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HOW FINNS LEARN MATHEMATICS: What is the Influence of 25 Years of

Research in Mathematics Education?

HOW FINNS LEARN MATHEMATICS: What is the Influence of 25 Years of

Research in Mathematics Education?

Erkki Pehkonen

University of Helsinki, Finland

Erkki Pehkonen

University of Helsinki, Finland

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IntroductionIntroduction

Today Finland is, because of the PISA reults, famous in the world as a country of excellent mathematics teaching.

In each PISA comparison (2000, 2003, 2006), Finland has been in the group of the top three (cf. Kupiainen & Pehkonen 2008).

This might be a reason why other countries are interested in our “secret weapon”, i.e. how the Finnish educational system functions and what might be the reasons for our success.

Today Finland is, because of the PISA reults, famous in the world as a country of excellent mathematics teaching.

In each PISA comparison (2000, 2003, 2006), Finland has been in the group of the top three (cf. Kupiainen & Pehkonen 2008).

This might be a reason why other countries are interested in our “secret weapon”, i.e. how the Finnish educational system functions and what might be the reasons for our success.

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In order to uncover our teaching system we produced a couple of years ago the book How Finns learn mathematics and science (Pehkonen, Ahtee & Lavonen 2007).

Furthermore, in a published paper (Pehkonen 2008) I gave background information on the development of the Finnish mathematics instruction and curricula within last 30 years.

And this presentation continues the same communication process.

In order to uncover our teaching system we produced a couple of years ago the book How Finns learn mathematics and science (Pehkonen, Ahtee & Lavonen 2007).

Furthermore, in a published paper (Pehkonen 2008) I gave background information on the development of the Finnish mathematics instruction and curricula within last 30 years.

And this presentation continues the same communication process.

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MATHEMATICS TEACHING IN FINNISH SCHOOLS

MATHEMATICS TEACHING IN FINNISH SCHOOLS

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The school systemThe school system

In Finland, we have a nine-year comprehensive school that begins at the age of seven.

After the comprehensive school, there are two options: the upper secondary school (grammar school) and vocational school.

In the comprehensive school, mathematics is taught with 3–4 lessons per week, and in the upper secondary school there are two selective courses: advanced mathematics and general mathematics.

The amount of mathematics taught in vocational schools varies according to the career, and it usually is combined with situations of the career in question.

In Finland, we have a nine-year comprehensive school that begins at the age of seven.

After the comprehensive school, there are two options: the upper secondary school (grammar school) and vocational school.

In the comprehensive school, mathematics is taught with 3–4 lessons per week, and in the upper secondary school there are two selective courses: advanced mathematics and general mathematics.

The amount of mathematics taught in vocational schools varies according to the career, and it usually is combined with situations of the career in question.

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Development of the mathematics curricula

Development of the mathematics curricula

A general picture of the development of the Finnish mathematics curricula from the 1960s to around 2000 is presented in Figure (below).

Changes adopted in the US curriculum played a central role in this development, with a delay of about 10 years.

However, the principles of each trend were not taken as such, but they were modified in the process of implementation to better fit the Finnish education system.

A general picture of the development of the Finnish mathematics curricula from the 1960s to around 2000 is presented in Figure (below).

Changes adopted in the US curriculum played a central role in this development, with a delay of about 10 years.

However, the principles of each trend were not taken as such, but they were modified in the process of implementation to better fit the Finnish education system.

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Development of trends in mathematics teaching in Finland and in the US

(according to Kupari 1999).

Development of trends in mathematics teaching in Finland and in the US

(according to Kupari 1999).

”Back to Basics”

”Problem Solving” ”Standards” ”New Math”

”New Math” ”Back to Basics”

”Problem Solving”

”National standards”

1960 1970 1980 1990 2000

1960 1970 1980 1990 2000

USA

Finland

New national curriculum 1972

New national curriculum 1985

New national curriculum 1994

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Changes in learning conceptionsChanges in learning conceptions

During the 1980s the established view on learning began to change, including mathematics teaching.

Cognitive psychology, emphasizing students’ own construction of knowledge and learning, began to replace the older behaviouristic paradigm.

Consequently, the focus of learning shifted to students’ activities and to their ways of perceiving and shaping the world around them (cf. Lehtinen 1989).

In the 1990s, responding to the new demand, a group of Finnish mathematics educators wrote a booklet on mathematics teaching (Halinen & al. 1991), presenting a view very similar to the later concept of mathematical literacy in PISA.

During the 1980s the established view on learning began to change, including mathematics teaching.

Cognitive psychology, emphasizing students’ own construction of knowledge and learning, began to replace the older behaviouristic paradigm.

Consequently, the focus of learning shifted to students’ activities and to their ways of perceiving and shaping the world around them (cf. Lehtinen 1989).

