Mitchell J. Nathan

Improving Students’ Learning With Effective Learning Techniques Promising Directions From Cognitive and Educational Psychology

Many students are being left behind by an educational system that some people believe is in crisis. Improving educational outcomes will require efforts on many fronts, but a central premise of this monograph is that one part of a solution involves helping students to better regulate their learning through the use of effective learning techniques. Fortunately, cognitive and educational psychologists have been developing and evaluating easy-to-use learning techniques that could help students achieve their learning goals. In this monograph, we discuss 10 learning techniques in detail and offer recommendations about their relative utility. We selected techniques that were expected to be relatively easy to use and hence could be adopted by many students. Also, some techniques (e.g., highlighting and rereading) were selected because students report relying heavily on them, which makes it especially important to examine how well they work. The techniques include elaborative interrogation, self-explanation, summarization, highlighting (or underlining), the keyword mnemonic, imagery use for text learning, rereading, practice testing, distributed practice, and interleaved practice. To offer recommendations about the relative utility of these techniques, we evaluated whether their benefits generalize across four categories of variables: learning conditions, student characteristics, materials, and criterion tasks. Learning conditions include aspects of the learning environment in which the technique is implemented, such as whether a student studies alone or with a group. Student characteristics include variables such as age, ability, and level of prior knowledge. Materials vary from simple concepts to mathematical problems to complicated science texts. Criterion tasks include different outcome measures that are relevant to student achievement, such as those tapping memory, problem solving, and comprehension. We attempted to provide thorough reviews for each technique, so this monograph is rather lengthy. However, we also wrote the monograph in a modular fashion, so it is easy to use. In particular, each review is divided into the following sections: General description of the technique and why it should work How general are the effects of this technique?  2a. Learning conditions  2b. Student characteristics  2c. Materials  2d. Criterion tasks Effects in representative educational contexts Issues for implementation Overall assessment The review for each technique can be read independently of the others, and particular variables of interest can be easily compared across techniques. To foreshadow our final recommendations, the techniques vary widely with respect to their generalizability and promise for improving student learning. Practice testing and distributed practice received high utility assessments because they benefit learners of different ages and abilities and have been shown to boost students’ performance across many criterion tasks and even in educational contexts. Elaborative interrogation, self-explanation, and interleaved practice received moderate utility assessments. The benefits of these techniques do generalize across some variables, yet despite their promise, they fell short of a high utility assessment because the evidence for their efficacy is limited. For instance, elaborative interrogation and self-explanation have not been adequately evaluated in educational contexts, and the benefits of interleaving have just begun to be systematically explored, so the ultimate effectiveness of these techniques is currently unknown. Nevertheless, the techniques that received moderate-utility ratings show enough promise for us to recommend their use in appropriate situations, which we describe in detail within the review of each technique. Five techniques received a low utility assessment: summarization, highlighting, the keyword mnemonic, imagery use for text learning, and rereading. These techniques were rated as low utility for numerous reasons. Summarization and imagery use for text learning have been shown to help some students on some criterion tasks, yet the conditions under which these techniques produce benefits are limited, and much research is still needed to fully explore their overall effectiveness. The keyword mnemonic is difficult to implement in some contexts, and it appears to benefit students for a limited number of materials and for short retention intervals. Most students report rereading and highlighting, yet these techniques do not consistently boost students’ performance, so other techniques should be used in their place (e.g., practice testing instead of rereading). Our hope is that this monograph will foster improvements in student learning, not only by showcasing which learning techniques are likely to have the most generalizable effects but also by encouraging researchers to continue investigating the most promising techniques. Accordingly, in our closing remarks, we discuss some issues for how these techniques could be implemented by teachers and students, and we highlight directions for future research.

A Study of Whole Classroom Mathematical Discourse and Teacher Change

This article presents a comparison of the first 2 years of an experienced middle school mathematics teachers efforts to change her classroom practice as a result of an intervening professional development program. The teachers intention was for her teaching to better reflect her vision of reform-based mathematics instruction. We compared events from the 1st and 2nd years whole class discussions within a multilevel framework that considered the flow of information and the nature of peer- and teacher-directed scaffolding. Discourse analyses of classroom videos served both as an analytic tool for our study of whole classroom interactions, as well as a resource for promoting discussion and reflection during professional development meetings. The results show that there was little change in the teachers specific goals and beliefs in light of a self-evaluation of her Year 1 practices, but substantial changes in how she set out to enact those goals. In Year 2, the teacher maintained a central, social scaffolding role, but removed herself as the analytic center to invite greater student participation. Consequently, student-led discussion increased manifold, but lacked the mathematical precision offered previously by the teacher. The analyses lead to insights about how classroom interactions can be shaped by a teachers beliefs and interpretations of educational reform recommendations.

Teachers' and Researchers' Beliefs About the Development of Algebraic Reasoning

Mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students. Predictions deviated systematically from algebra students’ performances but closely matched a view implicit in textbooks. Analysis of students’ problem-solving strategies indicates specific ways that students’ algebraic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Model of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede algebraic skills, was contrasted with the Verbal Precedence Model of development, which provided a better quantitative fit of students’ performance data. Implications of the findings for student and teacher cognition and for algebra instruction are discussed.

