Publication Type: Conference Paper/Unpublished Manuscript
Abstract: As recent studies in economic and financial sociology have underscored, calculation is central to economic practices. The paper, first, outlines how these practices of economic calculation can be studied empirically and from a cultural sociology perspective. In this part, the paper refers to the notion of “operative writing” with which the performative effects of equations and formulae can be taken into account. Second, the paper analyses practices of negotiation in risk management departments of big international banks. It is argued that this remote communication between subsidiaries and the headquarters of international banks is structured by written documents which serve as social prostheses and which are the outcome of calculation tools.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: The topic of my presentation will address the evolution of teaching an Introductory Statistics course to students enrolled in pharmacy and health sciences degree programs. I will show how, since I have begun teaching this course in the Spring of 2004, my goal has been to advance my students’ understanding of statistical concepts, in particular concepts needed to interpret and validate statistical analyses in medical research papers. To this end, my presentation will discuss ideas and methods I have explored in the classroom with various degrees of success, such as: raising awareness about misleading statistics used in the mass-media, different approaches to using hand-held calculators, incorporating data and examples from the real world, finding a balance between teaching students the basics - by requiring them to do calculations and to memorize theory - and encouraging them to engage in analytical thinking so that they could gain a deeper understanding of statistics and its applications.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: The students’ use of technology plays an important role in their learning of mathematics. Here we report the work shown by high school students who participated in problem solving activities using an algebraic calculator TI-92. Tasks proposed by the instructors are used to illustrate approaches that appeared during the students’ work. Each approach shows diverse mathematical processes and resources that helped them explore and solve the tasks at hand and the role played by the algebraic calculator as a mathematical instrument.

The curriculum framework proposed by the NCTM (2000) emphasizes the importance of organizing students’ learning activities in terms of mathematical content ––for instance numbers, geometry, algebra, data analysis–– and processes that appear in the practice of doing mathematics ––problem solving, reasoning, communication, connections, and representations. It is also recognized that the use of calculators and computers, in particular of algebraic calculators, is important to promote a mathematical way of thinking that is consistent with the practice of the discipline.
Electronic technologies –calculators and computers- are essential tools for teaching, learning, and doing mathematics. They furnish visual images of mathematical ideas, facilitate organizing and analysing data, and compute efficiently and accurately. They can support investigation by students in every area of mathematics… When technological tools are available, students can focus on decision-making, reflection, reasoning, and problem solving (NCTM, 2000, p.24).
Along students’ learning experiences, it becomes important to document methods and strategies that appear as fundamental in their problem-solving approaches. Research questions that guide this study include:
What type of mathematical thinking can be enhanced via the use of technology while learning the discipline?
To what extent is students’ thinking is compatible with approaches based on paper and pencil?
What features of mathematical proof are privileged via the use of technology?
We concentrate on investigating students’ explicit answers while working on a series of learning activities implemented throughout a ten weeks problem-solving seminar (with 12th grade students), that goes on with the work presented at PME25 (see Moreno & Santos, 2001). We are interested in characterizing the kind of tasks used during the course and highlighting aspects of mathematical thinking that emerged from their implementation. Students used the TI-92 algebraic calculator.
Conceptual Framework, Methods and General Procedures
We designed and organized the tasks around mathematical activities that involved:
(1) Generalisation and formalisation of patterns,
(2) Representation and examination of mathematical situations through the use of algebraic symbols, and
(3) Visual modelling to represent and analyse quantitative relationships.
Kaput (1999) recognizes that these types of activities are crucial in promoting students’ algebraic reasoning and should be present throughout the mathematical curriculum. A task that initially included the use of basic algebraic resources could be transformed into a platform to discuss ideas of variation and display the power of other mathematical representations. During the development of instruction, we followed a structure that included several phases:
(i) The instructor provides information regarding goals or aims of the task,
(ii) Students work individually and then share their ideas within a small group. Later, the solutions are presented to the whole group,
(iii) The students’ processes of searching for extensions and connections of the original problem, and
(iv) Concluding remarks where students identify themes and goals that appear during their interaction with the task.
Students were encouraged to share their ideas, to listen to other students, and to communicate their approaches (Schoenfeld, 1998). Students could show their approaches to the entire class (using a view-screen) and share files with other students. We present, in the following sections, a few examples that are representative of the collection of tasks that were implemented during the seminar. In the complete report we offer a detailed analysis of students’ work. The instructors’ interventions played an important role in orienting the students’ discussions.
SEARCHING FOR INVARIANTS: GENERALIZATION AND FORMALIZATION OF PATTERNS.
Students were asked to look for expressions that approximate the value of pi. A student brought into the class discussion the famous infinite product proposed by John Wallis (1616-1703).

