Concreteness Fading Research Articles

From Tiles to Worksheet: Exploring Concreteness Fading in Learning Vector Addition

From Tiles to Worksheet: Exploring Concreteness Fading in Learning Vector Addition

Promoting teaching improvement in elementary mathematics: A case with inverse relations

In this study, we examine two U.S. elementary teachers’ use of worked examples, representations, and deep questions in elementary mathematics lessons both before and after a project intervention designed to promote teaching improvement. This intervention was guided by a cognitive construct containing these three aspects, which were demonstrated via cross-cultural video lessons of both U.S. and Chinese teachers. We specifically analyze two teachers (Ann and Bea) who taught first-grade lessons on the topic of inverse relations. Although their lessons were rated similarly in year 1, the rating of their lessons in year 4, post-intervention, demonstrated different patterns of change. While Ann's use of worked examples and representations underwent no significant improvement, Bea spent more time unpacking a single worked example with a representational sequence fading from concrete to abstract (also called “concreteness fading”). Neither teacher showed improvement in asking deep questions. An analysis of the teachers’ responses to the project intervention revealed that their video noticing skills and personal takeaways were related to the areas of teaching where they improved. Other factors such as teachers’ prior knowledge, beliefs, and textbook resources, also played a role in teaching changes. These findings shed light on a recent endeavor called the “science of improvement” in mathematics teaching, which seeks to document teaching changes and the underlying mechanisms that might induce these changes. Implications for both research and practice are discussed.

Cultural Symbols Didactize Concreteness Fading in Basic Multiplication

Cultural Symbols Didactize Concreteness Fading in Basic Multiplication

Learning with multiple external representations in physics: Concreteness fading versus simultaneous presentation

AbstractMultiple external representations (MERs) are useful for teaching complex content in science education. An open question is whether there is an especially effective way to sequence MERs. On the one hand, the so‐called concreteness fading approach suggests starting instruction with more concrete representations and proceeding stepwise to more idealized representations. The effectiveness of this fading approach is, however, supported mainly by studies in mathematics education, while the results in physics are equivocal. On the other hand, presenting different representations simultaneously may support linking, that is, the comparison and contrast of representations, which may benefit learning. In an experimental classroom study (N = 187), we compared concreteness fading and simultaneous presentation of MERs for learning a challenging physics content in high school, namely, Faraday's law. We found no significant differences between conditions in posttest performance, and an equivalence test with bounds d = −0.5 to 0.5 showed that both approaches performed equally. The results align with previous findings questioning the superiority of concreteness fading over other ways of sequencing MERs. Therefore, facilitating students' understanding of a complex physics content may involve more than determining the optimal order of presenting MERs. We discuss limitations of the present study and implications for future research and practice.

Generalizing actions with the subtraction-compensation property: primary students’ algebraic thinking with tasks involving vertical towers of blocks

An important approach for developing children’s algebraic thinking involves introducing them to generalized arithmetic at the time they are learning arithmetic. Our aim in this study was to investigate children’s attention to and expression of generality with the subtraction-compensation property, as evidence of a type of algebraic thinking known as relational thinking. The tasks involved subtraction modelled as difference and comparing the heights of towers of blocks. In an exploratory qualitative study, 22 middle primary (9–11-year-old) students from two schools participated in individual videoed interviews. The tasks were designed using theoretical perspectives on embodied visualization and concreteness fading to provide multiple opportunities for the students to make sense of subtraction as difference and to advance their relational thinking. Twelve out of 22 students evidenced conceptual understanding of the comparison model of subtraction (subtraction as difference) and expression of the compensation property of equality. Four of these students repeatedly evidenced relational thinking for true/false tasks and open equivalence tasks. A proposed framework for levels of attention to/expression of generality with the subtraction-compensation property is shared and suggestions for further research are presented.

