Using Simulation to Bridge Teachers´ Content and Pedagogical Knowledge in Probability
Carmen Batanero, Rolf Biehler, Carmen Maxara
Joachim Engel and Markus Vogel
Probability is increasingly taking part in the school mathematics curriculum; yet most teachers have little experience with probability and share with their students a variety of probabilistic misconceptions. Probability is difficult to teach for various reasons, including disparity between intuition and conceptual development even as regards apparently elementary concepts. Since an education that only focuses on technical skills is unlikely to help teachers overcome their erroneous beliefs, it is important to find new ways to teach probability to them, while at the same time bridging their content knowledge and their pedagogical content knowledge.
In this paper we analyse the use of simulation to increase teachers' knowledge and help them confront their stochastic misconceptions while at the same time making them familiar with their students' misconceptions and providing models of didactical situations they can use in their own teaching. Results from our simulation experiences in the education of primary teachers in Spain and secondary and primary teachers in Germany will be analysed. We will also reflect our different institutional contexts in which different conceptions and traditions of bridging content knowledge, pedagogical content knowledge and pedagogical knowledge exist.
Using Simulation to Bridge Teachers´ Content and Pedagogical Knowledge in Probability
Carmen Batanero, University of Granada, Spain, batanero@ugr.es
Rolf Biehler and Carmen Maxara, University of Kassel, Germany
Joachim Engel and Markus Vogel, University of Ludwigsburg, Germany
Abstract
In this paper we analyse the role of simulation in providing teachers with stochastic experience and knowledge, helping them confront their probabilistic misconceptions and offering them examples of didactic situations to teach probability with a modelling approach. Examples of simulation activities used in the training of teachers are analysed.
1. Introduction
Probability is increasingly taking part in the school mathematics curriculum; yet most teachers have little experience with probability and share with their students a variety of probabilistic misconceptions (Stohl, in press). Since an education which only focuses on technical skills is unlikely to help these teachers overcome their erroneous beliefs, it is important to find new ways to teach probability to them, while at the same time helping to bridge conceptualization and pedagogy (as suggested by Ball, 2000). Moreover, since students build their knowledge in an active way, by solving problems and interacting with their classmates we should present future teachers with some activities based on a constructivist and social approach to teaching (Jaworski, 2001).
Simulation is today a fundamental tool, that even in primary education allows students to model and experiment with random phenomena and predict the long run behaviour, giving a reality to the frequency interpretation of probability which is hardly possible without fast random generators. In this paper we analyse the use of simulation to increase teachers’s knowledge, help them confront their stochastic misconceptions (Heaton, 2002) while at the same time making them familiar with their students’ misconceptions and providing models of didactical situations they can use in their own teaching. Results of simulation experiences in training primary teachers in Spain and secondary teachers in Germany are analysed (complementary experiences are described in Batanero, Godino & Roa, 2003).
2. Stochastic Reasoning and Intuition
Probability is a young area of mathematics and its formal development was linked to a large number of paradoxes (Batanero, Henry and Parszycs, in press). Counterintuitive results in probability are found even at very elementary levels, whereas in other branches of mathematics they only appear at advanced levels (Borovcnik & Peard, 1996) and are reflected in the many widespread probabilistic misconceptions (see Jones & Thornton, in press for an update). For example, although independence is mathematically reduced to the multiplicative rule, a didactical analysis of independence should include discussion of the relationships between stochastic and physical independence, and of psychological issues related to causal explanation that subjects often relate to independence.
To sum up, stochastics is difficult to teach, because we should not only present different models and show their applications, but we have to go deeper into wider questions, consisting of how to obtain knowledge from data, why a model is suitable and how to help students develop correct intuitions in this field.Below we are arguing that simulation plays a very important role in developing sound stochastic intuitions and in reifying probabilistic situations. In the probability education of future teachers we suggest that simulation is complemented with some formal mathematical analyses as well as reflection on the pedagogical component of the situation.
3. Experiences involving simulation
An important challenge is to develop an instructional design where teachers learn the main stochastic concepts and support sound intuitions. Fortunately, we count on simulation, where we substitute a real random situation by a different experiment, which is a model for the original but can easily be manipulated and analysed. Computersupported interactive simulation helps to build a simplified model, where irrelevant features are disregarded, and the phenomena is condensed in time and available to the students’ work (Engel & Vogel, 2004). Formal mathematics is reduced to a minimum allowing students to explore underlying concepts and experiment with varying setups.
Simulation provides a good opportunity to include a modelling approach in the teaching of probability (Biehler, 1991; Dantal, 1997). Between the domain of reality (the random situation we want to analyse) and the theoretical domain (a mathematical model) Coutinho (2001) locates the pseudoconcrete domain where we work with simulation. There, the student is out of reality and works with an abstract ideal situation. The didactical role of a pseudoconcrete model is to implicitly induce the theoretical model to the student, even when mathematical formalisation is not possible (Henry, 1997). The simulated experiment is at the same time a physical and algorithmic model of reality, since it allows an intuitive work on the model that facilitates later mathematical formalization.
3.1. Spanish experiences
The main aim of these experiences, carried out with future primary school teachers is introducing them to the use of simulation in solving and reflecting on counterintuitive probability problems. Our approach is based on the following steps:

