EXPLORATORY DATA ANALYSIS OF A EUROPEAN TEACHER TRAINING COURSE ON MODELLING

Transcription

1 EXPLORATORY DATA ANALYSIS OF A EUROPEAN TEACHER TRAINING COURSE ON MODELLING Richard Cabassut* & Jean-Paul Villette** *University of Strasbourg, France, LDAR, Paris 7 University **University of Strasbourg, France The European project LEMA experimented a teacher training course on modelling in four different countries. We present here an exploratory analysis to evaluate the effect of the training course. From the teachers' answers to a questionnaire on beliefs before the beginning of the training course, the teachers are split in different clusters. With the answers to the questionnaire after the training course, we observe how in every cluster the answers change. We will show how the national groups can be taken in account. Then we will point to questions and challenges related to comparative studies. THE EVALUATION OF A TEACHER TRAINING COURSE The LEMA project context and the focus of the paper The project LEMA [1] has developed from 2006 to 2009 a teacher training course on mathematical modelling. This project was not a research project trying to answer a research question. This project has tried to evaluate the training course. This course was implemented in some partnership countries. Before attending the course the teachers have answered a questionnaire. This questionnaire deals with their biography, with their interest in modelling, with their beliefs about mathematics, and with their beliefs about their ability to teach modelling. The same questionnaire had to be answered at the end of the attended training course. The aim of this paper is not to present the teacher training course, which does not mean that this presentation would not be interesting. A description of the LEMA project is made in (Cabassut & Mousoulides, 2009) and a description of the questionnaire is made in (Maaß & Gurlitt, 2009). The aim of this paper is not to discuss the relevancy of this questionnaire, which does not mean that this discussion is not of interest. A discussion on qualitative and quantitative methods in comparative research is made in (Cabassut, 2007). An example of qualitative evaluation of the teacher training course is offered in (Cabassut & Mousoulides, 2009) or (Schmidt, 2009). The main aim of this paper is to present an exploratory analysis of the data, produced by the answers to the questionnaires, in order to help to answer the questions: how to evaluate the LEMA teacher training course? How to integrate in this evaluation the comparative approach between countries? A secondary aim is to present a comparative method of analysis of data from different countries in order to contribute to the debate on comparative methods. Let us consider the theoretical framework related to this exploratory analysis.

2 The theoretical frameworks Different theoretical frameworks are used in this paper. For the study of modelling we use the theoretical framework of PISA, based on the mathematisation cycle inspired by the works of (Blum, 1996) and described in (Cabassut & Mousoulides, 2009). The questionnaire takes into consideration works of Grigutsch, Raatz and Törner (1998) on teacher's beliefs in school mathematics, of Bandura (1997) on teacher's self efficacy and of Kaiser (2006) on knowledge and beliefs on modelling: the theoretical framework of the questionnaire is well described in (Maaß & Gurlitt, 2009). We describe with more details the theoretical framework related to the main object of this paper: the exploratory data analysis. We propose to present this method of analysis because first we did not find so many studies in mathematical education using exploratory data analysis and secondly this data analysis looks as fitting with a comparative approach as illustrated later. The exploratory data analysis was developed in (Tukey, 1977) in order to analyse data, and to formulate hypotheses based on this analysis. These hypotheses could be confirmed with a confirmatory analysis. In the case of exploratory analysis, descriptive statistics is used to observe the data without preconceptions and without formulating hypotheses in advance. In our example, the data are composed of a whole population of 83 teachers and their answers to a questionnaire. The descriptive statistics on the whole population will allow an inductive approach to describe the population and to formulate hypotheses on the structure of the population depending on the results of the exploratory data analysis. It is possible to try to evaluate assumptions made on the population. On the contrary, confirmatory analysis uses inferential statistics to test, in a deductive approach, the hypotheses formulated in advance. Confirmatory analysis works on a sample of a population, using hypothesis tests and confidence interval estimation. The assumptions have to be accepted and are not testable: only the hypotheses are tested. Confirmatory data analysis is used in the LEMA project as reported in (Maaß & Gurlitt, 2009). In this confirmatory analysis two teachers' groups are compared: the intervention group has attended the teacher training course and the control group did not. The data analysis tests the research question: has the intervention group outperformed the control group over time (by comparing the responses to the questionnaire before the training course and after the training course). Means and standard deviations are used to summarize the answers to the different parts of the questionnaire (questions on beliefs, self-efficacy and pedagogical content knowledge) for the intervention group and for the control group. A two-factorial analysis of variance is used (factor intervention/control group and factor pre/post test). A F-test is used to test the hypotheses. Let us present now the methodology used in our exploratory data analysis.

