Abstract:

Grammar schools (“gymnasia”) in Slovakia are finished with school leaving examination. One of the subjects with external form is mathematics. Students are usually prepared to be able to solve typical, mostly low cognitive demand tasks. These tasks require minimum thinking or cognitive analysis, and rather focus on single, concrete answers that are solved using prior knowledge. Students are often “trained” to be successful. Therefore, our research question was, whether students prepared by the above-described way can solve a task concerning the length of the Koch curve. This task requires more than just direct application of a "trained" procedure to find the answer.

Our target group consisted of 36 students in the last year of study (19 years old) at a grammar school in Bratislava. The aim of the task was to find the length of the Koch curve if we are doing more and more iterations. We have made analysis a priori of the task (in a sense of Brousseau), we have pointed out several important concepts students should have been able to identify and then apply the learned procedure. By analyzing their solutions, we have identified five groups of typical students’ solving strategies (only one with the correct solution). The way of students' argumentation showed us the lack of doing mathematics (according to Van de Walle-Karp-Bay Williams). In the analysis a posteriori we are focusing on students’ argumentation, misconceptions, and the ways of solving the task in order to show that procedural knowledge is not sufficient.

The main reason for choosing this topic was discourse with secondary and high school teachers who think that “Students have to know the procedure. Why it works they will find out later.” (discourse with teachers, November 2018). Other reasons were: typical tasks in international testing like PISA, TIMMS, national tests T9 (15 years old students) and MATURITA (18-20 years old students) differ from the main goal of many mathematics lessons. The tasks in the international testing require reproduction, reflection, and connection, T9 demands lower levels of Revised Bloom Taxonomy (Remembering, Understanding and Applying), and MATURITA is somewhere in between the mentioned groups. Tasks used during many mathematics lessons are often requiring only the lowest level of Revised Bloom Taxonomy: The Knowledge dimension – Factual knowledge.

After the testing of the students, we asked them for their opinion on the given task. The answers differed from “Too demanding/complicated” to “I like it a lot and I want to know more about it”. Therefore, we suggested some changes in mathematics lessons at grammar school to prepare the students not only for one test in their life. Due to a strong connection between mathematics and informatics, we suggested to utilize this interdisciplinary connection and offer students complex and intertwined information.

Keywords:

Revised Bloom Taxonomy, Koch curve, strategies of solving, misconceptions and misinterpretations.