Students’ work focused on exploring particular cases associated with this expression. So the first goal was to introduce this formula into the calculator in a way that enabled the analysis of partial results. They noticed that numbers on the numerator were even, while those on the denominator were odd. With this information and after a period of deliberation, they proposed to use (2n)(2n) and (2n-1)(2n+1) to represent respectively the numerator and denominator of the expression. Afterwards, they explored partial products for different values of n. Thus students realized that when the value of n increases the value of the expression approaches the value of . It was clear that the use of the calculator helped students endow meaning to this expression and also became important to discuss ideas related to the concept of limit.
Similarly, students had the opportunity to examine Euler's expressions to approximate the value of e (sums of reciprocal of factorials).
Again, an important first step is entering this expression into the calculator. Eventually, students wrote the expression and explored the behaviour of partial sums for different values of n.

Mathematical themes and processes that appeared during the implementation of these and similar tasks include the idea of approximation, a certain notion of convergence, the use of algebraic symbols to obtain compact expressions, looking for patterns and invariants and the use of basic properties to operate symbols. Moreover, students realized that the constants pi and e has specific meanings and observed the behaviour of the formulas to compute them.
ALGEBRAIC REPRESENTATION OF MATHEMATICAL SITUATIONS
Calculators are powerful semiotic tools for dealing with regularity in general expressions of mathematical phenomena.
Determine a formula for the product of differences between unity and reciprocals of squeares .
A first goal for students was to represent each expression in a closed form. This process involves observing the patterns and transforming the products and sums into expressions that can be worked with the calculator. The corresponding expressions will be shown in the full version. It is important to mention that students verified their formulae by calculating and comparing results from the original expressions and formulae of various values of n.
An example that can be studied in terms of criteria of divisibility is: show that the expression n5-5n3+4n is divisible by 120 for all positive integers n. Here, students used their calculator to factor the expression and gave arguments to support their answers. For example, they observed that the factors of the expression are five consecutive numbers: n, n-1, n-2, n+1, and n+2. What can we say about five consecutive numbers? In this example, it was evident that the use of the calculator led students to achieve an expression that needed to be interpreted in terms of mathematical properties (divisibility criteria). Similarly, when students dealt with the problem “how many zeros appear at the end of 100! – factorial of n?”, they realized that it was not enough to factor 100! To solve the problem, but it was necessary to reflect on the meaning attached to what they had obtained.

REMARKS

Students’ growing familiarization with computational tools allows these tools to be transformed into mathematical instruments in the sense that computational resources are gradually incorporated into the student’s mathematical activity. We suggest then, that exploring with computational tools eventually allows students to realize how the mediational role of these tools helps them to re-organize their problem-solving strategies (Guin, & Trouche, 1999). Working with the virtual versions of mathematical objects provided by the algebraic calculator promotes students’ constructive activities. Indeed, these virtual versions produce the feeling of material existence, because we can manipulate them where they exist: On the screen. Without these tools, it is quite difficult for students to establish conjectures, or producing a formulation associated with their explorations and express it in the language of the computational medium wherein they are working. The computing environment is an abstraction domain (Noss & Hoyles, 1996), which can be understood as a scenario in which students can make it possible for their informal ideas to begin coordinating with their more formalized ideas on a subject.

References
Guin, D. & Trouche, L. (1999). The complex process of converting tools into mathematical instruments: the case of calculators. International Journal of Computers for Mathematical Learning vol. 3p.195-227.
Kaput, J. (1999). Teaching and learning a new algebra. In E.Fennema & T.A. Romberg (Eds.), Mathematics classroom that promote understanding, pp.133-155. Mahwah, NJ: Lawrence Erlbaum.
Moreno, L. & Santos, M. (2001). The Students’ Processes of Transforming the Use of Technology in Mathematics Problem Solving Tools. In van den Heuvel-Panhuizen, M. (ed.) PME25, vol. 2, Utrech, The Netherlands.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston VA: The Council.
Noss, R. &Hoyles, C. (1996). Windows on mathematical Meanings. Learning Cultures and Computers. Dordrecht: Kluwer Academic Publishers.
Schoenfeld, A., H. (1998). Reflections on a course in mathematical problem solving. Research in Collegiate Mathematics Education III., pp. 81-113.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: This paper investigates the ways in which Advanced Placement Calculus AB students use the graphing calculator when problem solving independently. When students were solving problems on their own they used the graphing calculator in four of the five tool modes suggested by Doerr & Zangor (2002). The students were prompted to use the calculator for five purposes: to skip a step, to get oriented in the problem, to save time, as a new approach, or to check a conjecture. Preliminary findings suggest that students are most often prompted to use the graphing calculator as a visualizing tool when they need a new approach for a problem.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: In the English criminal justice system, defendants charged with triable either-way offences such as actual bodily harm can decide whether to be tried by lay judges in the magistrates’ court or by jury in the Crown court. Statistics indicate that defendants are more likely to be convicted in the magistrates’ than Crown court. However, if convicted, the maximum penalty is greater in the Crown than magistrates’ court. We studied the decision strategies used by prisoners from two prisons when deciding whether to be tried by judge or jury. We examined their use of expected utility calculations and heuristic strategies.