Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.     

Prevailing educational practices for area measurement and students’ failure in measuring areas

The present research was carried out with the participation of 106 students in their last grade in Elementary School and revealed certain problems that these students faced in understanding the concept of area measurement. The students in the sample persisted on using measurement strategies that often led to failure.Our research plan comprises a comparison between the strategies for area measurement strategies used by two groups: the experimental group (E.G.) and the control group (C.G.). The experimental group attended a special teaching course, which stressed the conceptual characteristics of the area measurement process.The present research aims at revealing the students’ understanding, strategies, and misconceptions regarding area measurement. In addition to that, we examine whether the special teaching course and the use of different measurement tools may lead the two research groups to adopt different measurement strategies.     

Learning beginning algebra with spreadsheets in a computer intensive environment

This study is part of a large research and development project aimed at observing, describing and analyzing the learning processes of two seventh grade classes during a yearlong beginning algebra course in a computer intensive environment (CIE). The environment includes carefully designed algebra learning materials with a functional approach, and provides students with unconstrained freedom to use (or not use) computerized tools during the learning process at all times. This paper focuses on the qualitative and quantitative analyses of students’ work on one problem, which serves as a window through which we learn about the ways students worked on problems throughout the year. The analyses reveal the nature of students’ mathematical activity, and how such activity is related to both the instrumental views of the computerized tools that students develop and their freedom to use them. We describe and analyze the variety of approaches to symbolic generalizations, syntactic rules and equation solving and the many solution strategies pursued successfully by the students. On that basis, we discuss the strengths of the learning environment and the open questions and dilemmas it poses.     

The Interplay Between Fairness and Randomness in a Spatial Computer Game

This article describes how children use an expressive microworld to articulate ideas about how to make a game seem fair with the use of randomness. Our aim in this study is to disentangle different flavours of fairness and to find out how children used each flavour to make sense of potentially complex behaviour. In order to achieve this, a spatial computer game was designed to enable children to examine the consequences of their attempts to make the game fair. The study investigates how 23 children, aged between 5.5 and 8 years, engaged in constructing a crucial part of a mechanism for a fair spatial lottery machine (microworld). In particular, the children tried to construct a fair game given a situation in which the key elements happened randomly. The children could select objects, determine their properties, and arrange their spatial layout in the machine. The study is based on task-based interviewing of children who were interacting with the computer game. The study shows that children have various cognitive resources for constructing a random fair environment. The spatial arrangement, the visualisation and the manipulations in the lottery machine allow us gain a view into the children’s thinking of the two central concepts, fairness and randomness. The paper reports on two main strategies by which the children attempted to achieve a balance in the lottery machine. One involves arranging the balls symmetrically and the other randomly. We characterize the nature of the thinking in these two strategies: the first we see as deterministic and the latter as stochastic, exploiting the random collisions of the ball. In this article we trace how the children’s thinking moved between these two perspectives.

Embodying mathematics and science: Microworlds as representations

This paper examines the category of open-ended exploratory computer environments that have been labeled “microworlds.” The paper reviews the ways in which the term “microworld” has been used in the mathematics and science education communities, and analyzes examples of specific computer microworlds. Two definitions of microworld are proposed: a structural definition that focuses on design elements shared by the environments, and a functional definition that highlights commonalties in how students learn with microworlds. In the final section of the paper, the notion that computer microworlds can be said to “embody” mathematical or scientific ideas is addressed, within the context of a re-evaluation of the general concept of representation.     

A Conversation With Zoltan P. Dienes

In this article, we analyze the visual and symbolic strategies developed by students to express generalizations of number patterns and the connections they make between them. By analysis of a series of case studies, we compare the approaches adopted by students working through parallel task sequences, which integrate different computer tools in different ways. Finally, we make suggestions as to how students might be encouraged to exploit visual reasoning alongside the symbolic and draw out implications for curriculum design.     

Mixing Microworld and Cas Features in Building Computer Systems that Help Students Learn Algebra

We present the design principles for a new kind of computer system that helps students learn algebra. The fundamental idea is to have a system based on the microworld paradigm that allows students to make their own calculations, as they do with paper and pencil, without being obliged to use commands, and to verify the correctness of these calculations. This requires an advanced editor for algebraic expressions, an editor for algebraic reasoning and an algorithm that calculates the equivalence of two algebraic expressions. A second feature typical of microworlds is the ability to provide students information about the state of the problem in order to help them move toward a solution. A third feature comes from the CAS (Computer Algebra System) paradigm, consisting of providing commands for executing certain algebraic actions; these commands have to be adapted to the current level of understanding of the students in order to only present calculations they can do without difficulty. With this feature, such a computer system can provide an introduction to the proper use of a Computer Algebra System. We have implemented most of these features in a computer system called aplusix for a sub-domain of algebra, and we have done several experiments with students (mainly grades 9 and 10). We had good results, with positive feedback from students and teachers. aplusix is currently a prototype that can be downloaded from http://aplusix.imag.fr. It will become a commercial product during 2004. This revised version was published online in July 2006 with corrections to the Cover Date.     

