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The Influence of Perceptually Rich Manipulatives and Collaboration on Mathematic Problem- Solving and Perseverance

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Abstract

We conducted a two-part study to examine how the use of different manipulatives, levels of instructional guidance, and collaboration among college-aged students influenced their mathematics performance and perseverance. In Study 1, we manipulated different types of manipulatives (no manipulative, bland, and perceptually rich), and different contextual factors of instructional guidance (high vs. low) to identify how they influenced students’ ability to complete mathematical problems as well as impact students’ perseverance. Findings showed that participants’ use of bland manipulatives positively impacts their ability to complete word problems. Study 2, grounded in findings from Study 1, incorporated collaborative learning to lessen the negative effects of using perceptually rich manipulatives and enhance students’ perseverance in mathematics. The results from Study 2 aligned with the findings from Study 1 concerning the negative effect of perceptual richness for problem-solving and perseverance. Moreover, the collaboration of participants had a positive effect on students’ perseverance during problem-solving as students who collaborated with perceptually rich manipulatives persevered as much as those who collaborated with bland manipulatives as well as those who worked individually with bland manipulatives.

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This chapter summarizes research and theory concerned with the effects of learner expertise (prior knowledge) on multimedia learning principles. The expertise reversal principle is that, in many situations, design principles that are effective for novice learners may not be effective or even hinder learning for more knowledgeable learners. The main theoretical issue associated with this principle concerns the integration in working memory of instructional information with knowledge structures held in long-term memory. The major instructional implication is the need to tailor instructional formats and procedures to changing levels of expertise. Essential research directions include identifying instructional procedures that are optimal for learners with different levels of expertise, investigating appropriate means for balancing the degree of instructional guidance provided to learners, and developing viable diagnostic instruments for the real-time evaluation of levels of learner expertise to be used in adaptive multimedia learning.
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Multimedia learning environments combine multiple sources of information (e.g., text, diagrams, and simulations) to help students master cognitively challenging domains. However, in order to benefit from these environments, students need to make connections among the sources of information. One strategy for encouraging students to think deeply about and cognitively engage with the learning material is prompted self-explanation. Self-explanation is a constructive or generative learning activity that facilitates deep and robust learning by encouraging students to make inferences using the learning materials, identify previously held misconceptions, and repair mental models. In this chapter, we present a framework for categorizing the many forms of prompted self-explanation and highlight ways that self-explanation has been successfully incorporated into multimedia learning environments to improve student learning. In addition, we discuss specific forms of self-explanation that may be particularly well suited for multimedia learning environments. We end with a discussion of implications for cognitive theory and instructional design and ideas for future work.
Article
The purpose of this study was to investigate the effect of different types of worked examples on student learning and transfer of a problem-solving task. Four types of worked examples were examined: standard worked examples, worked examples with self-explanation prompts, worked examples with instructional explanations, and worked examples with a combination of instructional explanations and self- explanation prompts. Two hundred and five middle school students were randomly assigned to the treatment conditions and a control condition. All students studied a self-paced instructional program on using two comma rules. Students in each worked example condition received condition-specific example-problem pairs during practice, while students in the control group received problems without any worked examples. Learning, transfer, and time on task were measured. Analysis of the data indicated that using instructional explanations and self-explanation prompts with worked examples had a positive effect on learning.
Article
The worked example effect indicates that examples providing full guidance on how to solve a problem result in better test performance than a problem-solving condition with no guidance. The generation effect occurs when learners generating responses demonstrate better test performance than learners in a presentation condition that provides an answer. This contradiction may be resolved by the suggestion that the worked example effect occurs for complex, high element interactivity materials that impose a heavy working memory load whereas the generation effect is applicable for low element interactivity materials. Two experiments tested this hypothesis in the area of geometry instruction using students with different levels of prior knowledge in geometry. The results of Experiment 1 indicated a worked example effect obtained for materials high in element interactivity and a generation effect for materials low in element interactivity. As levels of expertise increased in Experiment 2, thus reducing effective complexity, this interaction was replaced by a generation effect for all materials. These results suggest that when students need to learn low element interactivity material, learning will be enhanced if they generate rather than study responses but if students need to learn high element interactivity material, study may be preferable to generating responses.
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Data from a study of a learning community program in an urban community college are used to explore the educational character of student persistence. Analyses reveal that classroom activities influence student persistence by changing the way students and faculty interact within and beyond the classroom setting. Implications for current theories of persistence are discussed and a modified theory proposed.
