Non-proportional Situations Research Articles

Middle School Students’ Proportional Reasoning Ability in Solving Proportional and Non-proportional Problem

The objective of this descriptive qualitative study is to describe students' proportional reasoning ability in solving each type of proportional problem, namely missing-value problem, numerical comparison problem, and qualitative comparison problem. A non-proportional problem was also included in this study to assess students' ability in one of the indicators used to measure proportional reasoning ability. The instrument used in this study consists of mathematical word problems on the proportion concept. The participants involved nine eighth-grade students, who were chosen from a total of 49 eighth-grade students at a school in Bandung City. These students were selected using a technique known as purposive sampling, by looking at the answers of students who best represent the answers of other students. The gathered data were analyzed using content analysis and narrative analysis techniques. According to the results, students appeared to have difficulty solving proportional problems of all types using proportional reasoning. Students are still struggling with distinguishing between proportional and non-proportional situations, as well as direct and inverse proportions. Furthermore, students often encounter difficulties when attempting to solve numerical comparison and qualitative comparison problems. This might be a consequence of students' lack of experience in solving these types of problems. Another tendency is the use of the cross-multiplication algorithm in solving missing-value problems without knowing the purpose of using the algorithm.

Proportional and Non-Proportional Situation: How to Make Sense of Them

<p style="text-align:justify">Teacher knowledge is one of the main factors in the quality of mathematics learning. Many mathematics teachers have difficulty using proportional reasoning. Proportional reasoning is one of the essential aspects of the middle school mathematics curriculum to develop students' mathematical thinking. Teachers should realize that developing proportional reasoning is not an easy task. In this study, we investigated how teachers give proportional reasoning about the concept of proportional and non-proportional situations, especially in making sense of them. The research subjects were mathematics teachers who had taught proportional-related material. Data was collected using task-based interviews outside the teacher's working hours. Data analysis and interpretation were completed using a framework meaning-based approach. The results of the data analysis showed that the teacher is careful in understanding information, is aware of multiple meanings, and knows key information in understanding the contextual structure of proportional and non-proportional situations. Furthermore, they are also able to identify additive and multiplication relationships, have flexibility in understanding proportional and non-proportional situations separately or collectively, and understand problem-solving systematics in detail.</p>

Incorrect Strategies Developed by Seventh- Grade Students to Solve Proportional Reasoning Problems

This study was conducted to determine the incorrect strategies developed by seventh-grade students to solve problems that require proportional reasoning and evaluate their solutions according to SOLO Taxonomy. This two-stage case study was conducted in a public school in the Central Black Sea region. The Proportional Reasoning Skill Test was administered to 33 seventh-grade students in the first stage. In the second stage, semi-structured interviews were held with 10 students by determining the 5 problems that students made mistakes in most. The students developed 5 different incorrect strategies when solving problems requiring proportional reasoning: additive relationships, data neglect, using numbers and no content, giving an emotional response, and failure to identify non-proportional situations. Students had difficulty identifying non-proportional situations and established additive relationships in problems requiring multiplicative relationships. Also, the levels of the participants were examined with the SOLO Taxonomy Rubric developed for proportional reasoning skills. Accordingly, the students with a high level of proportional reasoning were at abstract and relational structure levels, whereas those with moderate scores were at uni-structural and multi-structural levels. The students with low scores had low-level skills according to SOLO Taxonomy criteria.

The development and application of an interview structure on determining preservice mathematics teachers’ competence in proportional reasoning

Determining preservice teachers’ competence in proportional reasoning is a very complex task, and existing research methods do not provide effective tools in doing that. This study describes an interview structure and application of it in determining preservice teachers’ competence in proportional reasoning. The interview structure consists of in-depth questioning and cognitive conflicts and is situated in the knowledge-in-pieces epistemological perspective. The preservice teachers’ competence in proportional reasoning was determined by examining the knowledge resources that they drew upon when reasoning about proportional and nonproportional situations. Semi-structured interviews were conducted with six preservice teachers to demonstrate the effectiveness of the interview structure. The case analysis indicated the preservice teachers’ attention to the various fine-grained knowledge resources. However, they mostly drew upon the qualitative relationships, cross-multiplication, and across-multiplication. The preservice teachers’ over-attention to these three knowledge resources hindered their ability to distinguish proportional relationships from nonproportional relationships. Furthermore, the productivity of knowledge resources was easily influenced by the relationships presented in problems and participants’ past learning experiences. Implications of the application of the interview structure in teacher education programs and further research suggestions are discussed.

