**Chapter 2**

** **

**BACKGROUND FOR THE STUDY**

According to Romberg (Grouws, 1992), there is no
general agreement on the definition of learning, how learning takes place and
what constitutes reasonable evidence that learning has taken place. Some say it
is observable changes in behavior, others that it means acquiring new
knowledge, and other say that it is the creating of a disequilibrium.

Psychologists have made different philosophic
assumptions about the nature of the learning process. Those who hold that
learning is determined by the forming connections between the environment
stimuli and useful responses are called associationist. A representative of
this view, E.B. Thorndike (1922), recommended that in mathematics, for example,
students perform much drill and practice on correct procedures and facts to
strengthen correct mental bonds. Associationists also argued that curricula
should be structured to keep related concepts well separated, so that students
did not form incorrect ties.

By
1943, the behaviorists were maintaining that a real science of education could
be built only on direct observation. Absent from the research and discourse of
behaviorists were "thinking", "meaning" or other such
unobservable and possibly nonexistent phenomena. Though behaviorists, led by
B.F. Skinner, denied the theory of "mental bonds" that associationist
had put forth, their prescriptions for mathematics teaching were similar: drill
and practice, with reinforcement by reward for desirable behavior in the form
of correct answers and punishment for undesired behavior. The behaviorists
brought to the educational scene programmed learning curricula and new
standardized testing techniques. In the study of teaching
"process-product" researchers searched for types of teaching behavior
that led to greater student achievement.

During that same time there existed other views of knowledge
and learning. In 1916 Dewey said that "It is that reconstruction or
reorganization of experience which adds to the meaning of experience, and which
increases ability to direct the course of subsequent experience" (p.89).
In another occasion Dewey (1938) wrote that "I use the word understanding
rather than knowledge because ...knowledge to so many people means
'information.' Information is knowledge about things (it is static), and there
is no guarantee that understanding-the spring of intelligent action-will follow
from it"(p.48). Brownell (1935) maintained that although incidental
learning could help counteract the practice of teaching mathematics as an
isolated subject, it did not provide an organization in which "the
meaningful concepts and intelligent skills requisite to real arithmetical
ability" could be developed. Brownell wrote about a theory of instruction
in which making "sense" of what was learned was the central issue in
arithmetic instruction. Piaget and his coworkers who interviewed hundreds of
children, proposed that in learning, children pass through developmental stages
and that the use of active methods which gives scope to spontaneous research by
the child help him rediscover or reconstruct what is to be learned "not
simply imparted to him" (Piaget, 1973, p.23).

Piaget's research and theory, is called
developmental constructivism (Romberg, 1969), and maintains that children
acquire number concepts and operations by construction from the inside and not
by internalization. Piaget (1968) pointed out that every normal student is
capable of good mathematical reasoning if attention (and care) is directed to
activities of his interest, and if by this method the emotional inhibitions
that too often give him a feeling of inferiority in lessons in mathematics are
removed.

In contrast to Piaget's explanation of construction,
Vygotsky (1986) presented an alternate theory where imbalance and not
equilibrium is considered normal.

__Jean____ Piaget's Theory of Learning__

According to Jean Piaget (1979), human intellectual
development progresses chronologically through four sequential stages. The
order in which the stages occur have been found to be largely invariant,
however the ages at which people enter each higher order stage vary according
to each person's hereditary and environmental characteristics.

Piaget
defined intelligence as the ability to adapt to the environment. Adaptation
takes place through assimilation and through accommodation, with the two
processes interacting throughout life in different ways, according to the stage
of mental development.

In assimilation, the individual absorbs new
information, fitting features of the environment into internal cognitive
structures. In accommodation, the individual modifies those internal cognitive
structures to conform to the new information and meet the demands of the
environment. A balance is maintained through equilibration, as the individual
organizes the demands of the environment in terms of previously existing
cognitive structures. A child moves from one stage of cognitive development to
another through the process of equilibration, through understanding the
underlying concept so that the understanding can be applied to new situations.
Equilibration is a balance between assimilation and accommodation.

The stages of cognitive development that Piaget
distinguished are four: (Piaget, 1968)

Sensorimotor
(0-2 years of age) - children begin to use imitation, memory and thought. They
begin to recognize that objects do not cease to exist when they are hidden from
view. They move from reflex actions to goal-directed activity.

Preoperational
(2-7 years) - Children gradually develop language and the ability to think in
symbolic form. They are able to think operations through logically in one
direction and they have difficulty seeing another persons point of view.

Concrete
operational (7-11 years) - Children are able to solve concrete (hands-on)
problems in logical fashion. They understand the laws of conservation and are
able to classify and seriate. They also understand reversibility.

Formal
operational (11-15 years of age) - Children are able to solve abstract problems
in logical fashion. Their thinking becomes more scientific, they develop
concerns about social issues and about identity.

Piaget suggested that when children do not
understand or have difficulty with a certain concept, it is due to a too-rapid
passage from the qualitative structure of the problem (by simple logical
reasoning -e.g. a ball existing physically) to the quantitative or mathematical
formulation (in the sense of differences, similarity, weight, number, etc.).
Conditions that can help the child in his search for understanding according to
Piaget is the use of active methods that permit the child to explore
spontaneously and require that "new truths" be learned, rediscovered
or at least reconstructed by the student not simply told to him (Piaget,1968).
He pointed out that the role of the teacher is that of facilitator and
organizer who creates situations and activities that present a problem to the
student. The teacher must also provide counterexamples that lead children to
reflect on and reconsider hasty solutions. Piaget argued that a student who
achieves a certain knowledge through free investigation and spontaneous effort
will later be able to retain it. He will have acquired a methodology that
serves him for the rest of his life and will stimulate his curiosity without
risk of exhausting it.

