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Chapter 3 --Review of the Literature
This literature review will present findings concerning the history of the study of the
brain with regard to learning; proposals for an appropriate learning environment for LD students,
together with the importance of that environment; and the benefits of using a brain-compatible
approach to teach mathematical concepts.
Hart, (1983), points out that the human brain does not process information logically, but
instead reflects the illogical events and accidents of history, i.e., evolution. As it has evolved
over the past hundreds of millions of years, the brain has developed three parts--the so-called
“triune brain” (MacLean, 1978).
The oldest part of the brain is the “reptilian brain” (or R-complex), which consists largely
of the brain stem. Its purpose is related to actual physical survival and overall upkeep of the
body. Because the reptilian brain is basically concerned with physical survival, the behaviors
under its control are similar to the survival behaviors of animals. The R-complex behaviors are
automatic, have a ritualistic quality, and resist change.
The second brain to evolve was the limbic system, made up of the amygdala (which
associates events with emotion) and the hippocampus (the part of the brain dealing with
contextual memories). Our contextual memories are made up of internal information and
information received externally through our senses.
The third brain is our “thinking brain,” the neocortex. This part of the brain is
responsible for language, including speech and writing. Most of the processing of sensory data
occurs in the neocortex. It handles logical and formal operational thinking, and allows us to plan
for the future. (Hart, 1983)
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Hart (1983) refers to the collection of patterns and programs in the brain as a program
structure, or “proster”. His proster definition of learning is that learning is the “extraction of
meaningful patterns from confusion”. (Hart, 1983, p. 95) A person needs a collection of
programs for almost any action. One’s store of patterns and programs reflects his/her
experiences, and is not the same as intelligence. The “biasing” which affects one’s proster is the
total of what is stored in the brain: past experiences, plans, goals, fears, and older brain
influences. Much of this biasing is well below the conscious level. Since a teacher cannot alter
biases already stored in her students’ brains from their experiences, it becomes important for her
to change the setting or the situation (the specific circumstances immediately surrounding the
students). Learning in the classroom depends greatly on previous learning and biases stored in
the brains of each individual. Giving all students the same instruction without regard to what
they bring to the learning situation will almost guarantee a high rate of failure. (Hart,1983)
Caine & Caine (1991) continue Hart’s reasoning, namely, that the more meaningful and
challenging the events and activities in the classroom, the more opportunity exists for all children
to learn well. These authors make a distinction between rote memorization and “natural
memory”, which builds on students’ actual involvement and participation in activities around
them. They see the three brains operating together, giving them the capacity to re-program old,
outmoded ways of thinking or acting. In other words, the three layers of the brain interact.
Concepts and emotions are interconnected, and emotions energize our memories. Students’
enthusiasm prompts more solid learning and understanding. This is a way of looking at learning
as a “new survival principle”. (Caine et al., 1991, p. 61)
Some types of learning are positively strengthened by “relaxed alertness” and challenge,
but inhibited by perceived threat, which narrows our perceptual field. Hart (1983) calls that
perceptual narrowing “downshifting”. When a person downshifts (feels threatened), his
responses become more automatic and limited. He is not as able to notice environmental and
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internal cues. His ability to engage in open-ended thinking and questioning is reduced.
Downshifting appears to negatively affect many higher-order cognitive functions of the brain and
thus can prevent us from learning and discovering solutions for new problems. It also appears to
hinder our ability to see relationships and interconnectedness. Under stress, the ability of the
brain to find the proper programs and patterns is reduced. The brain’s short-term memory and
ability to form permanent new memories are inhibited. (Caine & Caine, 1991)
Making maximum connections in the brain requires the state of “relaxed alertness”, a
combination of low threat and high challenge. If the teacher wants students to understand what
the information presented means, to question it, and to make connections with what they already
know, then the teacher needs to provide a richly-stimulating, low-threat environment. Therefore,
the teacher’s goal is to create low-threat conditions for learning. The key is for educators to
appreciate what must actually take place for students to learn effectively, and then create
procedures which allow that to happen. (Caine & Caine, 1991)
Furner & Duffy (2002) relate how often a teacher’s covert behaviors (being hostile,
exhibiting gender bias, having an uncaring attitude, expressing anger, having unrealistic
expectations, or embarrassing students in front of peers) can cause anxiety in students and a
feeling of being threatened. Several teaching techniques cause math anxiety, such as assigning
the same work for everyone, teaching problem by problem out of the textbook, and accepting
only one way to solve a problem. Much of math anxiety could actually be test anxiety. It’s
important for teachers to incorporate strategies to prevent and reduce math anxiety, thereby
assisting students in becoming confident mathematical thinkers.
