Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra

The article deals with issues concerning the abductive-inductive reasoning of 42 preservice elementary majors on patterns that consist of figural and numerical cues. We discuss: ways in which the participants develop generalizations about classes of abstract objects; abductive processes they exhibit which support their induction leading to a generalization; ways they justify their generalizations in the abductive stage, and; the effects of figural and numerical cues in the manner they construct a plausible abductive generalization. Two types of abductions are explored, model-based and manipulative. A proposed abductive-inductive reasoning process for pattern sequences is presented and discussed in the concluding section.     

基于证据推理与最优指派策略的多属性单边匹配决策

针对匹配中某一方偏好失效的问题,提出一种基于证据推理和最优指派策略的单边匹配方法。一方主体采用多种数据类型描述由对方指定的多个属性信息;另一方给出关于各属性的权重信息;然后,使用证据推理组合规则递推合成多属性及权重信息,以此计算双方的匹配度。在此基础上,运用最优指派策略,建立匹配模型并求解得到匹配结果.实例表明该方法的可行性和有效性。     

Understanding the concepts of proportion and ratio among grade nine students in Malaysia

The purpose of the study was to investigate the concepts of ratio and proportion constructed by grade nine students by investigating grade nine students proportional reasoning schemes and procedures on three types of tasks: missing value, numerical comparison and qualitative reasoning. Comparisons among the different categories were made and the strategies used in solving these problems were identified. The relationship between student grades on a national examination and their knowledge of proportional reasoning was determined. The results of the quantitative analysis indicated that students performed generally well on the missing value tasks but their scores on the numerical comparison and qualitative tasks were much lower. The results indicate that only a small percentage of students who did well on the national exams were able to solve complex proportional problems and the grades obtained were not indicative of their knowledge of ratio and proportion. The difficulty experienced by the ninth graders indicated that students frequently used additive reasoning, that is a comparison of two numbers by subtraction rather than division. It appears that students cannot begin to understand the functional and scalar relationship inherent in a proportion until they first develop multiplicative reasoning.     

Middle school students’ patterning performance on semi-free generalization tasks

This longitudinal study empirically addresses the issue of structure construction and justification among a class of US seventh and eighth-grade Algebra 1 students (mean age of 12.5 years) in the context of novel semi-free pattern generalization (PG) tasks before and after a teaching experiment that emphasized a multiplicative thinking approach to patterns. We compared the students’ PG responses before and after the experiment and found that (1) one source of variability in their abduced structural processing was in part due to an initial conceptual preference toward thinking either in parts or in wholes and (2) a multiplicative understanding of structures significantly aided them in PG conversion (e.g., from the visual to the alphanumeric) and processing (e.g., from nonstandard to standard function-based formulas). Our findings provide both necessary and sufficient conditions for constructing, establishing, and justifying valid structures in the case of (semi-) free figural patterning tasks.     

Pattern Generalization with Graphs and Words: A Cross-Sectional and Longitudinal Analysis of Middle School Students' Representational Fluency

Cross-sectional and longitudinal data from students as they advance through the middle school years (grades 6-8) reveal insights into the development of students' pattern generalization abilities. As expected, students show a preference for lower-level tasks such as reading the data, over more distant predictions and generation of abstractions. Performance data also indicate a verbal advantage that shows greater success when working with words than graphs, a replication of earlier findings comparing words to symbolic equations. Surprisingly, students show a marked advantage with patterns presented in a continuous format (line graphs and verbal rules) as compared to those presented as collections of discrete instances (point-wise graphs and lists of exemplars). Student pattern-generalization performance also was higher when words and graphs were combined. Analyses of student performance patterns and strategy use contribute to an emerging developmental model of representational fluency. The model contributes to research on the development of representational fluency and can inform instructional practices and curriculum design in the area of algebraic development. Results also underscore the impact that perceptual aspects of representations have on students' reasoning, as suggested by an Embodied Cognition view.     

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.     

