Appropriate use of a CAS in the teaching and learning of mathematics

If the use of a computer algebra system (CAS) is to be meaningful and have an impact on students, then it must be grounded in good pedagogy and have some clearly defined goals. It is the authors' belief that an important goal for teaching mathematics with the CAS is that courses be designed so that students can become active participants in their learning experience, planning the problem-solving strategies and carrying them out. The CAS becomes an important tool and a partner in this learning process. To this end, here the authors' have linked the use of the CAS to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.     

Developing mathematical modelling skills: The role of CAS

Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters This is an important part of validating a mathematical model formulated to describe, a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.     

CAS in general mathematics education

A rational discussion of the use of Computer algebra systems (CAS) in mathematics teaching in general education needs an explicit image of (general) mathematics education, an explication of global perspectives and goals on mathematics teaching focusing on general education (chapter 1). The conception of general education according to the «ability of communication with experts» described in chapter 2 can be such an orientation for analysing, considering, classifying and assessing the didactical possibilities of using CAS. CAS are materialised mathematics allowing for more or less exhaustive outsourcing of operative (also symbolically) knowledge and skills to the machine. This frees up space of time as well as mental space for the development of those competences being in our view relevant for general mathematics education. In chapter 3 the idea of outsourcing and the role of CAS for it is discussed more detailed as well as consequences being possible for the CAS-supported teaching of mathematics. Beyond, CAS can be didactically used and reflected as a model of communication between (mathematical) experts and lay-persons (chapter 4). Chapter 5 outlines some research perspectives.     

Influences of CAS and GC in early algebra

In the last years, several studies have investigated the role of technology in teaching and learning mathematics. However, the specific role of computer algebra systems (CAS) in early algebra in contrast to graphic calculators (GC) is still unclear. The CAYEN project is researching this field by comparing 13-year-old pupils—one GC class and two CAS classes have been observed while acquiring elementary algebraic competences with nearly the same teaching sequence. The field of algebraic competences is split into syntactic abilities and symbol sense. The results of this explorative case study show that the development of symbol sense is influenced by the adoption of CAS in the learning process. Especially when transitioning from arithmetic to algebra, the pupils’ views of algebra as well as their conceptions of algebraic objects seem to be affected by the availability of CAS.     

Attitude towards mathematics: a bridge between beliefs and emotions

Recent research in the field of affect has highlighted the need to theoretically clarify constructs such as beliefs, emotions and attitudes, and to better investigate the relationships among them. As regards the definition of attitude, in a previous study we proposed a characterization of attitude towards mathematics grounded in students’ experiences, investigating how students express their own relationship with mathematics. The data collected suggest a three-dimensional model of attitude towards mathematics that includes students’ emotional disposition, their vision of mathematics, and their perceived competence. In this paper, we discuss the relationship between beliefs and emotions, investigating the interplay among the three dimensions in the proposed model of attitude, as emerging in the students’ essays.     

Pupils’ attitudes towards mathematics: a comparative study of Norwegian and English secondary students

Comparing English and Norwegian pupils’ attitude towards mathematics, in this article I develop a deeper understanding of the factors that may shape and influence ‘pupil attitude towards mathematics’, and argue for it as a socio-cultural construct embedded in and shaped by students’ environment and context in which they learn mathematics. The theoretical framework leans on work by Zan and Di Martino (The Montana Mathematics Enthusiast, Monograph 3, pp. 157–168, 2007) to elicit Norwegian and English pupils’ attitude of mathematics as they experience it in their respective environments. Whilst there were differences which could be seen to be accounted for by differently ‘figured’ environments, there are also many similarities. It was interesting to see that, albeit based on a small statistical sample, in both countries students had a positive attitude towards mathematics in year 7/8, which dropped in year 9, and increased again in years 10/11. This result could be explained and compared with other larger scale studies (e.g. Hodgen et al. in Proceedings of the British Society for Research into Learning Mathematics. 29(3), 2009). The analysis of pupils’ qualitative comments (and classroom observations) suggested seven factors that appeared to influence pupil attitude most, and these had ‘superficial’ commonalities, but the perceptions that appeared to underpin these mentions were different, and could be linked to the environments of learning mathematics in their respective classrooms. In summary, it is claimed that it is not enough to identify the factors that may shape and influence pupil attitude, but more importantly, to study how these are ‘lived’ by pupils, what meanings are made in classrooms and in different contexts, and how the factors interrelate and can be understood.     

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.     

Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work

The last decade has seen the development in France of a significant body of research into the teaching and learning of mathematics in CAS environments. As part of this, French researchers have reflected on issues of ‘instrumentation’, and the dialectics between conceptual and technical work in mathematics. The reflection presented here is more than a personal one – it is based on the collaboration and dialogues that I have been involved in during the nineties. After a short introduction, I briefly present the main theoretical frameworks which we have used and developed in the French research: the anthropological approach in didactics initiated by Chevallard, and the theory of instrumentation developed in cognitive ergonomics. Turning to the CAS research, I show how these frameworks have allowed us to approach important issues as regards the educational use of CAS technology, focusing on the following points: the unexpected complexity of instrumental genesis, the mathematical needs of instrumentation, the status of instrumented techniques, the problems arising from their connection with paper & pencil techniques, and their institutional management. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Application of a Computer Algebra System as a Tool in College Algebra

This study compared a traditional lecture-based college algebra course to an experimental algebra course. The experimental course stressed active student involvement and the use of the computer as a tool to explore mathematics. One hundred thirty-seven subjects were divided into an experimental group and a control group. Subjects in the experimental group scored significantly higher than the control group on a final measure of inductive reasoning, visualization, and problem solving while maintaining an equivalent level of manipulation and computation skills. However, the attitude of subjects in the experimental group towards the use of the computer in learning mathematics declined significantly.     

Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity.

The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt.     

Impacting the Science Attitudes of Minority High School Youth

A study involving urban, minority high school students in a supplemental education program was conducted during 1989 to test the null hypothesis that no relationship exists between exposure to program activities and changes in mathematics performance or attitude towards science. The treatment activities integrated science, with language arts, mathematics, computers and counseling and enabled students to discuss matters of concern and the relationship of these concerns to their academic work and to future success in careers based in science and mathematics. Mathematics performance data were analyzed using ANOVA (premath X group, postmath X group), and t-test/pairs (premath vs. postmath). Pre- and postreatment data on attitude towards science were rank ordered, paired and analyzed using the Wilcoxin Matched-Pairs Signed Ranks Test. The findings reveal a significantly positive treatment effect. In spite of the caution suggested by the limited sample, exposure to the treatment has resulted in an increased positive effect, not only upon attitude towards science, but also upon mathematics performance.     

Challenges students identified with a learning disability and as high-achieving experience when using diagrams as a visualization tool to solve mathematics word problems

This article addresses a much understudied topic and concern regarding how students of varying ability levels employ visualization as a strategy in mathematics learning. The importance of this topic can be found in its connection to students’ ability to solve mathematical word problems. Many students, particularly students with learning disabilities, often struggle to use visualization as a strategy and this impacts their mathematics performance. The purpose of this article is to present findings from a study that examined the challenges that students—those identified as learning disabled and high-achieving—displayed when using one visualization form, a diagram, to solve mathematics problems. Overall, nine challenges related to the use of diagram proficiency to solve problems were identified. Further, students with learning disabilities were found to be more likely than their high achieving peers to experience these challenges. Implications for practice are provided.     

Preservice Teachers' Beliefs and Attitude About Teaching and Learning Mathematics Through Music: An Intervention Study

This article presents exploratory research investigating the integration of music and a mathematics lesson as an intervention to promote preservice teachers' attitude and confidence and to extend their beliefs toward teaching mathematics integrated with music. Thirty students were randomly selected from 64 preservice teachers in a southern university. A 90‐minute mathematics lesson integrated with a music composition activity was taught by the first author. Pre‐ and postquestionnaires were provided to evaluate the change in preservice teachers' attitude and beliefs toward mathematics. The results demonstrated that the mathematics lesson integrated with music had a positive effect on preservice teachers' attitude and beliefs toward mathematics teaching and learning.     

What environmental studies students think about mathematics and what could be done about it

Mathematics is playing an increasing role in environmental studies, which causes problems for many students because of their low mathematical ability. A survey among first year environmental studies students revealed a significant number both of negative experiences in learning mathematics and negative feelings towards the subject. A proposal to design an alternative curriculum which aims to link the maths more closely with environmental studies is outlined as an answer to the problem of negative attitudes to the required mathematics.     

Teaching spline approximation techniques using maple

Using Computer Algebra Systems (CAS)-such as MAPLE-in teaching and learning mathematical concepts is a great challenge both from a didactical and a scientific point of view. We have to rewrite our traditional paper based teaching materials for interactive and living electronic worksheets, Only few statements and principles have to be acquired by the learner and the teacher from the CAS and after they can visualise, make animations modify quickly the program data, perform symbolic and numeric calculations step by step and in the whole, and verify deductions on their own. The author prepared Maple worksheets for teaching different types of function approximating techniques, such as interpolation-, least square-, spline and uniform approximation methods for post-graduate mechanical engineering students. In this paper we want to demonstrate how can we keep and improve the famous problem solving principles and rules given by G. Pólya and R. Descartes (Pólya 1962), when we use the capabilities of CAS. The education principles active learning, motivations and the successive phases are getting new meaning in the CAS. Our examples are always concerning with spline functions. Handling the formulas, calculating values and giving proofs are always in the form of Maple statements.     

