Tutkimuksen teoreettisen viitekehyksen perustan muodostaa geometrisen ajattelun kehittymistä kuvaava van Hielen teoria ja siihen liittynyt tutkimus. Työssä ko. Teoriaa verrataan eräisiin muihin uudempiin ja geometrian oppimisen kannalta keskeisiin matemaattisen ymmärryksen kasvua tarkasteleviin teorioihin. Tutkimuksessa konstruoidaan hypoteettinen malli geometrisen käsitetiedon kehittymisestä täydentämällä ns. van Hielen tasojen avulla esitettyä kuvausta geometrisen ajattelun kehittymisestä erityisesti prototyyppisten ja figuratiivisten käsitteiden muodostumisesta kertyneellä tiedolla. Tutkimuksen empiirisessä osassa koetellaan van Hielen tasojen käyttökelpoisuutta geometrisen ajattelun kehittymisen yleisenä viitekehyksenä. Tässä yhteydessä tutkitaan myös tärkeimpien visuaalis-geometrisen tiedon prosessointiin vaikuttavien yleisten taustatekijöiden yhteyttä tasojen syntyyn. Lisäksi aineiston avulla arvioidaan tutkimuksessa konstruoidun van Hielen teoriaa spesifimmän käsitetiedon oppimisen kuvauksen toimivuutta ja haetaan tietoa yläasteen oppilaiden geometrisen käsitetiedon olemuksesta ja kehittymisestä sinänsä. Empiirinen aineisto kerättiin yhden keskisuuren tamperelaisen yläasteen kaikilta luokkatasojen 7, 8 ja 9 oppilailta muutamaa oppilasta lukuun ottamatta. Kaikkiaan tutkimukseen osallistui 241 oppilasta. Useampien koulujen mukaan ottamista ei nähty välttämättömäksi, koska tutkimuksen pääasiallinen tarkoitus oli testata van Hielen teorian perushypoteeseja sekä konstruoitua geometrisen käsitetiedon mallia ja löytää riittävä esimerkkiaineisto tämän pohjaksi. Tällaisenaankin aineisto muodostui niin laajaksi, että se antaa viitteenomaista tietoa myös yleisemmin peruskoulun yläasteen oppilaiden geometrisen käsitetietouden laadusta. Tutkimuksessa alkuperäisten van Hielen tasojen vH0, vH1, vH2 ja vH3 muodostamaa rakennetta täydennetään tasojen vH1 ja vH2 väliin sijoittuvalla uudella tasolla vH1-2, joka tasojen hierarkiatarkasteluissa erottuu omaksi tasokseen yhtä hyvin kuin muutkin tarkastellut kehitystasot. Tutkimuksessa esitellään laajasti oppilaiden geometrisiin peruskäsitteisiin ja näiden keskinäisiin suhteisiin liittämiä merkityksiä sekä mm. oppilaiden tapaa määritellä käsitteitä. Näiltä osin tulokset ovat osin ristiriitaisia kirjallisuudessa yleisesti tasolle vH3 esitetyille tulkinnoille. Tälle tasolle sijoittuneiden oppilaiden määrittelytaidot ovat aineiston perusteella heikompia kuin tason kuvauksissa on edellytetty. Samoin oppilailta tutkitut tietorakenteet ovat jäsentymättömämpiä kuin niiden tällä tasolla on yleisesti oletettu olevan. Kaiken kaikkiaan oppilaiden geometrinen käsitetiedon kehitys näyttää tutkimuksen perusteella jäävän yläasteen aikana vähäiseksi arvioituna tässä tutkimuksessa tarkastelluilla piirteillä. Työn lopussa pohditaan käsitetiedollisten tavoitteiden heikon toteutumisen mahdollisia syitä ja esitetään joitakin pedagogisia keinoja tilanteen korjaamiseksi.

The theoretical framework of the study is constructed on four main themes: 1) an introduction to the van Hiele theory and a review of literature related to this theory, 2) comparison of the van Hiele theory with some more recent theories which in their examination of the development of mathematical understanding are essential from the viewpoint of learning geometry, 3) constructing a development model of conceptual knowledge in geometry, and 4) examining background factors influencing the processing of visual-geometrical knowledge.