Using RPG-Based Learning Environment to Increase Engagement and Motivation for Learning Higher Mathematics

Although game-based learning has been introduced in various fields, there appears to be a lack of engaging computer games that incorporate higher maths learning (Anagnostopoulou, 2021). This paper presents an exploration of how fantasy-based narrative and storytelling, through role-playing and game mechanics, can be used to increase engagement and motivation for learning higher mathematics in a role-playing game (RPG) based learning environment. As part of this research, the project developed an RPG with incorporated mathematical techniques. Players/learners access the fictional world, controlling in-game characters. Learning evolves via a storyline and six mechanics, progressively building knowledge of derivatives using puzzles with shapes, which gradually fade to the abstract notion of mathematical functions. An experiment was conducted to test the research hypothesis that an RPG based around a quest structure with concreteness fading can enable students to get a better understanding of higher mathematics in the area of differentiation without impeding their entertainment. Participants were recruited from various backgrounds and levels of mathematical knowledge and gaming experience. Data was collected through observations, interviews and two questionnaires: one to be completed before playing the game and the other after. This paper is mainly focused on the qualitative outcomes of the research. The results suggest that the method is effective, with serious learning taking place. The participants reported that they had found the game to be fun and a useful way to learn maths. Moreover, it is revealed that a PRG-based learning environment can increase engagement and motivation for learning higher maths, without invalidating entertainment. The findings have implications for the design, implementation, and evaluation of game-based learning environments for higher mathematics. The paper concludes that this approach has the potential to provide a more immersive, engaging learning experience and increase motivation for learning. Finally, recommendations on how to improve the current game as well as more bold and optimistic ideas for enhanced immersive game builds are suggested.

Transdisciplinary embodied education in elementary school: a real integrative approach for the science, technology, engineering, arts, and mathematics teaching

IntroductionConnecting academic disciplines and integrating knowledge is gaining popularity in elementary school. The relevant question is, how the targeted integration could be achieved? This research aimed to (a) evaluate the potential of teaching and learning Dynamic Systems Theory (DST) and Statistical Physics (ST) general concepts through embodied experiences in elementary school and, based on it (b) enable a far transfer analogical reasoning within and between different academic subjects.MethodsForty-eight elementary school students, aged 10.2 ± 0.82 y.o., followed an 8-week, 2-h/week intervention. The learning protocol contained a concreteness fading sequence of phases and consisted of four steps: (a) embodied experience, (b) reflective observation, (c) abstract conceptualization of DST/ST concepts, and (d) transfer of the DST/SP concepts to physical, sociological, biological and ecological phenomena. A validated questionnaire and an interview evaluated students’ knowledge and analogical reasoning.ResultsThe Wilcoxon Signed Ranks Test showed a general positive effect of the intervention on the understanding of DSP/SP concepts and on the far transfer competencies of students (Z = −5.98; p < 0.0001). There was no association between the previous and newly acquired competencies (Spearman’s ρ = 0.112; p = 0.441).DiscusionThe qualitative results showed that, in general, the suggested embodied learning protocol supports the acquisition of DST/SP general concepts and the far transfer competencies. However, students of this age are possibly highly sensitive to the balance of the focus between the embodied phases and the conceptualization/transfer phases of the learning protocol. The study points to the potential of learning the general DST/SP for elementary school students’ integrative and far transfer competencies. The DST/SP concept-based transdisciplinary embodied education may offer a truly integrative approach to STEAM teaching.

Teaching thermodynamics with augmented interaction and learning analytics

Learning thermodynamics concepts remains challenging to students through grades K-12. In this study, we presented a design-based research program that aims to design and examine an innovative application, Infrared Explorer, that organically integrates physical experiments with a virtual environment. Relying on the theory of concreteness fading, we use infrared imaging and multimodal analytics to augment students' interaction with physical materials. Through two iterations of design and testing, we analyzed a variety of data sources and found that students actively used the application during their physical experimentation process. This technological application can allow students to highlight salient information and remove confusing details such that students’ conceptual understanding of thermal concepts can develop further. Through this iterative development process, we also synthesized factors (e.g., multitasking and cognitive load) to consider when designing and implementing technologies to blend physical and virtual environments supporting learning. This research showed the promise of using concreteness fading to design science learning experiences through innovative technologies and identified important factors to consider during the design and implementation phases.

Do concreteness fading and guidance fading aid learning from perceptually rich visualizations? Changes in style lead to more cognitive load and interfere with learning

Do concreteness fading and guidance fading aid learning from perceptually rich visualizations? Changes in style lead to more cognitive load and interfere with learning

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.