The National Council of Teachers of Mathematics identified “equity” as their first
principle for school mathematics. All students have the right to learn math and feel confident in
their math abilities, and teachers must strive to see that “mathematics can and will be learned by
all students”.(NCTM, 2000, p.13) Their recommendations for reducing math anxiety include:
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acccommodate for different learning styles;
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teach the students to use “self-talk” to guide themselves positively to a solution
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create a variety of testing environments;
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design positive experiences in math classes;
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remove the importance of ego from classroom practice;
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emphasize that everyone makes mistakes in mathematics;
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make math relevant;
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let students have some input into their own evaluations;
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allow for different social approaches to learning mathematics;
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emphasize the importance of original, quality thinking rather than rote maniplation of
formulas;
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characterize math as a human endeavor;
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use writing in a math journal for thinking, expressing feelings, and solving problems;
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use of discussion and cooperative group work;
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use of manipulatives, calculators, computers, and all forms of technology
(NCTM, 2000)
Resnick, Greeno, and Collins (1996) discuss three different perspectives for
understanding the learning process. These perspectives are the behaviorist perspective, the
cognitive/rationalist perspective; and the situative/pragmatist perspective.
The behaviorist perspective looks at learning as a process by which associations and
skills are acquired. Transfer occurs to the extent that behaviors learned in one situation are
carried over to another situation. Motivation is extrinsic, provided in the form of incentives for
attending to the important aspects of the learning situation. This form of knowledge is often
expressed as behavioral objectives. Programmed instruction was such an approach.
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The cognitive/rationalist view incorporates the information-processing models of
reasoning and problem-solving. Piaget’s theory of logical structures impacts this cognitive
theory. Another important theme in the cognitive view of knowing is the concept of
metacognition, the capacity to reflect upon one’s own thinking, and thereby monitor and manage
it. Motivation is intrinsic. Developmental psychologists had originally thought that a reflective,
self-monitoring capacity discriminated more advanced children from their less-advanced peers.
The situative/pragmatist perspective focuses on the way knowledge is distributed in the
world among individuals, the books they use, and the communities and practices in which they
participate. From this point of view, one form of knowing stems from groups that carry out
cooperative activities. The practices of the community provide patterns which organize the
group’s activities and the participation of the individuals who are alert to those patterns.
Cognitive research has moved toward a concern with more naturalistic learning
environments. In this new, emerging theory, success in cognitive functions such as reasoning,
remembering, and perceiving is regarded as an achievement of a system, with contributions of all
the individuals who participate. Cooperative learning is one example of how this perspective can
be utilized in the classroom. Transfer of learning within this perspective often depends on the
manner in which solutions to problems are presented. If students understand the solution to a
problem as an example of a general method, and if they understand the general features of the
learning situation that are relevant to use of the method, the abilities they learn are more likely to
be transferred to new situations. The situative/pragmatist perspective involves students as
central to the functioning of the community, enhancing their sense of identity. The motivation
to learn the values and practices of the community establishes their identity as community
members. Mathematics classrooms are communities of practice in which students participate by
thinking about mathematical topics and discussing their ideas. (Resnick et al., 1996)
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Bryant, Bryant, and Hammill (2000) have attempted to identify specific behavioral
characteristics in the areas of arithmetic and word problem-solving. For instance, math
difficulties may be evidenced in problems with a) math fact automaticity; b) arithmetic
strategies; c) interpretation of word problem sentence construction; and d) word problem-solving
skills. Teachers need to task-analyze skills carefully and to provide students with strategies for
remembering and executing multiple steps to solve math problems. It appears that students may
require sequenced, explicit, systematic teaching with practice and corrective feedback, coupled
with activities that promote meaningful understanding of the steps inherent in solving math
problems. Instructional implications for the behavior “has difficulty with the language of math”
suggest that vocabulary (e.g., numerator, difference, sum, minuend) and abstract symbols (e.g., <,
>, +) specific to each math lesson should be identified and taught according to what we know
about effective instructional routines (e.g., explicit instruction, examples, and guided practice).
The variable, “has difficulty with multi-step problems”, is the single most important behavior for
predicting math weaknesses. Another variable is “makes re-grouping (re-naming) errors”.
