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A permutations problem. (A) A solution of the problem using a formal representation, numerical calculation. (B) A solution of the same problem using a grounded representation, outcome listing.
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The terms concreteness fading and progressive formalization have been used to describe instructional approaches to science and mathematics that use grounded representations to introduce concepts and later transition to more formal representations of the same concepts. There are both theoretical and empirical reasons to believe that such an approach...
Contexts in source publication
Context 1
... such formulas, representations such as verbal descriptions, pictures, concrete ma- nipulatives, tree diagrams, and outcome lists may also be used to conceptualize and solve combinatorics problems. Figure 1 shows examples of numerical calculation and outcome listing in the context of a permutations problem. ...
Context 2
... the terminology of the preceding section, numerical calcula- tions (see Figure 1A) constitute formal representations. Although outcome lists (see Figure 1B) appear at first glance rather abstract, they are in fact perceptually grounded in two respects. ...
Context 3
... the terminology of the preceding section, numerical calcula- tions (see Figure 1A) constitute formal representations. Although outcome lists (see Figure 1B) appear at first glance rather abstract, they are in fact perceptually grounded in two respects. First, they represent numbers by numerosities. ...
Context 4
... was hoped that fading would increase the benefits of grounded repre- sentations, relative to the findings of previous studies. The study focused on one type of combinatorics problem, permutations, and used outcome listing and numerical calculations (see Figure 1) as representatives of grounded and formal representations, respec- tively. It was proposed above that grounded and formal representations offer complementary benefits to mathematics learners. ...
Context 5
... listing and numerical calculation are amenable to making such connections due to their structural correspondences. For example, the numeral 3 in the numerical calculation of Figure 1A corresponds to the letters A, B, and C appearing in the far left position of the outcome list of Figure 1B, the numeral 2 to the sets of different letters appearing in the second position of the outcome list, and so on. These correspondences suggest that outcome lists and numerical calculations are good candidates for application of a fading approach, provided that learners are able to notice and understand the correspondences. ...
Context 6
... listing and numerical calculation are amenable to making such connections due to their structural correspondences. For example, the numeral 3 in the numerical calculation of Figure 1A corresponds to the letters A, B, and C appearing in the far left position of the outcome list of Figure 1B, the numeral 2 to the sets of different letters appearing in the second position of the outcome list, and so on. These correspondences suggest that outcome lists and numerical calculations are good candidates for application of a fading approach, provided that learners are able to notice and understand the correspondences. ...
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Depuis des années déjà, les chercheurs en informatique se sont emparés des questions de médiation scientifique et sont à l'origine de nombreuses initiatives qui visent à mettre à portée de tous, de manière souvent originale et ludique, des éléments de science informatique. Il ne s'agit pas ici d'enseigner mais bien de susciter la réflexion, de seme...
Citations
... Furthermore, returning to Fyfe et al.'s (2014) call for describing ways to optimize the fading technique, using the theory of levels of abstraction to guide CF seems to have great potential for optimizing its instructional use. As opposed to the representations used by Braithwaite and Goldstone (2013) in their concreteness-fading study of combinatorics with undergraduates (specifically, letter sequences followed by arithmetic explanations for factorials), we provide a very different and much elaborated interpretation of concrete versus abstract representation, as well as CF, for combinatorial reasoning. Indeed, our "fading" is accomplished by starting with small-number permutations of physical cubes and incrementally increasing this number so that actual manipulation of cubes becomes impractical and necessarily needs to fade into the background as symbolic representations are abstracted and come to the fore. ...
Over half a century has passed since Bruner suggested his three-stage enactive-iconic-symbolic model of instruction. In more recent research, predominantly in educational psychology, Bruner’s model has been reformulated into the theory of instruction known as concreteness fading (CF). In a recent constructivist teaching experiment investigating two undergraduate students’ combinatorial reasoning, we utilized an instructional approach that maintains the enactive-iconic-symbolic stages of CF, but through a gradual and much elaborated process. We found that our theory of levels of abstraction explicated the “fading” effect that is central to CF. In this theoretical report, we discuss how CF can be elaborated by our instructional approach and theoretical perspective.
... In a series of studies (Glenberg et al., 2007), third and fourth graders who used realistic figurines (i.e., those that resembled the characters and objects described in the text) solved more word problems correctly, used a larger number of correct strategies, and used less irrelevant numerical information in their solution attempts than those who did not use figurines. The figurines used in Glenberg et al.'s work can be called "grounded," or realistic, because they were instantiated in real-world contexts (Braithwaite & Goldstone, 2013). Overall, existing research suggests that realistic representations are effective for immediate performance and learning because their perceptual features can help students activate relevant prior knowledge (Belenky & Schalk, 2014), which in turn supports sense-making. ...