In the 1990s, responding to the new demand, a group of Finnish mathematics educators wrote a booklet on mathematics teaching (Halinen & al. 1991), presenting a view very similar to the later concept of mathematical literacy in PISA.

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New ideas for teachingNew ideas for teaching

Besides traditional teachers’ talk and pupils’ independent calculations, other means of teaching and learning mathematics were to be used: problem solving, exploration, discussions about mathematics, and dealing with problems rising from everyday life.

In implementing these ideas, two key points arose: understanding learning as an active endeavour, and mathematics as a skill to be used and applied in diverse situations.

Besides traditional teachers’ talk and pupils’ independent calculations, other means of teaching and learning mathematics were to be used: problem solving, exploration, discussions about mathematics, and dealing with problems rising from everyday life.

In implementing these ideas, two key points arose: understanding learning as an active endeavour, and mathematics as a skill to be used and applied in diverse situations.

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New ideas for teaching (cont.)New ideas for teaching (cont.)

The former meant that students should have ample time for learning and for deliberating on what they had learnt, while the latter emphasized the importance of using problems rising from everyday life.

This meant tasks where the level of mathematics was not necessarily so high, but where students could apply the mathematics learnt at school in situations that were familiar and meaningful to them.

The former meant that students should have ample time for learning and for deliberating on what they had learnt, while the latter emphasized the importance of using problems rising from everyday life.

This meant tasks where the level of mathematics was not necessarily so high, but where students could apply the mathematics learnt at school in situations that were familiar and meaningful to them.

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Mathematics teachingMathematics teaching

A typical Finnish mathematics lesson begins by checking and going through the last lesson’s homework.

Following this, the teacher introduces a new topic to be learnt, e.g. a new calculation method or a geometric concept, which will then be explored collectively with some examples.

Then the teacher assigns students some problems from the textbook to solve individually, in order to make sure that everything has been understood about the underlining idea.

At the end of the lesson he/she gives the students new homework from the textbook.

A typical Finnish mathematics lesson begins by checking and going through the last lesson’s homework.

Following this, the teacher introduces a new topic to be learnt, e.g. a new calculation method or a geometric concept, which will then be explored collectively with some examples.

Then the teacher assigns students some problems from the textbook to solve individually, in order to make sure that everything has been understood about the underlining idea.

At the end of the lesson he/she gives the students new homework from the textbook.

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This model was dominant in the 1980s and is still so today, despite the recurring curriculum reforms (cf. Maijala 2006; Savola 2008).

According to our experiences, this kind of textbook dependence is stronger in grades 1 to 6, i.e. for elementary teachers, than for the last three years of comprehensive school education with mathematics teachers.

This model was dominant in the 1980s and is still so today, despite the recurring curriculum reforms (cf. Maijala 2006; Savola 2008).

According to our experiences, this kind of textbook dependence is stronger in grades 1 to 6, i.e. for elementary teachers, than for the last three years of comprehensive school education with mathematics teachers.

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MATHEMATICS EDUCATION RESEARCH AND ITS

INFLUENCE

MATHEMATICS EDUCATION RESEARCH AND ITS

INFLUENCE

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DevelopmentsDevelopments

About 30 years ago (in 1974) in connection to the university study reform, elementary teacher program was moved from pedagogical high schools to universities.

At that time eight teacher education units (Helsinki, Joensuu, Jyväskylä, Oulu, Rovaniemi, Tampere, Turku, Vaasa) were established; typically there are a compound of department of education and department of teacher education.

About 30 years ago (in 1974) in connection to the university study reform, elementary teacher program was moved from pedagogical high schools to universities.

At that time eight teacher education units (Helsinki, Joensuu, Jyväskylä, Oulu, Rovaniemi, Tampere, Turku, Vaasa) were established; typically there are a compound of department of education and department of teacher education.

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In this connection new positions in mathematics education were established, both for professors and for lecturers.

Professor positions (as a matter of fact professorships for education of mathematical subjects) were established four: Helsinki, Jyväskylä, Oulu, Vaasa.

These positions have a research obligation, and therefore, research on mathematics education got much new power.

In this connection new positions in mathematics education were established, both for professors and for lecturers.

Professor positions (as a matter of fact professorships for education of mathematical subjects) were established four: Helsinki, Jyväskylä, Oulu, Vaasa.

These positions have a research obligation, and therefore, research on mathematics education got much new power.

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DissertationsDissertations

Here we will concentrate on dissertations done in Finnish school mathematics within the last 25 years (since 1984, altogether 34 studies).

Most of them are written in Finnish, there are only five dissertations in English, and two in Swedish.

