Articolo in inglese del 2009 sull’insegnamento della matematica in Finlandia che compara i punteggi di due test finlandesi simili effettuati rispettivamente nel 1981 e nel 2003 per conoscere l’evoluzione delle competenze in matematica apprese a scuola dagli studenti finlandesi . I risultati dei test ripercuotono i cambiamenti dei curricoli e della didattica della matematica in Finlandia. Articolo interessante per capire la sorprendente riuscita dei quindicenni finlandesi nell’indagine PISA e TIMSS.

Version imprimable de cet article Version imprimable

Degrado o cambiamento di rotta?

Cosa è cambiato in ventanni nell’insegnamento della matematica in Finlandia? Curricoli e didattica sono cambiati. Gli insegnanti sono stati all’altezza. I risultati non sono più gli stessi e nel 2012 la Finlandia non è più al vertice della classifica dell’indagine internazionale comparata PISA sulle competenze in matematica dei quindicenni. Inizio di un degrado ( ma rispetto a cosa?) oppure indizio di una svolta nell’impostazione dei curricoli e nella didattica? Probabilmente le due cose assieme. La seconda ipotesi è più pertinente. Lo si vedrà nel 2021 se si farà ancora allora l’indagine PISA imperniata sulla cultura matematica.

THE TEACHING OF MATHEMATICS

2009, Vol. XII, 2, pp. 51–56

 

LONG TERM EFFECTS IN LEARNING MATHEMATICS IN

FINLAND—CURRICULUM CHANGES AND CALCULATORS

 

Olli Martio

 

Abstract. Two similar tests to measure the skills of the Finnish school children in

mathematics took place in 1981 and 2003. The tests are compared to a test measuring

the knowledge of basic concepts in mathematics after the student examination. The

results of the tests reflect the changes in the mathematics curriculum and teaching

practices in Finland.

 

 

1. Introduction

 

 

Curricula changes in the Finnish school system have taken place in 8–10 year

intervals. The official curriculum texts are rather short in details. Schools are

free to choose their textbooks and there is neither an official inspection nor an

official approval of the textbooks in Finland. The free market principle prevails.

Hence textbooks and teaching practices should be studied in order to understand

the mathematics curriculum. A similar system is used in many countries. A rather

detailed description of mathematics and science teaching in Finland can be found

in [1]. This collection of articles also contains an account of the teacher training

system used in Finland.

Almost everybody finishing the high school (gymnasium) participates in the

matriculation (student) examination at the age of 18. Hence this test provides

an opportunity to study the final effectiveness of the Finnish school system. The

mathematics test is not obligatory although most students take it. The matriculation

test is 150 years old and its mathematics part has essentially remained

the same, except for the problems, for the last hundred years. In mathematics a

student may choose a basic or an advanced test independently of which courses

the student has followed at school. The basic test is more common. Both tests

consist of 15 problems written on an A4 sheet. A student can choose at most 10

problems out of 15. In practice, solving two problems, or slightly less, he or she

is able to pass the test. Eight or nine correctly solved problems is the standard

requirement for the highest grade but this varies annually. The students are graded

using seven grades whose distribution is the same each time. Because of this the

grades in matriculation tests cannot be used to compare changes in mathematical

skills of the students. The test problems have changed considerably during the

last decades. The problems are based on the aforementioned, rather loosely stated,

official curriculum.

A survey of the Finnish matriculation test in mathematics is in [2].

The purpose of this article is to study the changes in curriculum and teaching

practices that have had the most serious long term effects in learning mathematics

in Finland.

 

2. Mathematics curriculum—changes and effects

 

The changes in the mathematics curriculum in Finland have followed the international

trends. Since 1970 three major revisions have taken place. The first

was influenced by the so-called New Math. This created a lot of discussion but

had a relatively small effect. The second revision can be labelled “Back to basics”.

The last change “Problem solving” had a much greater impact. It was very much

influenced by the demand that the applications of mathematics are all important—

mathematics as such has little value. The influence of calculators was also profound.

It was thought unnecessary to teach those skills which can be performed by a calculator.

Similar changes were experienced in other OECD countries.