3 Methodology The population studied is composed of 83 teachers. The questionnaire is composed of questions (variables) with multiple-choice qualitative answers and questions with quantitative answers. All quantitative variables are reconditioned in two intervals by using the median to separate the classes. We split now the variables in three parts: the biographical variables are the questions related to the teachers' biography (country, age, gender, type of school... ) that do not change between the pre and the post questionnaires; the active variables are the questions of the pre-questionnaire, except the biographical variables; the variables of the post-questionnaire are considered as illustrative variables. We begin with a multiple correspondence analysis (MCA) on the actives variables (36 variables with a two choices answer what means 2 36 possibilities). Then we apply a hierarchical ascendant classification (HAC or cluster analysis) by using the distances measured on the first coordinates between teachers on the teachers' first coordinates on the factorial axes determined by the MCA. SPAD software is used. There is a cluster of three teachers who have too many no answers. We decide not to take in consideration these three teachers for the cluster analysis. After the cluster analysis every of these three teachers joins his nearest cluster. In this new cluster analysis there is a one teacher cluster. We repeat the precedent procedure and we get eventually 79 teachers offering with the active variables a cluster analysis with four clusters [2]. For every cluster we consider the splitting active variables. A splitting active variable is an answer for which the percentage of the answer in the cluster is very different than in the whole population. In the next paragraph, every cluster will be described through these splitting variables. In the post-test, the answers to the same questions are considered as illustrative-supplementary variables. Those for which the percentages are very different in the cluster than in the population are split by clusters. The biographical variables (gender, type of school, nationality, level of studies...) are also split by clusters. Let us describe the clusters. TEACHERS ' BELIEFS BEFORE THE TRAINING COURSE To describe the whole population, we observe a variety of positions about beliefs and self confidence: a majority can agree for some items, or disagree for other ones or being divided between agreement, disagreement or neutrality. To describe every cluster we look at the difference in the percentage of an answer between the studied cluster and the whole population. First cluster The first cluster contains 13 teachers with the following splitting answers, much more answered than in the whole population. They agree strongly that every student can create or recreate parts of mathematics, that there is usually more than one way to solve mathematical tasks and problems at school, that students with the right age are

4 able to solve the proposed modelling task and that this task does not take too much lesson time, that if students get to grips with mathematical problems they can often discover something new (connections, rules and terms). They disagree strongly that to solve a mathematical task at school, one has to know the one and only correct procedure or you are lost, that school mathematics is the memorizing and application of definitions, formulas, mathematical facts and procedures, that school mathematics is a collection of procedures and rules which determine precisely how a task is solved. The teachers looks less confident than the whole population for all the items, and specially to give effective verbal feedback to groups and individual students to assist them with modelling, or to support students in developing competencies in arguing related to modelling tasks. The teachers of this cluster look positive at the teaching of modelling, expressing a need to support students in modelling and having and open-minded view on school mathematics beliefs, with mostly strong positions on these points (strongly agree or disagree). Second cluster The second cluster contains 31 teachers with the following splitting answers, much more answered than in the whole population. Most of the teachers feel less confident than the whole population for all the items to teach modelling, and specially they feel less confident to be able to design modelling lessons that help students overcome difficulties in all modelling steps (e.g. problems in validating), to use students mistakes to facilitate their learning in modelling, to effectively assess students progress as they work on modelling tasks, to adapt tasks and situations in text books to provide realistic open problems, to design their own modelling tasks. In this cluster the teachers look less confident to teach modelling. Third cluster The third cluster contains 14 teachers with the following splitting answers, much more answered than in the whole population. They agree strongly that school mathematics is the memorizing and application of definitions, formulas, mathematical facts and procedures. They strongly disagree that school mathematics is useful in helping individuals to become critically aware citizens, that it is possible for students to discover and try out many things in school mathematics, that school mathematics helps to understand phenomena from various areas of our society. Much more than in the whole population, they are neutral to agree that mathematics is of general and fundamental use to society, that there is usually more than one way to solve mathematical tasks and problems at school or that school mathematics helps to solve daily tasks and problems. About self confidence, there is a variation depending on the items, sometimes they are less confident than the whole population, sometimes more confident, without strong difference. The teachers of this cluster look conservative on school mathematics and are less open to application of school mathematics in the life.