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Five studies examined how interacting with the physical environment can support the development of fraction concepts. Nine- and 10-year-old children worked on fraction problems they could not complete mentally. Experiments 1 and 2 showed that manipulating physical pieces facilitated children's ability to develop an interpretation of fractions. Experiment 3 demonstrated that when children understood a content area well, they used their interpretations to repurpose many environments to support problem solving, whereas when they needed to learn, they were prone to the structure of the environment. Experiments 4 and 5 examined transfer after children had learned by manipulating physical pieces. Children who learned by adapting relatively unstructured environments transferred to new materials better than children who learned with "well-structured" environments that did not require equivalent adaptation. Together, the findings reveal that during physically distributed learning, the opportunity to adapt an environment permits the development of new interpretations that can advance learning.
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Learners gain deep understanding in multimedia-based learning environments when they receive worked examples in initial cognitive skill acquisition. This is at least true if classic multimedia design principles and other instructional principles are taken into account. This chapter elaborates on the worked examples principle. More specifically, it i rst illustrates the tight connections between multimedia learning and worked examples by several exemplary cases before reporting about i ndings on the effectiveness of worked examples in multimedia learning. The effectiveness of worked examples is then theoretically explained. Furthermore, instructional principles are derived from i ndings on factors that moderate worked example effects. Finally, important questions to be addressed in further research (e.g., analyzing long-term effects in classroom learning) are proposed.
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Research in both cognitive and educational psychology has explored the effect of different types of external knowledge representations (e.g., manipulatives, graphical/pictorial representations, texts) on a variety of important outcome measures. We place this large and multifaceted research literature into an organizing framework, classifying three categories of external knowledge representations along a dimension of groundedness: (1) idealized, (2) grounded and including only relevant features, and (3) grounded and including irrelevant features. This organizing framework allows us to focus on the implications of these characteristics of external knowledge representations on three important educational outcomes: learning and immediate performance using the target knowledge, the degree to which that knowledge can transfer flexibly, and the interest engendered by the learning materials. We illustrate the framework by mapping a wide body of research from educational and cognitive psychology onto its dimensions. This framework can aid educators by clearly stating what the research literature says about these characteristics of external knowledge representations and how they activate and support the construction of internal knowledge representations. In particular, it will speak to how to best structure instruction using external knowledge representations with different characteristics, depending on the learning objective. Researchers will benefit from the analysis of the current state of knowledge and by the description of what open questions still remain.
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Prior research on conceptual change has identified multiple kinds of misconceptions at different levels of representational complexity including false beliefs, flawed mental models, and incorrect ontological categories. We hypothesized that conceptual change of a mental model requires change in the system of relations between the features of the prior model. To test this hypothesis, we compared instruction aimed at revising knowledge at the mental model level called holistic confrontation – in which the learner compares and contrasts a diagram of his or her flawed mental model to an expert model – to instruction aimed at revising knowledge at the false belief level – in which the learner is prompted to self-explain the expert model alone. We found evidence that participants who engaged in holistic confrontation were more likely to acquire a correct mental model, and a deeper understanding of the systems of relations in the model than those who were prompted to self-explain the expert model. The results are discussed in terms of their implications for science instruction.
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This study compared the effects of worked example and problem‐solving approaches in individual or group work settings on learning to solve geometry problems. One hundred and one seventh graders from Indonesia were randomly allocated to four experimental groups using a 2 (problem‐solving vs. worked examples) × 2 (individual vs. group study) design. Performance measures on numeric and reasoning abilities using both similar and transfer tasks were collected. The results indicated a significant superiority of the worked example approach in both the individual and group work settings. Supporting data revealed that students could understand the material more easily using worked examples than when solving problems. The experiment provided evidence that the advantage of using worked examples over solving problems extends to a group work context.
Article
Educators often use concrete objects to help children understand mathematics concepts. However, findings on the effectiveness of concrete objects are mixed. The present study examined how two factors-perceptual richness and established knowledge of the objects-combine to influence children's counting performance. In two experiments, preschoolers (N = 133; M(age) = 3;10) were randomly assigned to counting tasks that used one of four types of objects in a 2 (perceptual richness: high or low) × 2 (established knowledge: high or low) factorial design. Findings suggest that perceptually rich objects facilitate children's performance when children have low knowledge of the objects but hinder performance when children have high knowledge of the objects.
Article
A large body of research has shown that for novice learners, instruction that relies more heavily on worked examples than on problem solving, is more effective for learning as shown by higher test performance. Moreover, this beneficial effect is often obtained with less acquisition time and lower cognitive load during acquisition and test phase. However, most of this research has been conducted in laboratory settings with college or university students and a control condition consisting of problem solving without any additional support. The present study, using a quasi-experimental design, investigated the effects of implementing worked examples in an existing primary school mathematics curriculum in which a realistic mathematics teaching method is used, during a 3-week period. The results showed no significant differences in test performance or cognitive load; however, the worked examples group attained this level of performance with significantly less acquisition time.
