Carmen Batanero

Errors and difficulties in understanding elementary statistical concepts

This paper presents a survey of the reported research about students’ errors, difficulties and conceptions concerning elementary statistical concepts. Information related to the learning processes is essential to curricular design in this branch of mathematics. In particular, the identification of errors and difficulties which students display is needed in order to organize statistical training programmes and to prepare didactical situations which allow the students to overcome their cognitive obstacles. This paper does not attempt to report on probability concepts, an area which has received much attention, but concentrates on other statistical concepts, which have received little attention hitherto.

Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education

Since the mid-1980s, the International Commission on Mathematical Instruction (ICMI, www.mathunion.org/ICMI/) has investigated issues of particular significance to the theory or practice of mathematics education by organising specific ICMI studies on these themes. The 18 Study in this series has been organised in collaboration with the International Association for Statistical Education (IASE; www.stat.auckland.ac.nz/~iase/) and addresses some of the most important aspects of the teaching of statistics in schools by focussing on the education and professional development of teachers for teaching statistics. The Study included an IASE Roundtable Conference and is fully reported in the Proceedings of the Study Conference (www.ugr.es/~icmi/iase_study/) and in the Study book now published in the ICMI Study series by Springer.

Clarifying the Meaning of Mathematical Objects as a Priority Area for Research in Mathematics Education

The specific aim of mathematics education as a research field is to study the factors that affect the teaching and learning of mathematics and to develop programs to improve the teaching of mathematics. In order to accomplish this aim mathematics education must consider the contributions of several disciplines: psychology, pedagogy, sociology, philosophy, etc. However, the use of these contributions in mathematics education must take into account and be based upon an analysis of the nature of mathematics and mathematical concepts, and their personal and cultural development. Such epistemological analysis is essential in mathematics education, for it would be very difficult to efficiently study the teaching and learning processes of undefined and vague objects.

Training Teachers to Teach Probability.

In this paper we analyze the reasons why the teaching of probability is difficult for mathematics teachers, describe the contents needed in the didactical preparation of teachers to teach probability and analyze some examples of activities to carry out this training. These activities take into account the experience at the University of Granada, in courses directed to primary and secondary school teachers as well as in an optional course on Didactics of Statistics, which is included in the Major in Statistical Sciences and Techniques course since 1996. The aim is encouraging other colleagues to organize similar courses at their universities, either as part of their official programs or in their postgraduate training.

The Nature of Chance and Probability

The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account... It teaches us to avoid the illusions which often mislead us;... there is no science more worthy of our contemplations nor a more useful one for admission to our system of public education. (Laplace, 1986/1825, pp. 206-207).

WHAT IS THE NATURE OF HIGH SCHOOL STUDENTS' CONCEPTIONS AND MISCONCEPTIONS ABOUT PROBABILITY?

But it is important to stress that the relationship between intuition and logical structures plays an essential part in the domain of probability, perhaps more conspicuously and strikingly than it does in other domains of mathematics. (Fischbein, 1975, p.5) In high school, students are expected to determine the likelihood of an event by constructing probability distributions for simple s ample spaces, compute and interpret the expected value of random variables in simple ca ses, and describe sample spaces in compound experiments. They are also expected to learn to identify mutually exclusive and joint events, understand conditional probabilit y and independence, and draw on their knowledge of combinations, permutations, and counting principles to compute these different probabilities. By the end of high s chool, students should understand how to draw inferences about a population from random s amples; a process that involves understanding how these samples might be distributed. Such an understanding can be developed with the aid of simulations, that enable students to explore the variability of sample statistics from a known population and to ge nerate sampling distributions (NCTM, 2000; Pfannkuch, Chapter 11 in this book). Borovcnik and Peard (1996) remark that probabilisti c reasoning is different from logical or causal reasoning and that counterintuiti ve results are found in probability even at very elementary levels. By way of contrast, in o ther branches of mathematics counterintuitive results are encountered only when working at a high degree of

Randomness, its meanings and educational implications

An analysis of the different meanings associated with randomness throughout its historical evolution, and a summary of research concerning the subjective perception of randomness by children and adolescents is presented. Finally some teaching suggestions are included to help students to gradually understand the characteristics of random phenomena.

Teachers’ Attitudes Towards Statistics

Teachers’ attitudes towards statistics play a significant role in assuring success in implementing any new statistical curriculum. In this chapter, attitudes and their component factors are conceptualised, and the primary instruments available to assess attitudes are reviewed. Following this, the research on teacher attitudes towards statistics is summarised. Finally, some implications for training teachers in statistics are discussed.

Students and Teachers’ Knowledge of Sampling and Inference

Ideas of statistical inference are being increasingly included at various levels of complexity in the high school curriculum in many countries and are typically taught by mathematics teachers. Most of these teachers have not received a specific preparation in statistics and therefore, could share some of the common reasoning biases and misconceptions about statistical inference that are widespread among both students and researchers. In this chapter, the basic components of statistical inference, appropriate to school level, are analysed, and research related to these concepts is summarised. Finally, recommendations are made for teaching and research in this area.

Evaluación de sesgos en el razonamiento sobre probabilidad condicional en futuros profesores de educación secundaria

In this paper we present a study aimed at assessing biases in conditional probability reasoning in prospective secondary school and high school mathematics teachers. We analyze responses to 7 items taken from the CPR (conditional probability reasoning) in a sample of 196 last year mathematics students and Master students, finding a high incidence in both groups of the time axis fallacy, transposed conditional, confusion between conditional and compound probability, base rates fallacy, and incorrect conceptions of independence. We conclude that there is a need to prepare prospective teachers better to prevent them from transmitting these biases to their students.