Observing mathematical fluency through students’ oral responses

‘Procedural’ fluency in mathematics is often judged solely on numerical representations. ‘Mathematical’ fluency incorporates explaining and justifying as well as producing correct numerical solutions. To observe mathematical fluency, representations additional to a student’s numerical work should be considered. This paper presents analysis of students’ oral responses. Findings suggested oral responses are important vantage points from which to view fluency – particularly characteristics harder to notice through numerical work such as reasoning. Students’ oral responses were particularly important when students’ written (language) responses were absent/inconsistent. Findings also revealed the importance of everyday language alongside technical terms for observing reasoning as a fluency characteristic. Students used high modality verbs and language features, such as connectives, to explain concepts and justify their thinking. The results of this study purport that to gain a fuller picture of students’ fluency, specifically their explanations or reasoning, students’ oral responses should be analyzed, not simply numerical work.     

Learning to represent,representing to learn

This study explores how students learn to create, discuss, and reason with representations to solve problems. A summer school algebra class for seventh and eighth graders provided opportunities for students to create and use representations as problem-solving tools. This case study follows the learning trajectories of three boys. Two of the three boys had been low-achievers in their previous math classes, and one was a high achiever. Analysis of all three boys’ written work reveals how their representations became more sophisticated over time. Their small group interactions while problem-solving also show changes in how they communicated and reasoned with representations. For these boys, representation functioned as a learning practice. Through constructing and reasoning with representations, the boys were able to engage in generalizing and justifying claims, discuss quadratic growth, and collaborate and persist in problem-solving. Negotiating different student-constructed representations of a problem also gave them opportunities to act with agency, as they made choices and judgments about the validity of the different perspectives. These findings have implications for the importance of giving all students access to mathematics through representations, with representational thinking serving as a central disciplinary practice and as a learning practice that supports further mathematics learning.     

Students’ multilingual repertoires-in-use for meaning-making: Contrasting case studies in three multilingual constellations

Mathematics education for multilingual classrooms calls for instructional approaches that build upon students’ multilingual resources. However, so far, students’ multilingual resources and the interplay of their components have only partly been disentangled and rarely compared between different multilingual contexts. This article suggests a conceptualization of multilingual repertoires-in-use as characterized by (a) what students use of certain languages, registers, and representations as sources for meaning-making in mathematics classrooms and (b) their processes of how they connect certain languages, registers, and representations. This qualitative learning-process study compares students’ multilingual repertoires-in-use in three contexts: Spanish-speaking foreign language learners of German in Colombia, Turkish- and German-speaking students born in Germany, and Arabic-speaking German language beginners recently immigrated to Germany. The analysis reveals the biggest differences not only in what the students use, but how they connect languages, registers, and representations. Some of these differences can partly be traced back to different classroom cultural practices. These findings suggest extending the conceptual framework for multilingual repertoires-in-use and including it in a social theoretical perspective. Thus, these findings have important practical consequences for multilingual mathematics classrooms: The instructional approach of relating languages, registers, and representations needs to be applied more flexibly, taking into account students’ different starting points. When doing so, students’ connection processes should be supported and explicated more systematically in order to fully exploit the students’ repertoires.     

The impact of instructor pedagogy on college calculus students’ attitude toward mathematics

College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students’ attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students’ attitudes toward mathematics during a ‘mainstream’ calculus course while controlling for student backgrounds. The attitude measure combines students’ self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted ‘good teaching’ practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational ‘technology’ (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students’ attitudes towards mathematics. (3) ‘Ambitious teaching’ (e.g. group work, word problems, ‘flipped’ reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional ‘good teaching’ practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do not appear to have a reliable, positive impact on student attitudes toward mathematics.     

Assessing ‘functionality’ in school mathematics examinations: what does being human have to do with it?

This article analyses aspects of the process of developing ‘functional’ assessments of mathematics at the end of compulsory schooling in England. A protocol that was developed for scrutinising assessment items is presented. This protocol includes an indicator of the ‘authenticity’ of each assessment item. The data are drawn from scrutiny of 589 assessment items from thirty-nine formal unseen examinations taken by students aged sixteen, and the article illustrates ways that mathematics is presented in different contexts in examinations. We suggest that currently the ‘human face’ of the questions may serve to disguise routine calculations, and we argue that, in formal examinations, connections between mathematics assessments situated in context and functional mathematics have yet to be established.     

