The effect of explanations on mathematical reasoning tasks

Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Previous studies have shown that students that engage in creative mathematically founded reasoning to construct a solution method, perform significantly better in follow up tests than students that are given a solution method and engage in algorithmic reasoning. However, teachers and textbooks, at least occasionally, provide explanations together with an algorithmic method, and this could possibly be more efficient than creative reasoning. In this study, three matched groups practiced with either creative, algorithmic, or explained algorithmic tasks. The main finding was that students that practiced with creative tasks did, outperform the students that practiced with explained algorithmic tasks in a post-test, despite a much lower practice score. The two groups that got a solution method presented, performed similarly in both practice and post-test, even though one group got an explanation to the given solution method. Additionally, there were some differences between the groups in which variables predicted the post-test score.     

Analysis of proportional reasoning and misconceptions among students with mathematical learning disabilities

We investigated not only the effects of schema-based instruction (SBI) on the mathematical outcomes of seventh-grade students with mathematical learning disabilities (MLD), but also extended prior work to analyze students’ written explanations on open-ended items involving ratio and proportion situations—ratio, proportion, and percent of change problems— to understand the ability to reason about proportions and identify misconceptions. The sample of 338 students with MLD [scored below the 25th percentile on a proportional problem solving (PPS) pretest] was taken from Jitendra, Harwell, Im, et al. (2019), which randomly assigned classrooms to either the SBI or control condition. Students with MLD in SBI classrooms outperformed their counterparts in control classrooms on proportional problem solving and general mathematics problem solving. Similar results, favoring the SBI condition, were found on the open-ended items; however, overall mean scores across pretest, posttest, and delayed posttest were low. Findings provide evidence for the limited understanding of fractional representations of ratios and highlight students’ persistent use of numerical and additive reasoning in explaining their low performance on the open-ended items.     

The notion of motion: covariational reasoning and the limit concept

This paper investigates outcomes of building students’ intuitive understanding of a limit as a function's predicted value by examining introductory calculus students’ conceptions of limit both before and after instruction. Students’ responses suggest that while this approach is successful at reducing the common limit equals function value misconception of a limit, new misconceptions emerged in students’ responses. Analysis of students’ reasoning indicates a lack of covariational reasoning that coordinates changes in both x and y may be at the root of the emerging limit reached near x = c misconception. These results suggest that although dynamic interpretations of limit may be intuitive for many students, care must be taken to foster a dynamic conception that is both useful at the introductory calculus level and is in line with the formal notion of limit learned in advanced mathematics. In light of the findings, suggestions for adapting the pedagogical approach used in this study are provided.     

Smart case-based indexing in worsted roving process: Combination of rough set and case-based reasoning

Rough set data analysis (RSDA) has been primarily studied in order to obtain knowledge rules. Taking one group of data as an example, according to the simplified attributes and extractive rules a case was established; using case-based reasoning (CBR), another case was established based on the parameters influencing on the roving quality. In addition, RSDA was combined with CBR to build the case library. This allowed unimportant parameters to be removed and the case library to be simplified; in turn allowing easier, more efficient searching of attributes from the simplified case library. The union of Rule-Based Reasoning (RBR) and CBR means that complicated calculations around similar cases and the associated error can be avoided. Using RSDA one is able to reveal characteristic attributes and deduce knowledge rules associated with the model/problem in order to build a case library directly from historical data. So using the above procedure, allows machine settings to be defined and the prediction and control of the end-product quality to be easier.     

Happy and sad thoughts: An exploration of children's integer reasoning

The purpose of this study was to investigate elementary children's conceptions that might serve as foundations for integer reasoning. Working from an abstract algebraic perspective and using an opposite-magnitudes context that is relevant to children, we analyzed the reasoning of 33 children in grades K-5. We focus our report on three prominent ways of reasoning. We do this by describing and analyzing the responses of three particular children (in Grades 1, 3, and 5) who exemplify these ways of reasoning. We view each of the three ways of reasoning as rich and interesting, and we see relationships of each to formal integer reasoning. At the same time, we view these ways of reasoning in terms of increasing levels of sophistication, potentially belonging to a single learning trajectory. Thus, we see the roots of more sophisticated integer reasoning in children's early intuitions about opposite magnitudes.     