Article
The use of manipulatives in the classroom has been advocated for decades. However, the theoretical and empirical support for this practice is mixed. Some researchers suggest that manipulatives facilitate learning by (a) providing an additional channel for conveying information, (b) activating real-world knowledge, and/or (c) improving memory through physical action. However, there are at least two reasons to question the efficacy of manipulative use. First, manipulatives might lead students to focus on having fun at the expense of deep learning. Second, manipulatives might make learning more difficult because they require dual representation. Although these two criticisms are disparate in terms of their underlying rationale, both converge on the idea that teachers should reduce their use of manipulatives that are highly familiar and/or perceptually interesting. More generally, the manipulatives debate highlights the need for teachers and researchers to work together to evaluate the costs and benefits of various classroom practices.
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Dimensional analysis is traditionally one of the first topics covered in a general chemistry course. Chemists use dimensional analysis as a tool to keep track of units and guide them through calculations. Although unit conversions are taught in a variety of subjects over several grade levels, many students have not mastered this topic by the time they enter college. To properly equip beginning chemistry students, a collaborative active-learning activity was developed. This article describes the activity and reports data of the effects it had on students’ performance in a first-semester general chemistry course at a large research institution.Keywords: First-Year Undergraduate/General; High School/Introductory Chemistry; Chemical Education Research; Collaborative/Cooperative Learning; Hands-On Learning/Manipulatives; Mathematics/Symbolic Mathematics; Nomenclature/Units/Symbols
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The National Council of Teachers of Mathematics has set ambitious goals for the teaching and learning of mathematics that include preparing students for both the workplace and higher education. While this suggests that it is important for students to develop strong mathematical competencies by the end of high school, there is evidence to indicate that overall this is not the case. Both national and international studies corroborate the concern that, on the whole, US 12th grade students do not demonstrate mathematical proficiency, suggesting that students making the transition from high school to college mathematics may not be ready for its rigors. In order to investigate mathematical readiness of entering college students, this study surveyed mathematics faculty. Specifically, faculty members were asked their perceptions of average entering students' readiness related to relevant mathematical skills and concepts, and the importance of the same skills and concepts as foundations for college mathematics. Results demonstrated that the faculty perceived that average freshman students are generally not mathematically prepared; further, the skills and concepts rated as highly important — namely, algebraic skills and reasoning and generalization — were among those rated the lowest in terms of student competencies.
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Abstract— The use of “concrete manipulatives” in mathematics education is supported by research and often accepted as a sine qua non of “reform” approaches. This article reviews the research on the use of manipulatives and critiques common notions regarding concrete manipulatives. It presents a reformulation of the definition of concrete as used in educational psychology and educational research and provides a rationale of how, based on that reformulation, computer manipulatives may be pedagogically efficacious. The article presents 7 hypothesized, interrelated affordances of manipulatives and briefly reviews evidence for their empirical validity.
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A teaching strategy based on the use of manipulative materials as models and analogs of chemical entities, combined with structured peer interaction, was found to enhance learning of two chemistry concepts for both formal and nonformal operational students. Instruction in the control class was based on lectures, CHEM Study laboratories, and individual work. Both cognitive and affective outcomes were more positive in the experimental class.
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Abstract— A growing body of research suggests that the use of concrete materials is not a sure-fire strategy for helping children succeed in the classroom. Instead, concrete materials can help or hinder learning, depending on a number of different factors. Taken together, the articles in this issue highlight the complexities involved in using concrete materials in the classroom and warn educators and researchers that students’ learning from concrete materials can be derailed in a number of ways, such as (a) choosing the wrong types of materials, (b) structuring the environment in ways that do not support learning from concrete materials, and (c) failing to connect concrete representations to abstract representations. Each of these problems is discussed and some potential solutions are offered.
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Abstract— Mathematical concepts are often difficult to acquire. This difficulty is evidenced by failure of knowledge to transfer to novel analogous situations. One approach to this challenge is to present the learner with a concrete instantiation of the to-be-learned concept. Concrete instantiations communicate more information than their abstract, generic counterparts and, in doing so, they may facilitate initial learning. However, this article argues that extraneous information in concrete instantiations may distract the learner from the relevant mathematical structure and, as a result, hinder transfer. At the same time, generic instantiations, such as traditional mathematical notation, can be learned by both children and adults and can, in turn, allow for transfer, suggesting that generic instantiations result in a portable knowledge representation.
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Abstract— Researchers, teachers, teacher educators, and national organizations espouse the idea that children learn new concepts in concrete contexts and transfer these concepts to abstract situations. Although people can benefit from working with hands-on objects and do tend to solve problems more abstractly as they gain knowledge and experience in a domain, this does not imply that a concrete to abstract shift explains this development. In this article, an alternative theory of physically distributed learning (PDL) is presented. In PDL, learning involves changes in internal and external elements of cognitive systems. These changes occur as the internal and external elements coevolve, or change each other, over time.