Fostering prospective mathematics teachers’ proportional reasoning through a practice-based instruction

ABSTRACT The purpose of this study was to investigate prospective middle school mathematics teachers’ proportional reasoning before and after receiving a practice-based instruction based on proportional reasoning. The Proportional Reasoning Test, semi-structured interviews and observations of student teachings were used to collect data about the participants’ proportional reasoning. The results indicated that prospective teachers improved their proportional reasoning by completing a practice-based instruction. Before the instruction, they generally applied algebraic procedures without associating meaning and used a limited number of strategies to solve problems. Furthermore, they had difficulties in distinguishing proportional from nonproportional situations. However, by the end of the instruction, while they mostly preferred to use informal strategies, they relied less on formal strategies. Additionally, they utilized a broader range of strategies to solve problems and made sense of these strategies. Further, they could determine whether the quantities in a situation were related additively, multiplicatively, or in some other way.

Elementary School Students' and Prospective Teachers' Proportional Reasoning Skills

The purpose of this study is to investigate prospective elementary school mathematics teachers and elementary school students' ability to distinguish proportional situations from non-proportional situations. Participants of the study were 319 prospective elementary teachers and 320 elementary school students. Four problems consisting of two proportional and two non-proportional situations were used as a data collection tool. Participants' strategies were coded as multiplicative, additive, and other. Findings indicated that while elementary school students were dominantly additive reasoners, prospective teachers were multiplicative reasoners. While prospective teachers tended to display multiplicative errors, elementary school students inclined to make additive errors in choice of strategy. It was striking that less than half of the prospective elementary mathematics teachers and only one-third of the prospective primary teachers was able to apply appropriate strategies.

Preservice Mathematics Teachers’ Understanding of and Abilities to Differentiate Proportional Relationships from Nonproportional Relationships

This study investigated preservice middle school mathematics teachers’ (PSTs) understanding of proportional and nonproportional relationships and their abilities to differentiate these relationships from each other. The PSTs’ abilities to interpret and represent proportional and nonproportional situations and solution strategies were also investigated. Forty PSTs who attended a mathematics education course on fractions, ratios, and proportions participated in the study. The data included the PSTs’ written responses to four open-ended problems and semi-structured interviews conducted with six of the PSTs. The analysis of the data showed that the PSTs mostly attended to the simultaneous increases or decreases and constancy of the rate of change when determining relationships. The PSTs’ over attention to these specific features constrained their understanding of the proportional and nonproportional relationships. Therefore, they had difficulty differentiating proportional relationships from nonproportional relationships even after instruction on these relationships. Moreover, some PSTs still identified nonproportional relationships as proportional after determining correct additive relationships. In addition, the PSTs had difficulty representing and interpreting proportional and nonproportional relationships and relied on cross-multiplication and across-multiplication strategies when solving the given problems.

Interim monitoring using the adaptively weighted log-rank test in clinical trials for survival outcomes.

For testing treatment effect with survival data, the log-rank test has been the method of choice and enjoys an optimality property under proportional hazards alternatives. However, there can be significant loss of power in a variety of nonproportional situations. Yang and Prentice proposed an adaptively weighted log-rank test that improves the power of the log-rank test over a wide range of hazard ratio scenarios. In clinical trials, the data and safety monitoring board typically monitors the trial results periodically. Here, we develop interim testing procedures for the adaptively weighted log-rank test. We establish the asymptotic distribution of the test statistics at the interim looks and use a resampling method to obtain the stopping boundaries. Through examination of the test behavior under local alternatives, we show that the adaptively weighted log-rank test is fully efficient when the limiting censoring distributions in the two groups are equal. Unlike the case for the usual application of the log-rank test, critical values obtained from the resampling method are asymptotically valid without the assumption of equal censoring distributions in the two groups. Extensive simulation studies show that the new method improves the log-rank test for a wide range of treatment effect patterns, and that the resampling approach yields better control of the type 1 error rate than the originally proposed approach in the work of Yang and Prentice. The new procedures are illustrated in several clinical trial examples.