Concreteness fading in learning secondary school physics concepts

Concreteness fading has been proposed as a general instructional approach to support learning of abstract mathematics and science concepts. Accordingly, organizing external knowledge representations in a three-step concrete-to-idealized sequence should be more beneficial than the reverse, concreteness introduction, sequence. So far, evidence for the benefits of concreteness fading come mainly from studies investigating learning of basic mathematics concepts. Studies on learning natural science concepts are scarce and have not implemented the full three-step-sequence. In an experimental classroom study (N = 70), we compared concreteness fading and concreteness introduction in high school science education about electromagnetic induction using a detailed assessment. Furthermore, we explored whether these sequences differentially affect the use of the different representations during instruction. Both sequences were equally effective and there were no differences in using the representations. We discuss why our results question the proposed advantages of concreteness fading and highlight conceptual differences and learning goals across domains.

Student Development in Logical Reasoning: Results of an Intervention Guiding Students Through Different Modes of Visual and Formal Representation

Due to growing interest in twenty-first-century skills, and critical thinking as a key element, logical reasoning is gaining increasing attention in mathematics curricula in secondary education. In this study, we report on an analysis of video recordings of student discussions in one class of seven students who were taught with a specially designed course in logical reasoning for non-science students (12th graders). During the course of 10 lessons, students worked on a diversity of logical reasoning tasks: both closed tasks where all premises were provided and everyday reasoning tasks with implicit premises. The structure of the course focused on linking different modes of representation (enactive, iconic, and symbolic), based on the model of concreteness fading (Fyfe et al., 2014). Results show that students easily link concrete situations to certain iconic referents, such as formal (letter) symbols, but need more practice for others, such as Venn and Euler diagrams. We also show that the link with the symbolic mode, i.e. an interpretation with more general and abstract models, is not that strong. This might be due to the limited time spent on further practice. However, in the transition from concrete to symbolic via the iconic mode, students may take a step back to a visual representation, which shows that working on such links is useful for all students. Overall, we conclude that the model of concreteness fading can support education in logical reasoning. One recommendation is to devote sufficient time to establishing links between different types of referents and representations.

Improving Understanding of Mathematical Equivalence

Modify arithmetic problem formats to make the relational equation structure more transparent. We describe this practice and three additional evidence-based practices: (1) introducing the equal sign outside of arithmetic, (2) concreteness fading activities, and (3) comparing and explaining different problem formats and problem-solving strategies.

Review of the features of augmented reality application in the training of operators and maintenance staff

An overview of the use of augmented reality technology used to improve the training efficiency of operators and service personnel in the industry is presented. When considering the issues of building augmented reality systems, it was noted that in addition to technological issues, it is also essential to pay increased attention to the social aspect, especially if it is intended to apply augmented reality technology in multiplayer mode. A common environment in the interaction of several participants can disrupt the sense of identity of users, as well as potentially threaten the feeling of personal space and the sensation of physical objects. It is necessary to take into account the cognitive load theory and the schematic theoretical model of concreteness fading. The advantage of using augmented reality is to reduce human errors and improve the quality of training. For the use of augmented reality in the absence of a reliable network connection and the availability of only short communication sessions, the use of special tickets is proposed.

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.

Learning electric circuit principles in a simulation environment with a single representation versus “concreteness fading” through multiple representations

This study explored how learning in a simulation environment with a single representation compares to learning with concreteness fading in the domain of electric circuits in secondary school. Concreteness fading, an instructional technique where representations change from concrete to more abstract during a learning sequence, has been proposed as an approach to combine the strengths of each representation and to promote transfer, but the empirical evidence is not unequivocal. One reason may be that defining “concrete” and “abstract” for the purpose of concreteness fading is less straightforward in practice than in theory, as illustrated in the conceptualization of concreteness fading in this study. The results showed no differences in content coverage and total learning time between conditions, but the transition between representations temporarily increased learning time in the fading condition. An analysis of the learning revealed no differences between conditions on circuits similar in appearance to the simulation environment, but the single representation resulted in better application of the learned knowledge to realistic circuits. The discussion explores the results from the perspective of concreteness fading and alternative conceptualizations of the learning environment. One of these, resulting from the difficulty in positioning computer simulations within concrete–abstract dimensions, is that computer simulations can include concrete and abstract in a single representation, which may be easier (and more effective) in classroom settings than multiple representations. This suggests that a well-designed simulation can be a reckoned alternative to concreteness fading and to multiple representations in general in science education.

