

Regarding more complex mathematics we know even less. This is important because in mathematics education there is a large movement for students to engage in authentic mathematical practices – to learn to approach problems in a manner epistemologically consistent with the thinking of professional mathematicians (e.g., Common Core State Standards Initiative, 2012; Lampert, 1990; National Council of Teachers of Mathematics, 2000; Sfard, 1998). Designing for such learning requires an understanding of how mathematical experts complete mathematical tasks and how professional mathematicians practise their craft.

In particular, a desired outcome is for students to learn to solve non-routine problems (e.g., Common Core State Standards Initiative, 2012; National Council of Teachers of Mathematics, 2000; Schoenfeld, 1985), a process that often involves making conjectures and estimating their veracity. Such activity likely involves many behaviours and competences, one of which is constructing and interpreting mathematical representations. Both analysis of expert protocols (Koedinger & Anderson, 1990; Schoenfeld, 1985; Weber & Alcock, 2004) and mathematicians’ reflections on their own practice (Hadamard, 1945; Polya, 1957) indicate that mathematicians’ judgments about truth values often involve diagrams, and that their use of these can interact with symbolic reasoning in complex ways (e.g., Koedinger & Anderson, 1990). But there is a growing body of literature indicating that students’ propensities to use diagrams have little correlation with their mathematical achievement (Presmeg, 2006), and we do not yet know why students do not typically reap the same benefits from multiple representations as do mathematicians.

We also lack a clear understanding of what evidence mathematicians use to gain certainty (or high degrees of confidence) in mathematical conjectures. The traditional claim is that mathematicians rely on proofs – logically rigorous deductive arguments (e.g., Griffiths, 2000). But this only raises questions about how such proofs are produced and how mathematicians judge whether they are valid. Given that students usually lack proficiency in constructing and evaluating deductive proofs (e.g., Healy & Hoyles, 2000), even after completing curricula explicitly designed to teach these skills, better models of the proving process could have important implications for mathematics instruction.

Even in the absence of proof, mathematicians must still sometimes make subjective estimations about the plausibility of conjectures. Franklin (2013) argued that such judgments partially determine what problems a mathematician chooses to investigate, and Devlin (2002) proposed that the way mathematicians check arguments for correctness is determined by how much confidence or doubt they have in those arguments. However, there is almost no systematic research into how mathematicians do, or students should, make such judgments. Clearly mathematicians and students do experience varying levels of confidence and doubt about mathematical claims and arguments, so it is natural to ask how these arise and how they influence reasoning and behaviour; these are open questions.

We conclude this section by acknowledging the methodological difficulties inherent in investigating students’ and mathematicians’ mathematical practices. Developing conjectures and forming and evaluating arguments are time-consuming tasks that rely extensively on an individual’s background knowledge. Experts do not necessarily address these tasks in the same ways, and the obvious approaches to investigating their practices each have limitations. An investigator can ask individuals to describe their own reasoning processes, and many researchers have employed verbal protocol analyses in attempts to model students’ and mathematicians’ problem solving (e.g., Schoenfeld, 1985). But we know that individuals’ self-reports about their own reasoning are often inaccurate (e.g., Nisbett & Wilson, 1977). An alternative is to conduct a naturalistic study, but much mathematical reasoning is done in isolation and silence, so this might provide only limited information. Moreover, a naturalistic study incurs substantial time costs: even in school environments, developing and testing a single conjecture can take a class period or longer, and in expert practice, creating and testing a conjecture can span months or years. Naturalistic studies have taken on this challenge, observing mathematicians in meetings over several months (e.g., Greiffenhagen & Sharrock, 2011; Smith, 2012). But they necessarily have relatively small numbers of participants, making it difficult to generalize the findings to the broader population of mathematicians, especially those who work in different domains. And, of course, analyzing the data requires the investigators to be at least partially conversant with cutting-edge mathematics.

One more tractable methodological alternative is to present individuals with proxy tasks, simplified versions of the activities that we would like to investigate that can be completed in a relatively short time and that do not require extensive background knowledge. Using proxy tasks does not guarantee that behavioural patterns observed would be present in authentic situations, but such tasks do permit the investigator to competently analyze the data, and they are beginning to enable progress at least in confirming or debunking common claims. Recent eye-movement studies, for instance, have confirmed that undergraduate students pay less attention than mathematicians to words (as compared with symbols) in purported proofs, and that they are less likely to shift their attention around in a manner consistent with seeking deductive justifications (Inglis & Alcock, 2012). Other recent studies have revealed that mathematicians disagree on the validity of even fairly simple mathematical arguments (Inglis, Mejía-Ramos, Weber, & Alcock, 2013), that they respond in more nuanced ways than is traditionally believed to arguments based on diagrams (Inglis & Mejía-Ramos, 2009a) or presented as from an authoritative source (Inglis & Mejía-Ramos, 2009b), and that their sense of beauty in a mathematical proof is not systematically related to that proof’s perceived simplicity (Inglis & Aberdein, 2015). Several of these findings challenge conventional claims about the nature of mathematical expertise by revealing more than expected heterogeneity in the experiences and judgments of mathematicians. They thus constitute steps toward a more informed view of what mathematics students should learn.

