WGA1 Mathematics Education in Pre- and Primary School
Ingvill Merete Holden
Department of Mathematical Sciences/ Program for Teacher Education
Norwegian University of Science and Technology, N-7491 Trondheim, Norway
e-mail: firstname.lastname@example.org , fax: + 47 73 59 10 12
This study is about a playful approach to Mathematics in Primary school, based upon a constructuvistic and social learning environment. We are following two classes in Primary school where our goals are to give the students a positive attitude towards Mathematics, and a view of Mathematics as a subject where creativity, fantasy and originality are important tools for success. The research questions for the study has been:
Will the children´s attitudes towards Mathematics be more positive when Mathematics is presented in a fun way and in a playful context?
How will the children´s knowledge of basic skills develop when they are allowed to use their own methods and encouraged to share their thinking with the class?
The results so far tell us that the students do like Mathematics, and that it seems to be no disconnection between a playful, experimental approach to Mathematics, and the knowledge of basic skills.
The study is the beginning of a longitudinal study of how the students in these two classes develop their Mathematical understanding, knowledge and attitudes through primary and secondary school.
The context of the classroom
When the project started in 1998, the students were in third grade (8 years old). The school has 280 students and 40 teachers. It is a new and modern school with no closed classrooms. This means that there are no walls or doors between the two classes, only different "listening corners" where the students from each class can not see each other. In Anne-Gunn´s class there are 12 girls and 8 boys, and in Arne´s class there are 12 girls and 9 boys. The classes are normal Norwegian classes, which means that the students are at very different levels, and have different support from their parents. In each class there are two students that do not have Norwegian as their first language.
Norwegian Primary school teachers are responsible for all the subjects in the class. This means that they can be flexible about when to teach Mathematics. They can also decide to teach Mathematics for the whole day if that is necessary. They often do that when the student work with larger projects. In third grade the students are at school from 8.30 to 12.00 every day. In fourth grade they stay till 1.30pm twice a week.
Once a week they spend the whole day outdoors, and then they also do outdoor mathematics. Both Anne-Gunn and Arne are experienced teachers and have been more than willing to adopt my ideas and experiment with playful Mathematics.
This is a typical action research project, where the cooperation between the researcher and the involved teachers is of major importance. The two Mathematics teachers and the researcher form the research group. Together we plan the teaching, decide the focus of the research, collect data and discuss if we are getting any answers to our questions. The first year we met once a week for reflection and planning. The second year we had these meetings once a month, and the teachers sent their weekly reports and reflections to me by e-mail. I spend much time in the classroom, and the children think of me as an extra math teacher. In addition to finding answers to our research questions, we also have as a goal to improve practice in the classrooms.
We have used several methods for collecting data. Through classroom observations we get close to the children, listen to their mathematical discussions when they are working in small groups and when they have classroom discussions. I do informal interviews about what they think and how they reason. This is documented in my logg book. We have also done some video- and audiotape recording. More formal interviews with the students, teachers and some of the parents are also parts of the collected data. The interviews are all based upon real situations from the classrooms. In addition we have collected a large amount of students works and their workbooks. The parents have been asked about their own attitudes towards Mathematics and how they feel about their childs attitudes towards the subject. During the last half of fourth grade the students have been tested with the problems from the TIMSS test, population 1.
The teachers have written letters to all the students, and the students have written letters back. They all have a "letter book". This book has turned out to be a "gold mine" in many ways. The teacher can ask questions where the answers can show them how deep the student has understood a certain problem or concept. The student will be trained in showing his or her own thoughts in written form in a precise way, and get a chance to reflect upon his or her own learning. It is striking how well many of them can do this. We know that this kind of meta cognition is important for how the student will develop mathematically. I will give one example from a conversation in the letter book between the teacher, Arne, and the student, Jane.
Arne: Hi Jane, can you tell me what multiplication is?
Jane: Multiplication is the same as doing plusses. From Jane.
Arne: In away you are right. What pluss-problem is 5 x 3 the same as?
Jane: Hi Arne, 5 + 5 + 5 = 15
From Jane´s first answer Arne could not tell if she understood what multiplication is, but after checking with an example it is easy to tell that she does understand.
My role in the classroom has been as a participating observer. I have assisted the teachers in the classroom and worked in groups with the students. The first year of the study we were interested in getting hold of the students thoughts and ability to express themselves orally, and to what extent they would all participate in mathematical discussions in the classroom. The second year we have focused on how they develop their written mathematical language when they are allowed to use their own methods.
Examples of activities
The students have met Mathematics through many different activities and in combination with several other subjects. They have build and sent up kites, baked cakes, experimented with and calculated different ways of filling the trays for the oven with cakes, played several games, used drama to learn about geometry, practiced paperfolding to make two and three dimensional forms, and arranged Math Fair for another class among a lot of other things. We will here only have space to go into details with one example.
