Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.     

Limits and continuity: some conceptions of first-year students

A research method consisting of written tests and individual interviews was introduced to examine first-year university students' understanding of fundamental calculus concepts. Six hundred and thirty students from three South African universities were subjected to the tests pertaining to this study. Several misconceptions underlying students' understanding of calculus concepts were identified. This paper deals mainly with some of the common errors and misconceptions relating to students' understanding of ‘limit of a function’ and ‘continuity of a function at a point’.     

Integrating design problems in mathematics curriculum: an architecture college case study

This paper reports an attempt to improve results in the mathematics course in one of the architecture colleges in Israel through practise in applications. The effect of integrating structure design problems in the calculus curriculum on students' achievements and attitudes was examined. The applied topics in the curriculum were connected to calculus topics and studied through problembased learning activities. The integrated curriculum was implemented and the learning results in experimental and control groups were assessed by means of achievement tests, attitude questionnaires and student interviews. The learning achievements in the experimental group proved to be significantly higher than in the control group. The positive impact of learning applications on motivation, understanding, creativity and interest in mathematics is indicated.     

“LeActiveMath”—a new innovative European elearning system for calculus contents

This contribution gives, an overview of the project “LeActiveMath”. Within this project a new mathematics learning software has been developed. LeActiveMath is an innovative eLearning system for high school and college or university level classrooms which can also be used in informal contexts for self-learning, since it is adaptive to the learner and his or her learning context in many respects. Topics cover elements of basic knowledge like ‘linear equations’ as well as more sophisticated contents like ‘differential calculus’. This article describes some of the innovative components of the software that are meant to support the students' self-regulated learning. We conclude by reporting on the first evaluations in math classorooms in fall 2005.     

How Students Use Physics to Reason About Calculus Tasks

The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her “paradigmatic physical” context and begins to address the need for a theoretical basis for investigating learning and teaching in integrated mathematics and science classrooms. A case study design was used to investigate the different ways students use physics ideas as they worked through calculus tasks. Data were gathered through four individual interviews with each of 8 ICP students, classroom participant‐observation, and triangulation of the data through student homework and exams. The main result of this study is the Physics Use Classification Scheme, a tool consisting of four categories used to characterize students' uses of physics on tasks involving average rate of change, derivative, and integral concepts. Two of the categories from the Physics Use Classification Scheme are elucidated with contrasting student cases in this paper.     

Developing Student Understanding: Contextualizing Calculus Concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern stale. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.     

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.     

New directions in differential equations: A framework for interpreting students' understandings and difficulties

The purpose of this paper is to offer a framework for interpreting students' understandings of and difficulties with mathematical ideas central to new directions in differential equations. These new directions seek to guide students into a more interpretive mode of thinking and to enhance their ability to graphically and numerically analyze differential equations. The framework reported here is the result of investigating in depth six students' understandings through a series of task-based individual interviews and classroom observations. The two major themes of the framework, the function-as-solution dilemma theme and students' intuitions and images theme, extend previous research on student cognition at the secondary and collegiate level to the domain of differential equations and reflect the increased recognition of situating analyses of student learning within students' learning environment. For new areas of interest such as differential equations, mapping out students' understandings of important mathematical ideas can be an important part of curricular and instructional design that seeks to refine and build on students' ways of thinking.     

A Modeling Perspective on the Teaching and Learning of Mathematical Problem Solving

This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an intervention program consisting of a sequence of six modeling activities. To examine students' modeling abilities, the experimental and control groups completed a modeling abilities test on three occasions. Results showed that students' models improved as they worked through the sequence of problem activities and also revealed a number of factors, such as students' grade, experiences with modeling activities, and modeling abilities that influenced their modeling processes. The study proposes a three-dimensional theoretical model for examining students' modeling behavior, with ubsequent implications for the teaching and learning of mathematical problem solving.     

The Impact on Incorporating Collaborative Concept Mapping with Coteaching Techniques in Elementary Science Classes

The purpose of this research was to evaluate a collaborative concept‐mapping technique that was integrated into coteaching in fourth‐grade science classes in order to examine students' performance and attitudes toward the experimental teaching method. There are two fourth‐grade science teachers and four classes with a total of 114 students involved in the study. This study used a mixed method design, incorporating both quantitative and qualitative techniques. The findings showed that the two teaching methods obtained significant difference with respect to students' test scores. Using collaborative concept mapping to learn science could increase the opportunity of discussion between peers, thus fostering better organization and understanding the content. In addition, coteaching could enable teachers to share their expertise with one another. It could facilitate the implementation of collaborative concept mapping and the construction of student's concept mapping. Team teachers' attitude could affect the students' learning performance. However, some of the students had negative views on drawing concept maps because they found it was troublesome to write down many words, difficult to draw and arrange proposition, and time‐consuming. Coteachers' instant feedback and students' journal writing could guide and examine the students' concept maps to facilitate their cognitive learning.     