In Finland these trends had the following effects on the mathematics curriculum.

  • Mathematics at school became descriptive - exact definitions and proofs were largely omitted.
  • Geometry was neglected.
  • Computations were performed by calculators and numbers and not on a more advanced level.

Students also experienced difficulties when moving from elementary school

mathematics to secondary school mathematics and especially to high school mathematics.

Little has been done to ease this friction.

A rather recent test problem in a basic mathematics matriculation examination

demonstrates these effects. “Why is the sum of the angles in a triangle 180

degrees ?” Nobody knew although the problem was explained in some textbooks

(a line cuts two parallel lines in equal angles). This shows that teaching of mathematical

principles has declined, at least on the basic course, and replaced by a list

of facts given without reasoning. Many teachers are content to demonstrate this

property of all triangles with scissors and paper.

L. N¨averi [3] has studied the effects of the curriculum changes in Finland. Two

similar tests were performed in mathematics in 1981 and in 2003. Participants

belonged to the age group 15–16 year old (9. grade) ; this corresponds to the age

group in the PISA survey since the school starts at the age of seven in Finland. The

tests were participated by more than 350 students. The problems were identical

and supposed to be solved without a calculator. In the following only samples of

the test questions are presented.

The first samples of questions concerns multiplication and the percentages

show the correct answers.

 

Long term effects in learning mathematics in Finland 

 

Multiplication

1981

2003

5·5·5·5= 54

95.2%

90.1%

(3)2 =9

67.8%

47.5%

18·4·32·15=15·32·4·18

93.2%

85.9%

0,015·248=0,15·24,8

66.8%

62.3%

0·8436=0·0,536

79.0%

65.6%

 

In the questions concerning rational numbers the performance drop from 1981

to 2003 was the highest, 20%.

 

 

Rational numbers

1981

2003

26+17=

98.5%

89.8%

(1/2)·(2/3)=

56.4%

36.9%

(4/3)·5=

66.3%

44.4%

(1/6)·(1/2)=

56.5%

28.3%

(1/5):3=

49.2%

27.5%

1278/2=

55.1%

36.8%

 

 

Also in the algebra section the results did not give a healthy picture of the

effects of the curriculum changes.

Algebra

1981

2003

103·102=

72.5%

43.3%

x4·x5=

71.7%

47.3%

(592)3 =(593)2

61.1%

31.7%

 

 

 

If calculators were allowed in the test, the results would have most likely shown

different figures.

In the 2003 survey it was also asked : Explain with your own words the meaning

of (4/5) . 5. The results were as follows :

Correct 6.5%

Almost correct 5.4%

Correct computation but explanation incorrect 8.8%

No explanation but computation corect 31.5%

Incorrect computation and explanation 31.0%

No answer 16.8%

 

Rather few reliable international surveys have been made to compare the

changes of the students performance in elementary and secondary school mathematics

in the time scale of 2–3 decades. It is difficult to separate the effects that

are due to the changes in the curriculum from those which are due to changes in

teaching practices. The survey [3] certainly shows that these effects exist. It would

be interesting to survey the situation in other countries and to look for a general

pattern behind the results. There is at least one reason behind the above results.

It is the use of calculators.

Finland was the best country among the OECD countries in the PISA 2003 survey.

This survey concentrated to the 14 year old age group. The type of questions

asked in [3] were rare in the PISA test. In the TIMSS 1999 report the performance

of the Finnish pupils was also above average. In the latter test the problems were

closer to the questions asked in [3]. The reasons for the Finnish PISA success are

analyzed in [1]. A more critical discussion can be found in the Finnish electronic

journal Solmu [4] of school mathematics (http://solmu.math.helsinki.fi/)

where two special issues have been devoted to the Pisa survey.

 

3. After the matriculation examination

 

Students, who have passed the matriculation test, do not only go to universities

to study. Many of them go to professional schools (training schools for nurses,

various engineering colleges etc.)—usually, but not necessarily, they are students

who have got low grades in the matriculation test.