Charting Developmental Trajectories and Their Interactions

The questions in this section are:

In comparison with the questions discussed in the previous section, research pertaining to these is comparatively advanced. But it is by no means conclusive: the available data suggest that multiple potential pathways to mathematical success exist, but do not specify precisely what those pathways are. This is evident among children who have mathematical difficulties of known (e.g., Dennis et al., 2009) or unknown etiology (e.g., Geary, 2011; Kaufmann et al., 2013), the most easily differentiated of whom have readily classifiable developmental diagnoses such as fragile X syndrome, spina bifida, and 22q deletion syndrome (De Smedt, Swillen, Verschaffel, & Ghesquière, 2009; Mazzocco, Quintero, Murphy, & McCloskey, 2016). Some group differences, while disorder-specific, may provide models for variation in the general public if origins for and mechanisms underlying different pathways and trajectories were well understood. This makes inquiry into these trajectories particularly valuable.

To understand differences in developmental trajectories, researchers attend to relative deficits and assets – a neuropsychological approach that reveals potential dissociations of mathematics skills and that thus extends knowledge of basic cognitive processes and potential routes to both dyscalculia and typical mathematical achievement. For instance, school-age children with 22q deletion syndrome (22qDS) have intact fact retrieval skills but relatively impaired enumeration of quantities (see the review by De Smedt et al., 2009); they also exhibit intact word learning, but under-count enumeration errors for sets exceeding four (Quintero, Beaton, Harvey, Ross, & Simon, 2014). Whether these enumeration errors reflect difficulties with mapping of numbers to quantities (e.g., Dehaene, Piazza, Pinel, & Cohen, 2003), with spatial attention (Simon, Bearden, Mc-Ginn, & Zackai, 2005), or with other basic processes has important implications for the origins of developmental trajectories. But, in all possible cases, their existence demonstrates a dissociation between components of mathematical cognition.

Similar dissociations are evident across other conditions. For instance, mathematical performance remains comparatively unimpaired in children with 22qDS when tapping the verbal system (Mazzocco et al., 2016), but not when engaging in nonverbal tasks. In contrast, whereas children with spina bifida meningomyelocele show delayed counting principles in early childhood and poor numeracy across the lifespan, their mathematical fact retrieval performance is not correlated with presence or absence of a learning disability (English, Barnes, Taylor, & Landry, 2009) but rather with level of fine motor skills (Barnes et al., 2006). In children with Williams syndrome, the dissociation between numerical skills is not only in nature but also in timing: abstract representations of small quantity are age appropriate during infancy, as is the ability to track small item sets. But these skills do not remain intact throughout life, and impaired approximate number representations of large sets emerge even in early childhood. Remarkably, despite this profile, young children with Williams Syndrome (WS) outperform other young girls with fragile X syndrome on counting principles such as cardinality.

These systematic profile analyses are each guided by theories linked to specific biochemical, neuroanatomical, or neuropsychological characteristics of known conditions (as reviewed by Dennis et al., 2009). As such, principled approaches allow us to test theories of the origins of mathematical learning trajectories, or to ask when pathways diverge or become unique. Possible factors in divergence include impaired motor function that prevents motor exploration in infants and toddlers with spina bifida (and that limits finger use in older children), and disrupted development of the visuospatial processing system in 22qDS or WS. But these factors are not syndrome specific, so the mathematical assets observed in these and other populations have implications for potential compensatory mechanisms – routes via which we might promote mathematical learning in any children who share assets in these domains.

This argument about assets extends to domain-general skills like working memory (English et al., 2009), and of course developmental differences are also observed in children with typical mathematical achievement (LeFevre et al., 2010). To date, studies of typical populations have focused on numerical skills, fractions, fact retrieval, word problem solving, or general mathematical achievement, depending on the researcher’s discipline. Measured potential mediators or moderators frequently include executive functions (e.g., working memory, inhibitory control; Bull, Espy, & Wiebe, 2008), processing speed, and, more recently, indices of mathematics anxiety or motivational factors (Beilock et al., 2010; Wang et al., 2015). Importantly, most of the relationships to emerge from these studies are nonlinear. Outcomes may interact with problem type (Agostino, Johnson, & Pascual-Leone, 2010), stage of development (Ansari, 2010), motivation or mathematics anxiety (Beilock et al., 2010; Wang et al., 2015), and other known influences (e.g., reading disability, stereotype threat, motor function) work directly on mathematical outcomes or indirectly moderate known associations (for instance, children with a reading disability or moderate mathematics anxiety might be more or less responsive to an intervention).

Ideally, we would draw comparisons across studies that ask similar questions or address different disorders; this would help to identify common dissociations, alternative compensatory paths, and indicators of whether children are good candidates for specific compensation-based interventions. But, at present, an explicit comparison of this sort is problematic. At minimum it requires commonality in at least some of the measures used to assess skills (mathematical or otherwise), overlap in ages assessed, and similar methodological approaches to study design and data analysis. This returns us to the overarching questions about methodology raised in Section F: better consistency in design and reporting would contribute to both disorder-specific knowledge and knowledge of mathematical cognition in the broadest sense.