Concreteness Fading in Mathematics and Science Instruction: a Systematic Review

A bird's-eye view of research practices in mathematical cognition, learning, and instruction: Reimagining the status quo

Research on mathematical cognition, learning, and instruction (MCLI) often takes cognition as its point of departure and considers instruction at a later point in the research cycle. In this article, we call for psychologists who study MCLI to reflect on the "status quo" of their research practices and to consider making instruction an earlier and more central aspect of their work. We encourage scholars of MCLI (a) to consider the needs of educators and schools when selecting research questions and developing interventions; (b) to compose research teams that are diverse in the personal, disciplinary, and occupational backgrounds of team members; (c) to make efforts to broaden participation in research and to conduct research in authentic settings; and (d) to communicate research in ways that are accessible to practitioners and to the general public. We argue that a more central consideration of instruction will lead to shifts that make research on MCLI more theoretically valuable, more actionable for educators, and more relevant to pressing societal challenges.

What we mean when we say semantic: Toward a multidisciplinary semantic glossary

Tulving characterized semantic memory as a vast repository of meaning that underlies language and many other cognitive processes. This perspective on lexical and conceptual knowledge galvanized a new era of research undertaken by numerous fields, each with their own idiosyncratic methods and terminology. For example, "concept" has different meanings in philosophy, linguistics, and psychology. As such, many fundamental constructs used to delineate semantic theories remain underspecified and/or opaque. Weak construct specificity is among the leading causes of the replication crisis now facing psychology and related fields. Term ambiguity hinders cross-disciplinary communication, falsifiability, and incremental theory-building. Numerous cognitive subdisciplines (e.g., vision, affective neuroscience) have recently addressed these limitations via the development of consensus-based guidelines and definitions. The project to follow represents our effort to produce a multidisciplinary semantic glossary consisting of succinct definitions, background, principled dissenting views, ratings of agreement, and subjective confidence for 17 target constructs (e.g., abstractness, abstraction, concreteness, concept, embodied cognition, event semantics, lexical-semantic, modality, representation, semantic control, semantic feature, simulation, semantic distance, semantic dimension). We discuss potential benefits and pitfalls (e.g., implicit bias, prescriptiveness) of these efforts to specify a common nomenclature that other researchers might index in specifying their own theoretical perspectives (e.g., They said X, but I mean Y).

Are realistic details important for learning with visualizations or can depth cues provide sufficient guidance?

The optimal choice of the level of realism in instructional visualizations is a difficult task. Previous studies suggest that realism can overwhelm learners, but a growing body of research demonstrates that realistic details can enhance learning. In the first experiment (n = 107), it was assessed whether learning using realistic visualizations can be distracting and therefore particularly benefits from pre-training. Participants learned the anatomy of the parotid gland using labeled visualizations. While pre-training did not have an effect, a more realistic visualization enhanced learning compared to a schematic visualization. In the second experiment (n = 132), a schematic diagram was compared to a more realistic style featuring basic depth cues, and a highly realistic visualization containing a detailed surface. Regarding retention performance, no significant differences were found. However, an interesting pattern regarding subjective cognitive load ratings emerged: the schematic version received the highest cognitive load ratings, while the version featuring simplified shading was rated as least demanding. The version containing simplified depth cues also elicited lower cognitive load ratings than the detailed visualization. The two experiments demonstrate that fears concerning a detrimental effect of realistic details should not be over-generalized. While schematic visualizations may be easier to visually process in some cases, extracting depth information from contour drawings adds cognitive demands to a learning task. Thus, it is advisable that computer-generated visualizations contain at least simplified forms of shading, while the addition of details does not appear to have a strong positive effect.

The effects of gesture and action training on the retention of math equivalence

Introduction

Hand gestures and actions-with-objects (hereafter 'actions') are both forms of movement that can promote learning. However, the two have unique affordances, which means that they have the potential to promote learning in different ways. Here we compare how children learn, and importantly retain, information after performing gestures, actions, or a combination of the two during instruction about mathematical equivalence. We also ask whether individual differences in children's understanding of mathematical equivalence (as assessed by spontaneous gesture before instruction) impacts the effects of gesture- and action-based instruction.

Method

Across two studies, racially and ethnically diverse third and fourth-grade students (N=142) were given instruction about how to solve mathematical equivalence problems (eg., 2+9+4=__+4) as part of a pretest-training-posttest design. In Study 1, instruction involved teaching students to produce either actions or gestures. In Study 2, instruction involved teaching students to produce either actions followed by gestures or gestures followed by actions. Across both studies, speech and gesture produced during pretest explanations were coded and analyzed to measure individual differences in pretest understanding. Children completed written posttests immediately after instruction, as well as the following day, and four weeks later, to assess learning, generalization and retention.

Results

In Study 1 we find that, regardless of individual differences in pre-test understanding of mathematical equivalence, children learn from both action and gesture, but gesture-based instruction promotes retention better than action-based instruction. In Study 2 we find an influence of individual differences: children who produced relatively few types of problem-solving strategies (as assessed by their pre-test gestures and speech) perform better when they receive action training before gesture training than when they receive gesture training first. In contrast, children who expressed many types of strategies, and thus had a more complex understanding of mathematical equivalence prior to instruction, performed equally with both orders.