Errors in re-naming are often indicative of a conceptual misunderstanding of place value and its
application to subtraction problems. These variables in students indicate potentially serious
difficulties in mathematics. If behaviors persist despite intensive, individualized remedial efforts
in the general education classroom or with remedial specialists, and if other behaviors cited
earlier are present, it is quite possible that the student has a mathematics learning disability and
should be evaluated for that possibility. (Bryant et al., 2000)
Davis and Parr (1997) examined the characteristics of students with specific learning
disabilities in either reading and spelling, or arithmetic.
As a criterion for learning disability
eligibility, these students’ under-achievement was deemed to be not due to sensory handicap,
mental retardation, or cultural or environmental disadvantage.
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Two main groups of learning disabilities were included: disabilities in 1) linguisticphonological processing (associated with difficulties in reading or spelling; and 2) visual-spatial
analysis (associated with difficulties in arithmetic). (In the research done by Davis et al., 1997,
students in the first group were referred to as Group R-S; students in the second group were
referred to as Group A.) Students with underlying deficits in visual-spatial-organizational and
psychomotor skills have been noted to have weaknesses on tasks of mechanical arithmetic and
handwriting. Myklebust (1975) associated this latter group with lower scores on performance
tasks as compared to verbal tasks, poor math skills, poor spatial orientation, clumsiness, and poor
social perception and interpretation of social situtations.
Using scores from the Woodcock-Johnson (Revised), Davis et al. (1997) felt that students
who were identified as having specific academic deficiencies in arithmetic may be at relatively
high risk for emotional-behavioral problems at school. A cognitive weakness in mathematics
would be associated with a poor ability to process information involving non-verbal cues. Scores
on the WJ-R were compared with scores on the WISC-III.
On the WISC-III, the Perceptual Organization (PO) factor and the Verbal Comprehension
(VC) factor were used. The four subtests making up the PO composite were: Picture
Completion, Picture Arrangement, Block Design, and Object Assembly. The four subtests which
made up the VC composite were: Information, Similarities, Vocabulary, and Comprehension.
Group A scored significantly higher on the VC factor in relation to their PO score, as predicted.
Comparisons using measures from the Wechsler Intelligence Scale for Children-Third Edition
(WISC-III) indicated that Group A was weaker in nonverbal skills than Group R-S, despite
equivalent average IQ scores between the two groups. Group A students were more likely than
Group R-S students to have counseling provided as part of their Individualized Educational Plan,
suggesting greater socio-emotional difficulty among Group A students. Group A students had
more well-developed auditory-perceptual skills with deficits in visual-perceptual-organizational
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skills. Their performance IQs were significantly lower than their verbal IQs. Group A’s
strengths stemmed from underlying assets in phonemic discrimination, segmentation, and
blending which leads to a relatively strong ability to match phonemes and graphemes within a
system of codified rules (words). (Davis et al., 1997)
The neuropsychological deficiencies hypothesized to underlie the poor academic
achievement shown by Groups R-S and A would be expected to have far-reaching effects,
affecting not only the complex process of academic learning, but also other childhood tasks that
are non-academic in nature, such as social interactions. Group A students had greater difficulty
with nonverbal perception, interpretation, and expression. The difficulties with social interaction
that group A children had are believed to be due to their inadequate nonverbal reasoning
abilities, particularly in new situations where flexibility is needed. Group A children, identified
as the Nonverbal Learning Disabilities (NLD) group, tend to encounter increasing levels of
difficulty as the task demands become newer and more complex. In contrast, these children
exhibit well-developed auditory-perceptual skills.
Rivera and Pedrotty (1997) discuss trends in the fields of mathematics special education.
Studies seem to indicate that individuals with mathematics learning disabilities tend not to
perform at a level commensurate with their peers on the basic functional skills (e.g., telling time,
counting change) that are necessary for successful adult living. Investigations have shown that
these students may exhibit difficulties using metacognitive problem-solving strategies, memory
and retrieval processes and generalization skills. They also may have limited proficiency with
speed of processing, problem conceptualization, and use of effective calculation strategies.