The children have woodwork as part of their Arts and Crafts classes. Anne-Gunn is very fascinated by the connection between Mathematics and woodwork. One thing that she let the children do, was making games of wood that they would use in math class afterwards. An example of this is The Tower of Hanoi. To make the wooden tower they had to do a lot of measurements. The tower consists of a wooden plate with three poles evenly placed in a row. On one pole there are five wooden squares of different sizes, from small to big, to form a tower. This was a mathematical challenge for the students of one kind. Later, in math class they got a new mathematical challenge. They should try to solve the puzzle of Hanoi. The aim is to move the tower from one pole to one of the others. The rules are as follows:
The students practiced for some time, and tried to find out how many moves they would need to move the tower with three, four and five stories on the tower. Then I came to visit.
A group of students were gathered around me. They all agreed that they needed 15 moves with four stories. They also knew that with three stories they would need 7 moves. They were not sure about five stories. We put the results up in a table:
# stories least number of moves
I asked if they could guess what would be the number of moves for five stories.
Nicolai: The 15 should have been 13, sort of.
Researcher: Why do you say that?
Nicolai: Well, from 1 to 3 we add 2, and from 3 to 7 we add 4. Therefor I think we should add 6 to get the next number.
Researcher: that is a very good guess, but you have all tried several times, and all of you say that it is not possible to move the tower with less than 15 moves. Maybe we ought to look for another pattern.
After a little while Christopher said: It is 31!
Researcher: How do you see that?
Christopher: It is 1 more than the double.
Researcher: OK. Let us check it out.
Adrian was the best to see what moves to make to do it with the least number of steps. He did it, and the rest of the group counted. He did it in 31 moves! Everybody clapped their hands. They all knew the pattern now, and wanted Adrian to try with 6 stories. This time we need 63! Someone made an extra story out of cardboard. Adrian did it in 63 moves. They continued all the way up to 8 stories and 255 moves. They all felt they had discovered something big, and that the testing proved that their discovery was right.
The curriculum in Norway says that the students should be trained in scientific methods. I think this activity is a very good approach to this goal. They experienced that the first guess did not fit with the testing. Then they tried a new hypothesis and tested it as long as they thought nessesary. In a playful context they do important mathematical discoveries.
By the end of fourth grade, we tested the student with 45 of the problems from the TIMSS test, population 1. The problems were taken from all the areas of the test. The test was given in a different way than in the TIMSS test, since the students were given only a few problems by the end of each math class over a longer period. They had no time pressure as they were given the time they needed. Therefor, when we compare the results with the Norwegian and international results, we have not counted the students that did not reach the different problems in the TIMSS test.
With these reservations, we can say that the students in these two classes score exceptionally higher than both the international and the Norwegian students from the TIMSS test. Only on three problems they score lower than the international results, but about the same as the Norwegian results. The Norwegian researchers from the TIMSS test argue that on one of these problems the translation into Norwegian made the problem harder. The average percentage that give right answers to the problems is 68.5 for our classes, whereas it is 58.0 for the international and 52.0 for the Norwegian students.
One of the classes scored better than the other, and the girls did best. In addition to having a better score on the test, the students in the two classes did not skip any of the problems, with just a few exceptions. This is different from the Norwegian TIMSS students. When the problems were unfamiliar to them, they very often skipped them, most likely because they had not learned a method or an algorithm. Our students, however, are used to be given new and unfamiliar problems all the time. The problems come from concrete situations and a desire to find a solution to a problem. If the mathematics needed was unfamiliar, they would discuss it with each other and find their own methods to find a solution. So, even if they in the test had to work individually, they rather gave an answer than left it open.
The only logical explanation to the good scores, is the teaching and learning environment the students have experienced. When the students are asked what subject they like best, more than half of them answer Mathematics. This question was asked when the teachers had meetings with the students and parents about general school matters. They like mathematics because it is fun. This fits very well with my earlier study with older students; The most important motivation factor is to have fun with Mathematics. In addition to this the teachers enthusiasm is of major importance. The two teachers in the project have built up their own competence both in mathematics and mathematics education. Anne-Gunn has taken statistics and mathematics courses in addition to her teaching during the second year of the project. Arne has started to take teacher students to his classroom. They both show a great enthusiasm about their mathematics teaching. Their consciousness about teaching and learning has been improved by being given a chance to reflect upon their teaching and the students learning.
We conclude that students learn more and better, and get a more positive attitude towards mathematics in a playful learning environment. We also think that there is a positive effect from letting mathematics be a part of larger projects with more subjects involved. The children learn more when they use more senses, when they make their own material, use it and discuss problems and solutions with each other. The classroom discussions give students a chance to get ideas from other students and use this ideas on new problems if they are ready for it. This study will continue for five more years to see how the students develop mathematically. When the project is over, the students are ready for High School.
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