Discussion

These results demonstrate that action training, followed by gesture, can be a useful stepping-stone in the initial stages of learning mathematical equivalence, and that gesture training can help learners retain what they learn.

How teachers make connections among ideas in mathematics instruction

Conceptual understanding involves understanding connections among ideas within a domain. In this chapter, we consider how teachers support students in learning about connections among ideas in mathematics. We review research focusing on teachers' connection making in mathematics classrooms, and we consider several dimensions of variability in that connection making. Across three corpora of lessons that varied in students' grade levels (first grade to college), cultural settings (United States and China), and mathematics content, we found that all teachers produced linking episodes, but the frequency with which they did so varied substantially, raising new questions about the sources and consequences of that variability. Teachers of first-grade students in China routinely engaged their students in co-constructing links; teachers of middle schoolers and college students in the United States typically explained links to students. Linking episodes targeted many different types of connections, including connections between representations, connections between principles and exemplars, connections between procedures and concepts, and connections between concepts and real-world instantiations. Across all three corpora, teachers expressed linked ideas multimodally in a majority of linking episodes. Based on the findings, we present several hypotheses about how teacher behaviors may support students' understanding of connections among ideas, and we suggest directions for future work.

How do students reason about statistical sampling with computer simulations? An integrative review from a grounded cognition perspective

Interactive computer simulations are commonly used as pedagogical tools to support students' statistical reasoning. This paper examines whether and how these simulations enable their intended effects. We begin by contrasting two theoretical frameworks-dual processes and grounded cognition-in the context of people's conceptions about statistical sampling, setting the stage for the potential benefits of simulations in learning such conceptions. Then, we continue with reviewing the educational literature on statistical sampling simulations. Our review tentatively suggests benefits of the simulations for building statistical habits of mind. However, challenges seem to persist when more specific concepts and skills are investigated. With and without simulations, students have difficulty forming an aggregate view of data, interpreting sampling distributions, showing a process-based understanding of the law of large numbers, making statistical inferences, and context-independent reasoning. We propose that grounded cognition offers a framework for understanding these findings, highlighting the bidirectional relationship between perception and conception, perceptual design features, and guided perceptual routines for supporting students' meaning making from simulations. Finally, we propose testable instructional strategies for using simulations in statistics education.

The promise and pitfalls of a strength-based approach to child poverty and neurocognitive development: Implications for policy

There has been significant progress in understanding the effects of childhood poverty on neurocognitive development. This progress has captured the attention of policymakers and promoted progressive policy reform. However, the prevailing emphasis on the harms associated with childhood poverty may have inadvertently perpetuated a deficit-based narrative, focused on the presumed shortcomings of children and families in poverty. This focus can have unintended consequences for policy (e.g., overlooking strengths) as well as public discourse (e.g., focusing on individual rather than systemic factors). Here, we join scientists across disciplines in arguing for a more well-rounded, "strength-based" approach, which incorporates the positive and/or adaptive developmental responses to experiences of social disadvantage. Specifically, we first show the value of this approach in understanding normative brain development across diverse human environments. We then highlight its application to educational and social policy, explore pitfalls and ethical considerations, and offer practical solutions to conducting strength-based research responsibly. Our paper re-ignites old and recent calls for a strength-based paradigm shift, with a focus on its application to developmental cognitive neuroscience. We also offer a unique perspective from a new generation of early-career researchers engaged in this work, several of whom themselves have grown up in conditions of poverty. Ultimately, we argue that a balanced strength-based scientific approach will be essential to building more effective policies.

Examining the learning effects of concrete and abstract materials among university students using a two-dimensional approach

BACKGROUND

The debate on using concrete versus abstract materials in learning mathematics has been longstanding. For decades, research has focused on the physical characteristics of materials when defining them as concrete or abstract.

AIMS

This study extends the field by proposing a two-dimensional classification, which defines materials as concrete or abstract based on the two dimensions of representation, namely object (i.e., appearance) and language (i.e., label).

SAMPLE

A total of 120 university students participated in the study.

METHODS

Participants were randomly assigned to learn the concept of modular arithmetic with one of four types of learning materials: concrete object labelled with concrete language, concrete object labelled with abstract language, abstract object labelled with concrete language and abstract object labelled with abstract language. They were also divided into high and low maths anxiety groups.

RESULTS

Results showed that the students who learnt with abstract objects, regardless of the level of maths anxiety, outperformed their peers who learnt with concrete objects. However, for students with low maths anxiety only, those who learnt with materials labelled with abstract language showed better far-transfer performance compared with those who learnt with materials labelled with concrete language.

CONCLUSIONS

The findings offer a new direction in the conceptualization of concrete and abstract learning materials by specifying the dimensions of representation.

