Developing a Learning Progression for Curriculum,Instruction, and Student Learning: An Example from Mathematics Education

Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.     

An impoverished machine: Challenges to human learning and instructional technology

Many of the limitations to human learning and processing identified by cognitive psychologists over the last 50 years still hold true, including computational constraints, low learning rates, and unreliable processing. Instructional technology can be used in classrooms and in other learning contexts to address these limitations to learning. However, creating technological innovations is not enough. As part of psychological science, the development and assessment of instructional systems should be guided by theories and practices within the discipline. The technology we develop should become an object of research like other phenomena that are studied. In the present article, I present an informal account of my own work in assessing instructional technology for engineering thermodynamics to show not only the benefits, but also the limitations, in studying the technology we create. I conclude by considering several ways of advancing the development of instructional technology within the SCiP community, including interdisciplinary research and envisioning learning contexts that differ radically from traditional learning focused on lectures and testing.     

Combining Instructional Activities for Sense-Making Processes and Perceptual-Induction Processes Involved in Connection-Making Among Multiple Visual Representations

Abstract

Connection-making among multiple representations is a crucial but difficult competence in STEM learning. Prior research has focused on one type of learning process involved in connection-making: sense-making processes leading to conceptual understanding of connections. Yet, other research suggests that a second type of learning process is important: inductive learning processes leading to perceptual intuitions about connections. We investigate whether combining instructional activities designed to support sense-making processes for understanding of connections (understanding activities) and instructional activities that support inductive processes for perceptual intuitions about connections (perception activities) enhances students’ learning of chemistry knowledge. A laboratory-based experiment with 117 undergraduate students compared students in (a) a control condition that received only conventional activities that did not require connection-making; (b) a condition that received conventional and understanding-activities; (c) a condition that received conventional and perception-activities; and (d) a combined condition that received conventional, understanding-activities, and perception-activities. Results show that only the combined condition outperformed the control condition on a test of chemistry knowledge. Eye-gaze data and verbal reports show that understanding-activities and perception-activities have complementary effects on how students integrate information from multiple representations during the learning phase. Finally, we found that students’ spatial skills moderate their benefit from understanding-activities and perception-activities.     

Constraints on Constraints: Surveying the Epigenetic Landscape

It is now relatively commonplace to advocate the need for some sorts of constraints on learning and knowledge acquisition. The critical issues to cognitive science concern the sorts of constraints that are able to best model various phenomena of learning and development. Four types of constraints on learning are proposed to be used as an interpretative framework within which to: 1. Better understand the nature of current research: 2. Allow the exploration of alternative models of learning related phenomena, and 3. See more clearly needs for further research. Superficially similar learning phenomena can be modeled by very different configurations of underlying constraints with strong implications for the sorts of representational states that are involved. Each of the five papers in this issue (Brown, Gelman, Markman, Newport, and Spelke) is considered in terms of the configuration of constraints after which each author intends to model their phenomena and in terms of alternate configurations. The papers are construed as illustrating a diverse set of models of how constraints might guide learning, and while the evidence generally favors the configurations suggested by the authors, in each case alternative models are possible and motivate quite specific future research questions. More broadly, it is suggested that asking detailed questions about the sorts of Constraints types that could potentially model complex cases of natural knowledge acquisition helps motivate fundamental questions about learning and the nature of knowledge and that the five papers in this issue are superb examples of how adopting this kind of perspective has been fruitful research orientation.     

Thirty years of research on online learning

This paper presents a personal account of developments in research on online learning over the past 30 years. Research on how to design online instruction represents an example of applying the science of learning to education. It contributes to the science of learning (as exemplified by developments in cognitive load theory, the cognitive theory of multimedia learning, and incorporating metacognitive, motivational, and affective aspects of learning), the science of instruction (as exemplified by the continuing development of research‐based principles of instructional design), and the science of assessment (as exemplified by supplementing self‐report surveys and retention tests with multilevel transfer tests, log file data during learning, and cognitive neuroscience measures of cognitive processing during learning). Some recurring themes are that learning is caused by instructional methods rather than instructional media, so research should focus on features that are uniquely afforded by digital learning environments; instructional practice should be grounded in rigorous and systematic research, including value‐added experiments aimed at pinpointing the active ingredients in online instruction; research in online learning should identify boundary conditions under which instructional techniques are most effective; and research in online learning should test and contribute to learning theory.     

