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Daria Rovan, Goran Trupčević, Dubravka Glasnović Gracin

Faculty of Teacher Education, University of Zagreb, Croatia

Pre-service primary education teachers' beliefs and their approach to teaching mathematics – a longitudinal perspective

Every student's formal mathematics education begins with his primary education teacher, so it is very important that primary education teachers have good mathematics teaching skills, have positive attitudes toward mathematics and are motivated to teach it to their students. The research shows that mathematics teachers’ instructional decisions are significantly influenced by their beliefs (Cross, Rapacki & Eker, 2015). To encourage students' mathematical thinking, their teachers should hold beliefs that support the development of problem-centered, learner-oriented classroom environments (Cross, 2009).

In Croatia primary education teachers are trained as generalists and mathematics is only one of several different subjects that they teach. That means that when they chose their future profession, they were not necessarily drawn by their interest in becoming mathematics teacher, so we were interested to explore factors that could influence the development of their approach to teaching mathematics. Therefore, we designed a comprehensive study to explore pre-service primary education teachers' beliefs from a longitudinal perspective:

In their first year of studies, we collected data on their reasons for selecting teaching as a career and their experience with mathematics prior to entering university (the level of their mathematical competencies after high school, motivation for learning mathematics, mathematical epistemic beliefs, mathematics anxiety)

In their third year of studies, we collected data on their experience with mathematics during their studies (mathematics achievement, motivation for learning mathematics, mathematical epistemic beliefs, mathematics anxiety)

In their final year of studies, we collected data on their mathematics teaching efficacy beliefs, beliefs about the process of teaching mathematics (confidence in understanding students’ thinking, constructivist teaching, teacher-centered teaching), mathematical epistemic beliefs and mathematics anxiety.

Our results show that pre-service primary education teachers' motivation for learning mathematics during their studies is significantly related to their motivation for learning mathematics in high school, level of the state graduation exam taken and epistemic beliefs (Rovan, Trupčević & Glasnović Gracin, 2014). Their mathematics anxiety and engagement in mathematics during their studies are related to their motivation for learning mathematics in high school and during their studies. Mathematics anxiety is negatively related to self-efficacy and subjective value of mathematics. Engagement in mathematics is related to subjective value of mathematics. Pre-service teachers high in math anxiety have maladaptive achievement goals profile, while those high in engagement have adaptive achievement goal profile. Therefore, the development of motivational beliefs (especially self-efficacy and subjective value beliefs) should also become one of the goals of their mathematics education.

In the final year of their studies, pre-service primary education teachers hold adaptive beliefs on teaching mathematics – high mathematics teaching self-efficacy and student-centered approach to teaching (Rovan, Trupčević & Glasnović Gracin, 2018). Their mathematics teaching self-efficacy is related to their reasons for choosing teaching as a career, but even more to their previous experiences with mathematics. Their beliefs about the process of teaching mathematics are primarily related to mathematics teaching self-efficacy and also to the reasons for choosing teaching as a career.

Research results are in line with the assumption that both the previous experiences of learning mathematics and the reasons for selecting teaching career are important in the process of forming beliefs about teaching mathematics, but more complex research is needed to reveal possible interplay of those factors in developing pre-service teachers' approach to teaching mathematics.

Tatjana Hodnik Čadež, Vida Manfreda Kolar

Faculty of Teacher Education, University of Lubljana, Slovenia

Mathematical problem solving from different perspectives

Problem solving is a leading mathematical activity which stimulates mathematical thinking. From the theoretical point of view this activity is very complex due to different issues which describe what problem solving is and what is its role in the process of teaching and learning mathematics. Our interest regarding these issues mainly focuses on the following: what are basic characteristic of a mathematical problem, the nature (conceptual, procedural) and the role of representation (interplay between internal and external) of a mathematical problem, mental schemas for problem solving, heuristics as principles, methods and (cognitive) tools for solving problems, types of generalisations and reasoning (abductive, narrative, naïve, arithmetic, algebraic), problem solving as an important challenging activity for mathematically gifted students and role of teacher’s guiding of problem solving with an aim to teach students problem solving in the classroom.

Three studies on problem solving in mathematics are presented. The first study is focused on comparison of primary teacher students’ and mathematics teacher students’ competences in inductive reasoning. The students were posed a mathematical problem which provided for the use of inductive reasoning in order to reach the solution and make generalizations. Their results were analysed from two different perspectives: from the perspective of the problem solving depth and from the perspective of the applied strategies. We established that not all strategies were equally effective at searching for problem generalizations. Preserving the problem context turned out to be more effective than the mere operating with numbers at the transition into higher, more abstract stages of problem solving.

The second study deals with the teacher’s role during problem solving situations. A group of pupils (35 pupils between 10 and 19 years old) were given a geometrical problem that required them to define the number of parts created when a single plane was divided by straight lines. Each pupil tackled the problem individually, while primary teacher students observed and guided them. After analysing the students’ research reports on guiding pupils through the problem we came to the following conclusions: all the pupils needed guiding in order to make progress in problem solving towards general rule. Until presented with a problem that required a geometrical approach, the differences among the age groups in terms of successful problem solving were not that noteworthy, the difference among age groups was observed in examples of more complex problem solving where a shift towards an arithmetical approach was needed.