5 Fourth cluster The fourth cluster contains 21 teachers. For all the items of self confidence, these teachers are much more confident than the whole population, and specially to design modelling lessons that help students overcome difficulties in all modelling steps (e.g. problems in validating), to design their own modelling tasks, to effectively assess students progress as they work on modelling tasks, to develop detailed criteria (related to the modelling process) for assessing and grading students solutions to modelling tasks, to use students mistakes to facilitate their learning in modelling, to support students in developing competencies in arguing related to modelling tasks, to give effective verbal feedback to groups and individual students to assist them with modelling. They strongly agree to use the modelling approach in their future teaching. The teachers of this cluster look very self-confident to teach modelling. TEACHERS' BELIEFS AFTER THE TRAINING COURSE We observe that for the answers to self-confidence variables the average and median are increasing after the training course, what looks quite normal. For the variables on beliefs about mathematics there are small variations with some exceptions. Teachers agree much more that school mathematics is useful in helping individuals to become critically aware citizens. Maybe the training has offered examples or situations to illustrate this possibility. There is also a big increasing in the probability to use a type of modelling task in teaching, or to disagree that the students will not be able to solve a given modelling task. To analyse the change in every cluster, we look after the training course what are now the answers with the biggest differences between the percentage of the answer in the cluster and the percentage of the answer in the whole population. First cluster More than in the whole population the teachers strongly disagree that school mathematics is the memorizing and application of definitions, formulas, mathematical facts and procedures. They strongly agree that every student can create or recreate parts of mathematics and that there is usually more than one way to solve mathematical tasks and problems at school. They disagree that central aspects of school mathematics are flawless formalism and formal logic. For these split variables the difference with the whole population is smaller than it was before the training with the splitting variables. All these changes are positive to teach modelling. Similarly, before the training course, for all items of the self-confidence variables, the teachers were less confident than in the whole population. After the training we observe that the level of confidence has increased in the population and in the cluster and there are no more strong differences with the whole population for the selfconfidence variables. More precisely, for the two items where the self-confidence was splitting before the training course, after the training course we have a change of the confidence. Before the training course a majority of the cluster had a degree of

6 confidence under the median of the whole population for both items. It is the contrary after the training course, even if the median of the whole population has increased after the training course. More generally after the training course, the difference between this cluster and the whole population is weaker. Second cluster Much more than in the whole population, the teachers are neutral to think central aspects of school mathematics are flawless formalism and formal logic, disagree that doing mathematics at school involves innovative thinking and new ideas, and agree that there is usually more than one way to solve mathematical tasks and problems at school. These split differences are difficult to interpret because the differences observed are reduced. The case is particularly true for the confidence items. The teachers of the cluster keep less confident than in the whole population, even if the level of confidence has increased in the population and in the cluster. But there is no more big difference. After the training course, in this cluster, there are not so much split answers and the differences are considerably reduced. Third cluster More than in the whole population the teachers disagree that every student can create or recreate parts of mathematic, that school mathematics helps to understand phenomena from various areas of our society, that it is possible for students to discover and try out many things in school mathematics. For these three variables we observe after the training course a decreasing of strongly disagreement and an increasing of disagreement. The differences with the whole population are reduced in comparison with the splitting differences before the training course. The splitting variables of strong disagreement and agreement, present before the training course, are no more present as split variables after the training course. Even if the teachers of the cluster keep some conservative beliefs as illustrated above, their position is more moderated and less splitting than before the training course. Fourth cluster For all items of self-confidence the average has increased in the cluster and in the whole population. The median for all items has increased in the whole population. In the cluster for all items the majority of the answers keep in the interval over the median. Even if this percentage has decreased, for every item teachers are more confident in the cluster than in the whole population, and specially to develop detailed criteria (related to the modelling process) for assessing and grading students solutions to modelling tasks, to adapt tasks and situations in text books to provide realistic open problems, to support students in developing competencies in arguing related to modelling tasks, to effectively assess students progress as they work on modelling tasks and to select appropriate tasks suitable for a modelling approach to