During the last ten years teachers in professional schools, and not only mathematics

teachers, have complained on the level of the mathematical skills of the

new students. The following sample from [5] shows that these complaints are not

without basis. The test was performed for freshmen in an engineering college and

the figures indicate the percentages of those who correctly answered the problems

on the left hand side. “Basic test” and “Advanced test” mean students who have

passed the corresponding matriculation examination in mathematics. Calculators

were not allowed.

 

 

Basic test

Advanced test

32+42=

55%

78%

(1/31/7)/4=

25%

54%

a2(a+1)2+2a=

17%

50%

Find from the formula U=EIR

26%

68%

ln(x2)2lnx=

7%

34%

The test shows that formula handling, rational numbers, logarithms and algebraic

operations by hand are difficult for those who have passed the basic matriculation

examination. Among the students who have passed the advanced test and

Long term effects in learning mathematics in Finland 55

who have had much more mathematics lessons at school there are many who have

not learnt basic algebraic operations.

 

4. Conclusions

 

The most serious drawbacks in the Finnish mathematics curriculum are the

order and time allocated to different concepts and skills. It is outside the scope

of this report to analyze the situation in detail. Some typical examples can be

mentioned. In the advanced course probability is taught before the concept of an

integral and sequences and series are left to the very end. As the above reports

indicate there are serious defects in the secondary school mathematics curriculum.

In Finland a customer cannot any more ask for 3/4 kilogram meat in a butcher’s

shop since the meaning is not known to a shop-assistant. The right expression is

750 g since this can be fed to a computer. Although the changes in the mathematics

curriculum were made to help people to use mathematics in everyday life, this aim

has badly failed. The problems now considered at school are not those people meet

later on. Problem solving has been overestimated in all levels of the mathematics

curriculum. Teachers at professional schools have learnt this in a hard way.

From the studies [3] and [5] a serious defect in mathematics teaching emerges.

This is the incorrect use of calculators in teaching. Although number handling is

learnt by pen and paper at the elementary school, many students later completely

forget this skill because they have got used to calculators. This does not concern

so much the best 15–20% of the students as results show in the matriculation

examination. The use of calculators is overemphasized since nowadays their use is

extremely limited in everyday life. Professional users of mathematics almost never

use them. Hence the time spent with calculators does not follow the idea that the

skills obtained at school should have some practical value later.

Calculators came to schools in 1975–1995. After a slow beginning they are now

used more and more. No doubt this has been the most essential change in teaching

mathematics and the effects can be seen in the reports [3] and [5].

Mathematics does not concern professional mathematicians only. Mathematics

is used more and more in ordinary professions and the problems involved are

different from those in the PISA survey. In Finland, as in many countries, the

mathematics curriculum includes concepts and skills which once have been put

there because somebody has thought them useful. In most cases time has shown

that these special skills do not meet the demands of the society any more. The

Finnish curriculum architecture and teaching practices require considerable changes

to meet the challenge. Here Finland is not alone.

 

REFERENCES

 

[1] How Finns Learn Mathematics and Science, Editors : E. Pehkonen, M. Ahtee, J. Lavonen,

Sense Publishers 2007, 278 pp.

[2] Lahtinen, A., The Finnish Matriculation Examination in Mathematics, In : Nordic Presentations

(eds. E. Pehkonen, G. Brandell & C. Winslw), 2005, 64–68. University of Helsinki.

Department of Applied Sciences of Education. Research Report 262.

56 O. Martio

[3] N¨averi, L., Understanding computations, Dimensio 3/2005, 49–52 (in Finnish).

[4] SOLMU, 2 special volumes on the PISA survey 1/2005–2006, 2/2005–2006 : http://solmu.

math.helsinki.fi/2005/erik1/ (in Finnish) and http://solmu.math.helsinki.fi/2006/

erik2/ (in Finnish).

[5] Tuohi, R. et al., Fact or fiction—mathematical skills of new engineering students, Turun

ammattikorkeakoulun raportti 29, Turku 2004 (in Finnish).

Department of Mathematics and Statistics, P.O.Box 68 (Gustaf H¨allstr¨omink. 2b), FI 00014

University of Helsinki, Finland

E-mail : olli.martio@helsinki.fi

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