Abstract
The book focuses on so-called critical places in mathematics, that is, areas in which pupils fail repeatedly, in other words, which they have not mastered at a level that would enable their mathematical literacy to develop productively so that it could be used creatively in everyday life. These are usually the areas which teachers find challenging to teach, too.
The book draws on the research results of two research projects by GA ČR coordinated by the author and also on her experience from teacher education courses as well as on the results of national and, more importantly, international research. The first chapter summarises selected theoretical concepts which underlie considerations presented in the core of the book.
Attention is devoted to the concept of quality teaching in general and to the quality teaching of mathematics in particular. Characteristics of such teaching are related to the goals of teaching mathematics.
A concept development theory (the so-called theory of generic models) is described and illustrated, and extensively used in the other chapters of the book. The core of the chapter consists of the presentation of results of quality research, documenting effective practices which facilitate understanding in mathematics.
The theoretical notions from the first chapter are used as a springboard for the following three chapters, each of which focuses on one critical place: Chapter 3 (Word Problems), Chapter 4 (Algebra), Chapter 5 (Combinatorics). All the chapters have the same structure.
They begin with a theoretical section which summarises concepts pertinent to the topic in question. The following section is devoted to pupils' understanding and is divided into two subsections.
In the first one, concrete examples of pupils' understanding (from observations of lessons, task-based interviews with pupils, and literature) are presented with their short interpretation. In the second, the main research results in the topic in question are summarised and related to the above examples.
The next section of the chapter is devoted to the teaching aspect of the topic. In its first subsection, some teachers' statements about teaching the topic and observations from teaching the topic (observations of real lessons or videoed ones) are quoted and briefly interpreted.
The second subsection focuses on research results pertinent to teaching the topic (mostly results of intervention studies). Overall, the topics in question are presented in a very complex way, and thus the study of the book should ensure that teachers have sufficient information about the nature of pupils (mis)understanding of the topic in question and consequences of some teaching approaches.
This should in turn enable the teacher to design his/her teaching with a particular focus on pupils' understanding. Rather than advocating one particular teaching practice, the book seeks to present teaching practices which under some circumstances lead to pupils' understanding of the mathematics taught.