Iiris Attorps
University of Gävle, Sweden
DEFINITIONS AND PROBLEM SOLVING
In this paper I discuss the mechanisms governing student teachers’ concept acquisition. I describe what kind of conceptions the student teachers in mathematics have of the concept equation and how these conceptions are related to the prototype of the concept equation. Data was gathered from 75 student teachers by interviews and questionnaires. Both the quantitative and qualitative research methods were applied in the study. My research results indicate that student teachers interpret mathematical concepts operationally as processes even if the concepts were introduced structurally using definitions (Vinner and Dreyfus 1989; Sfard 1989). The majority of students do not use definitions when solving tasks because their everyday-life thought habits take over and they are unaware of the need to consult the formal definitions. In most cases referring to the concept image is successful. My investigation shows that students tend to identify the concept equation with one of the prototypical examples and that the concept image develops from one unique prototypical example to include more examples of increasing distance (in terms of different attributes) from the prototypical example. The results also indicate that the student teachers’ concept images include erroneous conceptions of the concept equation.
Olive Chapman
University of Calgary, Canada
FACILITATING PRESERVICE HIGH SCHOOL MATHEMATICS TEACHERS' DEVELOPMENT OF PEDAGOGICAL KNOWLEDGE ABOUT PROBLEM SOLVING
This paper discusses a way of using reflection to facilitate pre-service high school mathematics teachers' learning about problem solving and problem solving pedagogy. The approach, framed in a constructivist-oriented perspective of learning, focuses on reflection on personal experiences with genuine problems using narratives, participant observations, peer interactions, and flow charts. The goal of the study was to see what the participants would learn from this experience, so they were not provided with any theory about problem solving before or during the approach. The approach was first tested with a group of 12 participants and after minor modifications, was used with a second group of 12 a year later. Data consisted of written activities to capture their thinking about problem solving prior to the approach and all written work and field notes of their discussions during the approach. The findings showed that the participants were able to construct a deeper understanding of problems and problem solving that integrated cognitive and affective processes/factors and of the students' role and teacher's role in facilitating this process.
Chun Chor Litwin Cheng
Hong Kong Institute of Education
HOW CHILDREN USING PATTERN TO SOLVE MATHEMATICAL PROBLEMS
The paper is a result of a study conducted in primary schools in Hong Kong. Nonroutine mathematical problems were given to pupils of age 10 and 11 and by working on the first few particular case, children are asked to generate the solution of the problem. A pretest and post test on pattern recognition were conducted and it is found that ability in solving these combinatoric problems related to their ability of induction and pattern recognition. The problems include pissa toppings, finding the factors of an integer, finding the nunber of rectangles in a given figure. The study set up a framework on how children solve such problems and discuss on the possible barrier that chilren may encouter during their process of problem solving.
Avikm Gazit
The open University of Israel
HOW DO 10TH GRADE STUDENTS SOLVE CHALLENGING WORD PROBLEMS.
The aim of this research was to analyze the problem solving capacity of students from two achievement levels,10th grades'. Challenging problem has no algorithm in advance to work with,performing a new situation to the solver.The problems at the mathematics classroom ars often dull and routin.Teachers rarely give students atractive problems nor instructions how to solve them. The research question of this study was: What are the differnces in solving word Challenging problems between two levels of 10th grade students: High achievers(N=10)and low achievers(N=10). The research instruments were 3 riddles of proportional thinking and sets. Results: Low achievers, 10th grade, gave 14 correct answers out of 30 possible (46.7%). High achiever students gave 13 correct answers(43.3%). The results are contradicting to what we do expect.No differnces between the two achievment level students of 10th grade. We need to develop more problem solving capacity by using challenging problems and other materiales.
Günter Graumann
University of Bielefeld, Germany
In the last meetings of the ProMath group (September 2003) my focus was lying on problem fields in pure geometry. But also in connection with everyday life we can find interesting problem fields for mathematics education. Since the end of the 1970th I developed the conception of "PRactice Orientated Mathematics education (PROM)" where situations of real life are the departure for working with problems. In my presentation in Lahti I want to explain this conception and give three examples which range from grade three to ten.
