Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

Retention of differential and integral calculus: a case study of a university student in physical chemistry

This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly reconstruct the knowledge, given a few hints, this is just as good as retention. The study was conducted using a mixed method approach investigating students’ knowledge in three worlds of mathematics. The results showed that the student had a very low retention of concepts, even after hints. However, after completing the calculus course, the student had successfully used calculus in a physical chemistry study programme. Hence, using calculus in new contexts does not in itself strengthen the original calculus learnt; they appeared as disjoint bodies of knowledge.     

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

Similarity is a fundamental concept in the middle grades. In this study, we applied Vergnaud's theory of conceptual fields to answer the following questions: What concepts‐in‐action and theorems‐in‐action about similarity surfaced when students worked in a novel task that required them to enlarge a puzzle piece? How did students use geometric and multiplicative reasoning at the same time in order to construct similar figures? We found that students used concepts of scaling and proportional reasoning, as well as the concept of circle and theorems about similar triangles, in their work on the problem. Students relied not only on visual perception, but also on numeric reasoning. Moreover, students' use of multiplicative and proportional concepts supported their geometric constructions. Knowledge of the concepts and ideas that students have available when working on a task about similarity can inform instruction by helping to ground formal introduction of new concepts in students' informal prior experiences and knowledge.     

The role of the graphic calculator in mediating graphing activity

The PIGMI (Portable Information Technologies for supporting Graphical Mathematics Investigations) Project ¹ investigated the role of portable technologies in facilitating development of students' graphing skills and concepts. This paper examines the impact of a recent shift towards calculating and computing tools as increasingly accessible, everyday technologies on the nature of learning in a traditionally difficult curriculum area. The paper focuses on the use of graphic calculators by undergraduates taking an innovative new mathematics course at the Open University. A questionnaire survey of both students and tutors was employed to investigate perceptions of the graphic calculator and the features which facilitate graphing and linking between representations. Key features included visualization of functions, immediate feedback and rapid graph plotting. A follow-up observational case study of a pair of students illustrated how the calculator can shape mathematical activity, serving a catalytic, facilitating and checking role. The features of technology-based activities which can structure and support collaborative problem solving were also examined. In sum, the graphic calculator technology acted as a critical mediator in both the students' collaboration and in their problem solving. The pedagogic implications of using portables are considered, including the tension between using and over-using portables to support mathematical activity.     

Nascent function concepts in Nova Scientia

The origin of the function concept is usually traced to Galileo's work on motion. We argue that specific proto-function concepts appeared in the work of Tartaglia a century before the publication of Galileo's Two New Sciences. The study of Tartaglia's ideas can be used in the classroom as a historical introduction to various function concepts, and certain modern extensions of Tartaglia's optimal range problem and inverse range problem are sources of enrichment for undergraduate courses in analysis, mathematical modeling and computation.     

Chance and probability: What do they mean to university engineering students?

The great interest aroused by the incorporation of Statistics and Probability into curricular projects has been accompanied by considerable evidence of significant difficulties in the meaningful learning and application of the concepts. These difficulties have been the subject of many studies, mostly concerning secondary school students. This study seeks to investigate the level of understanding of random phenomena, in a group of university students, on the second year of a Technical and Industrial Engineering Degree at the University of the Basque Country, in Spain. The students, who had all undertaken an introductory course in Probability Theory, were tested through both written questionnaires and one-to-one interviews, where they were presented with situations that demanded the ability to apply the knowledge they had learnt, at a high level, to a range of situations and to justify their reasoning. The results show that the vast majority of students had a poor understanding of random phenomena, applying alternative ideas to the ones taught on their course in Probability. The need to improve teaching strategies in Probability and Statistics, so that students can develop skills in constructing probabilistic models, is highlighted.     

A conceptual pathway to confidence intervals

Finding ways for the majority of students to better understand conventional normal theory-based statistical inference seems to be an intractable problem area for researchers. In this paper we propose a conceptual pathway for developing confidence interval ideas for the one-sample situation only from an intuitive sense to bootstrapping for students from about age 14 to first-year university. We make the case that conceptual development should start early; that probability and statistical instruction should change so that both orientate students towards interconnected stochastic conceptions; and that the use of visual imagery has the potential to stimulate students towards such a perspective. We analyse our conceptual pathway based on a theoretical framework for a stochastic conception of statistical inference based on imagery and some research evidence. Our analysis suggests that the pathway has the potential for students to become conversant with the concepts underpinning inference, to view statistics probabilistically and to integrate concepts into a coherent comprehension of inference.     