Networks of probabilistic events in discrete time

The usual methods of applying Bayesian networks to the modeling of temporal processes, such as Dean and Kanazawa’s dynamic Bayesian networks (DBNs), consist in discretizing time and creating an instance of each random variable for each point in time. We present a new approach called network of probabilistic events in discrete time (NPEDT), for temporal reasoning with uncertainty in domains involving probabilistic events. Under this approach, time is discretized and each value of a variable represents the instant at which a certain event may occur. This is the main difference with respect to DBNs, in which the value of a variable V_i represents the state of a real-world property at time t_i. Therefore, our method is more appropriate for temporal fault diagnosis, because only one variable is necessary for representing the occurrence of a fault and, as a consequence, the networks involved are much simpler than those obtained by using DBNs. In contrast, DBNs are more appropriate for monitoring tasks, since they explicitly represent the state of the system at each moment. We also introduce in this paper several types of temporal noisy gates, which facilitate the acquisition and representation of uncertain temporal knowledge. They constitute a generalization of traditional canonical models of multicausal interactions, such as the noisy OR-gate, which have been usually applied to static domains. We illustrate the approach with the example domain of modeling the evolution of traffic jams produced on the outskirts of a city, after the occurrence of an event that obliges traffic to stop indefinitely.     

Three methods for robust grading

We propose three strategies by which a professor of a university course can assign final letter grades taking into account the natural uncertainty in students’ individual assignment and final numerical grades. The first strategy formalizes a common technique that identifies large gaps in the final numerical grades. For the second and third strategies, we introduce the notion of a borderline student, that is, a student who is close to, but below, the breakpoint for the next highest letter grade. Using mixed-integer linear programming and a tailor-made branch-and-bound algorithm, we choose the letter-grade breakpoints to minimize the number of borderline students. In particular, the second strategy treats the uncertainty implicitly and minimizes the number of borderline students, while the third strategy uses a robust-optimization approach to minimize the maximum number of borderline students that could occur based on an explicit uncertainty set. We compare the three strategies on realistic instances and identify overall trends as well as some interesting exceptions. While no strategy appears best in all cases, each can be computed in a reasonable amount of time for a moderately sized course. Moreover, they collectively provide the professor important insight into how uncertainty affects the assignment of final letter grades.     

Do students confuse dimensionality and “directionality”?

The aim of this research is to understand the way in which students struggle with the distinction between dimensionality and “directionality” and if this type of potential confusion could be a factor affecting students’ tendency toward improper linear reasoning in the context of the relations between length and area of geometrical figures. 131 9th grade students were confronted with a multiple-choice test consisting of six problems related to the perimeter or the area of an enlarged geometrical figure, then some interviews were carried out to obtain qualitative data in relation to students’ reasoning. Results indicate that more than one fifth of the students’ answers could be characterized as based on directional thinking, suggesting that students struggled with the distinction between dimensionality and “directionality”. A single arrow showing one direction (image provided to the students) seemed to strengthen the tendency toward improper linear reasoning for the area problems. Two arrows showing two directions helped students to see a quadratic relation for the area problems.     

基于神经网络的模糊推理

为了使模糊推理符合推理原则，目前已定义了１０多种模糊关系，但各种模糊关系定义都存在一定的缺陷。本文提出的基于神经网络的模糊推理，能很好地符合模糊推理原则。     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

An In‐depth Analysis of Achievement Gaps Between Seventh and Eighth Grades in the TIMSS Database

Middle school data from the Third International Mathematics and Science Study (TIMSS) are analyzed in this article to compare achievement difference between seventh and eighth grades. The statistical computing involves a selective subtraction of item scores in more than 40 countries. The research findings indicate that not all TIMSS items have resulted in a higher mean score at the upper grade level. Item features are discussed to characterize part of the released TIMSS instrument that generates a higher average score at the lower grade. This empirical study may help enrich understanding of the TIMSS benchmark among mathematics and science educators.     

Validating a concept inventory for measuring students’ probabilistic reasoning: The case of reasoning within the context of a raffle

Assessing students’ conceptions related to independence of events and determining probabilities from a sample space has been the focus of research in probability education for over 40 years. While we know a lot from past studies about predictable ways students may reason with well-known tasks, developing a diagnostic assessment that can be used by teachers to inform instruction demands the use of familiar and unfamiliar contexts. This paper presents the current work of a research team whose aim is to create a formative concept inventory with strong evidence of validity that uses a psychometric model to confidently predict whether a student exhibits one or more misconception across many items. We illustrate this process in this paper using a particular item with a context of a raffle aimed to measure whether a student reasons with misconceptions related to independence or equiprobability. The results of two aspects of the validity process: cognitive interviews to assess response processes on individual items, and a large-scale administration to examine internal structure of the concept inventory revealed difficulties in assessing students’ reasoning about these key probability concepts and trends in the prevalence of misconceptions across grades. Results can provide guidance for others aiming to develop assessments in mathematics education and also support further possibilities for research into understanding students’ reasoning about independence and sample space.     

