Metamodeling Knowledge: Developing Students' Understanding of Scientific Modeling

We argue that learning about the nature and utility of scientific models and engaging in the process of creating and testing models should be a central focus of science education. To realize this vision, we created and evaluated the Model-Enhanced ThinkerTools (METT) Curriculum, which is an inquiry-oriented physics curriculum for middle school students in which they learn about the nature of scientific models and engage in the process of modeling. Key components of our approach include enabling students to create computer models that express their own theories of force and motion, evaluate their models using criteria such as accuracy and plausibility, and engage in discussions about models and the process of modeling. Curricular trials in four science classes of an urban middle school indicate that this approach can facilitate a significant improvement in students' understanding of modeling. Further analyses revealed that the approach was particularly successful in clarifying and broadening students' understanding of the nature and purpose of models. The METT Curriculum also led to significant improvements in inquiry skills and physics knowledge. Comparisons of METT students' performance with that of prior ThinkerTools students suggest that the acquisition of metamodeling knowledge contributed to these gains. In particular, METT students wrote significantly better conclusions on the inquiry test and performed better on some of the far-transfer problems on the physics test. Finally, correlational results, including significant correlations of pretest modeling and inquiry scores with posttest physics scores, suggests that developing knowledge of modeling and inquiry transfers to the learning of science content within such a curriculum. Taken together, the findings suggest that an emphasis on model-based inquiry, accompanied by the development of metamodeling knowledge, can facilitate learning science content while also developing students' understanding of the scientific enterprise.     

Designing Computer Games to Help Physics Students Understand Newton's Laws of Motion

This research, explores, the design of a computer environment for helping science students to learn about Newtonian dynamics. The learning environment incorporates games set in the context of a Newtonian computer microworld, where students have to control the motion of a spaceship in order to achieve goals such as hitting a target or navigating a maze. The purpose of the games is to focus the students' attention on various aspects of the implications of Newton's laws. A set of general design principles guided the design of the games and microworld. These include: (I) represent the phenomena of the domain clearly; (2) eliminate irrelevant complexities from the computer microworld; (3) focus the students on as peers of their knowledge.that need revising; (4) facilitate the use of problem-solving heuristics; (5) encourage the application of relevant knowledge from other domains; and (6) encourage better ways of representing and thinking about the domain. A controlled study indicated that playing the games improved the students' ability to solve dynamics problems. The students utilized various components of their knowledge, including their intuitions concerning how forces affect motion and their partial understanding of the formal physics, to generate strategies for the games. The use of such knowledge is combined with the use of general problem-solving heuristics and feedback from the computer microworld to facilitate the evolution of the students' knowledge of Newtonian dynamics. An examination of the results in light of the original design principles suggested numerous improvements that could be made to the sequence of games and microworld. The design process could best be characterized as a series of successive refinements.     

Children's conceptions of physical events: Explicit and tacit understanding of horizontal motion

The conceptual understanding that children display when predicting physical events has been shown to be inferior to the understanding they display when recognizing whether events proceed naturally. This has often been attributed to differences between the explicit engagement with conceptual knowledge required for prediction and the tacit engagement that suffices for recognition, and contrasting theories have been formulated to characterize the differences. Focusing on a theory that emphasizes omission at the explicit level of conceptual elements that are tacitly understood, the paper reports two studies that attempt clarification. The studies are concerned with 6‐ to 10‐year‐old children's understanding of, respectively, the direction (141 children) and speed (132 children) of motion in a horizontal direction. Using computer‐presented billiards scenarios, the children predicted how balls would move (prediction task) and judged whether or not simulated motion was correct (recognition task). Results indicate that the conceptions underpinning prediction are sometimes interpretable as partial versions of the conceptions underpinning recognition, as the omission hypothesis would imply. However, there are also qualitative differences, which suggest partial dissociation between explicit and tacit understanding. It is suggested that a theoretical perspective that acknowledges this dissociation would provide the optimal framework for future research.     

Intuitive reasoning about abstract and familiar physics problems

Previous research has demonstrated that many people have misconceptions about basic properties of motion. In two experiments, we examined whether people are more likely to produce dynamically correct predictions about basic motion problems involving situations with which they are familiar, and whether solving such problems enhances performance on a subsequent abstract problem. In Experiment 1, college students were asked to predict the trajectories of objects exiting a curved tube. Subjects were more accurate on the familiar version of the problem, and there was no evidence of transfer to the abstract problem. In Experiment 2, two familiar problems were provided in an attempt to enhance subjects' tendency to extract the general structure of the problems. Once again, they gave more correct responses to the familiar problems but failed to generalize to the abstract problem. Formal physics training was associated with correct predictions for the abstract problem but was unrelated to performance on the familiar problems.     