Visual and Symbolic Reasoning in Mathematics: Making Connections With Computers?

In this article, we analyze the visual and symbolic strategies developed by students to express generalizations of number patterns and the connections they make between them. By analysis of a series of case studies, we compare the approaches adopted by students working through parallel task sequences, which integrate different computer tools in different ways. Finally, we make suggestions as to how students might be encouraged to exploit visual reasoning alongside the symbolic and draw out implications for curriculum design.     

The Emergence of Mathematical Collaboration in an Interactive Computer Environment

We present a case study of two students engaged in an investigation of number theory concepts in the computer environment, Geoboard. The two students struggle with significant problems to communicate mathematics, and the computer plays a vital role in helping them overcome these difficulties. In fact, the students constantly use the screen images to help them make sense of each other's vague ideas and incomplete utterances. Using the qualitative research methods of discourse analysis, we construct the learning profiles of the two students and show how the characteristics of each contributed to the shared process, and how they used the computer presence to enable this process. We trace this process through the students' construction of two mathematical concepts that arose in their activities, the n-star and common denominators.This revised version was published online in September 2005 with corrections to the Cover Date.     

Broadening the sense of ‘dynamic’: a microworld to support students’ mathematical generalisation

In this paper, we seek to broaden the sense in which the word ‘dynamic’ is applied to computational media. Focussing exclusively on the problem of design, the paper describes work in progress, which aims to build a computational system that supports students’ engagement with mathematical generalisation in a collaborative classroom environment by helping them to begin to see its power and to express it for themselves and for others. We present students’ strengths and challenges in appreciating structure and expressing generalities that inform our overall system design. We then describe the main features of the microworld that lies at the core of our system. In conclusion, we point to further steps in the design process to develop a system that is more adaptive to students’ and teachers’ actions and needs.     

Unfolding Meanings for Reflective Symmetry

In this paper, we analyse the processes through which students come to negotiate mathematical meanings for reflective symmetry. We describe a microworld, Turtle Mirrors, designed to provide tools to help students focus simultaneously on actions, visual relationships and symbolic representations. Through a detailed case study of one students thinking-in-change, we examine how the interactions with her partner and with the machine support a fusion of spontaneous and scientific concepts. Other examples of students work further illustrate how the microworld tools offer a means to supplement local understandings of symmetry with those with more explicit, mathematical formulations.This revised version was published online in September 2005 with corrections to the Cover Date.     

Computational Construction as a Means to Coordinate Representations of Infinity

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

Some didactical and Epistemological Considerations in the Design of Educational Software: The Cabri-euclide Example

We propose to use didactical theory for the design of educational software. Here we present a set of didactical conditions, and explain how they shape the software design of Cabri-Euclide, a microworld used to learn “mathematical proof ” in a geometry setting. The aim is to design software that does not include a predefined knowledge of problem solution. Key features of the system are its ability to verify local coherence, and not to apply any global and automatic deduction.     

A case study of dynamic visualization and problem solving

This paper reports an example of a situation in which university students had to solve geometrical problems presented to them dynamically using the interactive computerized environment of the ‘MicroWorlds Project Builder’. In the process of the problem solving, the students used ten different solution strategies. The unsuccessful strategies were then classified into three main categories: distracting, reducing and confusing. One student group had to solve the same problem in its non-dynamic version. The results received from both groups were compared and analysed. Analysis of the solution strategies and the process of the categorization revealed that the percentage of success in both groups was similar and in the case of the given problem, the dynamic visual mode of the problem distracted the students’ attention away from proper handling of the solution of the problem.     

Curriculum, Classroom Practices, and Tool Design in the Learning of Functions Through Technology-Aided Experimental Approaches

The paper starts from classroom situations about the study of a functional relationship with help of technological tools as a ‘transposition’ of experimental approaches from research mathematical practices. It considers the limitation of this transposition in existing curricula and practices based on the use of non-symbolic software like dynamic geometry and spreadsheets. The paper focuses then on the potentialities of classroom use of computer algebra packages that could help to go beyond this shortcoming. It looks at a contradiction: while symbolic calculation is a basic tool for mathematicians, curricula and teachers are very cautious regarding their use by students. The rest of the paper considers the design and experiment of a computer environment Casyopée as means to contribute to an evolution of curricula and classroom practices to achieve the transposition in the domain of algebraic activities linked to functions.     

