Making Students´ Thinking Explicit: Learning What They Know about Functions

During the transition to professional universities, it is normal that students bring preconceptions with them from many domains of knowledge as well as misconceptions and numerous other idiosyncrasies based on their previous learning experiences. Knowing how our students think will help us to understand their mathematical thinking skills, to make sense of their errors and to improve our instruction as a result. In this paper, we assert that even good students struggled as they encounter prior mathematical ideas and concepts in a new setting and faced various inhibitions in invoking their thinking powers. A study is carried out among fresh engineering undergraduates taking the first engineering mathematics course. They were assigned to do some tasks designed to make their thinking explicit, uncovering their conceptual appreciation and understanding of the basic concept of functions as they work collectively in class before the instruction on the course began. Their knowledge and thinking being observed in several aspects, namely their performance, awareness, mathematical sense making, and attitudes. The findings of this research revealed that the students´ preconceptions are inconsistent, illuminating their current understandings about the concepts they have learned previously, what they attended to and how they attended these in a mathematical manner are of concern. Understanding the human innate abilities, knowing the students´ current knowledge state and their prior experiences in ´functions´ informed us that even with high achievers, initiatives for conceptual accommodation to assimilate new concepts into existing knowledge are required.