A third type of knowledge that Piaget suggests is
social or conventional knowledge. He said that it is always through the
external educational action of family surroundings that the young child learns
language, which Piaget (1973) called is an "expression of collective
values." Piaget pointed out that without external social transmission
(which is also educational) the continuity of collective language remains
practically impossible.

There
are three types of feelings or emotional tendencies, according to Piaget, that
affect the ethical life of the child, that are first found in his mental
constitution. In the first place is the need for love, which plays a basic role
in development in various forms from the cradle to adolescence. There is a
feeling of fear of those who are bigger and stronger than himself, which plays
an important role in his conduct. The third is mixed, composed of affection and
fear at the same time. It is the feeling of respect that is very important in
the formation or exercise of moral conscience.

Noddings (1990a) points out certain characteristics
that constructivist teachers must have an ethical commitment to inquiry in
order to aid students in their investigations, and the receptivity and
responsiveness of an ethic of care which involves sharing and listening to
students, taking interest in their purposes as well as in those of the
teachers' truth.

A constructivist view of knowledge implies that
knowledge is continuously created and reconstructed so that there can be no
template for constructivist teaching (Peterson & Knapp, 1993). Since this
point of view holds that learning involves student's constructing their own
knowledge, this leads to a redefinition of the teachers' role to one of
facilitator. This also leads to teaching that emphasizes the importance of
listening to and valuing students' perception, even when their understanding
differs from conventional knowledge (Cochran, Barson & Davis, 1970).

__Mathematical____ understanding and number
sense__

Cognitive scientists and mathematics educators who favor the cognitive science approach have moved well beyond Piaget in describing the way the mind operates. There has been a shift from an organic language of Piaget to a language "highly colored" of computers, (Noddings 1990b) with words such as networks, connections, paths, frames, etc.

As a cognitive position, constructivism maintains
that all knowledge is constructed, as Piaget's theories hold. Not only are
intellectual processes themselves constructive but are themselves products of
continued construction. It can be said that the construction and subsequent
elaboration of new understandings is
stimulated when established structures of** **interpretation do not permit or accept a new situation or idea.
This clash (not understanding) produces a disequilibrium that lead to mental
activity and the modification of previously held ideas to account for the new
experience (Simon & Schifter, 1991).

Hiebert and Carpenter (1992) propose a framework for
considering understanding from the constructivist

perspective which would shed light on analyzing a "range of issues related to understanding mathematics." They make a

distinction
between the external and internal representation of mathematical ideas,
pointing out that, to think and

communicate
mathematical ideas, people need to represent them in some way. Communication
requires that the representations be
external, taking the form of spoken language, written symbols, drawings or
concrete objects.

Mathematical
ideas becomes tangible when people can express them. By learning to express
their ideas to one another,

students
can begin to appreciate the nuance of meaning that natural language often
masks, but that the precise language

of
mathematics attempts to distinguish (Lo, Wheatley, & Smith, 1994; Silver,
Kilpatrick & Schlesinger, 1990; Lesh,

Post
& Behr, 1987).

The __Curriculum and Evaluation Standards for
School Mathematics __of the National Council of Teachers of Mathematics
(1989) point out in the Standard on
Communication** **that
understanding mathematics can be defined as the ability to represent a
mathematical idea in multiple ways and to make connections among different
representations. In order to think about mathematical ideas these need to be
represented internally but these mental representations are not observable.
This has led cognitive science to consider mental representations as a field of
study (Ashcraft, 1982; Greeno, 1991; Hiebert & Carpenter, 1992) .

Connections between external representations of
mathematical ideas can be constructed by the learner (Hiebert & Carpenter,
1992) between different fortes of the same idea or between related mathematical
ideas. These connections are often based on relationships of similarity or of
differences. Connection within the same representation are formed by detecting
patterns and regularities.

The relationship between internal representations of
ideas constructs a network of knowledge. Understanding then is the way information is represented, so that a** **mathematical idea, procedure or fact
is understood if it is part of an internal network. Networks of mental
representations are developed gradually as new information is connected to the
network or new ties are constructed between previously disconnected
information. Understanding grows as the networks become larger and more
organized and can be limited if connections are weak or do not exist becoming
useless (Hiebert & Carpenter, 1992).

Although the image of adding to existing networks is
appealing in its simplicity, it may turn out that the image is too simple.
Studies have suggested that students in the act of building understanding
reveal a much more chaotic process (Hiebert, Wearne & Taber, 1991). There
have been a number of studies in which the process of learning and
understanding are of central interest: Cobb, Wood, Yackel, Nichols, Wheatley,
Trigatti & Perlwitz (1991); Carpenter, Fennema, Peterson, Chiang & Loef
(1989); Doyle (1988); Baroody, (1985); Hiebert & Wearne (1988).

Children are natural learners and the environment
both social and physical offers them many opportunities to acquire notions of
quantity. Even in very poor or diverse cultures, races or classes, children
have the opportunity to acquire quantitative notions (Gelman 1980; Ginsburg,
Posner & Russel1,1981; Ginsburg & Russell, 1981).

Each healthy human brain, no matter the age, sex,
race or culture, comes equipped with a set of unique features: the ability to
detect patterns and to make approximations, a capacity for various types of
memory, the ability to self-correct and learn from experience and external data
and self reflection, and an great capacity to create (Came & Caine 1994).
Because of this predisposition of the brain, children and adults constantly
search for ways to make sense and make connections. This can be translated into
a search for common patterns and relationships as Hiebert and Carpenter (1992)
propose.