Adaptive variability in children's conceptual models of division

This study examined whether different types of commonly used mathematical tasks affect how children think about whole number division problems. Prior research suggested that children tend to rely on the partitive model to understand whole number division, which is likely problematic when students transition to learning about fraction division. We assessed variability in correct whole number division problem-solving strategies among 63 elementary school children (41.5% female, 58.5% male, 0% nonbinary/gender expansive; 69.2% White, 10.7% multiracial, 6.1% Black, 4.6% Latino, 3.3% other/unidentified, 6.1% preferred not to answer). Each participant was asked to demonstrate four whole number division problems in each of three contexts (within participants): objects, story problems, and number lines. Most children displayed understanding of multiple conceptual models of division, but strategies varied by context. Story problems elicited partitive models, number lines elicited quotative models, and objects elicited both. Finally, elementary school children used strategies adaptively. Number line representations may afford conceptual connections between earlier-learned whole number concepts and analogous later-learned fraction concepts, supporting the integration of children's whole number and fraction knowledge.

Unlocking the Power of Gesture: Using Movement-Based Instruction to Improve First Grade Children's Spatial Unit Misconceptions

Gestures are hand movements that are produced simultaneously with spoken language and can supplement it by representing semantic information, emphasizing important points, or showing spatial locations and relations. Gestures' specific features make them a promising tool to improve spatial thinking. Yet, there is recent work showing that not all learners benefit equally from gesture instruction and that this may be driven, in part, by children's difficulty understanding what an instructor's gesture is intended to represent. The current study directly compares instruction with gestures to instruction with plastic unit chips (Action) in a linear measurement learning paradigm aimed at teaching children the concept of spatial units. Some children performed only one type of movement, and some children performed both: Action-then-Gesture [AG] or Gesture-then-Action [GA]. Children learned most from the Gesture-then-Action [GA] and Action only [A] training conditions. After controlling for initial differences in learning, the gesture-then-action condition outperformed all three other training conditions on a transfer task. While gesture is cognitively challenging for some learners, that challenge may be desirable-immediately following gesture with a concrete representation to clarify that gesture's meaning is an especially effective way to unlock the power of this spatial tool and lead to deep, generalizable learning.

Promoting spontaneous analogical transfer by idealizing target representations

Recent results demonstrate that inducing an abstract representation of target analogs at retrieval time aids access to analogous situations with mismatching surface features (i.e., the late abstraction principle). A limitation of current implementations of this principle is that they either require the external provision of target-specific information or demand very high intellectual effort. Experiment 1 demonstrated that constructing an idealized situation model of a target problem increases the rate of correct solutions compared with constructing either concrete simulations or no simulations. Experiment 2 confirmed that these results were based on an advantage for accessing the base analog, and not merely an advantage of idealized simulations for understanding the target problem in its own terms. This target idealization strategy has broader applicability than prior interventions based on the late abstraction principle because it can be achieved by a greater proportion of participants and without the need to receive target-specific information. We present a computational model, SampComp, that predicts successful retrieval of a stored situation to understand a target based on the overlap of a random, but potentially biased, sample of features from each. SampComp is able to account for the relative benefits of base and target idealization, and their interaction.

Mapping principles and worked examples for structural learning: effects of content complexity

Drawing connections between principles and worked examples is an approach to learning and instruction, but it is poorly understood. This study investigated the effects of principle and example complexity on learners' ability to map principles and worked examples. The complexity of a principle or example was determined based on the number of concepts and relationships involved. 138 college students were randomly assigned to one of the mapping conditions: principle-simple example, principle-complex example, simple example-simple example, simple example-complex example, and complex example-complex example. The participants studied related materials and completed a free-mapping and a guided-mapping task for a simple and a complex probability principle. The effects of the mapping activities were measured in terms of gains in structural and conceptual knowledge. For the simple principle, principle-example mapping led to fewer nonrelational comparisons (standalone concepts) than did example-example mapping and an equal number of relational comparisons (interconnected concepts). For the complex principle, principle-example mapping led to fewer nonrelational but more relational comparisons than example-example mapping did. Principle-example mapping of corresponding content was more difficult than example-example mapping was. However, principle-example mapping of noncorresponding content was as easy as or easier than example-example mapping. The two forms of mapping resulted in equivalent gains in structural and conceptual knowledge. The findings of this study expand the understanding of analogical reasoning and learning through mapping and comparison of abstract and concrete content. The findings indicate that principle-example mapping enables learners to overcome the obstacles of comprehending abstract or general information and to identify the interrelationships of the individual concepts in formal structures.

Socially intelligent machines that learn from humans and help humans learn

A hallmark of human intelligence is the ability to understand and influence other minds. Humans engage in inferential social learning (ISL) by using commonsense psychology to learn from others and help others learn. Recent advances in artificial intelligence (AI) are raising new questions about the feasibility of human-machine interactions that support such powerful modes of social learning. Here, we envision what it means to develop socially intelligent machines that can learn, teach, and communicate in ways that are characteristic of ISL. Rather than machines that simply predict human behaviours or recapitulate superficial aspects of human sociality (e.g. smiling, imitating), we should aim to build machines that can learn from human inputs and generate outputs for humans by proactively considering human values, intentions and beliefs. While such machines can inspire next-generation AI systems that learn more effectively from humans (as learners) and even help humans acquire new knowledge (as teachers), achieving these goals will also require scientific studies of its counterpart: how humans reason about machine minds and behaviours. We close by discussing the need for closer collaborations between the AI/ML and cognitive science communities to advance a science of both natural and artificial intelligence. This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.

