The effects of metacognitive training versus worked‐out examples on students' mathematical reasoning

Background: The present study is rooted in a cognitive‐metacognitive approach. The study examines two ways to structure group interaction: one is based on worked‐out examples (WE) and the other on metacognitive training (MT). Both methods were implemented in cooperative settings, and both guided students to focus on the problem's essential parts and on appropriate problem‐solving strategies. Aims: The aim of the present study is twofold: (a) to investigate the effects of metacognitive training versus worked‐out examples on students' mathematical reasoning and mathematical communication; and (b) to compare the long‐term effects of the two methods on students' mathematical achievement. Sample: The study was conducted in two academic years. Participants for the first year of the study were 122 eighth‐grade Israeli students who studied algebra in five heterogeneous classrooms with no tracking. In addition, problem‐solving behaviours of eight groups (N = 32) were videotaped and analysed. A year later, when these participants were ninth graders, they were re‐examined using the same test as the one administered in eighth grade. Method: Three measures were used to assess students' mathematical achievement: a pretest, an immediate post‐test, and a delayed post‐test. ANOVA was carried out on the post‐test scores with respect to the following criteria: verbal explanations, algebraic representations and algebraic solution. In addition, chi‐square and Mann‐Whitney procedures were used to analyse cooperative, cognitive, and metacognitive behaviours. Results: Within cooperative settings, students who were exposed to metacognitive training outperformed students who were exposed to worked‐out examples on both the immediate and delayed post‐tests. In particular, the differences between the two conditions were observed on students' ability to explain their mathematical reasoning during the discourse and in writing. Lower achievers gained more under the MT than under WE condition. Conclusions: The findings indicate that the kind of task and the way group interaction is structured are two important variables in implementing cooperative learning, each of which is likely to have different effects on mathematical communication and achievement outcomes.     

The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra

The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly. Furthermore, decreased solution time was accompanied by a decrease in the number of mathematical errors. Both of these findings were specific to problems identical in structure to the initial ones. It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practice and exposure to a wider range of schemas.     

Diagnosing students’ misconceptions in algebra: Results from an experimental pilot study

Computer-based diagnostic assessment systems hold potential to help teachers identify sources of poor performance and to connect teachers and students to learning activities designed to help advance students’ conceptual understandings. The present article presents findings from a study that examined how students’ performance in algebra and their overcoming of common algebraic misconceptions were affected by the use of a diagnostic assessment system that focused on important algebra concepts. This study used a four-group randomized cluster trial design in which teachers were assigned randomly to one of four groups: a “business as usual” control group, a partial intervention group that was provided with access to diagnostic tests results, a partial intervention group that was provided with access to the learning activities, and a full intervention group that was given access to the test results and learning activities. Data were collected from 905 students (6th–12th grade) nested within 44 teachers. We used hierarchical linear modeling techniques to compare the effects of full, partial, and no (control) intervention on students’ algebraic ability and misconceptions. The analyses indicate that full intervention had a net positive effect on ability and misconception measures.     

Rates of Specific Antecedent Instructional Practices and Differences Between Title I and non-Title I Schools

The use of effective instructional strategies is clearly emphasized in current educational reform, especially in the area of reading. The purposes of this study were to investigate the rates at which specific instructional practices (i.e., attention signals, prior knowledge supports, previews, instructor modeling, student modeling, organizational prompts) were utilized during literacy time in elementary schools, determine if there were relationships among the instructional variables, and explore if teachers in Title I schools and teachers in non-Title I schools differed in their use of specific practices. Participants included teachers and students from 35 classrooms who were each observed for 5 hours, resulting in a total of 175 observation hours. The Setting Factors Assessment Tool (SFAT) was used to measure the antecedent instructional variables. Main results included that teachers in non-Title I classrooms used significantly more prior knowledge references than teachers in Title I schools; the effect size for this finding was large. Several correlations among the instructional variables were significant. The article concludes with a discussion of the main findings, implications for future research and the limitations of this study.     

小学生数学问题解决中视觉空间表征的研究

本研究区分了两类数学应用题:非视觉化题目与视觉化题目,采用数学测验与个别访谈相结合的方法,考察了54名小学四、五、六年级不同学业水平学生的视觉空间表征。结果表明:图式表征在非视觉化题目与视觉化题目上都极大地促进了问题解决,图像表征妨碍非视觉化题目的解决但与视觉化题目的解决无关,并提出图式表征和图像表征在两类题目上有不同的含义。六年级学生的解题成绩及图式表征有显著的提高,但图像表征与年级因素无关。差生的图式表征能力很差,而在视觉化题目上使用图像表征显著地多于优生及中等生。在非视觉化题目的非视觉空间表征与图式表征之间的转换灵活性上,优生表现了明显的优势。     

The Microevolution of Mathematical Representations in Children's Activity

In this article, I discuss children's design of mathematical representations on paper, asking how material displays are constructed and transformed in activity. I show that (a) the design of displays during problem solving shapes one's mathematical activity and sense making in crucial ways, and (b) knowledge of mathematical representations is not simply recalled and applied to problem solving but also emerges (whether constructed anew or not) out of one's interactions with the social and material settings of activity. A detailed characterization of student-designed tables of values to solve problems about linear functions is also presented.     

Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.     

How to avoid inconsistent idealizations

Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of ocean waves typically involve the false assumption that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation.     

Handling mathematical objects: representations and context

This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on Riemann’s work on complex functions, I propose that mathematicians deal with systems of representations and that truth—or what we can prove—depends on available representations in some context where the problem can be solved.     

Narrative Representation and Phenomenological Knowledge

The purpose of this paper is to demonstrate that narrative representations can provide knowledge in virtue of their narrativity, regardless of their truth value. I set out the question in section 1, distinguishing narrative cognitivism from aesthetic cognitivism and narrative representations from non-narrative representations. Sections 2 and 3 argue that exemplary narratives can provide lucid phenomenological knowledge, which appears to meet both the epistemic and narrativity criteria for the narrative cognitivist thesis. In section 4, I turn to non-narrative representation, focusing on lyric poetry as presenting a disjunctive objection: either lucid phenomenological knowledge can be reduced to identification and fails to meet the epistemic criterion, or lucid phenomenological knowledge is provided in virtue of aesthetic properties and fails to meet the narrativity criterion. I address both of these problems in sections 5 and 6, and I close with a tentative suggestion as to how my argument for narrative cognitivism could be employed as an argument for aesthetic cognitivism.     

Who can escape the natural number bias in rational number tasks? A study involving students and experts

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth‐grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.     

Hyperformulas and Solid Algebraic Systems

Defining a composition operation on sets of formulas one obtains a many-sorted algebra which satisfies the superassociative law and one more identity. This algebra is called the clone of formulas of the given type. The interpretations of formulas on an algebraic system of the same type form a many-sorted algebra with similar properties. The satisfaction of a formula by an algebraic system defines a Galois connection between classes of algebraic systems of the same type and collections of formulas. Hypersubstitutions are mappings sending pairs of operation symbols to pairs of terms of the corresponding arities and relation symbols to formulas of the same arities. Using hypersubstitutions we define hyperformulas. Satisfaction of a hyperformula by an algebraic system defines a second Galois connection between classes of algebraic systems of the same type and collections of formulas. A class of algebraic systems is said to be solid if every formula which is satisfied is also satisfied as a hyperformula. On the basis of these two Galois connections we construct a conjugate pair of additive closure operators and are able to characterize solid classes of algebraic systems. Presented by Wojciech Buszkowski     

Why ‘scaffolding’ is the wrong metaphor: the cognitive usefulness of mathematical representations

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.

     

小学生视觉-空间表征类型和数学问题解决的研究

本研究考察并比较了四至六年级儿童的视觉－空间表征策略、数学问题解决和空间视觉化能力。结果表明：五、六年级儿童的解题正确率、使用图式表征策略的程度显著高于四年级儿童;使用图像表征策略的程度各年级无显著差异。将数学问题分成三个难度等级,发现年级差异主要表现在难度等级1的题目上。另外,六年级儿童的空间视觉化能力显著高于四年级儿童。     

In Defence of Naiveté: The Conceptual Status of Lagrangian Quantum Field Theory

I analyse the conceptual and mathematical foundations of Lagrangian quantum field theory (QFT) (that is, the ‘naive’ (QFT) used in mainstream physics, as opposed to algebraic quantum field theory). The objective is to see whether Lagrangian (QFT) has a sufficiently firm conceptual and mathematical basis to be a legitimate object of foundational study, or whether it is too ill-defined. The analysis covers renormalisation and infinities, inequivalent representations, and the concept of localised states; the conclusion is that Lagrangian QFT (at least as described here) is a perfectly respectable physical theory, albeit somewhat different in certain respects from most of those studied in foundational work.     

Student Thinking Processes While Constructing Graphic Representations of Textbook Content: What Insights Do Think-Alouds Provide?

This study examined the thinking processes students engage in while constructing graphic representations of textbook content. Twenty-eight students who either used graphic representations in a routine manner during social studies instruction or learned to construct graphic representations based on the rhetorical patterns used to organize textbook content produced think-aloud responses while constructing graphic representations. Responses indicated that both groups of students needed to restate text while writing, but knowledge of rhetorical patterns appeared to facilitate students' ability to discriminate between main ideas and details, understand relationships between ideas, and, generally, engage the text more deeply. Implications for instruction are discussed.     

The role of functionality in the mental representations of engineering students: some differences in the early stages of expertise

As engineers gain experience and become experts in their domain, the structure and content of their knowledge changes. Two studies are presented that examine differences in knowledge representation among freshman and senior engineering students. The first study examines recall of mechanical devices and chunking of components, and the second examines whether seniors represent devices in a more abstract functional manner than do freshmen. The most prominent differences between these 2 groups involve their representation of the functioning of groups of electromechanical components and how these groups of components interact to produce device behavior. Seniors are better able to construct coherent representations of devices by focusing on the function of sets of components in the device. The findings from these studies highlight some ways in which the structure and content of mental representations of design knowledge differ during the early stages of expertise acquisition.