On the shoulders of technology: calculators as cognitive amplifiers

This article presents teaching ideas designed to support the belief that students at all levels (preservice teachers, majors, secondary and elementary students) need exposure to non-routine problems that illustrate the effective use of technology in their resolution. Such use provides students with rapid and accurate data collection, leading them to sound conjectures, which is a precursor to learning mathematical proof. Students will therefore learn that while technology can be an effective tool for investigating problems, the onus of providing convincing arguments and proofs of their conjectures rests squarely on their shoulders. The paper describes how a diverse group of students took advantage of the power of the TI-92 to enhance their chances of reaching this final stage of proof. A series of mathematical problems are presented and analysed with a keen eye on the appropriate integration of the TI-92. A student survey was used to inform the results. To conclude, several challenging, yet accessible, non-routine problems were completed by students as undergraduate research projects, all using the TI-92 as a laboratory. Although most of the problems presented here have a discrete mathematics flavour, the authors' message is independent of the mathematical topic chosen.     

Didactic and theoretical-based perspectives in the experimental development of an intelligent tutorial system for the learning of geometry

This paper aims at showing the didactic and theoretical-based perspectives in the experimental development of the geogebraTUTOR system (GGBT) in interaction with the students. As a research and technological realization developed in a convergent way between mathematical education and computer science, GGBT is an intelligent tutorial system, which supports the student in the solving of complex problems at a high school level by assuring the management of discursive messages as well as the management of problem situations. By situating the learning model upstream and the diagnostic model downstream, GGBT proposes to act on the development of mathematical competencies by controlling the acquisition of knowledge in the interaction between the student and the milieu, which allows for the adaptation of the instructional design (learning opportunities) according to the instrumented actions of the student. The inferential and construction graphs, a structured bridge (interface) between the contextualized world of didactical contracts and the formal computer science models, structure GGBT. This way allows for the tutorial action to adjust itself to the competential habits conveyed by a certain classroom of students and to be enriched by the research results in mathematical education.     

Teaching mathematics using Steplets

This article evaluates online mathematical content used for teaching mathematics in engineering classes and in distance education for teacher training students. In the EU projects Xmath and dMath online computer algebra modules (Steplets) for undergraduate students assembled in the Xmath eBook have been designed. Two questionnaires, a compulsory student project and teaching in front of class show that using Steplets turn mathematics teaching from drill to understanding. The Steplets use algorithms developed for the Mathematica programming language.     

Advanced mathematics communication beyond modality of sight

This study illustrates how mathematical communication and learning are inherently multimodal and embodied; hence, sight-disabled students are also able to conceptualize visuospatial information and mathematical concepts through tactile and auditory activities. Adapting a perceptuomotor integration approach, the study shows that the lack of access to visual fields in an advanced mathematics course does not obstruct a blind student's ability to visualize, but transforms it. The goal of this study is not to compare the visually impaired student with non-visually impaired students to address the ‘differences’ in understanding; instead, I discuss the challenges that a blind student, named Anthony, has encountered and the ways that we tackled those problems. I also demonstrate how the proper and precisely crafted tactile materials empowered Anthony to learn mathematical functions.     

Using Culturally Responsive Stories in Mathematics: Responses From the Target Audience

This study examined how Black students responded to the utilization of culturally responsive stories in their mathematics class. All students in the two classes participated in mathematics lessons that began with an African American story (culturally responsive to this population), followed by mathematical discussion and concluded with solving problems that correlated to the story. The researcher observed and recorded responses by students during each part of these lessons with protocols. Students independently reflected weekly by answering five questions to share their perspective on the African American stories. The teacher reflected on each lesson as well, describing thoughts on how these students responded to the story in each lesson. This paper examines the analyzed data from the target audience: Black students. Results revealed that Black students responded to the use of African American stories with high self‐rated levels of engagement and enjoyment and that the stories helped them think about mathematics to varying degrees. Since students who are engaged and are thinking about mathematics are more likely to achieve mathematical understanding, the researcher concludes that this strategy should continue to be tested in diverse classrooms with an emphasis on student reflection to determine if the outcomes are transferable and generalizable.     