The Impacts of Three Educational Technologies on Algebraic Understanding in the Context of COVID-19

The current study investigated the effectiveness of three distinct educational technologies-two game-based applications (From Here to There and DragonBox 12+) and two modes of online problem sets in ASSISTments (an Immediate Feedback condition and an Active Control condition with no immediate feedback) on Grade 7 students' algebraic knowledge. More than 3,600 Grade 7 students across nine in-person and one virtual schools within the same district were randomly assigned to one of the four conditions. Students received nine 30-minute intervention sessions from September 2020 to March 2021. Hierarchical linear modeling analyses of the final analytic sample (N = 1,850) showed significantly higher posttest scores for students who used From Here to There and DragonBox 12+ compared to the Active Control condition. No significant difference was found for the Immediate Feedback condition. The findings have implications for understanding how game-based applications can affect algebraic understanding, even within pandemic pressures on learning.

Spatial Alignment Facilitates Visual Comparison in Children

Visual comparison is a key process in everyday learning and reasoning. Recent research has discovered the spatial alignment principle, based on the broader framework of structure-mapping theory in comparison. According to the spatial alignment principle, visual comparison is more efficient when the figures being compared are arranged in direct placement-that is, juxtaposed with parallel structural axes. In this placement, (1) the intended relational correspondences are readily apparent, and (2) the influence of potential competing correspondences is minimized. There is evidence for the spatial alignment principle in adults' visual comparison (Matlen et al., 2020). Here, we test whether it holds for children. Six- and eight-year-old children performed a same-different task over visual pairs. The results indicated that direct placement led to faster and more accurate comparison, both for concrete same-different matches (matches of both objects and relations) and for purely relational matches-evidence that the same structural alignment process holds for visual comparison in 6- and 8-year-olds as in adults. These findings have implications for learning and education.

Detailed bugs or bugging details? The influence of perceptual richness across elementary school years

Visualizations are commonly used in educational materials; however, not all visualizations are equally effective at promoting learning. Prior research has supported the idea that both perceptually rich and bland visualizations are beneficial for learning and generalization. We investigated whether the perceptual richness of a life cycle diagram influenced children's learning of metamorphosis, a concept that prior work suggests is difficult for people to generalize. Using identical materials, Study 1 (N = 76) examined learning and generalization of metamorphosis in first- and second-grade students, and Study 2 (N = 53) did so in fourth- and fifth-grade students. Bayesian regression analyses revealed that first and second graders learned more from the lesson with the perceptually rich diagram. In addition, fourth and fifth graders generalized more with the bland diagram, but these generalizations tended to be incorrect (i.e., generalizing metamorphosis to animals that do not undergo this type of change). These findings differ from prior research with adults, in which bland diagrams led to more correct generalizations, suggesting that the effect of perceptual richness on learning and generalization might change over development.

Grounding (fairly) complex numerical knowledge: an educational example

In this article, we contextualize and discuss an on-line contribution to this special issue in which a video-recorded lecture demonstrates the teaching of an abstract mathematical concept, namely regression to the mean. We first motivate the pertinence of this example from the perspective of embodied cognition. Then, we identify mechanisms of teaching that reflect embodied cognitive practices, such as the concreteness fading approach. Rather than a comprehensive review of multiple extensive literatures, this article provides the interested reader with several sources or entries into those literatures.

Two graphs walk into a bar: Readout-based measurement reveals the Bar-Tip Limit error, a common, categorical misinterpretation of mean bar graphs

How do viewers interpret graphs that abstract away from individual-level data to present only summaries of data such as means, intervals, distribution shapes, or effect sizes? Here, focusing on the mean bar graph as a prototypical example of such an abstracted presentation, we contribute three advances to the study of graph interpretation. First, we distill principles for Measurement of Abstract Graph Interpretation (MAGI principles) to guide the collection of valid interpretation data from viewers who may vary in expertise. Second, using these principles, we create the Draw Datapoints on Graphs (DDoG) measure, which collects drawn readouts (concrete, detailed, visuospatial records of thought) as a revealing window into each person's interpretation of a given graph. Third, using this new measure, we discover a common, categorical error in the interpretation of mean bar graphs: the Bar-Tip Limit (BTL) error. The BTL error is an apparent conflation of mean bar graphs with count bar graphs. It occurs when the raw data are assumed to be limited by the bar-tip, as in a count bar graph, rather than distributed across the bar-tip, as in a mean bar graph. In a large, demographically diverse sample, we observe the BTL error in about one in five persons; across educational levels, ages, and genders; and despite thoughtful responding and relevant foundational knowledge. The BTL error provides a case-in-point that simplification via abstraction in graph design can risk severe, high-prevalence misinterpretation. The ease with which our readout-based DDoG measure reveals the nature and likely cognitive mechanisms of the BTL error speaks to the value of both its readout-based approach and the MAGI principles that guided its creation. We conclude that mean bar graphs may be misinterpreted by a large portion of the population, and that enhanced measurement tools and strategies, like those introduced here, can fuel progress in the scientific study of graph interpretation.