Markus Hähkiöniemi
University of Jyväskylä, Finland
OPEN APPROACH TO ACQUIRING DIFFERENT REPRESENTATIONS OF THE DERIVATIVE
This paper concerns open approach to learning of the derivative and how does such approach help students in acquiring different representations of the derivative. Problems of the open approach, such as great needs of time and small sizes of groups, are also discussed. Open approach to the subject of the derivative in a high school course is presented. The planned approach is based on the theoretical framework and analysis of the teaching experiment conducted in autumn 2003. The teaching experiment is a part of the pre-research of the author’s dissertation. This pre-research focuses on with which kind of representations students can start their concept acquisition process of the derivative. In the five-hour teaching period the derivative was first introduced by using different representations and open approach. At first, the rate of change of the function was perceived from the graph. Moving hand along the curve, placing pencil as a tangent, looking how steep the graph is and the local straightness of the graph were used as embodied representations. Then, the average rate of change was calculated by different quotient and as the slope of the line. After that the students were given the following problem: How to determine the rate of change at a certain point? After the recorded lessons, the task based interviews were conducted to five students. The results of the analysis show that the students used different representations to examine the derivative, altogether five different methods to estimate the derivative of the function 2^x at point 1. They all used more than one method. All of them were also able to handle the derivative as an object and to determine, for example, the sign of the derivative from the graph of the function.
Raimo Kaasila & al.
University of Lapland, Finland
PRE-SERVICE ELEMENTARY TEACHERS’ DIVISION STRATEGIES
In this paper we will present some preliminary results of our research project on pre-service elementary teachers’ views of mathematics (the project financed by the Academy of Finland). During the project we will examine altogether 269 students from three different universities in Finland (Helsinki, Turku and Lapland). Here we concentrate in teacher students’ understanding of one division problem in the beginning of their studies in teacher education. In the mathematics achievement test teacher students had to solve problem: “We know, that 498 : 6 = 83. How could you conclude from this relationship (without using long division algorithm), what is 491: 6 = ?” Based on results we could conclude that most students rely on standard algorithms and can not reason the result for this unfamiliar tasks. Most do understand that the result will be roughly 82, but only one fifth of the students could solve the problem correctly. In our presentation we will describe what kind of strategies pre-service teachers used when they solved the problem and what kind of misconceptions they had.
Zekeriya Karadag & al.
University of Sakarya
A PATH FROM THE WRONG TO THE RIGHT
The aim of this specific study is to develop an educational software that uses alternative feedback approach to students’ possible wrong answers as an individualized learning opportunity. Students can get the feedback of what could have gone wrong via these feedback alternatives. They are given many chances to succeed and they learn to learn from their mistakes. This software may even be helpful for addressing the misconceptions in problem solving with specific topics. We learn from our mistakes if we know why/what we did wrong. Explanations accompanying wrong answers may promote learning instead of saying only “your answer is wrong”. In a typical classroom environment this aspect is mostly overlooked due to time constraints or teachers’ self-efficacy doubts. To achieve this in the virtual environment is easier than the classroom environment. In our study we propose an educational software in which for each question, the expected mistakes are stated via associated solution paths as different feedback to the students. These solution paths may contain not only short feedback but also in-depth explanations of prerequisite knowledge. This yields us to the individualized education.
Simo Kivelä
Helsinki University of Technology, Finland
HOW TO SOLVE MATHEMATICAL PROBLEMS IN THE COMPUTER ERA?
Problem solving is an essential part of mathematical studies. Traditionally, paper and pen has been the most important device, but tables for numerical calculations have also been used. About 35 years ago calculators of pocket size became available and they have replaced the tables. For 20 years we have had more and more sophisticated software to be used in desktop computers. The mathematical algorithms taught in mathematics courses in upper secondary schools and in university level basic courses have changed very little during this time period. At the moment, computers are used widely in real life problems and this should influence much more on the way mathematics is taught. In general, this means that the algorithms we are teaching should be seen from a new view point. At the same time, the understanding of the basic concepts becomes more important. This is because rather large calculations can be done very easily with the computer, but the user must understand the structure of the problem and be able to decide if he/she can trust on the results. In the talk, some computer based study material is presented and some examples of very traditional problems and their computer solutions are discussed.
Krzywacki Heidi
University of Helsinki, Finland
PROBLEM SOLVING IN MATHEMATICS CLUB
In this presentation I will tell about problem solving as a part of activities in mathematics club. One aim of the teacher education is to combine theory and practice, and also to give teacher students opportunities to develop themselves in practice. In this research I will focus on teacher students (N=13) conceptions of problem solving activities in mathematics clubs which the students had in autumn 2003. Those mathematics clubs were part of their minor subject studies in mathematics (30 ECTS credits). Students planned and were responsible for the implementation of mathematics club sessions which were for primary school pupils (age 9-11). They also reflected the process by writing for example about their conceptions, expectations and notices. Teacher students’ conceptions of problem solving and opinions of possibilities to use problem solving as teaching method will be emphasised.