Gestures,Speech, and the Sprouting of Signs: A Semiotic-Cultural Approach to Students' Types of Generalization

To improve our understanding of novice students' production of symbolic algebraic expressions, this article contrasts students' presymbolic and symbolic procedures in generalizing activities. Although a significant amount of previous research on the learning of algebra has dealt with students' errors in the mastering of the algebraic syntax, the semiotic cultural theoretical approach presented here focuses on the role that body, discourse, and signs play when students' refer to mathematical objects. Three types of generalizations are identified: factual, contextual, and symbolic. The results suggest that the passage from presymbolic to symbolic generalizations requires a specific kind of rupture with the ostensive gestures and contextually based key linguistic terms underpinning presymbolic generalizations. This rupture means a disembodiment of the students' previous spatial temporal embodied mathematical experience.     

Student Attitudes and Calculus Reform

This paper compares the attitudes about mathematics of students from traditionally taught calculus classes and those from a “reformed” calculus course. The paper is based on three studies, which together present a consistent picture of student attitudes about calculus reform. The reformed course appeared to violate students' deeply held beliefs about the nature of mathematics and how it should be learned. Although during their first months in the reformed course most students disliked it, their attitudes gradually changed. One and 2 years after, reform students felt significantly more than the traditionally taught students that they better understood how math was used and that they had been required to understand math rather than memorize formulas.     

Declining participation in post-compulsory secondary school mathematics: students’ views of and solutions to the problem

We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific variables. We used Ellis' actor-oriented generalisations framework to characterise how students generalised their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalisation – what students see as similar between ideas in multiple contexts. We found that students generalised their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our findings suggest that the domain and range of multivariable functions is a topic instructors should explicitly address.     

What to Trust: Reconciling Mathematical Work Done by Hand with Conflicting Graphing Calculator Solutions

This paper reports on a mixed‐methods study of 111 Advanced Placement calculus students' self‐reports of their graphing calculator use, comfort, and rationale for trusting a solution produced with or without a graphing calculator when checking written work. It was found that there was no association between gender, teacher‐reported mathematical ability, or comfort with the graphing calculator and students' trust in either a graphing calculator‐produced solution or a solution produced without a graphing calculator. Furthermore, regardless of solution choice, the same four categories were evident in students' rationale for their solution choice: (a) an awareness of the possibility of careless errors, (b) the importance of checking over work, (c) a recognition of the limitations or affordances of the graphing calculator, and (d) a confidence (or lack thereof) in their own mathematical abilities. These results have implications for mathematics teaching as graphing calculators are used extensively in middle and high school mathematics classrooms and standardized tests in the United States.     

What Matters Most: A Comparison of Expert and Novice Teachers' Noticing of Mathematics Classroom Events

In this study, we examined 10 expert and 10 novice teachers' noticing of classroom events in China. It was found that both expert and novice teachers, who were selected from two cities in China, highly attended to developing students' mathematics knowledge coherently and developing students' mathematical thinking and ability; they also paid attention to students' self‐exploratory learning, students' participation, and teachers' instructional skills. Furthermore, compared with novice teachers, expert teachers paid greater attention to developing mathematical and high‐order thinking, and developing mathematics knowledge coherently, but paid less attention to teachers' guidance. Moreover, we further illustrated the qualitative differences and similarities in their noticing of classroom events. Finally, we discussed the findings and relevant implications.     

What Can Be Learned From Informal Student Writings in a Science Context?

This exploratory study analyzed four informal science-related writing tasks produced by 374 seventh-grade students (172 boys and 202 girls) from two schools with different socioeconomic populations. The study demonstrates that students' informal writing in science contexts can provide a rich source of information regarding students' cognitive and attitudinal engagement with science. Students' writing reflects the level at which students understand previously learned science-related ideas and gives insight into themes and issues they would be interested in learning. This study further demonstrates how students organize and personalize science knowledge acquired inside as well as outside of school when given novel and unconventional (informal) science-related tasks. The study also demonstrates that informal writing tasks encourage students to express opinions, values, and attitudes associated with science and science learning. Examples are provided of similarities and differences in students' writing preferences and in the quality of writing produced by boys and girls. Suggestions for further studies for teachers and researchers are discussed.     

Effects of variation theory approach in teaching and learning of algebra on urban and rural students’ algebraic achievement and motivation

This study investigates the effect of utilizing variation theory approach (VTA) on students' algebraic achievement and their motivation in learning algebra. The study used quasi-experimental non-equivalent control group research design. It involved 114 Form Two students in four intact classes (two classes were from an urban school, another two classes from a rural school). The first group of students from each school learnt algebra in class which used the VTA, while the second group of students in each school learnt algebra through conventional teaching approach. Two-way analysis of covariance and two-way multivariate analysis of variance (MANOVA) were used to analyse the data collected. The result of this study indicated that the use of VTA has significant effect on both urban and rural students' algebraic achievement. There were evidences that VTA has significant effect on rural VTA students' overall motivation in its five subscales: attention, relevance, confidence, satisfaction and interest but it was not so for urban VTA students' motivation. This study provides further empirical evidence that utilization of variation theory as pedagogical guide can promote the teaching and learning of Form Two Algebra topics in urban and rural secondary school classrooms.