Caine and Caine (1994) argue that brain research
confirms that multiple complex and concrete experiences are essential for
meaningful learning and teaching. They add that the brain is designed as a
"pattern detector" and that the function of educators should be to
provide students with the kind of experiences that enable them to perceive
"the patterns that connect."

Children from a very young age are sensitive to
quantity. They perceive differences in number; they see correlation among
different numbers of events; their actions contain quantity and they use words
referring to basic mathematical events (Gelman, 1980; Ginsburg, 1989). Various
studies (Ginsburg & Baron, 1993; Starkey & Cooper, 1980; Van de Walle
& Watkins, 1993) have pointed out the importance of taking into account
children's informal mathematical connections as building block for formal
mathematics. Ginsburg (1989) suggests that students need to learn that it is
acceptable, "even desirable", for them to connect conventional
arithmetic with their own informal knowledge, intuition and invented
procedures.

In
a study by Carpenter, Ansell, Franke, Fennema and Weisbeck (1993), the results
suggest that children can solve a wide range of problems, including problems
involving multiplication and division, much earlier than is generally presumed.
With only a few exceptions, children's strategies could be characterized as
representing or modeling the action or relationships described in the problem.
These researchers conclude that young children's problem-solving abilities have
been seriously underestimated. They suggest that if from an early age children
are motivated to approach problem solving as an effort to make sense out of problem** **situations, they may come to believe
that learning and doing mathematics involves solving problems in a way that
always makes sense.

For years mathematics educators have advocated using
a variety of forms to represent mathematical ideas for students. Physical
three-dimensional objects are often suggested as especially useful. Despite the
intuitive appeal of using materials, investigations of the effectiveness of the
use of concrete materials have yielded mixed results (Bednarz & Janvier,
1988; Bughardt, 1992; Evans, 1991; Hestad, 1991; Hiebert, Wearne, & Taber,
1991; Simon, 1991; Thompson,J., 1992).

P. Thompson (1994) suggests that the apparent
contradictions in studies using manipulatives are probably due to aspects of
instruction and students' engagement to which the studies did not attend.
Evidently, just using concrete materials is not enough to guarantee success
according to Baroody (1989). The total instructional environment must be looked
into to understand the effective use of concrete materials. In a project by
Wesson (1992) for grades 1 and 2, which emphasized exploratory activities with
manipulatives, the results suggested that while a much wider range of content
than in standard books or tests was covered, there was no loss of arithmetic
skills.

Children understand when using concrete materials if
the materials are presented in a way that helps them connect with existing
networks or construct relationships that prompt a reorganization of networks.
It is important to consider then, the internal networks that students already
carry with them and the classroom activities that promote construction of
relationships between internal representations (Hiebert et al, 1991).
Manipulatives then can play a role in students' construction of meaningful
ideas. Clements and McMillan (1996) and others suggest they should be used
before formal instruction, such as teaching algorithms. Clements and McMillan
propose that concrete knowledge can be of two type: "sensory-concrete"
which is demonstrated when students use sensory materials to make sense of an
idea; and "integrated concrete" which is built through learning.
Integrated concrete thinking derives its strength from the combination of many separate ideas in an interconnected** **structure of knowledge. When children
have this type of interconnected knowledge, the physical objects, the actions
they perform on the objects, and the abstractions they make are all
interrelated in a strong mental structure.

Ross and Kurtz (1993) offers the following suggestions when planning a lesson involving the use of manipulatives. He suggests that the mathematics teacher should be certain that:

1. manipulatives have been chosen to support the
lesson's objectives;

2. significant plans have been made to orient
students to the manipulatives and corresponding classroom procedures;

3. the lesson involves the active participation of
each student;

4. the lesson plan includes procedures for
evaluation that reflect an emphasis on the development of reasoning skills.

In the last few years there have been studies about
the idea of students' constructing their own mathematical knowledge rather than
receiving it in finished form from the teacher or a textbook (Carpenter,
Ansell, Franke, Fennema, Weisbeck, 1993; Markovits & Sowder, 1994). A
crucial aspect of students' constructive processes is their inventiveness
(Piaget, 1973). Children continually invent ways of dealing with the world.
Many of the errors they make can be

interpreted
as a result of inventions (Ginsburg & Baron, 1993; Peterson, 1991).
Similarly, in school mathematics, students rely many times on invented
strategies to solve a variety of problems (Carpenter, Hiebert, & Moser, 1981;
Carraher & Schliemann, 1985; Ginsburg, 1989). Kamii and Lewis (1993) and
Madell (1985) have reported successful work in programs where children are not
taught algorithms, but are encouraged to invent their own procedures for the
basic operations. Treffers (1991) suggests a similar program in the Netherlands
and Baker & Baker (1991) in Australia.

Various studies have been made in the area of
invented strategies. Cook and Dossey's (1982) findings show that children learn
number facts easily and quickly and recall them better when using a strategy
approach than when using a learned algorithm, drill or practice approach.
Browne (1906); Howe and Ceci (1979); Kouba (1989); Rathmill (1978);

Sowder
and Wheeler (1989) have done studies on strategies used for calculation.
Carpenter and Moser (1984) found that children in the United States ordinarily
invent a series of abbreviated and abstract strategies to solve addition and
subtraction problems during their first four years in school. Romberg and
Collis (1987) found that even though some children are limited by their
capacity to handle information, most are able to solve a variety of problems by
inventing strategies that have not been taught. English (1991) observed that in
a study of young children's combinatoric strategies, a series of six
increasingly sophisticated solution strategies were identified. A significant
number of children independently adopted more efficient procedures as they
progressed on the task.

