**Chapter 1**

** **

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.