The initial treatment of the concept of function in the selected secondary school mathematics textbooks in the US and China

In order to provide insight into cross-national differences in students’ achievement, this study compares the initial treatment of the concept of function sections of Chinese and US textbooks. The number of lessons, contents, and mathematical problems were analyzed. The results show that the US curricula introduce the concept of function one year earlier than the Chinese curriculum and provide strikingly more problems for students to work on. However, the Chinese curriculum emphasizes developing both concepts and procedures and includes more problems that require explanations, visual representations, and problem solving in worked-out examples that may help students formulate multiple solution methods. This result could indicate that instead of the number of problems and early introduction of the concept, the cognitive demands of textbook problems required for student thinking could be one reason for differences in American and Chinese students’ performances in international comparative studies. Implications of these findings for curriculum developers, teachers, and researchers are discussed.     

Reflective abstraction, uniframes and the formulation of generalizations

In mathematics, generalizations are the end result of an inductive zigzag path of trial and error, that begin with the construction of examples, within which plausible patterns are detected and lead to the formulation of theorems. This paper examines whether it is possible for high school students to discover and formulate generalizations similar to ways professional mathematicians do. What are the experiences that allow students to become adept at generalization? In this paper, the mathematical experiences of a ninth grade student, which lead to the discovery and the formulation of a mathematical generalization are described, qualitatively analyzed and interpreted using the notion of uniframes. It is found that reflecting on the solutions of a class of seemingly different problem-situations over a prolonged time period facilitates the abstraction of structural similarities in the problems and results in the formulation of mathematical generalizations.     

A Study Comparing the Efficacy of a Mole Ratio Flow Chart to Dimensional Analysis for Teaching Reaction Stoichiometry

Reaction stoichiometry calculations have always been difficult for students. This is due to the many different facets the student must master, such as the mole concept, balancing chemical equations, algebraic procedures, and interpretation of a word problem into mathematical equations. Dimensional analysis is one of the main ways students are taught to solve these problems. However, this methodology does not provide all students with a complete understanding of how to solve these problems. Introduction of alternative problem solving techniques, such as proportional reasoning, can help to improve student understanding. The mole ratio flow chart (MRFC) is a logistical sequence of steps that incorporates molar proportions. Students are able to begin analysis of a problem from many different starting points using this MRFC method. Analyses of data collected indicate that MRFC users performed as well on exam problems covering reaction stoichiometry calculations as students using dimensional analysis. Further, class sections exposed to both dimensional analysis and MRFC methods scored as well on exam problems as class sections exposed only to dimensional analysis. These results indicate that the MRFC is a viable alternative method for teaching reaction stoichiometry calculations and for helping to create a more complete understanding of the subject.     

Student problems with a modelling exercise

In this paper an attempt is made to outline some of the problems facing the student who undertakes a course in mathematical modelling. A case study, considered by several cohorts of students, is used as a medium for introducing the various types of obstacles the students encounter. This approach also results in a demonstration of some of the errors that can be made in operating a modelling course and the paper concludes by offering suggestions on how to overcome certain difficulties that are exposed.     

Looking back to the roots of partially correct constructs: The case of the area model in probability

We use the notion Partially Correct Constructs (PaCCs) for students’ constructs that partially match the mathematical principles underlying the learning context. A frequent expression of partial construction of mathematical principles is that a student’s words or actions provide an inaccurate or misleading picture of the student’s knowledge. In this study, we analyze the learning process of a grade 8 student, who learns a topic in elementary probability. The student successfully accomplishes a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require nothing beyond his proven competencies, he encounters difficulties. Using the epistemic actions of the RBC model for abstraction in context as tracers, we analyze his knowledge constructing processes while working on the previous tasks, and identify some of his constructs as PaCCs that are concealed in these processes and explain his later difficulties. In addition, our research points to the complexity of the knowledge structures students are expected to deal with in their attempts to learn an elementary mathematical topic with understanding.     

