Patrick Johnson

Mathematical thinking: challenging prospective teachers to do more than ‘talk the talk’

This paper reports on a research project which aims to improve prospective mathematics teachers’ relational understanding and pedagogical beliefs for teaching in second-level Irish classrooms. Prospective mathematics teachers complete their teacher education training with varying pedagogical beliefs, and often little relational understanding of the mathematics they are required to teach at second level. This paper describes a course designed by the authors to challenge such beliefs and encourage students to confront and possibly transform their ideas about teaching, while simultaneously improving their subject knowledge and relational understanding. Both content and pedagogical considerations for teaching second-level mathematics are integrated at all times. The course was originally optional and was piloted and implemented in a third-level Irish university. Apart from offering an insight into the design considerations when creating a course of this type, this paper also addresses some of the challenges faced when evaluating such a course. Overall participant feedback on the course is positive and both qualitative and quantitative results are provided to support this and also highlight the efficacy of the programme.

Teaching Real World Mathematics: Some Background Theory

What do you think now? Our thinking led to what you are holding in your hand: this book! As you have seen as you worked your way through the book our approach is to start with a motivating example, offer ways to get started in the classroom before moving on to more challenging activities. Finally, at the end of this book we think that you might be interested in reading a little about the theoretical background that underpins our approach.

Motivation—Why Teach Applicable Mathematics?

We have been discussing the issue of motivation with mathematics teachers and mathematics pedagogues for many years, trying to find conclusive answers to these questions. We firmly believe that we have found some answers and we will summarise them for you in this chapter. Further, we believe the insights that we offer have a universal appeal. However, from our experience and many discussions with mathematics teachers, mathematicians and mathematics educators, we know that what is convincing to us is not necessarily so to others.

Introduction to Modelling

The most important feature of this book is that is tries to speak directly to you, the mathematics teachers. The authors attempt to draw you into a continuous dialogue about activities you are asked to engage in as learners. You are asked to do something, and through doing and reflecting you will gain first-hand experience of new approaches and materials. In this way, you can learn to teach applicable mathematics to your students using your own experience as learners of applicable mathematics, motivated and supported by the book.

Empirical Findings on Modelling in Mathematics Education

It would make life much easier for us, as mathematics teachers, if we knew in advance of trying something new, such as reality-based mathematics, that it would work out and result in increases in student motivation, understanding, and/or enjoyment of school mathematics. However, it is understood that forecasts of this nature are uncertain because the predicted events or outcomes lie in the future. Consequently, neither we nor anybody else can assure you that your first attempt, or subsequent lessons, will be a success. However, we do believe we can help improve your lessons and that is what this book is all about.

Further Tips for Teachers Who Want to Implement Applicable Mathematics Education

In this chapter, we ask you, the teachers, and your students to reflect on your learning journey as you progressed through the book. We invite learners to follow a structured approach based on the layout of the book and to question key facets of their journey. An approach to this practical, personal evaluation is presented in the first part of the chapter and deals with suggestions for the preparation of applicable mathematics lessons, followed by a section on finding new problems, and concludes with templates for appraising students’ progress.

An Example of a More Extensive Project

In this chapter we will present a project idea to illustrate how modelling projects can be implemented over a longer period of time. We have selected gambling as the theme for this project. Most people become aware of gambling through advertisements which state the odds or quotas for a certain event (usually sporting events) and therefore in order to understand gambling you first need to understand odds or quotas and the mathematical underpinnings of these concepts. This is addressed in the early stages of this chapter using two examples based on real life data. Once these mathematical ideas have been introduced and understood we then use modelling to explore a variety of different sports betting scenarios.

Tasks Derived from Everyday Occurrences

In the last chapter we considered textbook problems, and tried to see how realistic they were and how we could make them more up-to-date, relevant and interesting. We gave you plenty of suggestions on how this can be achieved. In this chapter, we will start from a different perspective, by creating small classroom units using everyday occurrences.

Adapting Textbook Problems to Create a More Reality-Based Mathematics Education

Historically mathematics textbooks have been a key resource for mathematics teachers. However, textbooks and other curricular materials must be aligned purposefully with the curriculum goals but this is not always the case when we compare traditional mathematics textbooks with the objectives of reality based mathematics education.

How Do Experts Model? Using This Knowledge and Understanding in the Mathematics Classroom

Are you aware of the fact that thousands of people earn their living by developing mathematical models? It is reasonable to think of these people as expert modellers. However, you may ask: What exactly do these people do? What types of models do they develop and what kind of results does their work produce? We consider these questions in this chapter and, as in previous chapters, we ask for your active participation.