7 teaching. The teachers of this cluster keep more confident than the whole population even if the difference is reduced. BIOGRAPHICAL VARIABLES AND CLUSTERS We can observe now how the biographical answers (gender, age, country...) are split in the clusters. We use the same method than for the answers after the training. We look what are the biggest differences between the percentage of the answer in the cluster and the percentage of the biographical answer in the whole population. We will try to interpret the relation between biography and clusters. But it is clear that you can find the same age or the same country or the same type of school split in different clusters. It means that we want to break the idea to define a cluster with the biography answer. A biography answer, for example a given country like France, can be an explanation factor but neither a necessary factor, neither a sufficient factor to belong to a cluster, it means to fill the profile of a cluster. First cluster Younger teachers and French teachers are more numerous in this cluster than in the whole population. On the contrary Hungarian teachers, secondary school teachers, older teachers, teachers with a high number of years as teacher are less numerous. The teachers of this cluster look positive to teach modelling, expressing a need to support students in modelling and having and open-minded view on school mathematics beliefs, with mostly strong positions on these points (strongly agree or disagree). Young teachers could be more sensible to open-minded beliefs because their teacher education was more focused on didactics and pedagogy than older teacher education In France problem-solving plays a main role in mathematics teaching focused on mathematics contents. The French teachers of the training course were from primary school where every day life problems are very important in the official syllabus (Cabassut & Wagner, 2009). Hungarian teachers are less present maybe because their school system is more traditional (Vancso & Ambrus, 2009). Secondary school teachers are also less present maybe because they are more focused on mathematics contents than on modelling activities. Second cluster Secondary school teachers, German teachers, older teachers are more numerous in this cluster than in the whole population. On the contrary primary school teachers, and younger teachers are less numerous. The teachers of this cluster look less confident to teach modelling, and moderately open-minded to modelling. The secondary school teachers are maybe more focused on mathematical contents than primary school teachers and have pressure to achieve their syllabus. German teachers have known a big change in their curriculum in 2009 where modelling becomes a leading idea (Garcia et al., 2007). This official change could make them open to modelling but less confident because it is a new idea in the curriculum. Older teachers could also feel uncomfortable to change their teaching if

8 modelling corresponds to a new teaching. Third cluster Older teachers, teachers with a high number of years as teacher, Hungarian teachers, teachers who have studied mathematics at university level, secondary school teachers are more numerous in this cluster than in the whole population. On the contrary German and Spanish teachers, teachers with a low number of years as teacher, younger teachers are less numerous. The teachers of this cluster look very conservative on school mathematics and are less open to application of school mathematics in the life. Older teachers with long experience could have more conservative behaviour than young and less experienced teachers. Hungary has a traditional and theoretical mathematics teaching that can explain this position (Vancso & Ambrus, 2009). Fourth cluster Younger teachers, Spanish teachers, primary school teachers are more numerous in this cluster than in the whole population. On the contrary older teachers, teachers with a high number of years as teacher, German teachers, secondary school teachers are less numerous. The teachers of this cluster look very self-confident to teach modelling. Younger teachers or teachers from Spain are maybe more confident (Garcia et al., 2007). Primary school teachers, used to multi-subject activities, are more confident to teach modelling. We have explained in the second cluster why German teachers, older teachers or secondary school teachers could be less confident to teach modelling. With the fourth cluster we observe that young teachers or primary school teachers seem to be more open to modelling. DISCUSSION Teachers can be split in different groups depending on their mathematics beliefs (conservative or open minded) and on the degree of confidence to teach modelling. The main difference after the training course is the increasing of the level of confidence. The consequence is that the level of confidence is no more a split variable for the two first clusters. It keeps to be a split variable for the fourth cluster gathering teachers much more confident than the whole population. After the training course, the variables on the beliefs about mathematics are no more strongly split variables: the training course seems to homogenize and to moderate the beliefs. The training course has effectively taken into consideration advantages and disadvantages, difficulties and interests, needs and potentialities of the teaching of modelling. The third cluster keeps on characterising the teachers by conservative points of view on mathematics teaching, even if they look more moderate after the training course. It means that a big change in the beliefs seems difficult to be achieved as expressed in (Maaß & Gurlitt, 2009). If one of the aims of a future course is to change the teachers' beliefs, the course will have to focus on this aim.