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.     

Understanding the integral: Students’ symbolic forms

Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the cognitive resources (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This paper presents cognitive resources of the integral that a sample of experienced calculus students drew on while working on pure mathematics and applied physics problems. This research provides evidence that students hold a variety of productive cognitive resources that can be employed in problem solving, though some of the resources prove more productive than others, depending on the context. In particular, conceptualizations of the integral as an addition over many pieces seem especially useful in multivariate and physics contexts.     

On variable length coding with side information for discrete memoryless sources

In this paper we consider the zero error variable length coding problem for discrete memoryless sources with side information at the decoder only. Using concepts of graph theory, we obtain a characterization of the minimum achievable rater for this coding problem. Furthermore, computable upper and lower bounds for the minimum achievable rater is given and a condition under which the upper and lower bounds are equal is found. Finally, this result is applied to some examples in which the minimum achievable rater can be evaluated.     

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

Using math magic to reinforce algebraic concepts: an exploratory study

An exploratory study was conducted to investigate the use of magic activities in a math course for prospective middle-school math teachers. This research report focuses on a lesson using two versions of math magic: (1) the 5-4-3-2-1-½ Magic involves having students choose a secret number and apply six arithmetic operations in sequence to arrive at a resultant number, and the teacher-magician can spontaneously reveal a student’s secret number from the resultant number; and (2) the Everyone-Got-9 Magic also involves choosing a secret number and applying arithmetic operations in sequence, but everyone will end up with the same resultant number of 9. These magic activities were implemented to reinforce students’ understanding of foundational algebra concepts like variables, expressions, and inverse functions. Analysis of students’ written responses revealed that (1) all students who figured out the trick in the first magic activity did not used algebra, (2) most students could apply what they learned in one trick to a similar trick but not to a different trick, and (3) many students were weak in symbolic representations and manipulations. Responses from a survey and a focus group indicate that students found the magic activities to be fun and intellectually engaging.     

The Approximation of the Optimal Control of Vibrations—A Geometrical Approach

In this paper we employ concepts from Banach space geometry in order to examine the problem of approximating the optimal distributed control of vibrating media whose motion is governed by a wave equation with a 2n-order self-adjoint and positive-definite linear differential operator. We show that this geometrical approach, arrived at via duality theory, provides the exact framework in which the approximation problem must be placed in order to get the correct convergence results, for it is here that the necessary and sufficient conditions for the approximate norm or time minimal control can be fully developed. Using the theory of Asplund, we are also able to improve the traditional weak^* convergence results for the more difficult case of L^∞ controls. Finally, we consider certain numerical examples which help illustrate our theoretical results.     

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.     

Learning of winning strategies for terminal games with linear-size memory

We prove that if one or more players in a locally finite positional game have winning strategies, then they can find it by themselves, not losing more than a bounded number of plays and not using more than a linear-size memory, independently of the strategies applied by the other players. We design two algorithms for learning how to win. One of them can also be modified to determine a strategy that achieves a draw, provided that no winning strategy exists for the player in question but with properly chosen moves a draw can be ensured from the starting position. If a drawing- or winning strategy exists, then it is learnt after no more than a linear number of plays lost (linear in the number of edges of the game graph). Z. Tuza’s research has been supported in part by the grant OTKA T-049613.     

A cognitive analysis of dragging practises in Cabri environments

Dragging in Dynamical Geometry Software (DGS) is described by introducing a hierarchy of its functions. This is suitable for classifying different attitudes and aims of students who investigate a geometric problem, such as exploring, conjecturing, validating and justifying. Moreover the hierarchy has cognitive features and can be used to describe the twofold modulities namely ascending and descending in which students interact with external representations (e.g. Cabri drawings). Switching from one modality to the other through dragging often allows them to produce fruitful conjectures and to pass from the empirical to the theoretical side of the question. The genesis of such different functions in students does not happen automatically but is the consequence of specific didactical interventions of the teacher in the pupils' apprenticeship of Cabri practises. A worked-out example illustrates the theoretical concepts introduced in the paper.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

An investigation of 6th graders’ solutions of Cartesian product problems and representation of these problems using arrays

Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).     

Vertex Colouring and Forbidden Subgraphs – A Survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions. Thus, one who wishes to obtain useful results from a graph coloring formulation of his problem must do more than just show that the problem is equivalent to the general problem of coloring a graph. If there is to be any hope, one must also obtain information about the structure of the graphs that need to be colored (D.S. Johnson [66]).Final version received: September 9, 2003