The dissertations can be roughly divided into six sections: learning requirements (6), teaching in elementary school (8), teaching in middle school (7), teaching in high school (4), university students (4), mathematics teachers (5).

Here we will concentrate on dissertations done in Finnish school mathematics within the last 25 years (since 1984, altogether 34 studies).

Most of them are written in Finnish, there are only five dissertations in English, and two in Swedish.

The dissertations can be roughly divided into six sections: learning requirements (6), teaching in elementary school (8), teaching in middle school (7), teaching in high school (4), university students (4), mathematics teachers (5).

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Finnish DissertationsFinnish Dissertations

Learning requirements Kallonen-Rönkkö (1984) Aitola (1989) Yrjönsuuri (1989) Malmivuori (2001) Linnanmäki (2002) Hannula (2004)

Teaching in e lementary school Vornanen (1984) Lindgren (1990) Sinnemäki (1998) Häggblom (2000) Niemi (2004) Räty-Zaborsky (2006) Leppäaho (2007) Tikkanen (2008)

Teaching in middle school Silfverberg (1999) Joki (2002) Hihnala (2005) Törnroos (2005) Attorps (2006) Hassinen (2006) Näveri (2009)

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Finnish Dissertations (cont.)Finnish Dissertations (cont.)

Teaching in hi gh school Repo (1996) Merenluoto (2001) Joutsenlahti (2005) Hähkiöniemi (2006)

University students Huhtala S. (2000) Kaasila (2000) Pietilä (2002) Viholainen (2008)

Mathematics teachers Kupari (1999) Huhtala M. (2002) Lilja (2002) Perkkilä (2002) Soro (2002)

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Research projectsResearch projects

Here I will focus on some research projects in mathematics education that have an established status e.g. by getting finance from the Academy of Finland, and that might have influenced mathematics teaching.

The red line in the research program of Erkki Pehkonen has been the use of open problem tasks in school; the program is a compound of three Academy projects.

Here I will focus on some research projects in mathematics education that have an established status e.g. by getting finance from the Academy of Finland, and that might have influenced mathematics teaching.

The red line in the research program of Erkki Pehkonen has been the use of open problem tasks in school; the program is a compound of three Academy projects.

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The 1st projectThe 1st project

The first project “Open tasks in mathematics” was implemented in the upper grades (grades 7–9) of the comprehensive school in 1989–92 in Helsinki area.

It was focused on how problem fields (a certain type of sequences of open tasks) could be used as enrichment of ordinary mathematics teaching and what kind of influences the use of the problem fields has (cf. Pehkonen & Zimmermann 1990).

The first project “Open tasks in mathematics” was implemented in the upper grades (grades 7–9) of the comprehensive school in 1989–92 in Helsinki area.

It was focused on how problem fields (a certain type of sequences of open tasks) could be used as enrichment of ordinary mathematics teaching and what kind of influences the use of the problem fields has (cf. Pehkonen & Zimmermann 1990).

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The 2nd projectThe 2nd project

The second project “Development of pupils’ mathematical beliefs” was implemented in 1996–98 in schools of Helsinki area.

In the first research project teachers’ and pupils’ beliefs were recognized as obstacles for change (cf. Hannula & al. 1996).

The second project “Development of pupils’ mathematical beliefs” was implemented in 1996–98 in schools of Helsinki area.

In the first research project teachers’ and pupils’ beliefs were recognized as obstacles for change (cf. Hannula & al. 1996).

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The 3rd projectThe 3rd project

The third project “Teachers’ conceptions on open tasks” that was implemented in 1998, concentrated on the second observed obstacle: teachers’ pedagogical knowledge (cf. Vaulamo & Pehkonen 1999).

The third project “Teachers’ conceptions on open tasks” that was implemented in 1998, concentrated on the second observed obstacle: teachers’ pedagogical knowledge (cf. Vaulamo & Pehkonen 1999).

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The other Academy projects by Erkki Pehkonen

The other Academy projects by Erkki Pehkonen

Research project “Understanding and Self-Confidence in School Mathematics”, financed 2001-03 by the Academy of Finland.

Research project “Elementary Teacher Students’ Mathematics”, financed 2003–06 by the Academy of Finland.

Research project “Understanding and Self-Confidence in School Mathematics”, financed 2001-03 by the Academy of Finland.

Research project “Elementary Teacher Students’ Mathematics”, financed 2003–06 by the Academy of Finland.

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Other Academy research projectsOther Academy research projects

Other research projects that were financed by the Finnish Academy were Erno Lehtinen’s Pythagoras project (University of Turku), and the bigM project by Simo Kivelä (Technical University, Espoo).

The first one focused on real number concept in upper secondary school (cf. Merenluoto 2001), and the second one developed virtual materials for the first-year mathematics students mainly in technical universities (cf. Kivelä & Spåra 2001).