Is a Preference for Realism Really Naive After All? A Cognitive Model of Learning with Realistic Visualizations

The use of realistic visualizations has gained considerable interest due to the proliferation of virtual reality equipment. This review is concerned with the theoretical basis, technical implementation, cognitive effects, and educational implications of using realistic visualizations. Realism can be useful for learners, but in several studies, more abstract illustrations have resulted in higher performance. Furthermore, a preference for realistic visualization has been declared as being based on misconceptions regarding the cognitive system. However, we argue that this perspective is unable to fully explain the conflicting results found in the literature. To fill this theoretical gap, we devised a model to describe and compare the various levels of realism found in visualizations. We define realism as a combination of three dimensions: geometry, shading, and rendering. By varying these dimensions, it is possible to create a variety of realistic graphics. Thus, when comparing different visualizations, the realism of each of these three dimensions needs to be considered individually. Based on this technical definition, we introduce a cognitive model of learning with realistic visualizations that includes three different stages: perception, schema construction, and testing. At these three stages, variables such as the perceptual load generated by the visualization, learner characteristics influencing how well details are processed, and test types that demand concrete or flexible representations can affect whether realism fosters or hinders learning. Using the cognitive model presented in this paper, more accurate predictions and recommendations concerning the use of realism can be formulated.

Finding formulas: Does active search facilitate appropriate generalization?

BACKGROUND

One criterion of adaptive learning is appropriate generalization to new instances based on the original learning context and avoiding overgeneralization. Appropriate generalization requires understanding what features of a solution are applicable in a new context and whether the new context requires modifications or a new approach. In a series of three experiments, we investigate whether searching for an algebraic formalism before receiving direct instruction facilitates appropriate generalization.

RESULTS

(1) Searching buffers against negative transfer: participants who first searched for an equation were less likely to overgeneralize compared to participants who completed a tell-and-practice activity. (2) Likelihood of creating a correct new adaptation varied by performance on the searching task. (3) Asking people to sketch alleviated some of the negative effects of tell-and-practice, but sketching did not augment the effect of searching. (4) When participants received more elaborate tell-and-practice instruction, the advantages of searching were less notable.

CONCLUSIONS

Searching for an algebraic formula prior to direct instruction may be a productive way to help learners connect a formula to its referent and avoid overgeneralization. Tell-and-practice instruction that only described the mathematical procedures led to the greatest levels of overgeneralization errors and worst performance. Tell-and-practice instruction that highlighted connections between the mathematical structure of the formula and the visual referent performed at similar or marginally worse levels than the search-first conditions.

Mathematics Meets Science in the Brain

Mathematics and science are highly integrated disciplines, but the brain association between mathematics and science remains unclear. The current study used functional magnetic resonance imaging (fMRI) scans of 34 undergraduates (17 males, mean age = 20.3±1.64 years old) while they completed mathematical, physical and chemical principles, arithmetic computation, and sentence comprehension. We examined neural activation level, neural activation pattern, and neural connectivity to investigate the neural associations between mathematics and science (including physics and chemistry). The results showed that mathematical, physical, and chemical principles elicited similar neural activation level and neural activation pattern in the visuospatial network (mainly in the middle frontal gyrus and inferior parietal lobule), which were different from those elicited by sentence comprehension; those three principles also elicited similar neural activation level and neural activation pattern in the semantic network (mainly in the middle temporal gyrus, angular gyrus, inferior frontal gyrus, and dorsomedial prefrontal cortex), in contrast to that elicited by arithmetic computation. Effective connectivity analyses showed stronger connectivity between the middle temporal gyrus and inferior parietal lobule for mathematical, physical, and chemical principles than for sentence comprehension. The results suggest that visuospatial and semantic networks were critical for processing both mathematics and science.

Enhancing spatial skills of preschoolers from under-resourced backgrounds: A comparison of digital app vs. concrete materials

Spatial skills support STEM learning and achievement. However, children from low-socioeconomic (SES) backgrounds typically lag behind their middle- and high-SES peers. We asked whether a digital educational app-designed to mirror an already successful, spatial assembly training program using concrete materials-would be as effective for facilitating spatial skills in under-resourced preschoolers as the concrete materials. Three-year-olds (N = 61) from under-resourced backgrounds were randomly assigned to a business-as-usual control group or to receive 5 weeks of spatial training using either concrete, tangible materials or a digital app on a tablet. The spatial puzzles used were an extension of items from the Test of Spatial Assembly (TOSA). Preschoolers were pretested and posttested on new two-dimensional (2D) TOSA trials. Results indicate that both concrete and digital spatial training increased performance on the 2D-TOSA compared to the control group. The two trainings did not statistically differ from one another suggesting that educational spatial apps may be one route to providing early foundational skills to children from under-resourced backgrounds.

Reconceptualizing Symbolic Magnitude Estimation Training Using Non-declarative Learning Techniques

It is well-documented that mathematics achievement is an important predictor of many positive life outcomes like college graduation, career opportunities, salary, and even citizenship. As such, it is important for researchers and educators to help students succeed in mathematics. Although there are undoubtedly many factors that contribute to students' success in mathematics, much of the research and intervention development has focused on variations in instructional techniques. Indeed, even a cursory glance at many educational journals and granting agencies reveals that there is a large amount of time, energy, and resources being spent on determining the best way to convey information through direct, declarative instruction. The proposed project is motivated by recent calls to expand the focus of research in mathematics education beyond direct, declarative instruction. The overarching goal of the presented experiment is to evaluate the efficacy of a novel mathematics intervention designed using principles taken from the literature on non-declarative learning. The intervention combines errorless learning and structured cue fading to help second grade students improve their understanding of symbolic magnitude. Results indicate that students who learned about symbolic magnitude using the novel intervention did better than students who were provided with extensive declarative support. These findings offer preliminary evidence in favor of using learning combination of errorless learning and cue fading techniques in the mathematics classroom.