Investigating Links from Teacher Knowledge,to Classroom Practice,to Student Learning in the Instructional System of the Middle-School Mathematics Classroom

Using data collected in 125 seventh-grade and 56 eighth-grade Texas classrooms in the context of the “Scaling Up SimCalc” research project in 2005–07, we examined relationships between teachers’ mathematics knowledge, teachers’ classroom decision making, and student achievement outcomes on topics of rate, proportionality, and linear function—three important and cognitively demanding prealgebra topics. We found that teachers’ mathematical knowledge was correlated with student achievement in only one study out of three. We also found a lack of correlations between teachers’ mathematical knowledge and critical aspects of instructional decision making. Curriculum and other learning resources (e.g., technology, student–student interactions) are clearly important factors for student learning in addition to, and in interaction with, teachers’ mathematical knowledge. Our results suggest that mathematics knowledge for teaching may have a nonlinear relationship with student learning, that those effects may be heavily mediated by other instructional factors, and that short-term content knowledge gains in teacher workshops may not persist in classroom instruction. We discuss a need in the field for richer models of how “mathematical knowledge for teaching” works in the context of complete instructional systems.     

An effective metacognitive strategy: learning by doing and explaining with a computer‐based Cognitive Tutor

Recent studies have shown that self‐explanation is an effective metacognitive strategy, but how can it be leveraged to improve students' learning in actual classrooms? How do instructional treatments that emphasizes self‐explanation affect students' learning, as compared to other instructional treatments? We investigated whether self‐explanation can be scaffolded effectively in a classroom environment using a Cognitive Tutor, which is intelligent instructional software that supports guided learning by doing. In two classroom experiments, we found that students who explained their steps during problem‐solving practice with a Cognitive Tutor learned with greater understanding compared to students who did not explain steps. The explainers better explained their solutions steps and were more successful on transfer problems. We interpret these results as follows: By engaging in explanation, students acquired better‐integrated visual and verbal declarative knowledge and acquired less shallow procedural knowledge. The research demonstrates that the benefits of self‐explanation can be achieved in a relatively simple computer‐based approach that scales well for classroom use.     

Spanning seven orders of magnitude: a challenge for cognitive modeling

Much of cognitive psychology focuses on effects measured in tens of milliseconds while significant educational outcomes take tens of hours to achieve. The task of bridging this gap is analyzed in terms of Newell's (1990) bands of cognition—the Biological, Cognitive, Rational, and Social Bands. The 10 millisecond effects reside in his Biological Band while the significant learning outcomes reside in his Social Band. The paper assesses three theses: The Decomposition Thesis claims that learning occurring at the Social Band can be reduced to learning occurring at lower bands. The Relevance Thesis claims that instructional outcomes at the Social Band can be improved by paying attention to cognition at the lower bands. The Modeling Thesis claims that cognitive modeling provides a basis for bridging between events on the small scale and desired outcomes on the large scale. The unit‐task level, at the boundary of the Cognitive and Rational Bands, is useful for assessing these theses. There is good evidence for all three theses in efforts that bridge from the unit‐task level to educational applications. While there is evidence for the Decomposition Thesis all the way down to the 10 millisecond level, more work needs to be done to establish the Relevance Thesis and particularly the Modeling Thesis at the lower levels.     

Eliciting explanations: Constraints on when self-explanation aids learning

Generating explanations for oneself in an attempt to make sense of new information (i.e., self-explanation) is often a powerful learning technique. Despite its general effectiveness, in a growing number of studies, prompting for self-explanation improved some aspects of learning, but reduced learning of other aspects. Drawing on this recent research, as well as on research comparing self-explanation under different conditions, we propose four constraints on the effectiveness of self-explanation. First, self-explanation promotes attention to particular types of information, so it is better suited to promote particular learning outcomes in particular types of domains, such as transfer in domains guided by general principles or heuristics. Second, self-explaining a variety of types of information can improve learning, but explaining one’s own solution methods or choices may reduce learning under certain conditions. Third, explanation prompts focus effort on particular aspects of the to-be-learned material, potentially drawing effort away from other important information. Explanation prompts must be carefully designed to align with target learning outcomes. Fourth, prompted self-explanation often promotes learning better than unguided studying, but alternative instructional techniques may be more effective under some conditions. Attention to these constraints should optimize the effectiveness of self-explanation as an instructional technique in future research and practice.     