Significant instructional strategies for these youngsters have been developed in recent
years. These strategies include explicit direct instruction, relevant practice, peer-mediated
instruction, and alternative algorithms that foster mathematical understanding and evaluative
thinking. Constructivist activities that support active student learning around problem-solving
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situations can be facilitated by the teacher’s guidance and questioning. Social settings and
classroom communities are now recognized as important factors in helping to develop
mathematical cognitions. (Rivera et al., 1997)
Rourke and Conway (1997) discuss the evidence that some brain systems are involved in
processes of calculation. They indicate that understanding brain-behavior relationships in
children who exhibit disabilities of arithmetic and mathematical reasoning requires a general
familiarity with some issues surrounding cerebral asymmetry. Cerebral asymmetry hypothesizes
that the left hemisphere specializes in the processing of routine behaviors, whereas the right
hemisphere is specialized for inter-modal integration, processing of new stimuli, and dealing
with informational complexity.
A neuropsychological approach to LD is oriented toward the full range of brain-behavior
relationships that may interact with or affect the arithmetic learning situation. It appears that
neuropsychological assessment can reveal patterns of assets and deficits that are predictive of
later academic performance, including arithmetic.
Desoete, Rowyers, and Buysse (2001) examined the relationship between metacognition
and mathematical problem solving in children with average intelligence, in order to help teachers
and therapists to gain a better understanding of contributors to successful math performance.
Flavell introduced the concept of metacognition in 1976, defining it as “one’s knowledge
concerning one’s own cognitive processes and products or anything related to them.”
Metacognition also refers to the active monitoring of these processes in relation to the cognitive
objects or data on which they bear. Executive control or metacognitive skills are the voluntary
control people have over their own cognitive processes.
Keeler and Swanson (2001) investigated the relationship between working memory
(WM) and math achievement in children with and without math disabilities. WM is the limited
capacity system that allows simultaneous storage and processing of temporary information. WM
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deficits underlie the difficulties of students with reading and mathematical disabilities. Students
with mathematical disabilities (MD) have shown memory difficulties related to higher order
skills (such as executive processing) and knowledge of algorithms. MD children who have
difficulty problem-solving use less efficient strategies for retrieving information than normallyachieving peers. Keeler et al., (2001) concluded from the results of their study that since one of
the main functions of working memory is retrieval of stored long-term knowledge, storage rather
than processing efficiency may account for the poor WM performance of children with MD.; and
consequently, one way to improve math achievement is to understand the factors that relate to
working memory deficits and to develop methods of improving students’ awareness of effective
strategies. (Keeler et al., 2001)
Multiple Intelligences theory gives a useful look at helping students retain what they learn.
Armstrong (2000) relates that Howard Gardner, a psychologist, views memory as “intelligencespecific”. [Gardner (1999) has identified eight intelligences: Linguistic, Logical-mathematical,
Spatial, Musical, Bodily-kinesthetic, Interpersonal, Intrapersonal, and Naturalist.] This new
perspective on memory means that students with “poor memories” may have poor memories in
only one or two of the intelligences. Consequently, the solution to the memory problem involves
helping those students use their “good” memories in other intelligences (e.g., musical, spatial,
etc.). Work involving memorization of material in any subject should be taught in such a way
that all eight “memories” (Armstrong, 2000) are activated. The teacher’s job is to help students
associate what they need to learn with representatives of the different intelligences: Linguistic
with words; Logical-mathematical with numbers; Spatial with pictures; Musical with musical
phrases; Bodily-kinesthetic with physical movements; Interpersonal with social interactions;
Intrapersonal with examining personal feelings; and Naturalist with natural phenomena.
Armstrong’s interpretation is that once students have been introduced to memory strategies from
all eight intelligences, these students will be able to use those strategies that work best for them,
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and then use them independently during their own study times. Kornhaber, Fierros, and
Veenema (2004) indicate that although a mathematician may require strong logical-mathematical
skills, he may also rely on spatial skills for visualizing relationships and tap into his interpersonal
intelligence to make mathematical ideas understood and interesting to others. These authors
believe that if schools were to engage a wider range of students’ strengths, more students would
succeed in school and on into their adult lives. Their book is based on a national investigation
(called SUMIT—Schools Using Multiple Intelligence Theory) of 41 diverse schools that
associate MI with improvements for students (Kornhaber, Fierros, & Veenema, 2004). It
identifies approaches that are successful across a wide variety of classrooms, schools, and
student groups. Their work illustrates a research-driven description of effective practices
involving MI.
Silver, Strong, and Perini (2000) believe that multiple intelligences and learning styles can
be easily integrated to account for different processes of thought and feeling. Putting those two
models together in a way that will maximize achievement necessitates paying attention to four
key principles of learning: comfort, challenge, depth, and motivation. These four key principles
are guided by what current brain research (Caine & Caine, 1991) tells us about getting the most
out of the learning process.