Multiaxial fatigue assessment of crankshafts by local stress and critical plane approach

ABSTRACT. For multiaxially-loaded parts several stress-based fatigue assessment concepts are applicable mostly taking uniaxial test results as basis. These approaches work well in case of proportional loading states, but on contrary, for non-proportional stress conditions, implying a change of the principal stress direction, deviations in the fatigue life estimation may occur. The aim of this study is to evaluate the cyclic multiaxial material behavior experimentally and to proof the applicability of stress-based methods to assess the fatigue strength. The investigated base materials incorporate the commonly applied crankshaft steels 50CrMo4 and 34CrNiMo6 without surface-layer post-treatments. Extensive fatigue tests with small-scale specimens are performed to evaluate the material behavior under cyclic loading. The experiments include basic uniaxial characterizations, such as notch stress sensitivity and effect of loading type, including tests under tension, rotating bending, and torsion loading. Additionally, combined loading tests with proportional and non-proportional situations are presented to reveal the fatigue resistance for multiaxial stress states. Significant loading conditions, such as proportional stress under rotating bending and torsion, and further on, non-proportional effects like phase shifts and varying frequency ratios are presented. The local fatigue strength assessment is performed on the basis of the critical plane approach, whereat the normal and shear stresses are transformed in numerous cutting planes. Equivalent stress hypotheses are applied and compared with the experiments showing that the Huber-Mises-Hencky criterion fits well to the test results in case of proportional rotating bending and torsion loading. KEYWORDS. Multiaxial fatigue assessment; Local stress concept; Critical plane approach; Crankshaft.

DEVELOPING AN INSTRUMENT FOR MEASURING TEACHERS’ MATHEMATICS CONTENT KNOWLEDGE ON RATIO AND PROPORTION: A CASE OF INDONESIAN PRIMARY TEACHERS

In this study, we developed an instrument for assessing teachers’ mathematics content knowledge (MCK) on ratio and proportion and examined the profile of Indonesian primary teacher’s MCK on this topic. The MCK items were administered to 271 Indonesian in-service primary teachers with a variety of educational backgrounds and teaching experiences. Teachers’ responses were analyzed by factor analysis and cluster analysis. The MCK instrument was found to have good acceptability in the reliability analysis with 3 factor components—meaning of proportional and non-proportional situations, number structures in situation, and figural representation—which was the main result of the study. With respect to the 3 factors, the teachers in the 3 assigned categories (“Good,” “Middle,” or “Low”) showed consistent performance on the items of the 3 factors. In particular, our results indicated that Indonesian in-service primary teachers had difficulty with the factor on figural representation, but they performed best on number structures in situation representing products of proportional reasoning.

Conocimiento de matemáticas especializado de los estudiantes para maestro de primaria en relación al razonamiento proporcional

Un dominio particular del conocimiento matemático para la enseñanza es el conocimiento de matemáticas especializado. Este estudio se centra en examinar el conocimiento de matemáticas especializado en el ámbito del razonamiento proporcional de un grupo de estudiantes para maestro de Educación Primaria. Los resultados muestran que los estudiantes para maestro tienen un conocimiento especializado sobre el razonamiento proporcional limitado puesto de manifiesto por la dificultad en identificar situaciones no proporcionales, en desarrollar formas de razonar en relación a la construcción de la unidad y en manejar el significado multiplicativo de la idea de operador.