Gender, technology and attitude towards mathematics: a comparative longitudinal study with Mexican students

In this paper the results of a comparative longitudinal study investigating changes in girls’ and boys’ attitudes towards mathematics, and self-confidence in mathematics are presented. A 5-point Likert scale, AMMEC, was used to measure attitudes towards mathematics (AM), computer-based mathematics (AMC), and self-confidence in mathematics (CM). A total of 430 students using technology for mathematics and 109 students not using it were monitored for 3 years. At the beginning of the study, the participants were aged about 13 years. The statistical analyses of the data showed few gender differences in the way students’ attitudes and self-confidence changed over the 3 years. Significant gender differences favouring boys were found in attitudes towards mathematics in grades 8 and 9 for the group using technology. For the group using technology, significantly more boys than girls got high scores in attitudes towards computer-based mathematics in grade 7. Significantly, more girls using technology than girls not using it got high scores in grade 8. The use of technology did not have a positive impact on students’ self-confidence. Regardless of whether they used computers or not, from grades 7 to 9, there was a decrease in the self-confidence in mathematics of both boys and girls. To enrich these results and detect possible gender differences in the way attitudes were constructed, 12 girls and 13 boys were interviewed at the end of the study. The analysis of the arguments they presented to explain and justify their attitudes towards mathematics, computer-based mathematics, and their self-confidence in working in mathematics provided evidence of important gender differences in the ways in which boys and girls construct their attitude, indicating how their constructions reflect the gender stereotypes within Mexican society.     

Students’ Activity in Computer-Supported Collaborative Problem Solving in Mathematics

The purpose of this study was to analyse secondary school students’ (N = 16) computer-supported collaborative mathematical problem solving. The problem addressed in the study was: What kinds of metacognitive processes appear during computer-supported collaborative learning in mathematics? Another aim of the study was to consider the applicability of networked learning in mathematics. The network-based learning environment Knowledge Forum (KF) was used to support students’ collaborative problem solving. The data consist of 188 posted computer notes, portfolio material such as notebooks, and observations. The computer notes were analysed through three stages of qualitative content analysis. The three stages were content analysis of computer notesin mathematical problem solving, content analysis of mathematical problem solving activity and content analysis of the students’ metacognitive activity. The results of the content analysis illustrate how networked discussions mediated mathematical knowledge and students’ questions, while the mathematical problem solving activity shows that the students co-regulate their thinking. The results of the content analysis of the students’ metacognitive activity revealed that the students use metacognitive knowledge and make metacognitive judgments and perform monitoring during networked discussions. In conclusion, the results of this study demonstrate that working with the networked technology contributes to the students’ use of their mathematical knowledge and stimulates them into making their thinking visible. The findings also show some metacognitive activity in the students’ computer-supported collaborative problem solving in mathematics.     

Integrating technology into mathematics teaching at the university level

The emergence of new computing technologies in the second half of the twentieth century brought about new potentials and promised the rapid transformation of the teaching and learning of mathematics. However, despite the vast investments in technology resources for schools and universities, the realities of schooling and the complexities of technology-equipped environments resulted in a much slower integration process than was predicted in the 1980s. Hence researchers, together with teachers and mathematicians, began examining and reflecting on various aspects of technology-assisted teaching and learning and on the causes of slow technology integration. Studies highlighted that as technology becomes increasingly available in schools, teachers’ beliefs and conceptions about technology use in teaching are key factors for understanding the slowness of technology integration. In this paper, I outline the shift of research focus from learning and technology environment-related issues to teachers’ beliefs and conceptions. In addition, I highlight that over the past two decades a considerable imbalance has developed in favour of school-level research against university-level research. However, several changes in universities, such as students declining mathematical preparedness and demands from other sciences and employers, necessitate closer attention to university-level research. Thus, I outline some results of my study that aimed to reflect on the paucity of research and examined the current extend of technology use, particularly Computer Algebra Systems (CAS) at universities, mathematicians’ views about the role of CAS in tertiary mathematics teaching, and the factors influencing technology integration. I argue that due to mathematicians’ extensive use of CAS in their research and teaching, documenting their teaching practices and carrying out research at this level would not only be beneficial at the university level but also contribute to our understanding of technology integration at all levels.