Correlation between performance in physics and prior mathematics knowledge

The final grade of 1403 students enrolled in the first semester of the introductory, pre-professional physics course has been correlated with performance on a precourse diagnostic test of mathematical skills. The students were from a total of eight different sections taught by six separate instructors over a three year time span. The student population has been separated into two groups, those who completed the course (913 students) and those who dropped (490 students). The drops were assigned a “projected final grade”based on performance up to date of withdrawal. The Pearson product-moment correlation for students who completed the course is 0.418 and correlation for the drops is 0.232. Both correlations are significant at the p < 0.001 level. This study suggests that prior mathematical ability is a primary influence on performance in the course, and has a secondary influence on the tendency to drop out of the course.     

“Can I do well in mathematics reasoning?” Comparing US and Finnish students’ attitude and reasoning via TIMSS 2011

This study investigated differences between the US and Finland in terms of how students’ attitude is related to mathematical reasoning skills through the Trends in International Mathematics and Science Study (TIMSS) 2011. Attitude towards mathematics was observed via 3 TIMSS contextual variables: liking mathematics, valuing mathematics, and confidence in mathematics. Scores for mathematical reasoning were collected from the TIMSS 2011 database. We used hierarchical linear modelling to construct multilevel models with interactions of the attitude variables. Findings showed that confidence in mathematics had the strongest positive relationships with mathematical reasoning in both countries. Finnish students generally reported stronger positive relationships between confidence in mathematics and reasoning than US students. Strong relationships between confidence and reasoning remained visible when examining valuing and liking mathematics. Findings provide important implications regarding the complex interactions between attitude towards mathematics and reasoning, critical for mathematics educators and policymakers to consider in an increasingly competitive international environment.     

Mathematical Reasoning Requirements in Swedish National Physics Tests

This paper focuses on one aspect of mathematical competence, namely mathematical reasoning, and how this competency influences students’ knowing of physics. This influence was studied by analysing the mathematical reasoning requirements upper secondary students meet when solving tasks in national physics tests. National tests are constructed to mirror the goals stated in the curricula, and these goals are similar across national borders. The framework used for characterising the mathematical reasoning required to solve the tasks in the national physics tests distinguishes between imitative and creative mathematical reasoning. The analysis process consisted of structured comparisons between representative student solutions and the students’ educational history. Of the 209 analysed tasks, 3/4 required mathematical reasoning in order to be solved. Creative mathematical reasoning, which, in particular, involves reasoning based on intrinsic properties, was required for 1/3 of the tasks. The results in this paper give strong evidence that creative mathematical reasoning is required to achieve higher grades on the tests. It is also confirmed that mathematical reasoning is an important and integral part of the physics curricula; and, it is suggested that the ability to use creative mathematical reasoning is necessary to fully master the curricula.     

Zwischen Schule und Universität: Argumentation in der Mathematik

Several studies show that university students in Germany still have problems in reasoning mathematically although this already should be fostered at high school since the implementation of standards for school mathematics. Mathematical argumentation is a core competence and highly important, especially in academic mathematics. To foster mathematical argumentation at the beginning of university studies, competence models are needed which give more detailed insights in the skills that are necessary for reasoning. As mathematical argumentation is a complex process, especially at the higher secondary level or at university, many little steps are needed to complete a competence model for argumentation at the secondary–tertiary transition gradually. A possible step can be to initially identify several aspects of mathematical argumentation competence that influence the reasoning quality. The empirical basis for identifying those aspects is a cross-sectional study with 439 engineering students who participate in a transition course in mathematics. We address the following questions: (1) how is the quality of student’s reasoning? (2) Which kind of arguments do students use? (3) What resources do students who reasoned correctly use for solving the problems? (4) Does the content of the tasks play an important role? The results show a great influence of the content on the reasoning quality, especially if the content is abstract or concrete. Argumentation quality of students decreases with an increasing level of abstraction of the content. Furthermore, the results reveal that students often use routines for solving the problems. That indicates that procedural approaches still play an important role in school mathematics. If procedures could be used for solving the tasks, students are more successful. Competence models for mathematical argumentation at the beginning of the tertiary level should, therefore, include these factors.     

Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students’ problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.     

A study of two measures of spatial ability as predictors of success in different levels of general chemistry

Preliminary data (Bodner and McMillen, 1986) suggested a correlation between spatial ability and performance in a general chemistry course for science and engineering majors. This correlation was seen not only on highly spatial tasks such as predicting the structures of ionic solids (r = 0.29), but also on tasks such as multiple-choice stoichiometry questions (r = 0.32) that might not be expected to involve spatial skills. To further investigate the relationship between spatial ability and performance in introductory chemistry courses, two spatial tests were given to 1648 students in a course for science and engineering majors (Carter, 1984) and 850 students in a course for students from nursing and agriculture (La-Russa, 1985) at Purdue. Scores on the spatial tests consistently contributed a small but significant amount to success on measures of performance in chemistry. Correlations were largest, however, for subscores that grouped questions that tested problem solving skills rather than rote memory or the application of simple algorithms, and correlations were also large for verbally complex questions thaty required the students to disembed and restructure relevant information.     