Development and application of a two-tier diagnostic instrument to assess middle-years students’ proportional reasoning

Proportional reasoning involves the use of ratios in the comparison of quantities. While it is a key aspect of numeracy, particularly in the middle years of schooling, students do not always develop proportional reasoning naturally. Research suggests that many students do not apply proportional methods appropriately and that they often erroneously apply both multiplicative and additive thinking. Further, students cannot always distinguish non-proportional situations from those that are proportional. Understanding the situations in which students mistakenly use additive or multiplicative thinking and the nature of the proportional reasoning that students apply to different problem types is important for teachers seeking to support their students to develop proportional reasoning in the classroom. This paper describes the development and use of a two-tier diagnostic instrument to identify situations in which students could and could not apply proportional reasoning and the types of reasoning they used. It presents data from an Australian study involving over 2000 middle-years students (Years 5 to 9) as a means of illustrating the use of the instrument for diagnosing students’ reasoning in different situations. The findings showed that the instrument was useful for identifying problem types in which students of different ages were able to apply correct reasoning. It also allowed identification of the types of incorrect reasoning used by students. The paper also describes useful applications of the instrument, including its use as a diagnostic instrument by classroom teachers and its use in the design of classroom activities included in teacher professional learning workshops.

Cognitive and Metacognitive Aspects of Proportional Reasoning

In this study we attempt to propose a new model of proportional reasoning based both on bibliographical and research data. This is impelled with the help of three written tests involving analogical, proportional, and non-proportional situations that were administered to pupils from grade 7 to 9. The results suggest the existence of a multi-component model of proportional reasoning, contributing in this way to the reformulation of the concept. In this model, the components of analogical reasoning, routine proportionality, and meta-analogical awareness take a constitutive part. Thus, proportional reasoning does not coincide exclusively with success in solving a certain restricted range of proportional problems, as routine missing-value and comparison problems (routine proportionality), but it also involves handling verbal and arithmetical analogies (analogical reasoning) as well as the awareness of discerning non-proportional situations (meta-analogical awareness), which is metacognitive in nature.

Remedying secondary school students’ illusion of linearity: a teaching experiment aiming at conceptual change

Already at a very young age, children experience the wide applicability and intrinsic simplicity of linear/proportional relations. In primary and secondary school mathematics education, moreover, extensive attention is paid to this type of relations. In the long run, students develop the misbelief that each relation can be quantified as proportional, called the “illusion of linearity”. The best-known misconception originating from such a “synthetic model of linearity” is that if a geometrical figure enlarges k times, its area and/or volume become k times larger too. This article reports and discusses a teaching experiment aimed at remedying this misconception in 8th graders. Ten experimental lessons were developed in order to obtain a conceptual change in these students. The learning results were tested by means of a pretest–post-test–retention test design with an experimental and control group. The problem-solving behaviour of control group students did not change. In the experimental group, the intervention was successful: students’ automatic use of proportional strategies for solving non-proportional geometry problems drastically decreased. Never theless, some students continued to reason proportionally for all types of problems, while others suddenly started to apply non-proportional strategies to proportional problems too. The linearity illusion was broken in most students, but this did not always result in a deep conceptual understanding of proportional and non-proportional situations and relations.

Estimation in hazard regression models under ordered departures from proportionality

Estimation of age-varying covariate effects in hazard regression models is considered, where covariate dependence is monotone rather than proportional. It is well-known that the proportionality assumption underlying the Cox proportional hazards model does not hold in many applications. Further, the effect of a covariate is often monotone, in the sense that the lifetime (or duration) conditional on a higher value of the covariate ages faster or slower than that conditional on a lower value. These ordered/monotone alternatives to proportionality constitute a popular model for covariate effects in the two-sample setup, and have recently been extended to the case of continuous covariates. Under these models, the ratio of hazard functions (or cumulative hazard functions) increases or decreases with duration. Ordered departures of this kind are common in applications, and the models provide useful and intuitively appealing descriptions of covariate dependence in non-proportional situations. Here, methods for estimating hazard regression models under monotone departures are proposed. In particular, it is shown how the histogram sieve estimators can be smoothed and order restricted estimation performed using biased bootstrap techniques such as adaptive bandwidth kernel estimators or data tilting. The performance of the estimators is compared using simulated data, and their use is illustrated with applications from biomedicine and economic duration data.