And Now Put Aside Your Pens and Calculators: On Mental Problem Solving in the High School Mathematics Classroom
Karp, Alexander, Focus on Learning Problems in Mathematics
The aim of this article is to draw attention to the use of mental mathematics--to promote its use, and to pose several questions connected with it. Unfortunately, the psychological problems associated with the reception of teaching materials, the teaching materials themselves, and the concrete instructional methodology used in their presentation are frequently studied independently of one another. Yet, what working teachers often need in order to solve problems that they encounter is precisely a comprehensive analysis of the details and specifics of one or another methodological approach. Researching on the use of mental mathematics is going to take us at the intersection of several important fields currently being studied by mathematics educators, including problem solving, classroom management, and the problems of mathematical communication (Karp, 2004). This article presents the results of certain observations and analyzes them.
Mental mathematics is often mentioned in connection with elementary and middle school (Beishuizen & Anghileri, 1998; Hope & Sherill, 1987; Reys, Reys & Hope, 1993; Sowder, 1990). Mental calculations are often welcomed, in this context, as a means of encouraging and stimulating students to develop their own computational strategies, as a counterweight to the memorized "paper and pencil algorithm." Stressing the importance of such self-developed procedures, Sowder, following Plunkett (1979), lists their main positive characteristics. She notes that they are variable, flexible, active, holistic, and constructive. Most important of all, probably, is the fact that they "require understanding all along" (Sowder, 1990, p.20). The student who multiplies 35 by 4 without using paper or a calculator indeed possesses a much better number sense than the student who mindlessly computes the product using the paper and pencil technique. As Sowder rightly notes, "One topic that cannot be replaced by the calculator is the development of understanding of what numbers are" (p.20). Mental calculations stimulate the development of such an understanding and therefore remain important in an age when the use of technology is pervasive.
Much of what is taught in high school can also be effortlessly accomplished using a calculator (just a slightly more expensive calculator than the one needed to solve elementary school problems). However, discussions of what is essential in the traditional curriculum, and which parts of it will persist into the age of technology, rarely touch on the role of mental mathematics.
One of the very few publications devoted to mental mathematics in high school (Rubenstein, 2001) tellingly contains three question marks in its title: mental mathematics beyond the middle school level appears as something quite unexpected. The author of the article, however, lists a number of reasons to show why mental mathematics is useful at high school level as well. As Rubenstein argues, mental mathematics
* is useful for workers, consumers, and citizens;
* facilitates learning many structural topics;
* allows students to become less calculator-dependant;
* is rewarding because students are challenged by the problems and feel proud of their accomplishments. (pp. 442-443)
It is evident that the use of mental mathematics problems can be analyzed from different viewpoints--in terms of purely methodological as well as social-pedagogical frameworks. The following crucial questions deserve further attention:
* Which problems can be used as mental mathematics problems in high school? How can these problems be useful in class? What guidelines must teachers use in selecting (or constructing) mental mathematics problems?
* What forms of lesson organization permit teachers to make use of mental mathematics?
* How does the use of mental mathematics problems develop the students' communication skills? What peculiarities of students' mathematical language emerge during work on mental mathematics problems? …