We examined the role of visual representations and knowledge of relational statements on children’s understanding of the structure of compare word problems. Second and third graders (N = 82) were randomly assigned to three representation conditions: (a) abstract pictures; realistic pictures that were (b) relevant, and (c) irrelevant to the story described in the problem. On the Problem Structure Task, children were presented a compare problem containing inconsistent language structures and judged whether a visual representation matched its structure. On the Relational Statement Task, children judged whether two statements describing an additive comparison between two sets were equivalent in meaning. Representation type had no effect on performance, whereas knowledge of relational language accounted for significant variance in children’s problem structure understanding.
... A number of researchers have sought ways to improve students' combinatorial thinking and activity. This has included designing tasks that will evoke particular combinatorial concepts or approaches (e.g., Eizenberg & Zaslavsky, 2004;Lockwood et al., 2015;Tillema, 2013;Tillema & Gatza, 2016), investigating the role of representations and listing (e.g., Braithwaite & Goldstone, 2013; Wasserman & Galarza, 2019), developing instructional interventions to teach combinatorial content or practices (e.g., CadwalladerOlsker et al., 2012;Lockwood et al., 2015;, and describing ways of thinking (e.g., Halani, 2012;Lockwood, 2014) that might help students solve counting problems more successfully. This has also included outlining potential learning trajectories in combinatorics across which students might progress (e.g., Maher et al., 2011;Tillema, 2020). ...
... In addition, Tillema (2013Tillema ( , 2018 has investigated various ways in which students might symbolize or represent outcomes they are trying to count, and he has emphasized the importance of such representations in developing students' combinatorial reasoning. Further, in a psychological study, Braithwaite and Goldstone (2013) explored the role of representations and formalisms among psychology students in a combinatorial context. Ultimately, Braithwaite and Goldstone summarize their findings by saying the following: ...
... This type of question poses challenges for students with a variety of experience with counting and computing, which is a point we will demonstrate in our data. As we aim to show, the mth element question necessitates attention to the set of outcomes that are being counted and their connections to the counting process, and it does not overemphasize formulas that might be applied without understanding (e.g., Braithwaite & Goldstone, 2013;Lockwood, 2014). ...
Research has shown that solving counting problems correctly can be difficult for students at all levels, and mathematics educators have sought to identify strategies and interventions to help students reason conceptually about combinatorial tasks. A set-oriented perspective (Lockwood, 2014) is a way of thinking about counting problems that emphasizes the importance of reasoning about the set of outcomes being counted. From a set-oriented perspective, one possible type of intervention is to have students focus on the sets of outcomes rather than formulas and expressions, and specifically to reason about the structure of the set of outcomes. Yet, reasoning about sets of outcomes is not sufficient for students to make connections between outcomes and counting processes. In this paper, we investigate tasks where students wrote computer code to enumerate the set of outcomes in a specific order by implementing listing processes, and they were then asked to determine a specific numbered outcome in their list by using the structure of their enumeration scheme. We clarify particular aspects of a set-oriented perspective that were productive for students, and we demonstrate that tasks that asked students to name a specific outcome in their list elicited meaningful connections between counting processes and sets of outcomes. Further, such tasks reinforce desirable mathematical practices such as leveraging structure and connecting representations.
... This work makes an important contribution to understanding how interventions that are successful in laboratory contexts with simple materials can extend to more ecologically valid mathematical problems. The current experiment used combinatorics story problems, which have been used in other work that has highlighted the difficulties learners face with this topic (Bassok et al., 1995;Braithwaite & Goldstone, 2013;Ross, 1987). The topic of combinatorics is important in statistics, which is increasingly common as a quantitative requirement instead of algebra at the college level (Lattimore & Depenbrock, 2017;Logue et al., 2016). ...
Previous research has demonstrated benefits of interleaved practice over blocked practice for learning mathematical formulas. This experiment tested whether the benefits from interleaved practice would generalize to more complex problems, where the problem type must be inferred from information in the problem. We compared delayed test performance of participants assigned to blocked practice to participants assigned to interleaved practice who had high or low practice performance. University students (Mage = 18.97, SDage = 1.50, 64% female) learned how to solve probability word problems in blocked practice, interleaved practice, or hybrid conditions that included both kinds of practice. Conditions that included some interleaved practice outperformed a condition that included only blocked practice at delayed test. Participants with high performance on interleaved practice problems outperformed participants assigned to blocked practice at delayed test. These results suggest that interleaved practice can confer learning advantages even for more complex problems.