Designing Exhibits to Support Relational Learning in a Science Museum

Science museums aim to provide educational experiences for both children and adults. To achieve this goal, museum displays must convey scientifically-relevant relationships, such as the similarities that unite members of a natural category, and the connections between scientific models and observable objects and events. In this paper, we explore how research on comparison could be leveraged to support learning about such relationships. We describe how museum displays could promote educationally-relevant comparisons involving natural specimens and scientific models. We also discuss how these comparisons could be supported through the design of a display—in particular, by using similarity, space, and language to facilitate relational thinking for children and their adult companions. Such supports may be pivotal given the informal nature of learning in museums.

What we count dictates how we count: A tale of two encodings

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults' reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults' encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.

One Instructional Sequence Fits all? A Conceptual Analysis of the Applicability of Concreteness Fading in Mathematics, Physics, Chemistry, and Biology Education

To help students acquire mathematics and science knowledge and competencies, educators typically use multiple external representations (MERs). There has been considerable interest in examining ways to present, sequence, and combine MERs. One prominent approach is the concreteness fading sequence, which posits that instruction should start with concrete representations and progress stepwise to representations that are more idealized. Various researchers have suggested that concreteness fading is a broadly applicable instructional approach. In this theoretical paper, we conceptually analyze examples of concreteness fading in the domains of mathematics, physics, chemistry, and biology and discuss its generalizability. We frame the analysis by defining and describing MERs and their use in educational settings. Then, we draw from theories of analogical and relational reasoning to scrutinize the possible cognitive processes related to learning with MERs. Our analysis suggests that concreteness fading may not be as generalizable as has been suggested. Two main reasons for this are discussed: (1) the types of representations and the relations between them differ across different domains, and (2) the instructional goals between domains and subsequent roles of the representations vary.

The Disappearing "Advantage of Abstract Examples in Learning Math"

When teaching a novel mathematical concept, should we present learners with abstract or concrete examples? In this experiment, we conduct a critical replication and extension of a well-known study that argued for the general advantage of abstract examples (Kaminski, Sloutsky, & Heckler, 2008a). We demonstrate that theoretically motivated yet minor modifications of the learning design put this argument in question. A key finding from this study is that participants who trained with improved concrete examples performed as well as, or better than, participants who trained with abstract examples. We argue that the previously reported "advantage of abstract examples" manifested not because abstract examples are advantageous in general, but because the concrete condition employed suboptimal examples.

Using a Virtual Manipulative Intervention Package to Support Maintenance in Teaching Subtraction with Regrouping to Students with Developmental Disabilities

To live independently, it is critical that students with disabilities maintain the basic mathematical skills they have acquired so they may apply these skills in daily life. To support maintenance of mathematical skills among students with developmental disabilities, the researchers used a multiple probe across participants design to examine the effectiveness of the VRA instructional sequence with fading support in teaching subtraction with regrouping to four students with developmental disabilities. A functional relation was found between the VRA instructional sequence with fading support and students' accuracy in solving the problems. Students also maintained the skill up to 6 weeks after the intervention.

Concreteness Fading Strategy: A Promising and Sustainable Instructional Model in Mathematics Classrooms

Conceptual understanding has been emphasized in the national curriculum and principles and standards across nations as it is the key in mathematical learning. However, mathematics instruction in classrooms often relies on rote memorization of mathematical rules and formulae without conceptual connections. This study considers the concreteness fading instruction strategy—starting with physical activities with manipulatives and gradually fading concreteness to access abstract concepts and representations—as a promising and sustainable instructional model for supporting students in accessing conceptual understanding in mathematics classrooms. The results from the case study support the validity of the concreteness fading framework in providing specific instructional strategies in each phase of concept development. This study implies the development of sustainable teacher education and professional development by providing specific instructional strategies for conceptual understanding.

Leveraging Multiple Analytic Frameworks to Assess the Stability of Students' Knowledge in Physiology

When a student explains a biological phenomenon, does the answer reflect only the product of retrieving knowledge or does it also reflect a dynamic process of constructing knowledge? To gain insight into students' dynamic knowledge, we leveraged three analytic frameworks-structures-behaviors-functions (SBF), mental models (MM), and conceptual dynamics (CD). To assess the stability of student knowledge, we asked undergraduate students to explain the same physiological phenomenon three times-once verbally, once after drawing, and once after interpreting a diagram. The SBF analysis illustrated fine-grained dynamic knowledge between tasks. The MM analysis suggested global stability between tasks. The CD analysis demonstrated local instability within tasks. The first two analyses call attention to differences between students' knowledge about the parts of systems and their organization. The CD analysis, however, calls attention to similar learning mechanisms that operate differently vis-à-vis external representations. Students with different mental models deliberated localization or where to locate the structures and mechanisms that mediate physiological responses, but students made these deliberations during different tasks and arrived at different conclusions. These results demonstrate the utility of incorporating dynamic approaches to complement other analytic approaches and motivate future research agendas in biology education research.