The third study deals with analysis of problem solving schemas on a group of primary teacher students. The results confirm the importance of a well-structured schema for the successful solving of a complex mathematical problem. The students applying unstructured or partially formed schemas had problems when addressing a complex problem, whereas the students who were able to solve it mostly accommodated or assimilated their knowledge, which again proves these processes to be a necessary prerequisite for successful problem solving.

Ljerka Jukić Matić

Department of Mathematics, University of Osijek, Croatia

University science and engineering students: learning outcomes and beliefs about mathematics

Mathematics is tightly interwoven with science and engineering, where it has numerous applications. In the educational context, there is an ongoing debate who should teach mathematics to non-mathematicians and how this mathematics should be taught. The knowledge gained in mathematics course is used in another course (mathematics, science or engineering), hence students should retain core concepts some time after learning. Beliefs that students have about mathematics influence on their learning, and consequently on the retained knowledge. This presentation will refer to the parts of the study which examined the retention of the students’ knowledge in core calculus concepts, namely, derivative and integrals two, six and ten months after the students had passed an calculus exam. Students’ beliefs about mathematics will also be presented and discussed. For instance, students showed positive beliefs about mathematics in their study program, but were not certain where this knowledge will be used later.

Maja Planinić

Department of Physics, University of Zagreb, Croatia

Does transfer of knowledge occur between mathematics and physics? An investigation of student understanding of line graphs

Joint work by Ana Sušac¹, Lana Ivanjek² and Željka Milin Šipuš³

¹Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia

²Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria

³Department of Mathematics, Faculty of Science, University of Zagreb, Croatia

Student understanding of graphs in physics and mathematics and the possible transfer of knowledge between mathematics and physics was investigated using questions from mathematics without context, physics (kinematics) and mathematics in contexts other than physics. Eight sets of parallel (isomorphic) mathematics, physics and other context questions about graphs were developed and administered to 385 first-year students at the Faculty of Science, University of Zagreb. Average difficulties of items in three domains (mathematics, physics, and other contexts) and over two concepts (graph slope and area under a graph) were computed and compared. Analysis suggests that the variation of average difficulty among the three domains is much smaller for the concept of graph slope than for the concept of area under the graph and that mathematics without context is the easiest domain for students. Adding either physics or other context to mathematical items generally seems to increase item difficulty. The same test was later administered to 417 first- year students at University of Vienna. Average difficulties of domains and conceptual areas followed the same trends in both groups of students. Student strategies of graph interpretation were analyzed and found to be largely context dependent and domain specific. In physics, the preferred strategy was the use of formulas, which sometimes seemed to block the use of other, more productive strategies which students displayed in other domains. Overall, the transfer of knowledge appeared weak, although some students have shown indications of transfer in the sense that they used techniques and strategies developed in physics for solving (or attempting to solve) other context problems. Students’ answers indicated the presence of well known student difficulties with graph interpretation, slope-height confusion and interval-point confusion, in all three domains. Students generally better interpreted graph slope than the area under a graph, although the concept of slope still seemed to be quite vague for many. Implications of the study findings for teaching on graphs and the question of how to promote transfer of knowledge between mathematics and physics are discussed.

Ana Sušac

Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia

Students' strategies in simple equation solving: insights from developmental and eye-tracking studies

Joint work by Andreja Bubić¹, Maja Planinić², Andrija Vrbanc², Jurica Kaponja², Marijan Palmović³

¹Chair for Psychology, Faculty of Humanities and Social Sciences, University of Split, Croatia

²Department of Physics, Faculty of Science, University of Zagreb, Croatia

³Laboratory for Psycholinguistic Research, Department of Speech and Language Pathology, University of Zagreb, Croatia

Simple equation rearrangement is an important skill required for problem solving in mathematics and science. The developmental trajectory of the students’ ability to solve simple algebraic equations was investigated using a computerized test administered to 311 primary and secondary school students (age 13–17 years). Younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In addition to behavioral data, we recorded eye movements of the university students while they were rearranging algebraic equations. The results indicated that the number of fixations represents a reliable and sensitive measure that can give valuable insights into participants’ flow of attention during equation solving. A correlation between the number of fixations and participants’ efficiency in equation solving was found, suggesting that the more efficient participants developed adequate strategies, i.e. “knew where to look”. The measures derived from eye-tracking data were found to be more objective and reliable than the participants’ reports. Overall, eye tracking provides insights into otherwise unavailable cognitive processes and may be used for exploring problem difficulty, student expertise, and metacognitive processes.

Lana Horvat Dmitrović, Ana Žgaljić Keko

Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia

The role of concept-based teaching in smooth transition to university mathematics

Teaching methods in primary and secondary mathematical education traditionally focus on procedural and computational knowledge. Consequently, in the mathematical courses at university level, the students stay focused on developing their computational skills and thus overlook the importance of conceptual thinking. One of basic engineering skills is being able to easily apply mathematical concepts in different contexts. We recognized the need to improve this skill by applying the concept-based approach to teaching mathematics. Our talk also considers other problems appearing in students’ transition to university-level mathematics: pre-knowledge issues ranging from misconceptions to deficient computational skills which originate in the lack of conceptual understanding, their resistance to learning mathematical theory, weak critical thinking and problem-solving ability.

Furthermore, we will give some examples on overcoming these issues in the students’ transition which have been applied to mathematical courses at the first year at the Faculty of Electrical Engineering and Computing in Zagreb. Primarily, we will present some different methods of employing concept-based approach through learning outcomes, materials, teaching and learning methods and feedback.