Henry Leppäaho
University of Jyväskylä, Finland
DEVELOPING OF MATHEMATICAL PROBLEM SOLVING AT COMPREHENSIVE SCHOOL
The problem situations concern both adults and children. Problems don’t limit only mathematics or natural science; problems appear in every school subject and in everyday life. The teaching problem solving was overshadowed often by routine tasks. Only the fast and the talented students have time to do the problem tasks as extra tasks without the teacher’s guide. The aims of this research are: How does problem solving teaching, which is supervised and integrated with different subjects, influence students’ problem solving skills? How does problem solving teaching could influence students’ attitudes to problem solving and to mathematical-scientific subjects? As a didactical challenge is to get students to work in problem solving situation so that they can solve problems by themselves. One purpose is to test a solving map— method with students. The research was carried out grade 6 students 20.10–11.12.2003. The experimental consisted of 19 and the control group consisted of 37 students. The experimental group was taught problem solving course (29 lessons). Lessons were recorded. Teaching was integrated in mathematics, Finnish, art, natural sciences, handicraft and technology. After this was analysed what kind of affects the problem solving course have had to students.
Kaarina Merenluoto
University of Turku, Finland
RESTRAINED BY BELIEFS IN ALGORITHMS AND DISCRETE NUMBERS: STUDENT TEACHERS' SOLUTION STRATEGIES IN OPEN AND CLOSED PROBLEMS USING RATIONAL NUMBERS
This study analyses the solution strategies of six student teachers who participated in problems solving interviews, where open and closed problems were used. These students had been high achievers in mathematics on secondary level. Each one worked with tasks involving sorting of decimal numbers and fractions, adding the density of numbers on the number line, identification of different representations of rational numbers and mechanical tasks involving multiplication and division between two fractions. In the tape-recorded interviews that lasted about 1.5 hours each, the participants were asked to think aloud while working with the tasks. The analysis of the think-aloud protocols showed that the students had a dominating strategy for working with the different tasks, but that most of them were able to change it to another after a while if the first one did not lead to a solution. However, the uses of strategies were clearly restrained by students' beliefs in the foremost use of algorithms ("Help! I've forgotten the formula/ rule"), rules of discrete numbers ("add one"), and by the belief that there is a numeric to every question in mathematics. The results also suggest that with using this method it is possible to observe on-line the processes of increasing sensitivity to the cognitive demands of the tasks, dealing with the ambiguity caused by a recognized cognitive conflict, and the dynamics of experienced certainty during the process.
Martti E. Pesonen & al.
University of Joensuu, Finland
SOLVING MATHEMATICAL PROBLEMS WITH DYNAMIC SKETCHES: AN EXPLORATORY STUDY
We report the results of a study concerning an online learning environment based on interactive animations and corresponding problems. The research purpose was to evaluate how students behave and learn with different kinds of tasks given through dynamic geometry sketches dealing with the function concept. It was tried to find out which were the specific advantages of the interactive representations and which misconceptions appear through the interactive representation form. Also, we wanted to know about the acceptance and usefulness of hints and help, for example hyperlinks to definitions, and how special properties of dynamic geometry sketches can be used sensibly in solving mathematical problems. In this study 42 first year university students (becoming teachers or research mathematicians) worked through this interactive computer-based test environment about the function concept. The learning behaviour of the students were recorded by a video capture software. The results of the video analysis and the test show that the features of dynamic sketches: (1) tracing, (2) dragging, (3) scaling and (4) animating can be sensibly used in solving mathematical problems. Hints and help must be designed very carefully, if we want the students to really use them.
Bernd Zimmermann
University of Jena, Germany
CHANGING THE MODE OF REPRESENTATION AS AN IMPORTANT TOOL FOR MATHEMATICAL PROBLEM SOLVING - ALREADY IN HISTORY OF MATHEMATICS
Relating to the well known educational psychologist Jerome Bruner many studies were carried out which proved that different modes of representation are important to support learning and understanding of mathematics.
The Canadian math educator Claude Janvier edited a quite interesting book on “Problems of Representation in the Teaching and Learning of Mathematics.” Representation is also a major issue in the “Principles and Standards for School Mathematics” of the NCTM.
New results from brain research carried out e. g. by Krause and Seidel et al. proved the importance of different modalities of representation connected with mathematical giftedness.
Stimulated by these clues we want to outline by presenting some examples, that changing the mode of representation has been a fundamental and very fruitful strategy in mathematical problem solving nearly since the first testimonies of mathematical activities in its history.
COUNTRY |
# PARTICIPANTS |
Australia |
1 |
Canada |
1 |
Finland |
7 |
Germany |
3 |
Hong Kong |
1 |
Israel |
1 |
Sweden |
1 |
Turkey |
1 |
USA |
1 |
∑ |
17 |