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Secondary school students in Singapore are expected to find an expression for the general or "nth" term of an arithmetic progression (AP) without using the AP formula T[subscript n] = a + (n-1)d, where "a" is the first term, "n" is the number of terms and "d" is the common difference between successive terms. This type of question is worth only one mark in the GCE O-Level Examination but the usual method taught in school is far too long and many students feel that it is not worth trying so hard for one mark. This led the author, when he was a teacher, to look for other shorter methods. In this article, he will discuss five methods to find an expression for the "nth" term of an AP, including arguably the fastest method that he has invented. He believes teachers will not only find the "fastest" method useful for their students but they can also challenge their students to be innovative and creative: find as many methods as possible to find the general term of an AP.

Educators usually mean different constructs when they speak of open tasks: some may refer to pure-mathematics investigative tasks while others may have authentic real-life tasks in mind; some may think of the answer being open while others may refer to an open method. On the other hand, some educators use different terms, e.g. open and open-ended, to mean the same construct, while others distinguish between these terms. It is difficult to hold a meaningful discussion or to define clearly an area of research on open tasks if the idea of what constitutes the construct of openness is vague. Moreover, what students learn depends on the types of tasks that they are given, and different kinds of tasks place differing cognitive demands on students. Thus, the objectives of this article are to clarify the types of mathematical tasks and develop a framework to characterise their openness based on five task variables: goal, method, task complexity, answer and extension; and to discuss how different types of tasks and openness may affect student learning. The openness framework can help teachers to design or select more appropriate tasks to cater to students with different abilities in order to develop in them various kinds of mathematical thinking processes, and it can also make it easier for researchers to study the interaction between different types of openness and student learning.

Most students love to play games. Ernest (1986) believed that games could be used to teach mathematics effectively in four areas: motivation, concept development, reinforcement of skills, and practice of problem-solving strategies. Fifteen is an interesting and thought-provoking game that helps students learn mathematics at the same time. Playing Fifteen is like playing tic-tac-toe on a 3 x 3 magic square in which the three numbers in each row, column, and diagonal add up to 15. The game can be used in the classroom to motivate and help students learn mathematics and develop heuristic skills such as examining all possible scenarios and systematic listing (Polya 1957), spatial visualization (Presmeg 2008), and thinking skills, including predicting, conjecturing, generalizing, and checking (Ainley 1990). (Contains 10 figures.)

Many educators believe that mathematical investigation involves both problem posing and problem solving, but some teachers have taught their students to investigate during problem solving. The confusion about the relationship between investigation and problem solving may affect how teachers teach their students and how researchers conduct their research. Moreover, there is a research gap in studying the thinking processes in mathematical investigation, partly because it is not easy to define these processes. Therefore, this article seeks to address these issues by first distinguishing between investigation as a task, a process and an activity; and then providing an alternative characterisation of the process of investigation in terms of its core cognitive processes: specialising, conjecturing, justifying and generalising. These will help to clarify the relationship between investigation and problem solving: an open investigative activity involves both problem posing and problem solving; but the problem-solving process entails solving by the process of investigation and/or by using "other heuristics". In other words, mathematical investigation does not have to be open. The article concludes with some implications of this alternative view of mathematical investigation on teaching and research. (Contains 2 figures.)