Using algorithmic thinking to design algorithms: The case of critical path analysis

Algorithmic thinking is emerging as an important competence in mathematics education, yet research appears to be lagging this shift in curricular focus. The aim of this generative study is to examine how students use the cognitive skills of algorithmic thinking to design algorithms. Task-based interviews were conducted with four pairs of Year 12 students (n = 8) to analyze how they used decomposition and abstraction to specify the projects, designed algorithms to solve scheduling problems by first devising fundamental operations and then using algorithmic concepts to account for complex and special cases of the problems, and tested and debugged their algorithms. A deductive-inductive analytical process was used to classify students’ responses according to the four cognitive skills to develop sets of subskills to describe how the students engaged these cognitive skills.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

Using dynamic software in mathematics: the case of reflection symmetry

This study was carried out to examine the effects of computer-assisted instruction (CAI) using dynamic software on the achievement of students in mathematics in the topic of reflection symmetry. The study also aimed to ascertain the pre-service mathematics teachers’ opinions on the use of CAI in mathematics lessons. In the study, a mixed research method was used. The study group of this research consists of 30 pre-service mathematics teachers. The data collection tools used include a reflection knowledge test, a survey and observations. Based on the analysis of the data obtained from the study, the use of CAI had a positive effect on achievement in the topic of reflection symmetry of the pre-service mathematics teachers. The pre-service mathematics teachers were found to largely consider that a mathematics education which is carried out utilizing CAI will be more beneficial in terms of ‘visualization’, ‘saving of time’ and ‘increasing interest/attention in the lesson’. In addition, it was found that the vast majority of them considered using computers in their teaching on the condition that the learning environment in which they would be operating has the appropriate technological equipment.     

Relationships between emotional dispositions and mathematics achievement moderated by instructional practices: analysis of TIMSS 2015

ABSTRACT

This research is a secondary analysis with Korean students’ data collected in the TIMSS 2015 to describe the moderation effects of instructional practices on the relationships between students’ emotional dispositions toward mathematics and mathematics achievement. From the TIMSS 2015 database, we collected mathematics achievement scores, a student-level contextual scale for students’ emotional disposition, and teacher-level contextual scales representing teachers’ instructional practices. We applied hierarchical linear modelling to construct multilevel models. The findings showed that the achievement gap between emotional dispositions – like and dislike – became smaller when teachers more frequently implemented certain instructional practices like asking students to complete challenging exercises, decide their own problem-solving procedures, and express their ideas in class. Students who disliked mathematics were likely to have higher scores as their teachers implemented each of those practices more frequently. Findings provide important implications to teachers regarding: It is important to encourage students to reason through instructional practices like asking them to decide their own problem-solving procedures and to solve challenging problems.     

Perceived parental influence and students' dispositions to study mathematically-demanding courses in Higher Education

We explore the influence of family on adolescent students' mathematical habitus by investigating the association between students' perceptions of parental influence and their dispositions towards mathematics. A construct measuring ‘perceived parental influence’ was validated using Rasch methodology on data from 563 Cypriot students on ‘core’ and ‘advanced’ mathematics pre-university courses, and was then used to predict students' dispositions towards future study of mathematically-demanding courses at university. In most of the regression models, perceived parental influence was not associated significantly with students' dispositions towards mathematics, when other variables were included in the models. However, further statistical analysis showed that perceived parental influence is mediated by (i) the mathematics course students are studying and (ii) their mathematical inclination. We suggest that family influences on students' dispositions are significantly accounted for by students' prior choice of mathematics course and the family's inculcation of their mathematical inclination; these are important factors influencing university choices.     

A reform in undergraduate mathematics curriculum : more emphasis on social and pedagogical skills

The paper describes the changes that are being made in the mathematics teachers' subject studies in the Department of Mathematics at the University of Joensuu, in order to provide our mathematics students both with a sufficiently deep knowledge of mathematics and science, and with present-day expertise in their profession as teachers. While the formal structure of the mathematics curriculum remains structured and taught as courses with mostly traditional names like algebra, analysis, and linear algebra, there are also totally new ‘professionally oriented’ courses. Some of the old courses—with rather traditional and rigorous contents—have been changed in a more student-driven direction. In these ‘pedagogically oriented’ courses students are encouraged, and even forced, to study co-operatively in social interaction, for example to negotiate how to solve a problem decently, or how to build a formal definition for a concept with certain wanted attributes. As an ultimate example of a pedagogical experiment we describe in more detail an abstract algebra course, where co-operative learning is combined with intensive programming in a mathematically oriented computer environment.     

Capturing and characterizing students’ strategic algebraic reasoning through cognitively demanding tasks with focus on representations

In mathematics education, it is important to assess valued practices such as problem solving and communication. Yet, often we assess students based on correct solutions over their problem solving strategies—strategies that can uncover important mathematical understanding. In this article, we first present a framework of competencies required for strategic reasoning to solve cognitively demanding algebra tasks and assessment tools to capture evidence of these competencies. Then, we qualitatively describe characteristics of student reasoning for various performance levels (low, medium, and high) of eighth-grade students, focusing on generating and interpreting algebraic representations. We argue this analysis allows a more comprehensive and complex perspective of student understanding. Our findings lay groundwork to investigate the continuum of algebraic understanding, and may help educators identify specific areas of students’ strength and weakness when solving cognitively demanding tasks.     