Contributions of domain-general cognitive resources and different forms of arithmetic development to pre-algebraic knowledge

The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n = 279, mean age = 7.59 years) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems at start of 2nd grade and on calculations, word problems, and pre-algebraic knowledge at end of 3rd grade. Multilevel path analysis, controlling for instructional effects associated with the sequence of classrooms in which students were nested across Grades 2-3, indicated arithmetic calculations and word problems are foundational to pre-algebraic knowledge. Also, results revealed direct contributions of nonverbal reasoning and oral language to pre-algebraic knowledge, beyond indirect effects that are mediated via arithmetic calculations and word problems. By contrast, attentive behavior, phonological processing, and processing speed contributed to pre-algebraic knowledge only indirectly via arithmetic calculations and word problems. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

Continuity in Representation Between Children and Adults: Arithmetic Knowledge Hinders Undergraduates' Algebraic Problem Solving

This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184) solved addition facts or participated in one of several control conditions. Those who solved addition facts were less likely to solve prealgebra equations (e.g., 6 + 8 +4 = 7 + __) correctly under speeded conditions. In a fourth experiment, the negative effects of solving arithmetic problems extended to undergraduates (N = 74) solving algebra problems with no time pressure. Taken together, results suggest that arithmetic activates knowledge that hinders performance on algebra problems. Thus, an operational view of equations, which is prevalent in children, does not seem to be revised or abandoned, even after years of experience with algebra.     

Assessing Students' Misclassifications of Physics Concepts: An Ontological Basis for Conceptual Change

Physics novices and experts solved conceptual physics problems involving light, heat, and electric current and then explained their answers. Novices were ninth-grade students with no background in physics; experts were two postgraduates in physics and two advanced physics graduate students. Problems were multiple choice, with one correct response and three alternative responses representing possible misconceptions. For each conceptual physics problem, an isomorphic material-substance problem was constructed by imagining a materialistic conception of the physics topic and creating the resulting version of the problem. In each physics problem, one of the incorrect choices corresponded to the correct choice in the isomorphic material-substance problem. The empirical question was whether novices would reason about the physics problem as if it were conceptually similar to the substance isomorph. This question was addressed by comparing subjects' responses in the problem pairs, as well as by examining their explanations concerning all problems. A content analysis of subjects' explanations revealed that physics novices were strongly inclined to conceptualize physics concepts as material substances, whereas expert protocols revealed distinctly nonmaterialistic representations. A theory of conceptual change involving ontologically distinct categories is substantiated by these findings.     

Developmental changes in children's understanding of horizontal projectile motion

This study investigated 5‐ to 13‐year‐old children's performance in solving horizontal projectile motion problems, in which they predicted the trajectory of a carried object released from a carrier in three different contexts. The results revealed that 5‐ and 8‐year‐olds' trajectory predictions were easily distracted by salient contextual features (e.g. the relative spatial locations between objects), whereas a proportion of 11‐ and 13‐year‐olds' performance suggested the engagement of the impetus concept in trajectory prediction. The impetus concept is a typical misconception of inertial motion that assumes that motion is caused by force. Children's performance across ages suggested that their naïve knowledge of projectile motion was neither well‐developed and coherent nor completely fragmented. Instead, this study presented the dynamic process in which children with age gradually overcame the influences of contextual features and consistently used the impetus concept across motion problems.     

Language and mathematical problem solving among bilinguals

Does using a bilingual's 1st or 2nd language have an effect on problem solving in semantically rich domains like school mathematics? The author conducted a study to determine whether Filipino-English bilingual students' understanding and solving of word problems in arithmetic differed when the problems were in the students' 1st and 2nd languages. Two groups participated-students whose 1st language was Filipino and students whose 1st language was English-and easy and difficult arithmetic problems were used. The author used a recall paradigm to assess how students understood the word problems and coded the solution accuracy to assess problem solving. The results indicated a 1st-language advantage; that is, the students were better able to understand and solve problems in their 1st language, whether the 1st language was English or Filipino. Moreover, the advantage was more marked with the easy problems. The theoretical and practical implications of the results are discussed.     

