Research Directions in the Dept of Mathematics

Research in Pure Mathematics

Research in Mathematics Education

Mathematics Education team

ass.prof. Madis Lepiklecturer Jüri Kurvitslecturer Tiiu Kaljasresearcher Kirsti Kislenko

Ph.D. studies:Kirsti KislenkoIndrek KaldoJüri KurvitsRegina Reinup

Research areas

Beliefs Development of mathematical knowledge Textbook studies Proof, reasoning, inquiry

Studies on mathematical beliefs

Teachers’ beliefs (Madis) Pupils’ beliefs (Kirsti) Students’ beliefs (Indrek)

Beliefs Beliefs are understood broadly as the conceptions, views, personal

ideologies and values that shape the teaching-learning practice.

Mathematics related beliefs include the implicitly and explicitly held subjective conceptions about mathematics, teaching and learning mathematics and the self as learner/ teacher of mathematics.

Beliefs- mathematical disposition. Beliefs have an important influence on learning and teaching mathematics.

As such, beliefs held by mathematics teachers and pupils provide an interesting window through which to study mathematics education and its change.

Teachers’ beliefs:NorBa project (2011-2014)

Granted by NordForsk, Eduko

The study incorporated teachers of mathematics in Baltic and Nordic countries.

It was agreed to focus on teachers’ beliefs about: general pedagogical approach (conceptions of teaching/ learning in

general); effective/good teaching of mathematics; their own classroom practice and textbook usage; the professional identity of a teacher.

Overall sample size was approximately 800 teachers.

Cross-cultural differences in teachers’ beliefs provide important information

regarding the scope of possible classroom practice and teachers’ inclination to different teaching approaches.

Moreover, knowledge of teachers’ beliefs may indicate to specificity of teaching approaches and thus inform teacher education or curricular reforms.

Pupils’ beliefs about mathematics:Kirsti Kislenko’s doctoral project

The aim of this study was to explore Estonian and Norwegian pupils’ beliefs about mathematics, mathematics learning and the self.

Research questions of this study:a) In what ways are these beliefs structured?b) What makes learning math a likable/ dislikable experience?c) Teachers’ role in forming pupils mathematical beliefs?

Students’ views of mathematics:Indrek Kaldo’s doctoral project

Indrek explores university students’ view of mathematics and learning of mathematics across the disciplines having at least one compulsory mathematics course.

Research questions of this study:a) What kind of structure can be identified to describe the construct view of mathematics?b) What are the general tendencies in the Estonian university students’ mathematics related view of mathematics as measured through Motivational orientation, Value of Mathematics, Competence beliefs, Perception of Teacher Role, and Cheating Behaviour?

Proof, reasoning, inquiry

Proof is a current issue in mathematics education and there is a renewed emphasis on proof in many countries.

In this project (Estonia, Finland, Sweden) we explore:

What is the status/role of proof and reasoning in the school curricula in the countries involved in the study?

How do secondary school teachers and pupils relate to proof and the teaching and learning of proof in these countries?

How is proof dealt within mathematics education?

Proof, reasoning, inquiry:design research and developmental project„Inquiry-based learning’’ refers to student-centered ways of teaching in which students raise questions, explore situations, and develop their own ways towards solutions. The focus is on student-centered work with cognitive activation and autonomous thinking.Inquiry-based teaching refers to teaching practice which foster students’ construction of their knowledge through inquiry, exploring, and finding their own path to solution. Further, it also supports collaborative work, during which students work together on ‘‘challenging’’ tasks.The aims of the project:-development of resources-development of instructional methods-teachers professional development

Development of mathematical knowledge: Proportional reasoning and rational numbers

Jüri focuses on students’ understandings of rational number different meanings (quotient, measure, ratio, part whole, operator) and representations (numerical and pictorial).

The object of this longitudinal study is transition from whole numbers to rational numbers, and misunderstandings that occur in this process. At the same time the development of pupils` proportional reasoning is also observed.

In the course of study, the same students were investigated during three academic years. Testing started with 5th grade students and was completed when the same students were about to finish the 7th grade. Altogether 62 students participated in all tests of the longitudinal study. All in all, 26 tests were carried out.