基于云模型和证据推理的生态环境安全评价

文章提出了一种生态环境安全综合评价模型。首先构建生态环境安全评价指标体系并划分等级区间, 通过云模型将各指标实际值转化为生态环境安全等级的关联度; 其次结合证据推理与熵权法动静态组合确定指标权重; 然后利用证据推理计算出各指标基本可信度分配函数, 再采用证据融合算法合成出生态环境安全综合评估概率分布; 最后根据“最大关联度准则”得出评价结果。以湖南省为例开展实例研究, 研究结果与模糊综合评价法评估等级结果一致, 验证了所提出方法具有合理性、可行性和有效性。     

Evidential reasoning based preference programming for multiple attribute decision analysis under uncertainty

Multiple attribute decision analysis (MADA) problems having both quantitative and qualitative attributes under uncertainty can be modelled and analysed using the evidential reasoning (ER) approach. Several types of uncertainty such as ignorance and fuzziness can be consistently modelled in the ER framework. In this paper, both interval weight assignments and interval belief degrees are considered, which could be incurred in many decision situations such as group decision making. Based on the existing ER algorithm, several pairs of preference programming models are constructed to support global sensitivity analysis based on the interval values and to generate the upper and lower bounds of the combined belief degrees for distributed assessment and also the expected values for ranking of alternatives. A post-optimisation procedure is developed to identify non-dominated solutions, examine the robustness of the partial ranking orders generated, and provide guidance for the elicitation of additional information for generating more desirable assessment results. A car evaluation problem is examined to show the implementation process of the proposed approach.     

Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement

The purpose of this study was to examine middle school students’ proportional reasoning, solution strategies and difficulties in real life contexts in the domain of geometry and measurement. The underlying reasons of the difficulties were investigated as well. Mixed research design was adopted for the aims of the study by collecting data through an achievement test from 935 sixth, seventh and eighth grade students. The achievement test included real life problems that required proportional reasoning, and were related to the measurement of length, perimeter, area and volume concepts. In addition, task-based interviews were conducted on 12 of these students to collect more comprehensive data and to support the findings of the achievement test. Findings revealed that although students were mostly successful in giving correct answers, their reasoning lacked a clear argument of the direct and indirect proportional relationships between the variables and that they approached the problems by superficial characteristics of the problems.     

“Approximate” multiplicative relationships between quantitative unknowns

Three 18-session design experiments were conducted, each with 6–9 7th and 8th grade students, to investigate relationships between students’ rational number knowledge and algebraic reasoning. Students were to represent in drawings and equations two multiplicatively related unknown heights (e.g., one was 5 times another). Twelve of the 22 participating students operated with the second multiplicative concept, which meant they viewed known quantities as units of units, or two-levels-of-units structures, but not as three-levels-of-units structures. These students were challenged to represent multiplicative relationships between unknowns: They changed the given relationship, did not think of the relationship as multiplicative until after concerted work, and used numerical values in lieu of unknowns. Our account for these challenges is that students needed to simplify the involved units coordinations. Ultimately students abstracted the relationship as multiplicative, but the exact relationship was not certain or had to be constituted in activity. Implications for teaching are explored.     

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.     

The group consensus based evidential reasoning approach for multiple attributive group decision analysis

Many multiple attribute decision analysis problems include both quantitative and qualitative attributes with various kinds of uncertainties such as ignorance, fuzziness, interval data, and interval belief degrees. An evidential reasoning (ER) approach developed in the 1990s and in recent years can be used to model these problems. In this paper, the ER approach is extended to group consensus (GC) situations for multiple attributive group decision analysis problems. In order to construct and check the GC, a compatibility measure between two belief structures is developed first. Considering two experts’ utilities, the compatibility between their assessments is naturally constructed using the compatibility measure. Based on the compatibility between two experts’ assessments, the GC at a specific level that may be the attribute level, the alternative level, or the global level, can be constructed and reached after the group analysis and discussion within specified times. Under the condition of GC, we conduct a study on the forming of group assessments for alternatives, the achievement of the aggregated utilities of assessment grades, and the properties and procedure of the extended ER approach. An engineering project management software selection problem is solved by the extended ER approach to demonstrate its detailed implementation process, and its validity and applicability.