From fractions to proportional reasoning: a cognitive schemes of operation approach

Four seventh grade students participated in a constructivist teaching experiment in which manipulatives within a computer microworld were used to solve fractional reasoning tasks followed by tasks that involve concepts of rate, ratio and proportion. Through a retrospective analysis of video tapes, their thinking processes were analyzed from the perspective of the types of cognitive schemes of operation used as they engaged in the given problem situations. One result of the study indicates that the modifications of the students’ available schemes of operation when solving the fractional reasoning tasks formed a basis for the cognitive schemes of operation used in their solutions of tasks involving proportionality.     

Using the Tower of Hanoi puzzle to infuse your mathematics classroom with computer science concepts

This article suggests that logic puzzles, such as the well-known Tower of Hanoi puzzle, can be used to introduce computer science concepts to mathematics students of all ages. Mathematics teachers introduce their students to computer science concepts that are enacted spontaneously and subconsciously throughout the solution to the Tower of Hanoi puzzle. These concepts include, but are not limited to, conditionals, iteration, and recursion. Lessons, such as the one proposed in this article, are easily implementable in mathematics classrooms and extracurricular programmes as they are good candidates for ‘drop in’ lessons that do not need to fit into any particular place in the typical curriculum sequence. As an example for readers, the author describes how she used the puzzle in her own Number Sense and Logic course during the federally funded Upward Bound Math/Science summer programme for college-intending low-income high school students. The article explains each computer science term with real-life and mathematical examples, applies each term to the Tower of Hanoi puzzle solution, and describes how students connected the terms to their own solutions of the puzzle. It is timely and important to expose mathematics students to computer science concepts. Given the rate at which technology is currently advancing, and our increased dependence on technology in our daily lives, it has become more important than ever for children to be exposed to computer science. Yet, despite the importance of exposing today's children to computer science, many children are not given adequate opportunity to learn computer science in schools. In the United States, for example, most students finish high school without ever taking a computing course. Mathematics lessons, such as the one described in this article, can help to make computer science more accessible to students who may have otherwise had little opportunity to be introduced to these increasingly important concepts.     

Impact of computer animations in cognitive learning: differentiation

In mathematic courses, construction of some concepts by the students in a meaningful way may be complicated. In such circumstances, to embody the concepts application of the required technologies may reinforce learning process. Onset of learning process over daily life events of the student's environment may lure their attention and may enable them to gain from the preliminary knowledge. Therefore, a good initiation may be realized in the course of meaningful learning. The underlying meaning of the abstract concepts by computer animations may be accomplished in class environments. That study is conducted searching out to discover the effects of animations over the learning process in mathematic courses. The study was performed over the 58 university freshman students selected randomly. Thirty-two students constituted the experiment group and 26 students constituted the control group. Computer animations-aided instruction model in constructive form were applied on the experiment group and non-computer-aided instruction model in constructive form were implemented on the control group. Student academic success via a test method developed by explored group with confidence rate .819 (Cronbach's alpha) revealed that data were evaluated by two-way variance analyses. The findings provided from the final test shows that the experiment group students were significantly higher according to the control group students in terms of academic success average scores. Computer animations were observed to be significant to assimilate the derivative concept in a discrete way over the students, to appeal their attention, animations of real life events observed to transform the abstract meanings in the events to a concrete manner. Students of whom the concrete stage is constructed meaningfully found to be tactful in reaching to semi-abstract and abstract stages.     

Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5

We examined ways of improving students’ unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student’s responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.     

Computer Tools for Interactive Mathematical Activity in the Elementary School

The computer tools for interactive mathematical activity (TIMA) were designed to provide children a medium in which they could enact their mathematical operations of unitizing, uniting, fragmenting, segmenting, partitioning, replicating, iterating and measuring. As such, they are very different from the drill and practice or tutorial software that are prevalent in many elementary schools. The TIMA were developed in the context of a constructivist teaching experiment focused on children's construction of fractions. They were used to promote cognitive play that could be transformed into independent mathematical activity. Teaching interventions were often critical in bringing forth mathematical activity. Students' interactions were also important provocations for mathematical reasoning with the TIMA. The TIMA do not, by themselves define a microworld. Rather it is the children's activity and their interpretations of the results of that activity, while interacting with others, that bring forth a microworld of mathematical operations. Designers of computational environments for children need to take into account the contributions children need to make in order to build their own mathematical structures. For teachers to make effective use of software such as the TIMA they need to understand (and share) the views of learning that shaped the development of the software.This revised version was published online in September 2005 with corrections to the Cover Date.