Solving some classical test items used in research on probabilistic reasoning where most future teachers are likely to have a particular misconception. In our experiments we considered the following misconceptions: neglect of sample size, equiprobabiliy bias, law of large numbers, base rate fallacy and misperception of randomness and independence (for a description see Jones & Thornton, in press).

Small sample simulation with random number tables.

Large sample simulations with professional statistics software (Statgraphics), and describing how regularity is achieved in the progressive increase of sample size.

A decisive step is an organised debate following the activity, where future teachers have to explain why experimental results contradicted their previous expectation, and find the mathematical solution to the problem with the help of a tree diagram. A written report should summarise their conclusions regarding probabilistic misconceptions, how they can be assessed and what type of teaching experiences might help students overcome these misconceptions.
3.2. German experiences
The main aim of these experiences is to introduce future secondary teachers to applied mathematics and modelling, following a fivestep process as described in Engel (2002) and similar in Biehler (2003a): 1) Introduction of a “realworld” problem involving some data analysis; activity to experience the dynamics of the phenomena; 2) Building a simulation model in a technological environment (Fathom, LipStat); 3) Generating and analysing data, including simulationbased inferences; 4) Critical evaluation of conclusions and reflecting on the impact of our assumptions; and 5) Mathematical analysis based on probability and mathematical statistics. At the same time, future teachers can experience activity based methods of teaching that can be extremely valuable for teaching statistics. Examples of content include: patterns in coin tossing series, sampling distribution; random walk; regression effect; estimation, stratified versus simple random sampling, capturerecapture; randomised response, central limit theorem, curve fitting and time series.
In Kassel, we integrated this simulation approach into a whole obligatory course on elementary stochastics for primary and secondary teachers, using Fathom as a tool for simulation (Biehler, 2003b). The competence the student teachers developed was studied with several methods. The weekly homework assignments required them to submit annotated simulation models. We use these student productions to analyse their cognitive change. Moreover pre and post test results and a qualitative study on students’ working at the computer on modelling problems are available. They give interesting insights into the profiles of pedagogically enriched content knowledge the students have constructed and into which preconceptions are most difficult to overcome.
4. Final Remarks
As stated by Ball (2000, p. 242), understanding the subject matter is essential for teachers listening flexibly to others and being inventive in creating learning experiences for a variety of students. Results from simulation were shocking for future teachers in the Spanish experience – as they contradicted their strongly held beliefs; but this contradiction helped to involve them in the mathematical activity and overcome different misconceptions (e.g. neglect of sample size) at the end of these experiences. Future teachers were also able to develop sound mathematical arguments to explain the different probabilities in the events involved and to recognise the potential usefulness of the didactical approach in teaching probability to their own students and in assessing their students’ misconception.
Given the widespread presence of probabilistic misconceptions in future teachers, these experiences have shown that simulation can help to train teachers simultaneously in probability content and its pedadogy, since it helps to improve the teachers’ probabilistic knowledge, while making them conscious of incorrect intuitions within their students and themselves. In the case of future secondary teachers which have a stronger previous mathematical background computerbased simulation provides ample opportunities to meet all the major demands of modern mathematical instruction (NCTM, 2000) such as representing and analyzing real situations mathematically, problem solving, decision making based on mathematical reasoning, communication of one’s own thinking and making connections.
We should, however be not too optimistic as regards the difficulties of simulation, since results from small sample simulations might confirm future teachers in their incorrect intuitions. On the other hand, some future teachers in our experiences confused the estimation of probability given through simulation with the real theoretical value of probability which is only accessible by formal calculus. Moreover, since simulation only provides the problem solution and not the reason why this solution is valid it has no explanatory power. Consequently, simulation should precede and complement some formal study of probability. Of course formal mathematics is only explanatory for the professional mathematician, but accepting the power of mathematical modelling will be facilitated if future teachers have previously built correct probability intuitions that might be supported by simulation which provides pseudo concrete models or reality.
Acknowledgement: Research supported by the grants HA20020069 and SEJ200400789.
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