Abstract:

Algebra is in many aspects considered to be the generalization of arithmetic. Many studies conducted over the last two decades have confirmed that by using an appropriate approach young students can integrate algebraic concepts and representations into their thinking. Current mathematical education literature offers a lot of activities and approaches for developing students´ functional thinking on elementary levels. In some sense, functional thinking includes identifying and generalizing patterns, so these approaches often utilize a sequence of figures containing certain elements. The figures and their respective elements change from one to the next and it is possible to define the relationship between elements of the figure and its position (step number). This study aims to understand how pre-service secondary mathematics teachers solve problems by involving the generalization of visual patterns. The research is focused on students’ difficulties with the generalizations while identifying the strategies used and the role of visualization in their reasoning.

In the Didactics of Mathematics course, students were asked to solve a sequence of tasks involving growing visual patterns. Since they have had completed several courses of higher mathematics prior to this exercise, we have assumed that they are able to continue any pattern when presented by algebraic expressions or diagrams in routine or non-routine tasks. There were 30 students involved in our pilot study. We have focused on identifying the level of functional thinking of future mathematics teachers on both, bachelor and master study level (11 bachelors, 19 masters). We have applied Statistical Implicative Analysis on their results by using the C.H.I.C software. The aim of this method was to define a way to answer the question: “If an object has a property, does it also have another one?” (R. Courtier, 2008). We have looked for several categories of algebraic generalization based on the type of the algebraic generalization and the algebraic formula (additive/multiplicative, constructive/deconstructive, and standard/nonstandard) and we have also been observing whether students are predominantly figural or numerical generalizators.

The results show that making a T-chart was not the most important part of their solutions in several tasks. Their organization of data varied from solution to solution. Although the most common approach was numerical generalizations, several solutions were dominated by figural generalizations. Students using the multiplicative scheme usually gave their answers in standard form and students using the additive scheme usually gave their answers in a non-standard form. There was a task in which nobody used figural generalization. The type of solution was strongly influenced by the context of the task. While several students were able to write down high mathematical structures, they were unable to solve it, with some of them failing to find the figure for lower step number and others misunderstanding the question.

The students used in our sample are due to become teachers in 4 or 1 year. We have expected a higher score and a deeper understanding of functions. Our survey leads to a new question - whether future mathematics teachers should be educated differently (offered more tasks for developing algebraic and functional thinking), or they should start with this education much sooner (such as at primary school).

Keywords:

Functional thinking, pattern generalization, visual templates, Statistical Implicative Analysis.