A study by Markovits and Sowder (1994) examined the
effect of an intervention in the instruction of seventh grade students for the
purpose of developing number sense. Instruction was designed to provide diverse
opportunities for exploring numbers, number relationships, and number
operations and to discover rules and invented algorithms. Measures taken
several months later revealed that after instruction students seem more likely
to use strategies that reflected number sense and that this was a long-term
change. Rathmill (1994) suggests that planning for instruction that promotes
the development of children's thinking and reasoning about mathematics not only
helps them make sense of the content they are studying, but also helps them
learn ways of thinking that later will enable them to make sense of new content.
Lampert (1986) proposed that "a sense-making" atmosphere is necessary
and that arithmetic should make sense in terms of children's own experience.
Reynolds'(1993) study suggests that children's imaging activity is at the heart
of their sense making and problem solving. Silver, Shapiro and Deutsch (1993)
found that students' performance was adversely affected by their dissociation
of sense making from the solution of school mathematics problems and their
difficulty in providing written accounts of their thinking and reasoning.

In the __Everybody Counts __document from the
National Research Council (1989) the major objective of elementary school
mathematics is the development of "number sense". The __Curriculum and
Evaluation Standards for School Mathematics__ (NCTM 1989) also includes
number sense as a major theme throughout its recommendations. Greeno (1991)
interprets number sense as "a set of capabilities for constructing and
reasoning with mental models." This perspective he argues, provides
reasons that support considering various aspects of number sense as features of
students' general condition of knowing in the area of numbers and quantities,
rather than skills that should be given specific instruction. The term number
sense refers to several important but elusive capabilities according to Greeno,
capabilities including flexible mental computation, numerical estimation and
quantitative judgment. Flexible mental computation according to Greeno involves
recognition of equivalence among objects that are decomposed and recombined in
different ways.

Reys et al. (1991) describe number sense in the
following manner:

Number sense refers to an intuitive
feeling for numbers

and their various uses and
interpretations; an

appreciation for various levels of
accuracy when

figuring; the ability to detect
arithmetical errors,

and a common-sense approach to using
numbers... Above

all, number sense is characterized by a
desire to make

sense of numerical situations (pp.3-4)

Sowder and Schappelle (1994) suggest that there are common**
**elements found in classrooms that help children acquire good number sense:

1.
Sense-making
is emphasized in all aspects of mathematical learning and instruction.

2.
The
classroom climate is conducive to sensemaking. open discussions about
mathematics occurs both in small groups and with the whole class.

3.
Mathematics
is viewed as the shared learning of an intellectual practice. This is more than
simply the acquisition of skills and information. Children learn how to make
and defend mathematical conjectures, how to reason mathematically and what it
means to solve a problem.

Mental computation according to Trafton (1986)
refers to nonstandard algorithms for computing exact answers. It is also
referred to as the process of calculating an exact arithmetic result without
the aid of an external computational or recording aid. (Hope, 1986; Reys,
1986). It is recognized as both important and useful in everyday living as well
as valuable in promoting and monitoring higher-level mathematical thinking
(Reys et al., 1995). It has been recognized in the __Curriculum and Evaluation
Standards for School Mathematics __(NCTM, 1989) that increased attention
should be given to mental computation. A __National Statement on Mathematics
for Australian Schools __(Australian Education Council and the Curriculum Corporation,** **1991) was released in 1991 recommending substantial change in
emphasis among mental, written and calculator methods of computation and
between approximate and exact solutions. A major objective is to redirect the
computational curriculum in schools to reflect a balance in the emphasis on
methods of solution. Before the Statement, the curriculum was divided as: 75%
written computation, 25% Calculator, Estimation, Mental Computation. With the
new statement it would be 25% for each method of computation.

According to Boulware (1950) mental arithmetic has
its origin during the second quarter of the nineteenth century. The idea of building
a broader foundation of meaning and understanding in arithmetic gave rise to
Mental Arithmetic as it was known in the middle of the nineteenth century with
Warren Colburn (1841) considered as pioneer in the field of mental arithmetic.
Before his time, arithmetic had reached a point of extreme abstraction
according to Boulware. The second half of the century witnessed the decline in
interest and understanding of the purpose of mental arithmetic. With the coming
of more writing paper, cheap pencils, with the rise of industry and its
accompanying needs for persons skilled in computation, the practical or
computational phase of arithmetic took on importance around the turn of the
century. The emphasis in arithmetic at that time was the teaching of isolated
facts, followed by drill upon these facts. High among the purposes stated for
the study of arithmetic many authors of the time placed speed, memory and
accuracy by mechanical rules. There was an emphasis in arithmetic on drill for
perfection and automatic response at the expense of meaning and understanding.
In 1950, a dissertation by Boulware is representative of the quest for the
development of "meaning" in mental computation stirred by Brownwell
(1935), who urged that meaning and seeing sense in what is being learned should
be the central focus of arithmetic instruction. Boulware's conception of mental
computation is as follows: Mental arithmetic deals with number as a unified,
consistent system, and not as an aggregate of unrelated facts. [It] consists of
methods of dealing with numerical situations whereby a clear concept of the
number system may be conceived and utilized in quantitative thinking. It
proceeds to the analysis of number combinations by processes of meaningful
experiences with concrete numbers, reflective thinking in number situations,
seeing relationships, and discovery of new facts as an outgrowth of known facts
(pp.7-8). In 1960, in an article by Sister Josefina there seems to begin
interest in mental computation and in the 1978 NCTM yearbook on computational
skills there appears an article by Trafton (1978) where the need for including
proficiency with estimation and mental arithmetic as goals for the study of
computation is presented. A good number of studies and articles about mental
computation appeared in the period of the 1980s (e.g. Reys, R.E., 1984, 1985;
Reys, B. J., 1985a, 1985b; Madell, 1985; Hope, 1985, 1986, 1987; Reys &
Reys, 1986; Langford, 1986; Markovits and Sowder, 1988; Baroody, 1984, 1985,
1986, 1987, 1989 and others) leading up to the statement of the inclusion of
mental computation as an area where increased attention is needed in school
mathematics by the NCTM (1989). With the increase of studies in cognitive
skills and number sense (e.g.Simon, 1979; Resnick, 1986; Silver, 1987;
Schoenfeld, 1987; Greeno, 1980; Sowder, 1988) and more recent studies mentioned
in this chapter, mental computation is suggested to be related to number sense,
needed for computational estimation skills and considered a higher order thinking
skill.