Teachers’ responses to student mathematical thinking in Chinese elementary mathematics classrooms

Recent research on teachers’ use of student mathematical thinking (SMT) and recommendations for effective mathematics instruction claim that how teachers respond to SMT has great impact on student mathematical learning in the classroom. This study examined some Chinese mathematics teachers’ responses to student in-the-moment mathematical thinking that emerged during whole class discussion. The findings of this study revealed that the majority of Chinese elementary mathematics teachers in the data involved the whole group of students to make sense of in-the-moment SMT. They either invited students to digest SMT involved in the instance or provided an extension of the instance to further develop student mathematical understanding.     

Capturing student mathematical engagement through differently enacted classroom practices: applying a modification of Watson's analytical tool

This study examined student mathematical engagement through the intended and enacted lessons taught by two teachers in two different middle schools in Indonesia. The intended lesson was developed using the ELPSA learning design to promote mathematical engagement. Based on the premise that students will react to the mathematical tasks in the forms of words and actions, the analysis focused on identifying the types of mathematical engagement promoted through the intended lesson and performed by students during the lesson. Using modified Watson's analytical tool (2007), students’ engagement was captured from what the participants’ did or said mathematically. We found that teachers’ enacted practices had an influence on student mathematical engagement. The teacher who demonstrated content in explicit ways tended to limit the richness of the engagement; whereas the teacher who presented activities in an open-ended manner fostered engagement.     

Enhancing student engagement to positively impact mathematics anxiety,confidence and achievement for interdisciplinary science subjects

Contemporary science educators must equip their students with the knowledge and practical know-how to connect multiple disciplines like mathematics, computing and the natural sciences to gain a richer and deeper understanding of a scientific problem. However, many biology and earth science students are prejudiced against mathematics due to negative emotions like high mathematical anxiety and low mathematical confidence. Here, we present a theoretical framework that investigates linkages between student engagement, mathematical anxiety, mathematical confidence, student achievement and subject mastery. We implement this framework in a large, first-year interdisciplinary science subject and monitor its impact over several years from 2010 to 2015. The implementation of the framework coincided with an easing of anxiety and enhanced confidence, as well as higher student satisfaction, retention and achievement. The framework offers interdisciplinary science educators greater flexibility and confidence in their approach to designing and delivering subjects that rely on mathematical concepts and practices.     

Using components of mathematical ability for initial development and identification of mathematically promising students

Kruteskii's work on the mathematical abilities of school children is a seminal work on the nature of mathematical ability. However, the task of developing methods for the practical application of his work is still a significant problem in mathematics education. The authors have developed a practical application of Kruteskii's approach to the important problem of initially developing components of mathematical ability in student and thereafter identifying mathematically promising students. Examples of problems that were designed to develop ability to generalize, flexibility and reversibility of mental processes are presented. A practical guide for determining the level of development of components of mathematical abilities in individual students, in terms of specified observables, is presented as a set of structured reference tables. The authors set out a practical application protocol that combines use of the tables and sets of specially developed problems for initial development of mathematical abilities prior to identification of mathematically promising students in the general classroom. A significant motivation for this work is the desire to avoid time-consuming and resource intensive practices such as interviews and summer schools which therefore have been used successfully because these practices are now out of reach for all but very wealthy countries or highly ideologically driven systems. On the other hand, special examinations heavily depend on the level of preparedness of the students for the particular examination, and therefore some students with high abilities but with fewer opportunities to prepare could be overlooked.     

Mathematical Remediation with Dignity

This paper demonstrates one way to approach mathematical remediation with dignity for faculty and students alike. It illustrates specific ways in which student deficiencies can be improved while simultaneously developing creative, significant and stimulating approaches to teaching basic mathematical concepts. In order to aid students who have not mastered even the addition and multiplication of rational numbers, examples will be provided to show how these deficiencies can be improved within mathematical formats that lead naturally to the principles of algebra, progressions, powers and roots, symbolism, number theory, probability theory, linear programming, information theory, etc. An integral part of this approach to mathematical remediation involves the selection of appropriate research problems by the students.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.