Improving children’s understanding of mathematical equivalence via an intervention that goes beyond nontraditional arithmetic practice.

Elementary school children (ages 7–11) struggle to understand mathematical equivalence, a foundational prealgebraic concept. Some manipulations to the learning environment, including well-structured nontraditional arithmetic practice alone, have been shown to improve children’s understanding; however, improvements have been modest. The goal of this study was to test an iteratively developed supplemental intervention for second grade that was designed to yield widespread mastery of mathematical equivalence. The intervention included three components beyond nontraditional arithmetic practice: (a) lessons that introduce the equal sign outside of arithmetic contexts, (b) “concreteness fading” exercises, and (c) activities that require children to compare and explain different problem formats and problem-solving strategies. After the development process, a small, randomized experiment was conducted with 142 students across eight second grade classrooms to evaluate the effects of the intervention on children’s solving of mathematical equivalence problems, encoding of mathematical equivalence problems, and defining of the equal sign. Classrooms were randomly assigned to the intervention or to nontraditional arithmetic practice alone. Analyses at the classroom level demonstrated that the intervention classrooms performed better than the active control classrooms both in terms of pre-to-post change in understanding of mathematical equivalence (g = 1.87) and accuracy on transfer problems at posttest (g = 2.07). Nonparametric analyses led to the same conclusions. Results suggest that the comprehensive intervention improves children’s understanding of mathematical equivalence to levels that surpass equivalent levels of well-structured arithmetic practice alone, as well as business-as-usual benchmarks from previous studies. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

Concreteness fading process of elementary school students based on mathematical ability

First grade students generally have difficulty in converting concrete to abstract thinking. This qualitative research that is concerned with concreteness fading process of students. The study aimed at describing the concreteness fading process of 1st grade elementary school students. In order to find out the concreteness fading process students can be seen based on enactive, iconic, and symbolic stages are known through learning, tasks and task-based interviews. The results generally show that students are still having difficulty solving mathematical equivalence problems in a symbolic stage objects. In general, first year students still have difficulty. Difficulties experienced that solve problems related to the sum of the natural numbers more precisely in a problems of mathematical equality.

Transition from Textbook to Classroom Instruction in Mathematics: The Case of an Expert Chinese Teacher

This study reports how an expert Chinese teacher implements mathematics textbook lessons in enacted instruction. Our video analysis indicates that both textbook and enacted teaching included only one worked example; however, the teacher engaged students in unpacking the example in great depth. Both the textbook and the enacted teaching showed “concreteness fading” in students’ use of representations. However, the Chinese teacher incorporated students’ self-generated representations and facilitated students’ active modeling of quantitative relationships. Finally, the Chinese teacher asked a greater number of deep questions than were suggested by the textbook. These deep questions often occurred as clusters of follow-up questions that were either concept-specific or promoted comparisons which facilitated connection-making between multiple representations and solutions.

Concreteness fading fosters children's understanding of the inversion concept in addition and subtraction

This study examined the effectiveness of various instructional strategies that aimed to enhance children's understanding of the inversion concept. One hundred and forty kindergartners were randomly assigned to each of the groups namely: (a) concrete-only, (b) abstract-only, (c) concreteness fading, (d) abstract-to-concrete, (e) control. They participated in a pre-test, two training sessions, an immediate post-test, and an 8-week delayed post-test. All the intervention groups showed significantly greater progress than the control group in solving the inversion problems in the post-tests. Concrete representations were more effective than abstract representations to promote knowledge transfer for children with lower prior knowledge. The superior benefits of concreteness fading appeared more prominent in the delayed post-test for children with lower prior knowledge. This study suggests that (1) concrete representations should not be avoided in teaching mathematics to children and (2) the order of presentation for various representations is a key for successful learning.