In a study by Reys, Reys and Hope (1993) they argued
that the low mental computation performance reported in this study most likely
reflected students' lack of opportunity to use mental techniques they
constructed based on their own mathematical knowledge. The study of Reys, Reys,
Nohda and Emori (1995) assessed attitude and computational preferences and
mental computation performance of Japanese students in grades 2, 4, 6, and 8. A
wide range of performance on mental computation was found with respect to all
types of numbers and operations at each grade level. The mode of presentation
(visual or oral) was found to significantly affect performance levels, with
visual items generally producing higher performance. The strategies used to do
mental computation were limited, with most subjects using frequently a mental version of a learned
algorithm.

In a study by G.W. Thompson (1991) about the effect
of systematic instruction in mental computation upon fourth grade students'
arithmetic, problem-solving and computation ability, a significant difference
favored the group taught mental computation, with girls improving more than
boys.

According to Markovits and Sowder (1994) it would
seem reasonable that if children were encouraged to explore numbers and
relations through discussions of their own and their peers' invented strategies
for mental computation, their intuitive understanding of numbers and number
relations would be used and strengthened. Okamoto (1993) found that children's
understanding of the whole number system seemed to be a good predictor of their
performance on word problems.

Cross-cultural
research has identified a variety of mental computation strategies generated by
students, (e.g. Hope & Sherill, 1987; Markovits & Sowder, 1988) and the
difference in mental computation in an out of school and in-school context
(e.g. Ginsburg, Posner, & Russell, 1981; Pettito & Ginsburg, 1982).
Sribner (1984) points out that individuals develop invented procedures suited
to the particular requirements of their particular occupations.

In a study on individuals who are highly skilled in
mental arithmetic (Stevens 1993), forty-two different mental strategies were
observed. Efficient, inefficient and unique strategies were identified for each
of five groups (grade** **8). Dowker (1992) describes in a study
the strategies of 44 academic mathematicians on a set of computational
estimation problems involving multiplication and division of a simple nature.
Computational estimation was defined as making reasonable guesses as to
approximate answers to arithmetic problems, without or before actually doing
the calculation. Observing people's estimation strategies, Dowker suggests, may
provide information not only about estimation itself, but also about people's
more general understanding of mathematical concepts and relationships. From
this perspective Dowker concludes that estimation is related to number sense.
Sowder (1992) who agrees with this position points out that computational
estimation requires a certain facility with mental computation.

In a study by Beishuizen (1993), he investigated the
extent to which an instructional approach in which students use of the hundreds
board supported their acquisition of mental computation strategies. In the
course of his analysis, he found it necessary to distinguish between two types
of strategies for adding and subtracting quantities expressed as two digit
numerals as follows:

1. 1010 strategy - 49 + 33 -> 40 + 30 -> 9 + 3
= 12

70 + 12 = 82

2. N10 strategy - 49 + 33 -> 49 + 30 -> 79 + 3
= 82

Beishuizen's analysis indicates that N10 strategies
are more powerful, but that many weaker students used only 1010 strategies. The
study's findings also suggest that instruction involving the hundreds board can
have a positive influence on a student's acquisition of N10 strategies. Fuson
and Briars (1990) and others have also identified these strategies.

Hope (1987) points out that because most written
computational algorithms seem to require a different type of reasoning than
mental algorithms, an early emphasis on written algorithms may discourage the
development of the ability to calculate mentally. Lee (1991) recommends that
perhaps it is time to investigate changing our traditional algorithms for
addition and subtraction to left-to-right procedures.

According to Reys et al. (1995) there have been many
studies that suggest the benefit of developing mental computation strategies.
Mental computation has also been highlighted in the __Curriculum and
Evaluation Standards for School Mathematics __(NCTM, 1989).

Mental computation can be viewed from the
behaviorist perspective as a basic skill that can be taught and practiced. But
it can also be viewed from the constructivist view in which the process of inventing
the strategy is as important as using it. In this way it can be considered a
higher-order thinking skill (Reys et al., 1995).

__Addition____, subtraction and teaching
strategies__

The __Curriculum and Evaluation Standards for
School Mathematics __(NCTM 1989) recognizes that addition and subtraction
computations remain an important part of the school mathematics curriculum and
recommends that an emphasis be shifted to understanding of concepts. Siegler
(1988) indicated how important it is for children to have at least one accurate
method of computation. In a study by Engelhardt and Usnick (1991) while no
significant difference between second grade groups using or not using
manipulatives was found, significant differences in the subtraction algorithm
favored those taught addition with manipulatives. Usnick and Brown (1992) found
no significant differences in achievement between the traditional sequence for
teaching double-digit addition, involving nonregrouping and then regrouping,
and the alternative, in which regrouping was introduced before non-regrouping
examples in second graders.