Biological knowledge is more tentative than physics knowledge: Taiwan high school adolescents' views about the nature of biology and physics

Many educational psychologists believe that students' beliefs about the nature of knowledge, called epistemological beliefs, play an essential role in their learning process. Educators also stress the importance of helping students develop a better understanding of the nature of knowledge. The tentative and creative nature of science is often highlighted by contemporary science educators. However, few previous studies have investigated students' views of more specific knowledge domains, such as biology and physics. Consequently, this study developed a questionnaire to assess students' views specifically about the tentative and creative nature of biology and physics. From a survey of 428 Taiwanese high school adolescents, this study found that although students showed an understanding of the tentative and creative nature of biology and physics, they expressed stronger agreement as to the tentativeness of biology than that of physics. In addition, male students tended to agree more than did females that physics had tentative and creative features and that biology had tentative features. Also, students with more years of science education tended to show more agreement regarding the creative nature of physics and biology than those with fewer years.     

Connectionist and rule-based representations of expert knowledge

Research in computer science has led to the development of two broad classes of models of knowledge representation: rule-based and connectionist systems. Both techniques solve the same abstract problem, that is, the assignment of cases to classes. Connectionist modeling techniques have been applied to three classification situations in which one would expect rule-based models to be applicable. Two of the situations involved the diagnosis of problems in and operation of a fictitious power plant. The third situation involved the classification of misconceptions held by elementary physics students. Connectionist modeling developed adequate simulations of behavior in all three cases.     

An investigation of the partial-assignment completion effect on students' assignment choice behavior

This study was designed to investigate the partial assignment completion effect. Seventh-grade students were given a math assignment. After working for 5 min, they were interrupted and their partially completed assignments were collected. About 20 min later, students were given their partially completed assignment and a new, control assignment that contained the same number of equivalent problems that were incomplete on their partially completed assignment. Students were told that they would have to complete an assignment but could choose which assignment they completed. Significantly more students chose their partially completed assignment. Theoretical and applied implications and directions for future research are discussed.     

Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

Developmental Analysis of Understanding Language About Quantities and of Solving Problems

We present information-processing models of different levels of knowledge for understanding the language used in texts of arithmetic word Problems, for forming semantic models of the situations that the texts describe, and for making the inferences needed to answer the questions in the problems. In the simplest cognitive models, inferences are limited to properties of sets that exist in a semantic model. In more complex cognitive models, relations between sets are represented internally and support more complex reasoning. Performance on three sets of problems by kindergarten through third-grade students was used to test the models. Global tests provided support for the models. These included measures of scalability and frequencies of individual children's patterns of solutions that agreed with predictions of the models. Performance on problems involving combinations and changes of sets was explained better by the cognitive models than performance on problems involving comparisons. Comparisons may require more advanced understanding of numbers as values of operators rather than only as cardinalities of sets.     

Pogge on Poverty: Contribution or Exploitation?

Thomas Pogge argues that affluent people in the developing world have contribution‐based duties to help protect the poor. And it follows from Pogge's most general thesis that affluent people are contributing to most, if not all, instances of global poverty. In this article I explore two problems with Pogge's general thesis. First, I investigate a typical way in which affluent people would be contributing to global poverty according to Pogge: that affluent countries use their superior bargaining power to get poor countries to accept trading schemes that are unduly favourable to the affluent. I suggest that this type of relation is best understood as exploitation, and that Pogge's general thesis is better understood as a thesis about how affluent people exploit poor people rather than about how they contribute to poverty. Second, I argue that the exploitation does not have the normative content of doing harm. Although exploiting people is often morally wrong, it is not at all clear how demanding exploitation‐based duties are.     

Self‐regulated learning of basic arithmetic skills: A longitudinal study

Background. Several studies have examined young primary school children's use of strategies when solving simple addition and subtraction problems. Most of these studies have investigated students’ strategy use as if they were isolated processes. To date, we have little knowledge about how math strategies in young students are related to other important aspects in self‐regulated learning. Aim. The main purpose of this study was to examine relations between young primary school children's basic mathematical skills and their use of math strategies, their metacognitive competence and motivational beliefs, and to investigate how students with basic mathematics skills at various levels differ in respect to the different self‐regulation components. Sample. The participants were comprised of 27 Year 2 students, all from the same class. Method. The data were collected in three stages (autumn Year 2, spring Year 2, and autumn Year 3). The children's arithmetic skills were measured by age relevant tests, while strategy use, metacognitive competence, and motivational beliefs were assessed through individual interviews. The participants were divided into three performance groups; very good students, good students, and not‐so‐good students. Results. Analyses revealed that young primary school children at different levels of basic mathematics skill may differ in several important aspects of self‐regulated learning. Analyses revealed that a good performance in addition and subtraction was related not only to the children's use of advanced mathematics strategies, but also to domain‐specific metacognitive competence, ability attribution for success, effort attribution for failure, and high perceived self‐efficacy when using specific strategies. Conclusions. The results indicate that instructional efforts to facilitate self‐regulated learning of basic arithmetic skills should address cognitive, metacognitive, and motivational aspects of self‐regulation. This is particularly important for low‐performing students.     

