LENGTH OF THE KOCH CURVE AS AN EXAMPLE OF NON-TRIVIAL MATHEMATICAL TASK Mária Slavíčková, Michaela Vargová Comenius University in Bratislava (SLOVAKIA) Abstract 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. Keywords: Revised Bloom Taxonomy, Koch curve, strategies of solving, misconceptions and misinterpretations. 1 INTRODUCTION High schools without specializations (like to be a plumber, waiter, cook etc) in Slovakia are finished with leaving exams. Mathematics is one of the subjects with an external form of exams. Students who choose this subject are usually prepared to be able to solve typical, mostly low cognitive demand tasks. "Low cognitive demand tasks involve stating facts, following known procedures, and solving routine problems.” [1] These tasks require minimum thinking or cognitive analysis, and rather focus on single, specific answers that are solved using prior knowledge. These tasks can be broken down into two different types: memorization and procedures without connections. According to [2]: • Memorization tasks: involve pulling facts and formulas from prior memory in order to solve the equation. These tasks are quick, and sometimes with a time limit, resulting in the inability to use procedures to find an answer; require no connections to the meaning of the information that is being learned. • Procedures without connections: tasks are algorithmic, follow a specific procedure from prior learning. They require little thinking of how to complete the task; has no connections to concepts or to why a procedure is done, require no explanation or mathematical understanding. On the other hand, there are high demand tasks, which could be characterized as those which require students to think abstractly and make connections to mathematical concepts. These tasks can use procedures, but in a way that builds connections to mathematical meanings and understandings. [3] “When completing higher demanding tasks students are engaged in a productive struggle, which challenges them to make connections to concepts and to other relevant knowledge.” [1] This kind of tasks students cannot solve them mindlessly. They must engage students with conceptual ideas, meaning the task triggers the procedure that is needed to complete the task, and develop understanding. [2] Like, low cognitive demand tasks, high-level tasks can be separated into two types: procedures with connections and doing math. According to [2]: • Procedures with connections: emphasize the use of procedures, in order to develop a students’ deeper level of understanding of math concepts and ideas. Opposite of a standard algorithm, these tasks, suggest pathways for students to follow; require some degree of thinking; students cannot solve them mindlessly. They must engage students with conceptual ideas, meaning the task triggers the procedure that is needed to complete the task, and develop understanding. • Doing math: these tasks require multifaceted thinking. They are not algorithmic, meaning they are not predictable, and the exact plans to solve the task are not clearly proposed in the instructions. Doing math requires students to comprehend and understand mathematical connections; require students to monitor their own process of thinking, while using applicable Proceedings of ICERI2019 Conference 11th-13th November 2019, Seville, Spain ISBN: 978-84-09-14755-7 7282