Differences in children’s thinking and learning during attentional focus instruction

Considerable evidence supports the motor learning advantage associated with an external focus of attention; however, very few studies have investigated attentional focus effects with children despite individual functional constraints that have the potential to impact use of instructional content. Thus, the purpose of this study was to determine the effect of attentional focus instruction on motor learning in children. Participants (n = 42) aged 9–11 years were randomly assigned to one of three gender-stratified groups: (1) control, (2) internal focus, or (3) external focus. Following initial instructions and task demonstration, participants performed 100 modified free throws over two days while receiving additional cues respective to their attentional focus condition and returned approximately 48 h later to perform 20 additional free throws. Results revealed no significant learning differences between groups. However, responses to retrospective verbal reports suggest that the use of external focus content during practice may have contributed to some participants’ superior performance in retention. Future research should continue to examine attentional focus effects across a variety of ages and incorporate retrospective verbal reports in order to examine children’s thoughts during attentional focus instruction.     

On the cognitive effects of learning computer programming

This paper critically examines current thinking about whether learning computer programming promotes the development of general higher mental functions. We show how the available evidence, and the underlying assumptions about the process of learning to program, fail to address this issue adequately. Our analysis is based on a developmental cognitive science perspective on learning to program, incorporating developmental and cognitive science considerations of the mental activities involved in programming. It highlights the importance for future research of investigating students' interactions with instructional and programming contexts, developmental transformations of their programming skills, and their background knowledge and reasoning abilities.     

Cognitive Evaluation of Machine Learning Agents

Advances in applying statistical Machine Learning (ML) led to several claims of human-level or near-human performance in tasks such as image classification & speech recognition. Such claims are unscientific primarily for two reasons, (1) They incorrectly enforce the notion that task-specific performance can be treated as manifestation of General Intelligence and (2) They are not verifiable as currently there is no set benchmark for measuring human-like cognition in a machine learning agent. Moreover, ML agent’s performance is influenced by knowledge ingested in it by its human designers. Therefore, agent’s performance may not necessarily reflect its true cognition. In this paper, we propose a framework that draws parallels from human cognition to measure machine’s cognition. Human cognitive learning is quite well studied in developmental psychology with frameworks and metrics in place to measure actual learning. To either believe or refute the claims of human-level performance of machine learning agent, we need scientific methodology to measure its cognition. Our framework formalizes incremental implementation of human-like cognitive processes in ML agents with an implicit goal to measure it. The framework offers guiding principles for measuring, (1) Task-specific machine cognition and (2) General machine cognition that spans across tasks. The framework also provides guidelines for building domain-specific task taxonomies to cognitively profile tasks. We demonstrate application of the framework with a case study where two ML agents that perform Vision and NLP tasks are cognitively evaluated.     

A reinforcement learning approach to personalized learning recommendation systems

Personalized learning refers to instruction in which the pace of learning and the instructional approach are optimized for the needs of each learner. With the latest advances in information technology and data science, personalized learning is becoming possible for anyone with a personal computer, supported by a data-driven recommendation system that automatically schedules the learning sequence. The engine of such a recommendation system is a recommendation strategy that, based on data from other learners and the performance of the current learner, recommends suitable learning materials to optimize certain learning outcomes. A powerful engine achieves a balance between making the best possible recommendations based on the current knowledge and exploring new learning trajectories that may potentially pay off. Building such an engine is a challenging task. We formulate this problem within the Markov decision framework and propose a reinforcement learning approach to solving the problem.     

Computer-based systems for applying psychological knowledge

Computer-based systems have long been seen as a means for applying psychological knowledge to instructing. However, from many years of research and development on instructional design and on using computers for instructing, we learned that the current instructional techniques are based on an inadequate scientific knowledge base and are poorly implemented. In order to do better, improved techniques based on modern cognitive science must be developed, and computer-based aids are needed to assist their application. Furthermore, a standardization and distribution system is needed to identify effective instructional programs and make them available. This seems especially important if we hope to have broad impact on education and training.     

