Abstract:

The use and impact of digital technologies in solving and expressing mathematical problems is scarce. Our research has been addressing this research gap and has proposed a descriptive model of Mathematical Problem-Solving with Technology (MPST model).

We have focused on non-routine mathematical problems that elicit the development of conceptual models of the solutions. To achieve a solution, the solver must devise a productive way to address the problem, by resorting to mathematical and technological knowledge. While digital tools are incorporated in the problem-solving process, it becomes challenging to establish a boundary between the actual solving phase and the reporting phase; therefore ‘solving-and-expressing’ represents the synchronous process of mathematization and expression of mathematical thinking. Previous studies have shown that middle school students can combine mathematical knowledge with the affordances of digital tools to solve and express mathematical problems. Such efficacy has been conceptualized as techno-mathematical fluency. However, little is known about the processes that teachers engage in when solving and expressing mathematical problems with digital tools.

Our current aim is to understand the processes developed by elementary and secondary school mathematics teachers when solving-and-expressing a mathematical non-routine problem using the digital tools of their choice. The study followed a qualitative research approach and was conducted during the Covid-19 pandemic. Five teachers were intentionally selected and invited to an online task-based interview, taking place through a Zoom meeting. The teachers were asked to choose one of four non-routine word problems, invited to solve the problem using the digital tools of their choice, and were asked to verbalize their thinking and actions, following a think aloud protocol. The teachers shared their screen with the interviewer, which enabled the video recording of the actions and utterances. Hence, data collection included the video recording of the whole meeting and the digital files produced by the teachers. The interviews were transcribed, the segments were coded using NVivo software, resorting to the ten processes defined by the MPST model.
In this paper we focus our analysis on the processes of problem-solving with technologies developed by two experienced mathematics teachers, who usually resort to digital tools in their classrooms.

Both teachers chose to solve the same problem using a spreadsheet, although one of them also used a dynamic geometry software. In line with previous results focusing on students’ activity, the MPST model allowed to describe and understand the role of the techno-mathematical resources used by the teachers. The solutions developed by the teachers are self-explainable, which means that solving-and-expressing-with-technology summarizes the whole process in which they engaged in. The MPST model allowed to identify critical moments in the teachers’ activity, in which the core processes are ‘integrate’ and ‘explore’. This means that the combined use of technological and mathematical resources precedes major advancements in achieving a techno-mathematical solution. The teachers’ techno-mathematical fluency includes the recognition of affordances in the digital tools to: interpret the situation from a techno-mathematical point of view; explore a conceptual model; and produce the techno-mathematical solution.

Keywords:

Mathematical problem solving, Solving-and-expressing, Mathematics teachers, Techno-mathematical fluency, Digital technologies, MPST model.