9 We have offered this example of comparative study to contribute to the discussion on comparative studies in mathematics education. We do not know why the exploratory data analysis is not so popular in mathematical education in comparison with other fields. A reason could be related to the fact that mathematical education is more controlled by mathematicians than other fields of knowledge. For a mathematician confirmatory data analysis uses a deductive approach and inferential statistics while exploratory analysis uses an inductive approach and descriptive statistics. n the exploratory analysis, clusters enable to zoom in the whole population by aggregating individuals through their answers and not through their biographical values. They enable to build types of teachers to whom targeted training can be addressed. In some international comparative studies, a descriptor aggregates the answers of a country in order to compare it with other countries, ranking the countries with the value of the descriptor. For example the performance of Germany at PISA 2000 shows that Bavaria is significantly above the OECD average and Brandenburg is significantly below. Considering the German performance is a zoom-out where the difference between Bavaria and Brandenburg is not taken in consideration. This zoom-out leads to identify for every country a coherent body of practices. The hit is to define the best country - Finland for example in PISA and to propose this country as a model for the others. The aim of the example developed in this study is to show that other ways are possible. The diversity of the practices in a country shows that the country contributes to different clusters and that teachers from different countries can be gathered in the same cluster. The information got from a zoom-in can be as much interesting as that from a zoom-out. The challenge is to develop the comparisons among different countries taking in account the diversity of the practices and of the beliefs inside every country. NOTES 1. LEMA means Learning and Education in and through Modelling and Applications. This project is funded by the European Union and is described on the project site: 2. The data analysis (MCA and HCA) is made for the statistical part under the control of Villette, J.- P., and for the didactical interpretation of the clusters under the control of Cabassut, R. REFERENCES Bandura A. (1997). Self-efficacy: The exercise of control. New York: Freeman and Company. Blum W. (1996). Anwendungsbezüge im Mathematikunterricht in der didaktischen Diskussion. Mathematische Semesterberichte, 32(2), Kaiser, G. (2006). The mathematical beliefs of teachers about application and modelling - results of an empirical study. Paper presented at the 30 th Conference of the International Group for the Psychology of Mathematics Education, Prague.

10 Cabassut R., & Wagner A. (2009). Roles of knowledge in the teaching of modelling at primary school through a French-German comparison. Paper presented at the 14 th International Conference on the Teaching of Mathematical Modelling and Applications. University of Hamburg. Cabassut R., & Mousoulides N. (2009). Theoretical considerations for designing and implementing a teacher training course on mathematical modeling: Insights from a French-Cypriot Comparison. In A. Gagatsis et al. (Eds.), Cyprus and France research in mathematics education. Lefkosia: University of Cyprus. Cabassut R. (2007). Examples of comparative methods in the teaching of mathematics in France and in Germany. Proceedings of the 5 th Congress of the European Society for Research in Mathematics Education. Larnaca, Cyprus. Cabassut R. (2009). The double transposition in mathematisation at primary school. Proceedings of the 6 th Congress of the European Society for Research in Mathematics Eucation. Lyon, France. Garcia F.J., Wake G., & Maaß K. (2007). Theory meets practice: working pragmatically within different cultures and traditions. Paper presented at the 13 th International Conference on the Teaching of Mathematical Modelling and Applications. University of Hamburg. Grigutsch, S., Raatz, U., & Törner, G. (1998). Einstellungen gegenüber Mathematik bei Mathematiklehrern. Journal für Mathematikdidaktik, 19(98), Maaß K., & Gurlitt J. (2009). Designing a teacher-questionnaire to evaluate professional development about modelling. Proceedings of the 6 th Congress of the European Society for Research in Mathematics Education. Lyon, France. Schmidt B. (2009). Modeling in the classroom motives and obstacles from the teacher s perspective. Proceedings of the 6 th Congress of the European Society for Research in Mathematics Education. Lyon, France. Tukey J. (1977). Exploratory Data Analysis. Reading, Massachusetts: Addison- Wesley. Vancsó Ö., & Ambrus G. (2009). Teaching mathematical modeling in Hungarian schools based on some national traditions. Paper presented at the 14 th International Conference on the Teaching of Mathematical Modelling and Applications. University of Hamburg.