The labyrinth of experts' minds: Some reasoning strategies and their pitfalls

A conceptual framework is proposed for understanding and analysing the cognitive processes involved in the intuitive generation of scenarios. Four stages are distinguished: The activation of relevant problem knowledge, the constitution of a mental model, the simulation of the mental model, and the selection of inferences. The approach is illustrated with examples from a project on the future impacts of new information and communication technology on private life.This contribution is largely based on a paper entitled The use of causal knowledge for inferential reasoning, published in: J.L. Mumpower, L.D. Phillips, O. Renn & V.R.R. Uppuluri (eds.),Expert Judgment and Expert Systems (Springer, New York, 1987).     

A mathematical theory of communication: Meaning,information, and topology

This article proposes a new mathematical theory of communication. The basic concepts of meaning and information are defined in terms of complex systems theory. Meaning of a message is defined as the attractor it generates in the receiving system; information is defined as the difference between a vector of expectation and one of perception. It can be sown that both concepts are determined by the topology of the receiving system. © 2010 Wiley Periodicals, Inc. Complexity 16: 10–26, 2011     

Hybridizing principles of the Electre method with case-based reasoning for data mining: Electre-CBR-I and Electre-CBR-II

Electre is an important outranking method developed in the area of decision-aiding. Data mining is a vital developing technique that receives contributions from lots of disciplines such as databases, machine learning, information retrieval, statistics, and so on. Techniques in outranking approaches, e.g. Electre, could also contribute to the development of data mining. In this research, we address the following two issues: a) why and how to combine Electre with case-based reasoning (CBR) to generate corresponding hybrid models by extending the fundamental principles of Electre into CBR; b) the effect on predictive performance by employing evidence vetoing the assertion on the base of evidence supporting the assertion. The similarity measure of CBR is implemented by revising and fulfilling three basic ideas of Electre, i.e. assertion that two cases are indifferent, evidence supporting the assertion, and evidence vetoing the assertion. Two corresponding CBR models are constructed by combining principles of the Electre decision-aiding method with CBR. The first one, named Electre-CBR-I, derives from evidence supporting the assertion. The other one, named Electre-CBR-II, derives from both evidence supporting and evidence vetoing the assertion. Leave-one-out cross-validation and hold-out method are integrated to form 30-times hold-out method. In financial distress mining, data was collected from Shanghai and Shenzhen Stock Exchanges, ANOVA was employed to select features that are significantly different between companies in distress and health, 30-times hold-out method was used to assess predictive performance, and grid-search technique was utilized to search optimal parameters. Original data distributions were kept in the experiment. Empirical results of long-term financial distress prediction with 30 initial financial ratios and 135 initial pairs of samples indicate that Electre-CBR-I outperforms Electre-CBR-II and other comparative CBR models, and Electre-CBR-II outperforms the other comparative CBR models.     

Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example

In this paper we examine the roles that metonymy may play in student reasoning. To organize this discussion we use the lens of a structured derivative framework. The derivative framework consists of three layers of process-object pairs, one each for ratio, limit, and function. Each of the layers can then be illustrated in any appropriate context, for example graphically as slope, verbally as rate of change, or kinesthetically as velocity. We will illustrate three main cases of metonymy using data from student reasoning with derivative-paradigmatic metonymy, individual metonymy and part-part metonymy. Metonymies may function both in powerful and problematic ways for students as they come to understand and work with a complicated concept such as the derivative. It is the goal of this paper to illuminate and clarify how metonymy may function in student reasoning in the hopes of providing insights to teachers and researchers.     

An ecological reading of mathematical language in a Grade 3 classroom: A case of learning and teaching measurement estimation

In our work in teacher education and professional development, we aim to help teachers to learn to participate in, and create, classroom ecologies that support students’ learning. In this article we focus on the challenges of developing a classroom ecology that provides mathematical sustenance for students. We pay particular attention to the ways in which classroom language can impede the development of a classroom ecology—one where all students are heard and where knowing is understood as participatory. We present recommendations for teaching practice drawn from an ecological reading of the classroom discourse during a series of lessons on measurement in a Grade 3 classroom.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND COMPRESSION COSTS ： PROOF OF THEOREMS

This report is virtually the appendix part of the author‘s previous paper which ineludes the proofs for the theorems and lemmas.