The impact of the pre-instructional cognitive profile on learning gain and final exam of physics courses: a case study

The case study described in this paper investigates the relationship among some pre-instructional knowledge, the learning gain and the final physics performance of computing engineering students in the introductory physics course. The results of the entrance engineering test (EET) have been used as a measurement of reading comprehension, logic and mathematics skills and basic physics knowledge of a sample of 47 Computing Engineering freshmen at the University of Palermo (Italy). These data give a significant picture of the initial knowledge status of a student choosing engineering studies. The students' physics learning gain has been calculated using a standardized tool in mechanics: the force concept inventory (FCI). The analysis shows that mathematical and physical background contribute to achieve a good final preparation in physics courses of engineering faculties; however the students' learning gain in physics is independent of students' initial level of mathematics skills and physics knowledge. Initial logic skills and reading comprehension abilities are not significant factors for the learning physics gain and the performance on physics courses.     

Spatial ability and achievement in introductory physics

This research was undertaken to clarify the nature of the relationship between visual-spatial abilities and achievement in science courses. A related purpose was to determine what influence visual-spatial abilities have on the high attribution rate characteristic of many introductory college-level science courses. Three sections of introductory college level physics (S = 136) and one nonscience liberal arts section (S = 52) received pre- and postmeasures of visual-spatial ability in the areas of perception, orientation, and visualization. Increases in visual-spatial abilities were greatest with an experimental section that received a spatial intervention. These gains were related to test items that utilized graphical form and to laboratory work. Substantial gains in visual-spatial ability were also registered by a placebo and by control sections. These increases suggest that taking introductory physics improves visual-spatial abilities. Although students who withdrew from the course demonstrated mathematics skills comparable to those of students who completed the course, their scores on perception tests were appreciably lower. Visual-spatial scores of the liberal arts group were lower than those of the physics sections, suggesting that visual-spatial ability influences course selection.     

Success of students in a college physics course with and without experiencing a high school course

High school students with high ability were enrolled in a standard college physics course for each of two summers with the same professor, same course outline, same textbook, same laboratories, and the same examinations. Half of each group had completed a high school physics course; half had not. Dormitory counselors were available for assistance and support. In addition, tutors were available in the laboratories to provide any help necessary with interpretation of lectures and performances in the laboratory, and with mathematical computation. Pre- and posttest measures concerning course content and attitude were given. After the eight-week summer instruction, the students who had not completed high school physics performed as well on the final course examination; there were no differences with respect to course grade or attitude toward physics. The group that had not completed high school physics used the tutors provided far more frequently than did students who had completed the high school course. When high-ability students are enrolled in college physics with tutors made available for needed assistance, there appears to be no advantage for students to complete the standard high school physics course.     

A CROSS-CURRICULAR INTEGRATED LEARNING EXPERIENCE IN MATHEMATICS AND PHYSICS

Since April, 1996, Kanazawa Technical College has offered a cross-curricular course for first-year and second-year students using a TI-83 (Graphing Calculator) and a Computer Based Laboratory (CBL). The goal of the course is for students to learn about the connection between mathematics and physics through hands-on activities. Students conduct experiments on the motion of a person walking, the dropping of an object, the cooling rate of water, the motion of a swinging pendulum, and sound waves. This is the first time that the TI-83 calculator and the CBL have been used in Japan to explore physical phenomena in a classroom situation based on the Japanese curriculum for mathematics and physics. The following findings were obtained: Most students (a) replaced their naive assumptions regarding the laws of physics with scientific concepts; (b) independently made connections between the results of experiments and their previous mathematical knowledge; (c) reported that their level of interest in physical phenomena and science had either not decreased or had improved, (d) valued mathematics more, and (e) realized the importance of cooperative work. The use of CBLs and TI-83s changed not only the authors teaching style but also students attitudes. Students had ownership of their experiments, and they engaged in higher-order thinking skills such as making predictions, analyzing data, and modeling data with equations. As a result, students became more interested in learning mathematics and science.     

Mathematical development: the role of broad cognitive processes

This study investigated the role of broad cognitive processes in the development of mathematics skills among children and adolescents. Four hundred and forty-seven students (age mean [M] = 10.23 years, 73% boys and 27% girls) from an elementary school district in the US southwest participated. Structural equation modelling tests indicated that calculation complexity was predicted by long-term retrieval and working memory; calculation fluency was predicted by perceptual processing speed, phonetic coding, and visual processing; problem solving was predicted by fluid reasoning, crystallised knowledge, working memory, and perceptual processing speed. Younger students’ problem solving skills were more strongly associated with fluid reasoning skills, relative to older students. Conversely, older students’ problem solving skills were more strongly associated with crystallised knowledge skills, relative to younger students. Findings are consistent with the theoretical suggestion that broad cognitive processes play specific roles in the development of mathematical skills among children and adolescents. Implications for educational psychologists are discussed.     