Internal Models in Physics Problem Solving

In this paper, we discuss the relations among internal mental models, attentional cues, and knowledge structure in solving elementary physics problems. The notion of semantic sensitivity is introduced as a qualitative measure of problem-solver receptivity to cues for generating, or making a shift to, a new internal model that may lead to the correct answer. Using three physics problems and their variations, the following results were obtained. First, think-aloud protocol data of two experts and one novice, working on those problems, were analyzed to provide empirical examples of internal models, and also to show that generation and shift of internal models generally results from the interplay of two kinds of processes, model construction and model development. Second, protocol data from 15 novices and seven professionals were used to present a more extensive empirical taxonomy of internal models for those problems. Third, seven experiments were used, along with the results from the protocol data, to suggest the following conditions on semantic sensitivity. (1) If the problem solver's present internal model is independent of his knowledge of physics principles underlying the given problem, but if attentional cues are related directly to the knowledge, then his present model is semantically sensitive to the cues, that is, it can be shifted easily to another model that is based on the knowledge. (2) His present model is semantically insensitive, if given cues are encoded based on the same knowledge as used for generating the model. (3) The relevance of cues to knowledge structure is critical for semantic sensitivity of the presently encoded model: Which cues to attend to is guided by the problem solver's knowledge, and if he can focus on those most relevant to the physics principles underlying the problem, then his present model is semantically sensitive to those cues, that is, he is more able to shift the present model to another that would lead to the correct answer.     

When being a girl matters less: Accessibility of gender‐related self‐knowledge in single‐sex and coeducational classes and its impact on students' physics‐related self‐concept of ability

Background . Establishing or preserving single‐sex schooling has been widely discussed as a way of bringing more girls into the natural sciences. Aims . We test the assumption that the beneficial effects of single‐sex education on girls' self‐concept of ability in masculine subjects such as physics are due to the lower accessibility of gender‐related self‐knowledge in single‐sex classes. Sample . N = 401 eighth‐graders (mean age 14.0 years) from coeducational comprehensive schools. Methods . Random assignment of students to single‐sex vs. coeducational physics classes throughout the eighth grade. At the end of the year, students' physics‐related self‐concept of ability was measured using a questionnaire. In a subsample of N = 134 students, the accessibility of gender‐related self‐knowledge during physics classes was assessed by measuring latencies and endorsement of sex‐typed trait adjectives. Results . Girls from single‐sex physics classes reported a better physics‐related self‐concept of ability than girls from coeducational classes, while boys' self‐concept of ability did not vary according to class composition. For both boys and girls, gender‐related self‐knowledge was less accessible in single‐sex classes than in mixed‐sex classes. To the extent that girls' feminine self‐knowledge was relatively less accessible than their masculine self‐knowledge, their physics‐related self‐concept of ability improved at the end of the school year. Conclusions . By revealing the importance of the differential accessibility of gender‐related self‐knowledge in single‐ and mixed‐sex settings, our study clarifies why single‐sex schooling helps adolescents to gain a better self‐concept of ability in school subjects that are considered inappropriate for their own sex.     

Impetus beliefs as default heuristics: Dissociation between explicit and implicit knowledge about motion

We examined the extent to which findings from the literature on naive physics and representational momentum studies are consistent with impetus beliefs postulating imparted internal energy as a source of motion. In a literature review, we showed that, for situations in which impetus theory and physical principles make different predictions, representational momentum effects are consistent with impetus beliefs. In three new experiments, we examined people’s implicit and explicit knowledge of the effect of mass on the rate of ascending motion. The results suggest that implicit knowledge is consistent with impetus theory and is unaffected by explicit knowledge. Expert physicists, whose explicit knowledge is in accord with Newtonian principles, exhibited the same implicit impetus beliefs as novices when asked to respond in a representational momentum paradigm. We propose that, in situations in which an immediate response is required and one does not have specific contextual knowledge about an object’s motion, both physics experts and novices apply impetus principles as a default heuristic.