... The current study's emphasis on cross-notation knowledge aligns well with research that emphasizes the importance of understanding and flexibly using multiple external representations (MERs) when learning and doing math (Ainsworth, 2006). Previous research on MERs has largely emphasized connections between symbolic and graphical representations or between multiple graphical representations (e.g., Acevedo Nistal, Van Dooren, & Verschaffel, 2014;Ainsworth, 2006;Braithwaite & Goldstone, 2013) such as between fractions and number lines, between fractions and pie charts, and between number lines and pie charts (Rau & Matthews, 2017). The current findings suggest that connections between multiple symbolic representations may also play an important role in math education. ...
Understanding fractions and decimals requires not only understanding each notation separately, or within-notation knowledge, but also understanding relations between notations, or cross-notation knowledge. Multiple notations pose a challenge for learners but could also present an opportunity, in that cross-notation knowledge could help learners to achieve a better understanding of rational numbers than could easily be achieved from within-notation knowledge alone. This hypothesis was tested by reanalyzing three published datasets involving fourth-to eighth-grade children from the United States and Finland. All datasets included measures of rational number arithmetic, within-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. fractions and decimals vs. decimals), and cross-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. decimals). Consistent with the hypothesis, cross-notation magnitude knowledge predicted fraction and decimal arithmetic when controlling for within-notation magnitude knowledge. Furthermore, relations between within-notation magnitude knowledge and arithmetic were not notation specific; fraction magnitude knowledge did not predict fraction arithmetic more than decimal arithmetic, and decimal magnitude knowledge did not predict decimal arithmetic more than fraction arithmetic. Implications of the findings for assessing rational number knowledge and learning and teaching about rational numbers are discussed.
... Although prior work suggests that it is generally beneficial to begin instruction with concrete representations and then gradually fade to more abstract or symbolic representations for most learners (Braithwaite & Goldstone, 2013;McNeil et al., 2012), the concrete representations used should be chosen carefully. Uttal, Scudder, and DeLoache (1997) demonstrated that concrete representations with features that are irrelevant to the concept to be learned can sometimes be harmful to learning; for example, children struggled to understand that toy-like manipulatives can be used as representations of an abstract concept. ...
Mathematics difficulties (MD) are widespread. Higher mathematics achievement is associated with college success and greater job opportunities. In this chapter, we address the intersection between cognitive research on the science of learning and the education of students with MD. First, a conceptual framework for mathematics learning is provided, including consideration of individual differences in domain general and domain specific learning processes. Next, learning principles that promote successful learning in mathematics are described, along with examples of how the principles can be translated into educational practice for students with MD. We argue that in addition to providing developmentally appropriate instruction in content knowledge, such as whole number and fraction knowledge, application of more general learning principles validated by research will lead to deeper and more durable learning for struggling students. These learning principles include studying and comparing correct as well as incorrect worked out problem-solutions; using integrated visual and verbal models to reduce splitting attention; interleaving or varying practice with problems of different types; providing frequent cumulative practice or review that is spaced out over time; connecting and integrating concrete and symbolic representations; presenting arithmetic problems in different formats; using physical movements and gestures to promote learning; and incorporating activities with number lines to increase learning of whole numbers and fractions.
... We provide an overview of concreteness fading, identify its design dimensions, and summarize the key findings about each dimension. Our analysis spans the research areas of math education [10,54,55], environmental health education [75], geography education [63], medical education [23], engineering education [83], science education [51,71], and computing education [3,43,82]. In summary, our research contributes: ...
... An experiment conducted by Kaminski et al. [42] showed that when teaching undergraduate students a complex mathematical concept, keeping the representation abstract led to greater learning than concreteness fading, providing empirical evidence for the only-abstract argument. Braithwaite and Goldstone [10] also reported similar results with teaching mathematics to undergraduate students. But since they were undergraduate students, who are capable of abstract reasoning according to cognitive development theory [37], these results are not surprising. ...
Over the years, concreteness fading has been used to design learning materials and educational tools for children. Unfortunately, it remains an underspecified technique without a clear guideline on how to design it, resulting in varying forms of concreteness fading and conflicting results due to the design inconsistencies. To our knowledge, no research has analyzed the existing designs of concreteness fading implemented across different settings, formulated a generic framework, or explained the design dimensions of the technique. This poses several problems for future research, such as lack of a shared vocabulary for reference and comparison, as well as barriers to researchers interested in learning and using this technique. Thus, to inform and support future research, we conducted a systematic literature review and contribute: (1) an overview of the technique, (2) a discussion of various design dimensions and challenges, and (3) a synthesis of key findings about each dimension. We open source our dataset to invite other researchers to contribute to the corpus, supporting future research and discussion on concreteness fading.
... Children's improved performance on posttest assessments, in which fraction strips were not available, indicates that they made the transition successfully. The present study with children joins previous successful implementations of concreteness fading with adults for instruction in combinatorics (Braithwaite & Goldstone, 2013), modular arithmetic (McNeil & Fyfe, 2012), and complex systems concepts (Goldstone & Son, 2005). Concreteness fading provides a promising approach to instruction in other areas as well. ...