Mathematics for teaching and deep subject knowledge: voices of Mathematics Enhancement Course students in England

This article reports an investigation into how students of a mathematics course for prospective secondary mathematics teachers in England talk about the notion of ‘understanding mathematics in depth’, which was an explicit goal of the course. We interviewed eighteen students of the course. Through our social practice frame and in the light of a review of the literature on mathematical knowledge for teaching, we describe three themes that weave through the students’ talk: reasoning, connectedness and being mathematical. We argue that these themes illuminate privileged messages in the course, as well as the boundary and relationship between mathematical and pedagogic content knowledge in secondary mathematics teacher education practice.     

Associations between Fine Motor and Mathematics Instruction and Kindergarten Mathematics Achievement

Research Findings: This study investigates the role of fine motor and mathematics instruction in mathematics achievement in an international sample of kindergarteners from the United States and China. Multilevel modeling was used to assess the interaction between students’ entering skills and classroom time spent on basic math, higher-order math and fine-motor instruction. For American children, the effect of basic math and higher-order math instruction on student achievement depended on entering skills; however, fine motor instruction had negative average effects on student achievement and did not depend on students’ entering skills. Instruction time was not a significant predictor of achievement for Chinese students. Practice or Policy: Though fine motor skills have a robust correlation with mathematics achievement, a causal link has not been established. Our study indicates that time spent in fine motor instruction does not advance mathematics achievement in kindergarten and in fact may weaken mathematics achievement, given the limited time in the instructional day. American teachers in our sample who spent more time in fine-motor instruction tended to spend less time on basic math and higher-order mathematics instruction. Educators should weigh instructional trade-offs carefully and work to tailor instruction to students’ skill levels.     

数学物理方程课程研究性教学探索

数学物理方程课程是数学与应用数学专业的一门基础课,是研究物理学、工程学以及其它自然科学、工程技术中产生的一些典型偏微分方程的课程。它是数学联系实际的一个重要桥梁,但素来以“繁”、 “难”的特点让学生生畏,学生处于被动接受状态,主动探索的空间狭小。本文探讨数学物理方程课程研究性教学的一些尝试、见解和体会。     

Mathematical thinking of kindergarten boys and girls: similar achievement, different contributing processes

The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However, mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other forms of mathematics communication and teaching.     

Assessing the cognitive abilities of culturally and linguistically diverse students: Predictive validity of verbal,quantitative, and nonverbal tests

Verbal and quantitative reasoning tests provide valuable information about cognitive abilities that are important to academic success. Information about these abilities may be particularly valuable to teachers of students who are English‐language learners (ELL), because leveraging reasoning skills to support comprehension is a critical aptitude for their academic success. However, due to concerns about cultural bias, many researchers advise exclusive use of nonverbal tests with ELL students despite a lack of evidence that nonverbal tests provide greater validity for these students. In this study, a test measuring verbal, quantitative, and nonverbal reasoning was administered to a culturally and linguistically diverse sample of students. The two‐year predictive relationship between ability and achievement scores revealed that nonverbal scores had weaker correlations with future achievement than did quantitative and verbal reasoning ability scores for ELL and non‐ELL students. Results do not indicate differential prediction and do not support the exclusive use of nonverbal tests for ELL students. © 2012 Wiley Periodicals, Inc.     

Teachers attending to students’ mathematical reasoning: lessons from an after-school research program

There is a documented need for more opportunities for teachers to learn about students’ mathematical reasoning. This article reports on the experiences of a group of elementary and middle school mathematics teachers who participated as interns in an after-school, classroom-based research project on the development of mathematical ideas involving middle-grade students from an urban, low-income, minority community in the United States. For 1 year, the teachers observed the students working on well-defined mathematical investigations that provided a context for the students’ formation of particular mathematical ideas and different forms of reasoning in several mathematical content strands. The article describes insights into students’ mathematical reasoning that the teachers were able to gain from their observations of the students’ mathematical activity. The purpose is to show that teachers’ observations of students’ mathematical activity in research sessions on students’ development of mathematical ideas can provide opportunities for teachers to learn about students’ mathematical reasoning.     

Effects of Metacognitive Skills and Nonverbal Ability on Academic Achievement of High School Pupils

Two experiments were conducted with senior high school pupils in the North‐West Province of South Africa to examine relationships of metacognitive strategies and nonverbal reasoning ability to their performance in tests of mathematics and English comprehension. The analyses of data revealed that both metacognitive ability and nonverbal reasoning ability have significant positive association with mathematics and English achievement scores. Significant sex differences in mathematics performance were also found. The findings of the two experiments suggested that some intervention programmes to teach metacognitive strategies to students, who lack such skills, may improve their academic attainment