Ohlsson, Ernst, and Rees (1992) used a computerized
model to measure the relative difficulty of two different methods of
subtraction, with either a conceptual or a procedural representation. The
results of the use of the model suggested that regrouping is more difficult to
learn than an alternative augmented method, particularly in a conceptual
representation, a result that contradicts current practice in American schools.
Dominick's (1991) study with third grade students suggested that students'
confusion with the borrowing algorithm centered around a misunderstanding of
what was being traded. Evans (1991) found that groups taught with pictorial
representations or by rote learned to borrow in significantly less time than
did a group using concrete materials in grades 2 and 3.

Sutton and Urbatch (1991) recommended the use of
base-ten blocks, beans and bean sticks or beans and bean cups to serve as
manipulatives to use for trading games and with the "transition
board". (A modified version of the base ten board). They also emphasized
that attempting to teach addition and subtraction without initially preparing
the student with trading games could be counterproductive and result in lack of
understanding due to lack of preparation. In a study which analyzed individual
children's learning of multidigit addition in small groups in the second grade,
results suggested that rarely did a child spontaneously link the block trades
with written regrouping (Burghardt, 1992). Fuson and Briars (1990) and P.W.
Thompson (1992) found that the base ten blocks could be a helpful support for
children's thinking, but many children do not seem spontaneously to use their
knowledge of blocks to monitor their written multidigit addition and
subtraction. The Fuson and Briars study suggested that frequent solving of
multidigit addition or subtraction problem accompanied by children's thinking
about the blocks and evaluating their written marks procedure, might be a
powerful means to reduce the occasional trading errors made by children. The
study also suggested that counting methods that use fingers, are not
necessarily crutches that later interfere with more complex tasks.

Fuson and Fuson (1992) found that in all of the
groups studied, children were accurate and fast at counting up for subtraction
as at counting on for addition. This contrasts with the usual finding that
subtraction is much more difficult than addition over the whole range of
development of addition and subtraction solution strategies. Sequence counting
on and counting up according to Fuson and Fuson are abbreviated counting
strategies in which the number words represent the addends and the sum. In both
strategies the counting begins by saying the number word of the first addend.
For example: 7 + 5, a child would say 7 pause 8, 9, 10, 11, 12 (up to five
numbers, the last number of the sequence is the answer) and 12 - 5 would be, 5,
pause 6, 7, 8, 9, 10, 11, 12, seven numbers were counted, which is the answer.
Thornton's (1990) study provides evidence that children who were given an
opportunity to learn a counting up meaning for subtraction as well as counting
down (counting back from minuend), preferred the counting up meaning.

In a series of studies by Bright, Harvey and Wheeler
(1985) they defined an instructional game as a game for which a set of
instructional objectives has been determined. These instructional objectives
may be cognitive or affective and are determined by the persons planning the
instruction, before the game is played by the students who receive the
instruction in it. The results of the studies suggest that:

- games can be effective
for more than drill and practice and for more than low level learning of
skills and concepts,
- games can be used along
with other instructional methods to teach higher level content such as
problem solving,
- games should probably
be used relatively soon before or after instruction planned by the teacher
for the same material,
- the use of more challenge,
fantasy or curiosity might enhance the effectiveness of instructional
games.

Hestad (1991) found that the use of a card game was effective for third grade students in introducing new mathematical concepts and maintaining skills.

In a study by Cobb (1995) the use of the hundreds
board by second graders' in a classroom where instruction was broadly
compatible with recent reform recommendations (NCTM, 1989, 1991) was
investigated. The role played by the use of the hundreds board over a 10-week
period in supporting the conceptual development of four second graders was
studied.

Particular attention was given to the transition
from counting on to counting by tens and ones. The hundreds board

is a
ten-by-ten grid from either 0 to 99 or 1 to 100. The results indicated that the
children's' use of the hundreds board did not support the construction of
increasingly sophisticated concepts of ten. However, children's use of the
hundred board did appear to support their ability to reflect on their
mathematical activity once they had made this conceptual advance. The utility
of the hundreds table in teaching computation has been also recognized by
Beishuzen (1993);Hope, Leutzinger, Reys and Reys (1988); Thornton, Jones and
Neal (1995) and Van de Walle and Watkins (1993) .

__Teachers____' Pedagogical Beliefs about
Mathematics Teaching Learning and Assessment__

We can learn more about how invisible components in
the teaching and learning situation can contribute to or detract from the
quality of the mathematical learning that takes place by focusing on the
culture according to Nickson (1992). It is important, he points out, in
exploring the mathematics classroom from the perspective of the culture, it
generates, to remember that we are concerned with the people in the setting and
what they bring to it. Nickson adds that we must increase our sensitivity to
the importance of their hidden knowledge, beliefs, and values for mathematics
education.

One of the major shifts in thinking in relation to
teaching and learning of mathematics in recent years has been with respect to
the adoption of differing views about the nature of mathematics as a
discipline. The view of mathematics that has informed and historically transfixed
most mathematics curriculum has been, according to Lakatos, (1976) one of
considering that mathematics as consisting of "immutable truths and
unquestionable certainty". Such a view does not take into account how
mathematics changes and grows and is waiting to be discovered (Nickson, 1992).
Brown and Cooney (1982) note that the intensity of the teachers' beliefs is
very important in the classroom culture. The traditional detachment of
mathematics content from shared activity and experience, so that it remains at
an abstract and formal level, constructs barriers around the subject, according
to Nickson, that sets it apart from others areas of social behavior. The
message conveyed is that is has to be accepted unquestioningly and from which
no deviation is permitted. The classroom culture will mirror this unquestioning
acceptance. The visibility and acceptance of what is done or not done in
mathematics are factors in stopping teachers from engaging in activities that
they may instinctively feel are appropriate but might challenge the supposedly
inviolable essence of mathematics as they themselves were taught.