Spatial visualization and measurement of area: A case study in spatialized mathematics instruction

The purpose of this study was to explore the influence of spatial visualization skills when students solve area tasks. Spatial visualization is closely related to mathematics achievement, but little is known about how these skills link to task success. We examined middle school students’ representations and solutions to area problems (both non-metric and metric) through qualitative and quantitative task analysis. Task solutions were analyzed as a function of spatial visualization skills and links were made between student solutions on tasks with different goals (i.e., non-metric and metric). Findings suggest that strong spatial visualizers solved the tasks with relative ease, with evidence for conceptual and procedural understanding. By contrast, Low and Average Spatial students more frequently produced errors due to failure to correctly determine linear measurements or apply appropriate formula, despite adequate procedural knowledge. A novel finding was the facilitating role of spatial skills in the link between metric task representation and success in determining a solution. From a teaching and learning perspective, these results highlight the need to connect emergent spatial skills with mathematical content and support students to develop conceptual understanding in parallel with procedural competence.     

Socio-economic status and mathematics achievement in China: a review

Mathematics education is a cultural-specific social activity. China, as a developing country with a long history and a unique culture, has the largest number of teachers and students in the world. Hence, it is of significance to explore the issue of the impact of socio-economic status (SES) on mathematics education within the Chinese context. However, investigations aiming to address this issue are relatively rare. This study was designed to examine the relationship between Chinese students’ SES and their mathematics achievements. Results reveal that Chinese students’ SES exerts significant influence on their mathematics achievements, and several important constituents of SES, such as parents’ education and family income, stand out among others. In this paper, the cultural causes of the influence are discussed, together with a general introduction to the social and educational context. These could partly explain the empirical results, along with factors such as the values of education in traditional Chinese culture and the current important status of mathematics in modern society as well as Chinese school curriculum materials’ effect on students’ mathematics achievements. The economic and social situation in China, especially the imbalanced distribution of educational resources between and within the urban and rural areas, could magnify the role of SES in mathematics achievements. Finally, the future direction of measuring and interpreting the SES’s influence on mathematics achievement in the Chinese context is also discussed.     

Mental Computation of Students in a Reform-Based Mathematics Curriculum

Traditional school instruction in mathematics has generally produced students who are poor at mental computation and exhibit a weak sense of number and mathematical operations. In this study, fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems. Baseline data with students in traditional mathematics curricula were used as a comparison. The students in this reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. Additionally, a student preference survey indicated that students in the reform curriculum were more likely to consider the calculator as an option than were the baseline group. They were also more able to recognize problems that did not lend themselves to mental computation. Individual interviews indicated that experiences in the primary grades with “invented” algorithms and discussing alternative solutions led to a better ability to compute mentally and a stronger number sense.     

Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed.     

Can peer assisted learning be effective in undergraduate mathematics?

We report the implementation and evaluation of a ‘peer assisted learning’ (PAL) scheme designed to reduce the so-called ‘cooling off’ phenomenon in undergraduate mathematics. ‘Cooling off’ occurs when mathematics undergraduates lose motivation and interest in their studies, despite having previously actively chosen to study it at higher levels. We found that, despite concerns about the novel didactic contract inherent in PAL schemes, a majority of students chose to engage with the scheme, and that the student leaders of the PAL sessions were generally capable of implementing a student-centred pedagogy. Furthermore, we found that students who attended the PAL sessions had higher achievement in their final examinations, even after controlling for their lecture attendance and prior attainment. We conclude by arguing that PAL may provide a useful mechanism for reducing the prevalence of the ‘cooling off’ phenomenon in some – but not all – groups of mathematics students.     

Mathematics-related teaching competence of Taiwanese primary future teachers: evidence from TEDS-M

This paper draws on data from the international TEDS-M study, organized by the IEA, and utilizes a conceptual framework describing the Taiwanese perspective of mathematics and mathematics teaching competences (MTCs) with regard to investigating the uniqueness and patterns of Taiwanese future primary teacher performance in the international context. The framework includes content-oriented and thought-oriented categories of mathematics competence. The latter category contains subcategories adopted and revised from (3rd Mediterranean conference on mathematical education. Hellenic Mathematical Society, Athens, 2003) the competence approach by Niss. Hsieh’s (Research on the development of the professional ability for teaching mathematics in the secondary school level (3/3). Taiwan: National Science Council, 2009) model is also adopted and revised to serve as an analytical framework, including four categories relating to MTCs, representations, language, and misconceptions or error procedures. This paper shows that in thought-oriented mathematics competences Taiwan and Singapore share a unique pattern of higher percent correct in competences related to formalization, abstraction, and operations in mathematics than in those related to the way of thinking, modelling and reasoning in and with mathematics. The paper also addresses weak teaching competences claimed in domestic studies, which conflict with the TEDS-M results. Namely, in contrary to the international trend, Taiwanese future primary teachers are weak at judging mathematics competences required by students to learn mathematical concepts or solve problems, and superior at diagnosing and dealing with student misconceptions and error procedures.