The Curriculum and Evaluation Standards (1989) indicate that although computation is important in mathematics and in daily life, our technological age requires that we rethink how computation is done today. Despite the obvious advantage that hand-held calculators hold over other forms of calculation (Hope 1986) many situations exist in which either exact or approximate mental calculation is the preferred form of calculation.
Important benefits of the development of mental computation in school have also been pointed out by reports in mathematics education such as the Mathematical Sciences Education Board, 1990; the Conference Board of Mathematical Sciences, 1989 and the National Council of Teachers of Mathematics, 1991.
Traditionally in schools, proficiency in pencil and paper algorithms is equated to success in mathematics; however, research has shown that students who normally
perform poorly with pencil and paper are sometimes capable of impressive feats of mental calculation (Ginsberg, 1989).
The Hope and Sherill (1987) study of skilled and unskilled mental calculators with secondary school students suggests that whole number sense, exhibited by the flexible use of the structure of the whole number system, is highly related to skill in mental computation. The study found that skilled calculators had a variety of strategies at their disposal that allowed them to avoid the high memory requirements of pencil and paper algorithms used by their unskilled peers.
Researchers like Markovits and Sowder (1994), Baroody (1987) and Silver, Kilpatrick and Schlesinger (1990) point out that if students are encouraged to explore numbers and number relations through discussions of their own invented strategies and those of their peers, their intuitive understanding of numbers and number relations would be used and strengthened. These discussions can also lead to the discovery of important relationships. Markovits and Sowder (1994) have classified students' strategies in the following manner:
1. Standard: The student uses a mental version of a pencil and paper algorithm.
2. Transition: The student is somewhat bound to the standard algorithm. However, more attention is given to the numbers computed and less to the algorithmic procedure. (38+45=13+70=83)
3. Nonstandard with no reformulation: The left to right process is used. (38+45=30+40+13)
4. Nonstandard with reformulation: the numbers are reformulated to make the computation easier (38+45= 40+45-2).
Caine and Caine (1994) propose that as educators we must assist students in their search for how to make sense of things. This "search" as they point out is for "common patterns and relationships."
Number sense according to Sowder (1988) is a well organized conceptual network of number information that enables a person to relate numbers and operations to solve problems in flexible and creative ways.
The National Council of Teachers of Mathematics (1989) and the Mathematical Sciences Education Board (1990) have addressed the importance of teaching children to compute in a variety of ways in order to develop a good number sense.
Many researchers believe that mental calculation is one of the best means of developing and deepening a student s understanding of numbers and their properties. According to Baroody (1987), "mental arithmetic can serve to foster quantitative thinking, check written calculations and solve everyday problems." Mental arithmetic can also lead to the discovery of patterns, properties and structure of the number system (Reys, 1984).
Sowder (1992) suggests that mental computation is closely related to number sense. She indicates that although in current studies there has been a great deal of interest in number sense, number sense has not been a focus in instruction.
Van de Valle and Watkins (1993) point out that little research has been done on effective teaching strategies to develop computational skills in early grades. Sowder concurs that there is a need for research on how mental computation and the study of number sense should be incorporated into the curriculum.
Even though importance has been placed on mental computation as a curriculum goal, information related to student performance in this area according to Reys et al. (1993) is still limited. Elementary text books in the last few years have been including mental computation strategies, yet the curriculum of elementary school has been slow in its incorporation.
The purpose of this study is to determine the effect of mental computation instruction on third grade students. Questions to be answered by this study include:
1. What is the effect of instruction upon student's achievement in mental computation as measured in a pre/posttest?
2. What are the different strategies used by third grade students for mental computation classified as: standard, transition, nonstandard with no reformulation and nonstandard with reformulation (Markovits and Sowder, 1994)?
3. Does the use of the instructional materials help to increase number sense in the students
as related to the flexibility in the use of different computational strategies?
4. Does the use of the instructional materials have any effect on the teachers general pedagogical knowledge and beliefs about mental computation, learning strategies and assessment?
The relevant literature on mathematical thinking, mental computation, addition and subtraction, number sense, and teachers' beliefs about learning, and assessment was reviewed for the study. For the project design, mental computation strategies for addition and subtraction subskills were identified. These subskills were used for the test construction and refinement process. The test was used as a pre and post-treatment measure. The instructional activities that were developed for the treatment were paired with the subskills. Clinical interviews and questionnaires were designed and given to the teachers participating in the study. A clinical interview was also designed for the students as a post-treatment measure.
Data accumulated from the study was analyzed and prepared for the final report. The final report was then prepared including a summary, findings, recommendations and suggestions for further studies.
The report consists of five chapters. Chapter 1 is an introduction to the study and includes information pertaining to the need for the study, the purpose of the study, a summary of the procedures and a report plan which is an overview of the content of each chapter, the type of statistics used and data reported in the study.
The review of the literature is presented in Chapter 2. This review includes the theories of mathematical thinking that are the framework of the project. Studies pertaining to number sense, mental computation, addition and subtraction, and teacher's pedagogical beliefs about mathematics teaching and learning and assessment are also reviewed.
The methodology used in the study is reported in Chapter 3. The setting for the study and subjects are described. The theoretical framework for the instruments and treatment materials, their design, construction and application is also described in this chapter. This chapter also includes a detailed description of the procedures of the study.
The statistical reports which are found in Chapter 4 consist of: an analysis of pre/post test means for each group that participated in the study; an analysis of the variance between group types on pre and post mental computation examination; an item by item analysis of the increase or descrease in items answered correctly on pre and post mental computation examination; and the frequency and average percentage of strategies used by students in clinical interview. A summary of the teachers shared beliefs found in clinical interviews and three questionnaires administered to them is also presented.
Chapter 5 includes a summary of the research, conclusions and recommendations.