In investigating the relationship between what
teachers believe about how children learn mathematics and how those teachers
teach mathematics, A. Thompson (1992) points out that studies have examined the
congruence between teachers' beliefs and their practice and findings have not
been consistent. Researchers such as Grant (1984) and Shirk (1973) have
reported a high degree of agreement between teachers' professed views of
mathematics teaching and their instructional practice, where as others have
reported sharp contrasts (e.g. Carter, 1992; Cooney, 1988; Shaw, 1989;
Thompson, 1984).

It
has been argued (Nickson, 1988; Ball, 1993) that bringing teachers into the
arena of research activity can be an important step in increasing their
understanding of research processes and results and their relation to classroom
practice. Each mathematics classroom will vary according to the actors within
it. The unique culture of each classroom is the product of what teachers bring
to it in terms of knowledge, beliefs, and values, and how these affect the
social interactions within that context. The daily experiences of students in
mathematics classes of teachers with positive attitudes were found to be
substantially different from those of students in classrooms of teachers with
negative attitudes in a study by Karp (1991). Overall, teachers with negative
attitudes toward mathematics employed methods that fostered dependency and
provided instruction which was based on rules and memorization, relied on an
algorithmic presentation, concentrated on correct answers and neglected
cognitive thought processes and mathematical reasoning, whereas teachers with
positive attitudes were found to encourage student initiative and independence.
Swetman (1991) found no significant relationship between teachers' mathematics
anxiety and students' attitude toward mathematics in grades 3 to 6. Attitude
toward mathematics however, became more negative as grade increased in teachers
and students.

At the time of a study by Good and Grouws, (1977),
comparatively few studies had included observational measures that detail how
the teacher functions as an independent variable in order to influence student
achievement. Teacher effectiveness (as operationally defined in their study)
appeared to be associated strongly with the following clusters: student
initiated behavior; whole class instruction, general clarity of instruction,
and availability of information as needed, a non-evaluative and relaxed
learning environment which is task focused; higher achievement expectations;
classrooms that are relatively free of major behavioral disorders. Brophy's
(1986) study found that most investigative efforts had focused on curricular
content and students' learning without careful consideration of teachers'
instructional practices. Loef (1991) found that more successful teachers (in
grade 1) represented differences among addition and subtraction problems on the
basis of the action in the problem and the location of the unknown, and they
organized their knowledge on the basis of the level of the children's
understanding of the problem in context. Hiebert and Carpenter (1992) note that
it seems evident that procedures and concepts should not be taught as isolated
bits of information, but it is less clear what connections are most important
or what kind of instruction is most effective for promoting these connections.

__Teachers' influence on class content__

Even though some researchers have concluded that
textbooks determine the content addresses in classrooms (Barr, 1988; Barr &
Dreeben; 1983) others provide evidence to challenge that assertion (Freeman
.& Porter, 1989; Stodolsky,1989) as Sosniak and Stodolsky (1993) have
pointed out. In mathematics, Barr (1988) found that seven out of nine
fourth-grade teachers used their textbooks by moving lesson by lesson through
the book. In contrast, Freeman and Porter (1989) and Stodolsky (1989) found
most mathematics teachers to be selective in their use of textbook lessons,
problem sets, and topics, although topics not included in the texts were only
occasionally added to the instructional program.

Research
suggests that teachers are "gatekeepers" (Thornton, 1991) who make
their own decisions about which parts of a textbook to use and which ways to
use them (Barr & Sadow, 1989) and such decisions may not necessarily lead
according Brophy (1982) to close adherence to the textbook material.

Sosniak and Stodolsky (1993) found in a study of
four fourth-grade teachers that the influence of textbooks on teachers'
thinking and on instruction was somewhat less than the literature indicates.
Their results suggest that patterns of textbook use and thinking about these
materials were not necessarily consistent across subjects even for a single
teacher, and that the conditions of elementary teachers' work encouraged
selective and variable use of textbook materials.

In
a study by Stigler, Fuson, Ham, and Myong (1986), an analysis is made of
addition and subtraction word problems in American and Soviet elementary
mathematics textbooks. The data suggests that American children entering first
grade can solve the simple kinds of addition and subtraction word problems on
which American texts spend so much time.

Another
study on text books is one by Ashcraft and Christy (1995) in which they study
the frequency of arithmetic facts in elementary texts. The study tabulated the
frequency with which simple addition and multiplication facts occur in
elementary school arithmetic texts for grades 1-6. The results indicated a
"small-facts bias" in both addition
and** **multiplication.
"Large" facts, with operands larger than 5, occurred up to half as
frequently as those with operands in the 2-5 range. As was also found in an
earlier tabulation for grades K-3, facts with operands of 0 and 1 occurred
relatively infrequently, except for patterns like 1+2 and 1x3 which had a high
frequency. The small facts bias in the presentation of basic arithmetic, at
least to the degree observed, probably works against a basic pedagogical goal,
mastery of simple facts. It may also provide a partial explanation of the
widely reported problem size or problem difficulty effect, that children's and
adults' responses to large basic facts are both slower and more error prone
than their solutions to smaller facts.