Construct-A-Vis: Exploring the Free-Form Visualization Processes of Children

Building data analysis skills is part of modern elementary school curricula. Recent research has explored how to facilitate children's understanding of visual data representations through completion exercises which highlight links between concrete and abstract mappings. This approach scaffolds visualization activities by presenting a target visualization to children. But how can we engage children in more free-form visual data mapping exercises that are driven by their own mapping ideas? How can we scaffold a creative exploration of visualization techniques and mapping possibilities? We present Construct-A-Vis, a tablet-based tool designed to explore the feasibility of free-form and constructive visualization activities with elementary school children. Construct-A-Vis provides adjustable levels of scaffolding visual mapping processes. It can be used by children individually or as part of collaborative activities. Findings from a study with elementary school children using Construct-A-Vis individually and in pairs highlight the potential of this free-form constructive approach, as visible in children's diverse visualization outcomes and their critical engagement with the data and mapping processes. Based on our study findings we contribute insights into the design of free-form visualization tools for children, including the role of tool-based scaffolding mechanisms and shared interactions to guide visualization activities with children.

Can a relational mindset boost analogical retrieval?

BACKGROUND

Memory retrieval is driven by similarity between a present situation and some prior experience, but not all similarity is created equal. Analogical retrieval, rooted in the similarity between two situations in their underlying structural relations, is often responsible for new insights and innovative solutions to problems. However, superficial similarity is instead more likely to drive spontaneous retrieval. How can we make analogical retrieval more likely? Inducing a relational mindset via an analogical reasoning task has previously been shown to boost subsequent relational thinking. In this paper, we examined whether inducing a relational mindset could also boost analogical retrieval.

RESULTS

We find that a relational mindset can increase analogical retrieval if induced before information is encoded in the first place, amplifying the effect of a clearly labelled relational structure. On the other hand, inducing a relational mindset at the time of retrieval did not increase analogical retrieval.

CONCLUSION

This work further demonstrates the central importance of high-quality relational encoding for subsequent relation-based analogical retrieval, and that inducing a relational mindset can improve those encodings.

When calculators lie: A demonstration of uncritical calculator usage among college students and factors that improve performance

Calculators are often unnecessary to solve routine problems, though they are convenient for offloading cognitively effortful processes. However, errors can arise if incorrect procedures are used or when users fail to monitor the output for keystroke mistakes. To investigate the conditions under which people's attention are captured by errant calculator outputs (i.e., from incorrectly chosen procedures or keystroke errors), we programmed an onscreen calculator to "lie" by changing the answers displayed on certain problems. We measured suspicion by tracking whether users explicitly reported suspicion, overrode calculator "lies", or re-checked their calculations after a "lie" was presented. In Study 1, we manipulated the concreteness of problem presentation and calculator delay between subjects to test how these affect suspicion towards "lies" (15% added to answers). We found that numeracy had no effect on whether people opted-in or out of using the calculator but did predict whether they would become suspicious. Very few people showed suspicion overall, however. For study 2, we increased the "lies" to 120% on certain answers and included questions with "conceptual lies" shown (e.g., a negative sign that should have been positive). We again found that numeracy had no effect on calculator usage, but, along with concrete formatting, did predict suspicion behavior. This was found regardless of "lie" type. For study 3, we reproduced these effects after offering students an incentive for good performance, which did raise their accuracy across the math problems overall but did not increase suspicion behavior. We conclude that framing problems within a concrete domain and being higher in numeracy increases the likelihood of spotting errant calculator outputs, regardless of incentive.

A cluster-randomized controlled trial of the effectiveness of the JUMP Math program of math instruction for improving elementary math achievement

Students in many western countries struggle to achieve acceptable standards in numeracy despite its recognition as an important 21^st century skill. As commercial math programs remain a staple of classroom instruction, investigations of their effectiveness are essential to inform decision-making regarding how to invest limited resources while maximizing student gains. We conducted a cluster randomized-controlled trial of the effectiveness of JUMP Math, a distinctive math program whose central tenets are empirically supported, for improving elementary math achievement (clinical trial.gov no. NCT02456181). The study involved 554 grade 2 (primary) and 592 grade 5 (junior) students and 193 teachers in 41 schools, in an urban-rural Canadian school board. Schools were randomly assigned to use either JUMP Math or their business-as-usual, problem-based approach to math instruction. We tracked student progress in math achievement on standardized and curriculum-based measures of computation and problem solving, for 2 consecutive school years. Junior students taught with JUMP Math made significantly greater progress in computation than their non-JUMP peers but the groups did not differ significantly in problem solving. Effects took hold relatively quickly, replicating the results from an earlier pilot study. Primary students in the non-JUMP group made significantly greater gains in problem solving and computation in year 1. But those taught with JUMP Math made significantly greater gains in problem solving and the groups did not differ in computation, in year 2. The positive effects of JUMP Math are noteworthy given that the JUMP Math teachers were likely still adjusting to the new program. That these positive findings were obtained in an effectiveness study (i.e. in real-world conditions), suggests that JUMP Math may be a valuable evidence-based addition to the teacher’s toolbox. Given the importance of numeracy for 21^st century functioning, identifying and implementing effective math instruction programs could have far-reaching, positive implications.