Other research projects that were financed by the Finnish Academy were Erno Lehtinen’s Pythagoras project (University of Turku), and the bigM project by Simo Kivelä (Technical University, Espoo).

The first one focused on real number concept in upper secondary school (cf. Merenluoto 2001), and the second one developed virtual materials for the first-year mathematics students mainly in technical universities (cf. Kivelä & Spåra 2001).

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Other big research projectsOther big research projects

One of other bigger and long-lasting research project was Lenni Haapasalo’s MODEM project.

He began the project in the 1980’s at the University of Jyväskylä.

It focused i.a. to teach the concept of straight line for an eight-grader using computers (cf. Haapasalo 1994).

One of other bigger and long-lasting research project was Lenni Haapasalo’s MODEM project.

He began the project in the 1980’s at the University of Jyväskylä.

It focused i.a. to teach the concept of straight line for an eight-grader using computers (cf. Haapasalo 1994).

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Influence of research on mathematics teaching

Influence of research on mathematics teaching

Changes happening within 20 years, and the meaning of research for these changes

The authors have presented results of their dissertation studies both in Finnish teacher journals, and during the in-service training days of the Mathematics Teachers’ Union (MAOL).

The meaning of the Association for Research in Mathematics and Science Teaching

Changes happening within 20 years, and the meaning of research for these changes

The authors have presented results of their dissertation studies both in Finnish teacher journals, and during the in-service training days of the Mathematics Teachers’ Union (MAOL).

The meaning of the Association for Research in Mathematics and Science Teaching

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ConclusionConclusion

Although Finland ranked well in all three PISA comparisons (2000, 2003, 2006), a closer look at the results shows that the Finnish achievement level in many basic tasks of the PISA tests was only 50–70 % or less (cf. Kupiainen & Pehkonen 2008, 130).

The fact that the other countries’ achievements were still worse, does not make the Finnish achievement good.

It only shows that the level of mathematics teaching in all countries should be raised, also in Finland.

Although Finland ranked well in all three PISA comparisons (2000, 2003, 2006), a closer look at the results shows that the Finnish achievement level in many basic tasks of the PISA tests was only 50–70 % or less (cf. Kupiainen & Pehkonen 2008, 130).

The fact that the other countries’ achievements were still worse, does not make the Finnish achievement good.

It only shows that the level of mathematics teaching in all countries should be raised, also in Finland.

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Perspectives in FinlandPerspectives in Finland

Now we can ponder, to which direction and how far we are moving on a short time interval.

In Finnish mathematics teaching the direction seems to be to more individualizing in the comprehensive school, and mass teaching in the secondary schools.

Teachers try to balance between large teaching groups and those children who demand special attention.

Even more such children are coming to school who are accustomed to have the unshared attention of their parents and who have difficulties in their social relationships.

Now we can ponder, to which direction and how far we are moving on a short time interval.

In Finnish mathematics teaching the direction seems to be to more individualizing in the comprehensive school, and mass teaching in the secondary schools.

Teachers try to balance between large teaching groups and those children who demand special attention.

Even more such children are coming to school who are accustomed to have the unshared attention of their parents and who have difficulties in their social relationships.

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My evaluationMy evaluation

The direction to emphasize problem-solving and self-initiativeness seems to be a correct one.

But problem-solving should be used as a teaching method, and not only to solve separate problems.

All new information should not be given in a “ready form”, but the teacher should lead pupils via self-initiative thinking to learning objectives.

Problem posing is in a near connection to such a teaching style.

The direction to emphasize problem-solving and self-initiativeness seems to be a correct one.

But problem-solving should be used as a teaching method, and not only to solve separate problems.

All new information should not be given in a “ready form”, but the teacher should lead pupils via self-initiative thinking to learning objectives.

Problem posing is in a near connection to such a teaching style.

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The concluding noteThe concluding note

Now we can say e.g. in the case of problem solving in Finnish schools using the language proposed by the published paper Schroeder & Lester (1989):

Most teachers are in the teaching problem solving in the first phase (teaching about problem solving), i.e. they deal with separate problems, mathematical puzzles, in order to develop their pupils’ thinking skills.

Only a few teachers are in the phase 3 (teaching via problem solving), i.e. using problem solving as a teaching method.

Now we can say e.g. in the case of problem solving in Finnish schools using the language proposed by the published paper Schroeder & Lester (1989):

Most teachers are in the teaching problem solving in the first phase (teaching about problem solving), i.e. they deal with separate problems, mathematical puzzles, in order to develop their pupils’ thinking skills.

Only a few teachers are in the phase 3 (teaching via problem solving), i.e. using problem solving as a teaching method.