The Development of Proportional Reasoning The development of proportional reasoning is comparatively slow and the

researchers have shown that it has not fully developed in the majority of adults. Thus, proportional reasoning does not develop itself in the course of years and it cannot be taught as a concrete topic. It comprises the comprehension of different topics and its development requires a lot of time.

Overview of skills and topics that the proportional reasoning consists of:

The Development of Proportional Reasoning Researchers believe that proportional reasoning develops from qualitative to

multiplicative reasoning.

Qualitative reasoning is based on intuitive knowledge about ratios without its numerical quantification.

Qualitative level is followed by pre-proportional level, where a child is able to notice simple multiplicative relationships between quantities, but he/she generally uses strategies which are additive in their nature.

Proportional reasoning is developed when a student can recognize multiplicative relationships within a ratio and between ratios. The corresponding relationships can be integer or non-integer.

Development of mathematical knowledge: Proportional reasoning and rational numbers Regina’s PhD project is in teaching and learning of percentages. It investigates students' conceptual understanding and attitudes towards this

topic. The aim is to develop more understandable and emotionally gripping

learning materials on this topic.

Developed material, tips and (didactical) suggestions in teaching/learning of percentages are partly gathered into a book “Väike protsendiraamat” (Maurus, 2014)

Research work continues with analysing data/results of a test (N=261) with 7th grade students on skills in percentage calculation, and an in-depth study (test and interview) with an 7th grade boy

Regina and Jüri - cooperation

Estonian Grade 6 and 7 Students’ Preferences in Conversions between Fractions, Decimals and Percents Research question: Which types of conversions between numerical

representations of rational numbers do Estonian grade 6 and 7 students perceive to be easy, and which do they find difficult?

Our hypothesis is that the students will prefer conversions that they are the most competent in, i.e. in which they have better skills.

In theory-part we propose to harmonize terminology in an international level: translation is a change between the registers of different representational

systems (for example, placing the fraction or decimal on the number line, or expressing a shaded shape as a fraction or percent);

conversion is a change from one register to another within the same representational system (for example, 1/4=0.25=25%);

treatment is change within the same register (for example, 1/4=2/8=3/12 or 0.4=0.40=0.400);

transformation is a more general concept, i.e. a synonym for change.

F

MANIPULATIVEMODELS

WRITTEN SYMBOLS(NUMERICAL REPRESENTATIONS)

1

3

STATICPICTURES

REAL SCRIPTS

SPOKEN LANGUAGES

P

D2

Three levels of changes: 1. translations (dark continuous arrows); 2. conversions (dark dashed arrows); and 3. treatments (white arrows). F - fractions, D - decimals, P - percents.

Results (1)

Conversions needed from students in the first stage of a

percentage calculation.

Students’ a) preferences and b)

skills in conversions.

F D

P

a.

D

P

F

b.

F

P

D

Results (2). Change in the students’ performances.

a) representing 1/4 b) representing 0.4

Summary of this study:

The students feel more confident and prefer to start conversions from a source number whose appearance bears a closer resemblance to a whole number – i.e. percents and decimals. Also, their results were best in conversions between percents and decimals. Although the conversions from fractions to percents are the most important in part-whole relation subconstruct exercises, this sort of conversion was the most poorly performed and was the least preferred conversion for the Estonian 6th and 7th grade students in our study.

The hypothesis on the students’ preferences being a reflection of their competence was not confirmed. Instead, the students seemed to start their conversions from familiar looking number forms, even when their skills were not so good, and avoided some conversions that they were quite competent in.

Comparative study on textbook use in mathematics classes NordForsk grant, 2011- 2014 From many studies it is well documented that the textbook is one of the

most influential element for pupils’ mathematical learning. In the Nordic and Baltic countries the mathematics textbook is dominating in the teaching and teachers are heavily dependent on textbooks.

Some of aspects explored:-how teachers use textbooks-how pupils use textbooks-how textbooks influence pupils’ learning of mathematics-patterns of textbook use in math classes

Thank you!