Using the virtual-representational instructional sequence to support the acquisition and maintenance of mathematics for students with intellectual disability

Mathematics instruction - and interventions to support mathematics teaching - for students with intellectual disability is important yet underexamined. This study explored a graduated instructional sequence referred to as the virtual-representational (VR) as a mathematical intervention. Researchers taught four students with disabilities what multiplication or division means and how to solve the problems first via virtual manipulatives and subsequently via pictorial representations. The researchers found a functional relation between student accuracy in solving multiplication or division problems and the intervention of the VR instructional sequence for three of the four students; for the fourth student, the researchers added the system of least prompts (SLP) to the virtual subphase of intervention. Researchers also found the students were able to maintain their accuracy in solving multiplication and division problems at rates exceeding baseline when instruction did not precede.

Ten quick tips for creating an effective lesson

We present 10 tips for building effective lessons that are grounded in empirical research on pedagogy and cognitive psychology and that we have found to be practically useful in both classroom and free-range settings.

Mitigating the Attraction Effect with Visualizations

Human decisions are prone to biases, and this is no less true for decisions made within data visualizations. Bias mitigation strategies often focus on the person, by educating people about their biases, typically with little success. We focus instead on the system, presenting the first evidence that altering the design of an interactive visualization tool can mitigate a strong bias - the attraction effect. Participants viewed 2D scatterplots where choices between superior alternatives were affected by the placement of other suboptimal points. We found that highlighting the superior alternatives weakened the bias, but did not eliminate it. We then tested an interactive approach where participants completely removed locally dominated points from the view, inspired by the elimination by aspects strategy in the decision-making literature. This approach strongly decreased the bias, leading to a counterintuitive suggestion: tools that allow removing inappropriately salient or distracting data from a view may help lead users to make more rational decisions.

Recombinant Enaction: Manipulatives Generate New Procedures in the Imagination, by Extending and Recombining Action Spaces

Manipulation of physical models such as tangrams and tiles is a popular approach to teaching early mathematics concepts. This pedagogical approach is extended by new computational media, where mathematical entities such as equations and vectors can be virtually manipulated. The cognitive and neural mechanisms supporting such manipulation-based learning-particularly how actions generate new internal structures that support problem-solving-are not understood. We develop a model of the way manipulations generate internal traces embedding actions, and how these action-traces recombine during problem-solving. This model is based on a study of two groups of sixth-grade students solving area problems. Before problem-solving, one group manipulated a tangram, the other group answered a descriptive test. Eye-movement trajectories during problem-solving were different between the groups. A second study showed that this difference required the tangram's geometrical structure, just manipulation was not enough. We propose a theoretical model accounting for these results, and discuss its implications.

Structure Mapping and Vocabularies for Thinking

While extremes tend to capture attention, the ordinary is often most of the story. So it may be with the structure-mapping process. The structure-mapping process can account for such pinnacles of thinking as analogy and metaphor, which can lead to overlooking the mundane, incremental use of structure mapping. Consequently, the current discussion shifts focus to the value of close comparisons between literally similar items for the development of knowledge. The intent is to foster greater integration between process and content as well as between individuals and collectives. The payoff is identifying some undue simplifications and some promising new directions.

Effects of Instructional Guidance and Sequencing of Manipulatives and Written Symbols on Second Graders’ Numeration Knowledge

Concrete objects used to illustrate mathematical ideas are commonly known as manipulatives. Manipulatives are ubiquitous in North American elementary classrooms in the early years, and although they can be beneficial, they do not guarantee learning. In the present study, the authors examined two factors hypothesized to impact second-graders’ learning of place value and regrouping with manipulatives: (a) the sequencing of concrete (base-ten blocks) and abstract (written symbols) representations of the standard addition algorithm; and (b) the level of instructional guidance on the structural relations between the representations. Results from a classroom experiment with second-grade students (N = 87) indicated that place value knowledge increased from pre-test to post-test when the base-ten blocks were presented before the symbols, but only when no instructional guidance was offered. When guidance was given, only students in the symbols-first condition improved their place value knowledge. Students who received instruction increased their understanding of regrouping, irrespective of representational sequence. No effects were found for iterative sequencing of concrete and abstract representations. Practical implications for teaching mathematics with manipulatives are considered.

The effect of abstract versus concrete framing on judgments of biological and psychological bases of behavior

Human behavior is frequently described both in abstract, general terms and in concrete, specific terms. We asked whether these two ways of framing equivalent behaviors shift the inferences people make about the biological and psychological bases of those behaviors. In five experiments, we manipulated whether behaviors are presented concretely (i.e. with reference to a specific person, instantiated in the particular context of that person's life) or abstractly (i.e. with reference to a category of people or behaviors across generalized contexts). People judged concretely framed behaviors to be less biologically based and, on some dimensions, more psychologically based than the same behaviors framed in the abstract. These findings held true for both mental disorders (Experiments 1 and 2) and everyday behaviors (Experiments 4 and 5), and yielded downstream consequences for the perceived efficacy of disorder treatments (Experiment 3). Implications for science educators, students of science, and members of the lay public are discussed.