Expert Blind Spot Among Preservice Teachers

This study (N = 48) examined the relationship between preservice secondary teachers’ subject-matter expertise in mathematics and their judgments of students’ algebra problem-solving difficulty. As predicted by the “expert blind spot” hypothesis, participants with more advanced mathematics education, regardless of their program affiliation or teaching plans, were more likely to view symbolic reasoning and mastery of equations as a necessary prerequisite for word equations and story problem solving. This view is in contrast with students’ actual performance patterns. An examination across several subject areas, including mathematics, science, and language arts, suggests a common pattern. This article considers how teachers’ developmental views may influence classroom practice and professional development, and calls into question policies that seek to streamline the licensure process of new teachers on the basis of their subject-matter expertise.

Knowledge and Situational Feedback in a Learning Environment for Algebra Story Problem Solving

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Rethinking Formalisms in Formal Education

I explore a belief about learning and teaching that is commonly held in education and society at large that nonetheless is deeply flawed. The belief asserts that mastery of formalisms—specialized representations such as symbolic equations and diagrams with no inherent meaning except that which is established by convention—is prerequisite to applied knowledge. A formalisms first (FF) view of learning, rooted in Western dualist philosophy, incorrectly advocates the introduction of formalisms too early in the development of learners’ conceptual understanding and can encourage a formalisms-only mind-set toward learning and instruction. I identify the prevalence of FF in curriculum and instruction and outline some of the serious problems engendered by FF approaches. I then turn to promising alternatives that support progressive formalization, problem-based learning, and inquiry learning, which capitalize on the strengths of formalisms but avoid some of the most costly problems found in FF approaches.

To Disagree, We Must Also Agree: How Intersubjectivity Structures and Perpetuates Discourse in a Mathematics Classroom.

Learning in a socially mediated context like a classroom places emphasis on the ability of learners to communicate their ideas to others, and for members of a class to achieve shared meaning or intersubjectivity (IS). We take a participatory view of IS, where both consensual agreement and disagreement are regarded as aspects of a common set of processes that mediate collective activity. Interlocutors need not demonstrate convergence toward a common idea or solution to exhibit IS and, indeed, they appear to need a shared understanding to express substantive disagreement through divergent views. Multilevel, multimodal analyses of videotape of a middle school mathematics classroom, including speech, gestures, drawing, and object use, reveal a discourse that is organized into recurrent sequences of event triads. The dynamics toward and away from convergent ideas appears to be instrumental in fostering sustained and engaging discourse and influencing the representations that students propose during problem solving. Participants frequently exhibited IS, but, as allowed for in the participatory view, the interactions did not seem to convert many students from their initial interpretations. Instead, disagreements and a desire to establish common understanding appeared to lead participants to express their divergent views in more refined and accessible ways. Advancement of our understanding of the role that IS serves in socially mediated learning has the potential to inform both educational theory and emerging areas in embodied cognition and cognitive neuroscience that addresses imitation and empathy, and thus help to bridge research between brain function and social cognition.

How Teachers Link Ideas in Mathematics Instruction Using Speech and Gesture: A Corpus Analysis

This research investigated how teachers express links between ideas in speech, gestures, and other modalities in middle school mathematics instruction. We videotaped 18 lessons (3 from each of 6 teachers), and within each, we identified linking episodes—segments of discourse in which the teacher connected mathematical ideas. For each link, we identified the modalities teachers used to express linked ideas and coded whether the content was new or review. Teachers communicated most links multimodally, typically using speech and gestures. Teachers’ gestures included depictive gestures that simulated actions and perceptual states, and pointing gestures that grounded utterances in the physical environment. Compared to links about new material, teachers were less likely to express links about review material multimodally, especially when that material had been mentioned previously. Moreover, teachers gestured at a higher rate in links about new material. Gestures are an integral part of teachers’ communication during mathematics instruction.

Conducting Research in Schools: A Practical Guide

Cognitive development unfolds in many contexts, and one of the most important of these contexts is school. Thus, understanding the school context is critical for understanding development. This article discusses some of the reasons why cognitive developmental researchers might wish to conduct research in schools, describes how to get started conducting research in schools, and offers advice to help make school-based research proceed more smoothly.

A Framework for Understanding and Cultivating the Transition from Arithmetic to Algebraic Reasoning

As organizers of this special issue and investigators in the Supporting the Transition from Arithmetic to Algebraic Reasoning (STAAR) project, we both felt we would be remiss if we did not acknowledge the singular importance of Jim Kaput’s influence on this body of work. From the earliest meetings of the project team, Jim’s deep thinking about the nature of algebra and of algebraic reasoning and instruction served as a guide. Jim contributed substantially to each of the tiers. But, more significantly, Jim appreciated the need and challenge to address each of these areas of algebra education research in a systemic manner. He agreed unflinchingly when asked to join our advisory board, and the project benefited immensely from his writings and from various discussions on the work. It is a patent understatement that his early death is a loss for the mathematics education community in general and for the community of algebra researchers in particular. Yet we are proud to say his ideas and contributions will continue in projects such as this. Therefore, it is only fitting that we dedicate this set of papers to Professor James J. Kaput. Algebraic reasoning stands as a formidable gatekeeper for students in their efforts to progress in mathematics and science, and to obtain economic opportunities (Ladson-Billings, 1998; RAND, 2003). Currently, mathematics education reMATHEMATICAL THINKING AND LEARNING, 9(3), 179–192 Copyright