Beyond subgoaling: A dynamic knowledge generation framework for creative problem solving in cognitive architectures

In this paper we propose a computational framework aimed at extending the problem solving capabilities of cognitive artificial agents through the introduction of a novel, goal-directed, dynamic knowledge generation mechanism obtained via a non monotonic reasoning procedure. In particular, the proposed framework relies on the assumption that certain classes of problems cannot be solved by simply learning or injecting new external knowledge in the declarative memory of a cognitive artificial agent but, on the other hand, require a mechanism for the automatic and creative re-framing, or re-formulation, of the available knowledge. We show how such mechanism can be obtained trough a framework of dynamic knowledge generation that is able to tackle the problem of commonsense concept combination. In addition, we show how such a framework can be employed in the field of cognitive architectures in order to overcome situations like the impasse in SOAR by extending the possible options of its subgoaling procedures.     

The role of honour in interpersonal,intrapersonal and intergroup processes

In this article, we review research in psychology and other related social science fields that has adopted an honor framework to examine intrapersonal, interpersonal, and intergroup processes taking a culture-comparative or individual differences approach. In the sections below, we will first review research on the role of honor in interpersonal processes focusing primarily on interpersonal aggression including in close relationships, non-aggressive ways of responding to threats (e.g., forgiveness), and reciprocity. Next, we move onto reviewing research on the role of honor in intrapersonal processes, specifically in the domains of emotional responses to honor-threatening situations, mental, and physical health. Finally, we review research emerging from social and political psychology and political science that have utilized the honor framework to understand and explain group processes and intergroup relations at different level of analyses (e.g., social groups, nations). Given the limited space, our goal was to emphasize major and emerging areas of research on honor and provide food for thought for future research.     

Declarative working memory: A bio-inspired cognitive architecture proposal

Memory is considered one of the most important functions since it allows us to code, store and retrieve knowledge. These qualities make it an indispensable function for a virtual creature. In general, memory can be classified based on the durability of the stored data in working memory and long-term memory. Working memory refers to the capacity to maintain temporarily a limited amount of information in mind, which can then be used to support various abilities, including learning, reasoning, planning and decision-making. Unlike short-term memory, working memory is not only a storage site, but it is also a framework of interacting processes that involve the temporary storage and manipulation of information in the service of performing complex cognitive activities. Declarative memory is a type of long-term memory related with the storage of facts and events. This research focuses on the development of a cognitive architecture for the type of working memory that maintains and manipulates declarative information. The construction of the model was grounded in theoretical evidence taken from cognitive sciences such as neuroscience and psychology, which gave us the components and their processes. The model was evaluated through a case study that covers the encoding, storing, and retrieval stages. Our hypothesis is that a virtual creature endowed with our working memory model will provide faster access to the information needed for the ongoing task. Therefore, it improves the planning and decision-making processes.     

Individual differences in the preference for worked examples: Lessons from an application of dispositional learning analytics

Worked-examples have been established as an effective instructional format in problem-solving practices. However, less is known about variations in the use of worked examples across individuals at different stages in their learning process in student-centred learning contexts. This study investigates different profiles of students' learning behaviours based on clustering learning dispositions, prior knowledge, and the choice of feedback strategies in a naturalistic setting. The study was conducted on 1,072 students over an 8-week long introductory mathematics course in a blended instructional format. While practising exercises in a digital learning environment, students can opt for tutored problem solving, untutored problem solving, or call worked examples. The results indicated six distinct profiles of learners regarding their feedback preferences in different learning phases. Finally, we investigated antecedents and consequences of these profiles and investigated the adequacy of used feedback strategies concerning ‘help-abuse’. This research indicates that the use of instructional scaffolds as worked-examples or hints and the efficiency of that use differs from student to student, making the attempt to find patterns at an overall level a hazardous endeavour.     

The extended cognition thesis: Its significance for the philosophy of (cognitive) science

While the extended cognition (EC) thesis has gained more followers in cognitive science and in the philosophy of mind and knowledge, our main goal is to discuss a different area of significance of the EC thesis: its relation to philosophy of science. In this introduction, we outline two major areas: (I) The role of the thesis for issues in the philosophy of cognitive science, such as: How do notions of EC figure in theories or research programs in cognitive science? Which versions of the EC thesis appear, and with which arguments to support them? (II) The potentials and limits of the EC thesis for topics in general philosophy of science, such as: Can naturalism perhaps be further advanced by means of the more recent EC thesis? Can we understand “big science” or laboratory research better by invoking some version of EC? And can the EC thesis help in overcoming the notorious cognitive/social divide in science studies?