The empirical part of the study focuses on the content of the conceptual geometric knowledge of pupils in the upper level of the comprehensive school and the development of this knowledge during the three school years. The research uses both a cross-sectional and a longitudinal study in the examination of the data. The tests used in data collection are derived from the development model constructed in the theoretical part. The goals of the empirical part are threefold. Firstly, the study provides information on the essential character and general development of the conceptual geometric knowledge of pupils in the upper level of the comprehensive school. Secondly, with the help of empirical observations, it is possible to test the validity of the van Hiele theory as a description of the general level of geometrical thinking. Thirdly, the data enable an evaluation of the feasibility and validity of the constructed model of conceptual learning, which describes the process of learning more specifically than the van Hiele theory.

The empirical data were collected from the pupils in forms 7, 8 and 9 of the upper level of a medium-sized Tampere comprehensive school. As most of the pupils took part in the study, the total number of participants was 241. It was not considered necessary to include more schools because the main objective was to test the constructed model of conceptual geometric knowledge and to find enough example material for the study. The collected data proved large enough to provide implicative information even more generally on the quality of the pupils' conceptual knowledge in the upper level of the comprehensive school and the development of this knowledge during the three school years. By comparing the conceptual knowledge of pupils in the different school forms it was possible to obtain information on form-specific differences in conceptual knowledge and implications of the development of this type of knowledge during the three last years of comprehensive school. The features of the development of conceptual knowledge in geometry which were identified by a cross-sectional analysis were then verified by a longitudinal study. Part of the pupils (n = 85) were tested in the sixth as well as the ninth form, which made it possible to evaluate the development of the same pupils' conceptual knowledge in geometry. Exploring the pupils' knowledge structures also meant that some new research methods had to be generated. The study challenges many of the basic assumptions of the van Hiele theory. One of these is the hierarchy assumption related to the van Hiele levels. Testing the hierarchy of the levels only concerned levels vH0, vH1, vH2 and vH3, for these were considered the most relevant in terms of geometry instruction in the upper level of the comprehensive school. As an experiment, a new intermediate level, vH1-2, was constructed between levels vH1 and vH2. In it, in accordance with Piaget's notion of empirical abstraction, the common geometrical characteristics of the figures examined were interpreted as generalisations of that group of figures - and only that - which the drawings in each test item concretely depicted. At the same time, in accordance with Piaget's notion of reflective abstraction, level vH2 was interpreted so that pupils placed on this level are able to perceive the characteristics of the figures as common features of the whole figure category. The study distinguished between two different interpretations of the possible hierarchy, namely strong and weak hierarchy. The results do not support the assumption of a strong hierarchy, which would mean that the order in which the levels are achieved is constant, as maintained by the original theory of van Hiele. In other words, level vH1 is achieved before level vH1-2, and vH1-2 before vH2, etc. Instead, the levels seem to form a weak hierarchy which means that the pupils' geometric skills can develop simultaneously on more levels than one. However, features which according to van Hiele are on a lower level will normally precede the thinking characteristics of the higher van Hiele levels. Level vH1-2, which was added to the structure, can be distinguished as a level of its own as clearly as the other developmental levels observed, and in terms of hierarchy, it is located on that hierarchy level to which it belongs on the basis of its character and the theoretical examination of the development of conceptual knowledge. In this respect, the existence of the addeThe results of the study also challenge some interpretations of level vH3 presented in literature. On this level, the pupils' ability to form definitions was weaker than assumed by the descriptions of the level, and the pupils' knowledge structures were less systematic than expected.

The research shows that the pupil's spatial thinking and logical deduction skills as well as visual memory capacity were quite clearly connected with the development of his/her geometrical thinking depicted by the van Hiele level. The result is problematic, for the tests that were used to measure spatial and logical thinking skills as well as the capacity of the visual short-term memory require hardly any geometrical pre-knowledge. The van Hiele theory, on the other hand, has greatly emphasised the fact that the levels described by this theory specifically reflect a radical change in geometrical thinking which happens through learning.

It can be stated that the theoretical model, constructed to describe conceptual knowledge, functioned well both in the development of tests and in the interpretation of the results of the study. The development of the pupils' conceptual knowledge in geometry seemed to remain limited during the upper level of the comprehensive school when assessed with the features examined in this study. The possible reasons for the weak realisation of the objectives set for learning conceptual knowledge in geometry are also discussed and some pedagogic methods for improving the situation are suggested.

Key words: Conceptual knowledge, geometry, learning, mathematics, comprehensive school, secondary school, van Hiele levels, van Hiele theory