Learning fractions is a critical step in children’s mathematical development. However, many children struggle with learning fractions, especially fraction arithmetic. In this article, we propose a general framework for integrating understanding of individual fractions and fraction arithmetic, and we use the framework to generate interventions intended to improve understanding of both individual fractions and fraction addition. The framework, Putting Fractions Together (PFT), emphasizes that both individual fractions and sums of fractions are composed of unit fractions and can be represented by concatenating them (putting them together). To illustrate, both “3/9” and “2/9 + 1/9” can be represented by concatenating three 1/9s; similarly, 2/9 + 1/8 can be represented by concatenating two 1/9s and one 1/8. Interventions based on the PFT framework were tested in 2 experiments with fourth, fifth, and sixth grade children. The interventions led to improved performance on number line estimation and magnitude comparison tasks involving individual fractions and sums of fractions with equal and unequal denominators. Especially large improvements were observed on relatively difficult unequal-denominator fraction sum problems. The findings suggest that viewing individual nonunit fractions and sums of fractions as concatenations of unit fractions provides a sound conceptual foundation for improving children’s knowledge of both. We discuss implications of the research for teaching and learning fractions, children’s numerical development, and mathematics education in general.
... The present study focuses on concreteness fading (or progressive formalization), a special case of learning with multiple representations that has received notable attention in the STEM education literature (Braithwaite and Goldstone 2013;De Bock et al. 2011;Fyfe et al. 2014Fyfe et al. , 2015Goldstone and Son 2005;Jaakkola and Veermans 2015;Kaminski et al. 2008;McNeil and Fyfe 2012;Moreno et al. 2011;Siler and Klahr 2016). In concreteness fading, learning tasks start with concrete representations, which are afterwards replaced by more abstract representations. ...
... Alternatively, it might be that despite its theoretical basis, concreteness fading in particular is something that works only in the domain used by Goldstone and Son (2005). The literature however shows that concreteness fading has also been productive in other fields, particularly in various domains in mathematics (Braithwaite and Goldstone 2013;Fyfe et al. 2015;Kaminski et al. 2008;McNeil and Fyfe 2012), which provided additional empirical evidence against domain specific benefits of concreteness fading. ...
The present study investigates the effects that concreteness fading has on learning and transfer across three grade levels (4–6) in elementary school science education in comparison to learning with constantly concrete representations. 127 9- to 12-years-old elementary school students studied electric circuits in a computer-based simulation environment, where circuits remained concrete (bulbs) throughout the learning or faded from concrete to abstract (bulbs to resistors). The most important finding was that the outcomes seemed to be influenced by a developmental factor: the study found a significant interaction between condition and grade level in relation to learning outcomes, suggesting that the outcomes generally improved as a function of grade level, but that there were notable differences between the conditions regarding the improvement of outcomes across the three grades. According the results, learning with constantly concrete representations either took less time or resulted in better learning compared to concreteness fading. Because transfer is one of the central arguments for concreteness fading, a somewhat surprising finding was that the concrete condition succeeded at least as well as the fading condition on transfer tasks. The study also discusses why the results and issues related to the conceptualisation and operationalisation of central concepts in the study call for caution towards generalization and for more research with young learners across different grades.
... More recent work on concreteness fading has not been tied directly to Bruner's three stages; rather, it has been used more broadly as a term that encompasses any progression from concrete to abstract, without requiring the initial form to be a physical object, without requiring three distinct stages, and without requiring mastery at each stage (e.g. Braithwaite and Goldstone 2013;Ding and Li 2014;Johnson, Reisslein, and Reisslein 2014). For example, Goldstone and Son (2005, 70) define concreteness fading as "the process of successively decreasing the concreteness of a simulation with the intent of eventually attaining a relatively idealised and decontextualised representation that is still clearly connected to the physical situation that it models. ...
To promote learning and transfer of abstract ideas, contemporary theories advocate that teachers and learners make explicit connections between concrete representations and the abstract ideas they are intended to represent. Concreteness fading is a theory of instruction that offers a solution for making these connections. As originally conceived, it is a three-step progression that begins with enacting a physical instantiation of a concept, moves to an iconic depiction and then fades to the more abstract representation of the same concept. The goals of this paper are: (1) to improve the theoretical framework of concreteness fading by defining and bringing greater clarity to the terms abstract, concrete and fading; and (2) to describe several testable hypotheses that stem from concreteness fading as a theory of instruction. Making this theory of instruction more “concrete” should lead to an optimised concreteness fading technique with greater promise for facilitating both learning and transfer.
