Principles and Standards for School Mathematics (NCTM 2000) calls for instructional programs that emphasize problem solving and that have the goal of helping students develop sophistication with such mathematical processes as representation, mathematical reasoning, abstraction, and generalization. In particular, the Problem Solving Standard suggests that teachers should choose problems that further the mathematical goals of the class. Problem solving can be viewed as a process through which teachers can help students think mathematically, which Schoenfeld (1985, 1992) defines as developing a mathematical point of view. It includes valuing the processes of representation and abstraction and having the predisposition to generalize them. In this article, I describe my attempt to implement problem solving as a teacher of ninthgrade algebra. I had two explicit goals in mind. The first goal was to use carefully chosen problemsolving situations as a setting for an extended mathematical investigation that leads to the discovery of Steiner triple systems. The second goal was to use problem-solving situations to help students think mathematically, that is, to construct representations and to engage in mathematical reasoning, abstraction, and generalization.
The third edition of the Handbook of International Research in Mathematics Education (henceforth, HIRME) comes at an interesting time for the community of mathematics education researchers because it tackles two essential problems for the community, namely, (a) what constitutes “great challenges” for the field, in the opening chapter, and (b) how scalable mathematics education research is, in the concluding chapter.
Any typical narrative of the history of probability attributes the beginnings of this field of science to the needs of 17th century “gamers” in France for accurately computing chances in high-stakes gambling. Although the quantitative science of probability emerged in Pascal's time, mathematics historians (e.g., Franklin, 2001) have pointed out that qualitative probabilistic thinking has been a staple of human decision making well before the time of Pascal as seen in judiciary, theological, and philosophical arguments in ancient cultures east and west. That is, although the mathematical science of probability only emerged since Pascal's time, probabilistic reasoning was always an aspect of our thinking.
Bharath Sriraman and Lyn D. English
For many students, combinatorics is associated with negative experiences calculating permutations and combinations, often confusing one with the other. What exactly is combinatorics? Combinatorics can be defined as the art of counting, or more specifically, as “an area of mathematics in which we study families of sets (usually finite) with certain characteristic arrangements of their elements or subsets, and ask what combinations are possible and how many there are” (Rusin 2002).
Bharath Sriraman and Kyeong Hwa Lee
A review of Alan Schoenfeld's book How We Think: A Theory of Goal-Oriented Decision Making and Its Educational Applications.