In a study by Porter (1989) elementary school
mathematics is used as a context for considering what could be learned from
careful descriptions of classroom content. Teachers log and interviews show
that large numbers of mathematics topics are taught for exposure with no
expectation of student mastery: much of what is taught in one grade is taught
in the next, skills typically receive 10 times the emphasis compared to either
conceptual understanding or application, and depending on school and teacher
assignments, mathematics instruction a student receives may be doubled or
halved. Porter argues that "ultimately teachers must decide what is best
for their students and within the limits of their own knowledge, time and
energy." (p.15)

__Teaching practices
and their effects__

Koehler and Grouws (In Grouws, 1992) have suggested that teachers' behavior is influenced by their knowledge of: the mathematics content being taught, how students might learn or understand that particular content and of the methods of teaching of that particular content. Also influencing teachers' behaviors are teachers' attitudes and beliefs about teaching and mathematics.

Bush (1991) in a study about factors related to
changes in elementary student's anxiety found that mathematics anxiety tended
to decrease as teachers in grades 4-6 spent more time in small group
instruction, had more years of experience, and took more post-bachelor's
mathematics courses. According to a study by Tangretti (1994), findings
indicated that the elementary teachers that participated in the study were not
adequately prepared to meet NCTM expectations. Their teaching focus was found
to be an algorithmic approach with emphasis on numeration and computation. Lack
of confidence in content areas beyond arithmetic were reported as contributing
to the lack of preparedness of elementary teachers to implement innovative
curriculum. Wood, Cobb and Yackel (1991) report that after participating in a
study, changes occurred in a teacher's (second grade) beliefs about the nature
of mathematics (from rules and procedures to meaningful activity), about
learning (from passivity to interacting) and about teaching (from transmitting
information to guiding students' development of knowledge). A similar result
was reported in a study by Zilliox (1991). In-service elementary school
teachers felt they were teaching more and better mathematics lessons, were more
comfortable with student use of hands-on materials and with managing small
groups, and had a different sense of student capabilities and different
expectations for student behavior after participating in the study.

A study by Carpenter, Fennema, Peterson, Chiang, and
Loef (1989) investigated teachers' use of knowledge from research on children's
mathematical thinking and how their students' achievement is influenced as a
result. Although instructional practices were not prescribed, the teachers that
participated in the treatment activities taught problem solving significantly
more and number facts significantly less than did control teachers. Treatment
teachers encouraged students to use a variety of problem-solving strategies,
and they listened to processes their students used significantly more than did
the control group teachers and knew more about individual students
problem-solving processes.

__Teachers and assessment issues__

A view of learners as passive absorbers of facts,
skills, and algorithms provided by teachers is a basis for much of the most
current use of measures (Stenmark, 1991). Standard achievement tests according
to Kamii and Lewis (1991) measure students' abilities to recall and apply facts
and routines presented during instruction. Some items require only the
memorization of detail; other items, although designed to assess **higher-level learning outcomes, **often
require little more than the ability to recall a formula and to make the
appropriate substitutions to get the answer (Lambdin, 1993). Test items of this
type are consistent with the view of learning as a passive, receptive process,
a process which is additive and incremental. The

practice
of scoring answers to items of this type (right or wrong) is consistent with
the view that "bits" of knowledge or skills are either present or
absent in the learner at the time of testing. Under this approach, diagnosis is
simply a matter of identifying the missing pieces of knowledge in the
student,thereby creating a need for remedial teaching. In the constructivist
view of teaching, the student is a participant in building his own
understanding. The learner does not absorb new ideas and data but rather
constructs his own version and relates it to existing information (Wilson,
1992). In order to help the student construct firm connections in the sense of
the constructivist theory, the teacher can contribute by facilitating time for
auto-evaluating, reflection on processes and ideas, auto-monitoring procedures
like journals, portfolios, rechecking work (Sanford, 1993; Stenmark, 1989,
1991; NCTM, 1995). These are all metacognitive processes that can be
strengthened through these practices. Metacognition refers to one's knowledge
of one's own cognitive processes and products, and of the cognition of others.
It also refers to self-monitoring, regulation and evaluation of the cognitive
activity (Silver & Marshall, 1990). According to Beyer (1988),
metacognition involves thinking about how one thinks as well as thinking to
make meaning. For assessment and monitoring of student learning, an implication
of the constructivist view is that teachers must measure understanding and
models that individual students construct for themselves during the learning
process (Webb & Romberg, 1992) . Accordingly, achievement could be better
defined and measured not in terms of number facts and procedures that the
student can reproduce, but in terms of best estimates of his or her level of
understanding of key concepts and interrelated underlying principles (Wilson,
1992) .

A survey to investigate teacher awareness of
alternative assessment of students in mathematics of (n=126) public school
teachers in primary (K-2), elementary (3-5), middle school (6-8) and high
school (9-12) showed that significant differences in awareness of alternative
assessment practices exists among the four levels (Drury, 1994). In another
study, (Watts, 1993) which is a description of the implementation of the __Curriculum
Standards for School Mathematics __(NCTM, 1989) in grades K-3, it was found
that teachers used in their tests, knowledge level items significantly more
frequently than higher level items, and items with manipulative materials
significantly more than items without manipulatives. Alternative assessment
formats were considered significantly more difficult to use.

In a study on the influence of district standardized
testing on mathematics instruction for grades 3 and 8 (Kolitch, 1993) it was
reported that in two school districts, the curriculum was aligned to test
content; in a third district with an innovative mathematics program, the
district test had little influence on mathematics instruction, but the program
was in jeopardy because of decreasing computation scores. Kamii and Lewis (May
1991) also report a similar finding of achievement testing in primary
mathematics as perpetuating lower-order thinking. According to an achievement
test, traditional instruction produced results as good as or better than, a
constructivist program in second grade. Such tests were created within a
framework of mathematics which Kamii and Lewis argue does not measure
understanding.