A cognitive approach to mathematics education
Meir Ben-Hur
International Renewal Institute, Chicago, USA
Abstract
The early gaps in children’s cognitive abilities may not necessarily be large, but
inattention on the part of mathematics teachers to those gaps bears severe consequences
in the children’s future learning. What makes it difficult is that known fact that it is
always possible that the difficulties students have with mathematics are masked by the
appearance of rote learning, because students who are not developmentally ready to
learn certain concepts have little recourse but to memorize and reproduce what they are
told. It is therefore critical that teachers recognize the importance of a cognitive
approach to mathematics education and regularly analyze their students’ errors and their
students’ learning. In this paper I first analyze the cognitive challenges of mathematical
thinking, particularly at the primary school levels. Then I offer the example of number
sense to illustrate how the number sense is cognitively challenged.
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The importance of cognition in mathematics cannot be underestimated. From
kindergarten on, the typical mathematics curriculum involves six critical cognitive
operations and structures.
a. conservation
b. representation;
c. allocentric thinking;
d. spatial abilities and the sense of time,
e. projection of virtual relationships, and
f. self-regulation, focused, and systematic exploratory behavior;
These six critical and highly interdependent cognitive operations and structures
develop in children gradually, through mediated manipulation and movement among
objects, from infancy and beyond childhood. These operations and structures are briefly
defined below:
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Conservation
Conservation refers to the cognitive ability to judge changes through logical deduction
rather by appearance1. It is manifested in the conception of number that is flexible enough
to include all the possible variations of what is countable, how it is arranged, and how it is
counted: “all fours are 4”. This concept of numbers is involved not only in quantifying
objects, but also in equating different denominations of money, measurement, and time.
Another example of conservation involves naming geometrical forms, and understanding
“volume” without association to shape. Conservation and, eventually, the related schema
of transformation lay the foundation of all the reversible operations of mathematics.
Representation
Conservation cannot exist without representation. The cognitive tools of representation
include beside visualization, also language, concepts, and symbols. Thus, representation
frees our thinking about objects from our perception. What Numeration involves symbols;
geometry involves language and icons etc. Furthermore, representation permits the
thinker to discriminate input information, manipulate symbols rather than reality itself,
and communicate.
Allocentric (non-egocentric) thinking
Alocentric- thinking is manifest in the ability to differentiate between self and the world
to represent the world from different points of view. It permits us to relate objects in time
and space using an objective frame of reference. In fact it allows us to think
mathematically, because mathematical thinking is essentially allocentric. Terms such as
“more”, “less” etc., are truly relative only when they are conceived of objectively.
Clearly, allocentric thinking is a necessary condition for conservation.
Spatial and temporal sense
The sense of space and the sense of time depend upon conservation, representation, and
allocentric thinking. Young children first make sense only of the topological properties in
1
Cockcroft, K., Hook, D. & Watts, J. (2002). Developmental Psychology. UCT Press:
Cape Town.
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space (enclosure, inside space, outside space, and proximity)2. Later they start to develop
the Euclidean sense of space as they use allocentrical terms of relative size to describe the
relationships among. Only at the end of this developmental journey (normally around the
second and fourth grade) children can preserve the properties of distance, angles, and
straight parallel vertical, horizontal and diagonal lines3.
Children’s early concept of time is normally confused with space. They do not
attribute the longer distance accomplished by moving objects to high velocity, but to
longer time of the movement, and they understand “old” as synonymous with “big”. For
them time is discontinuous and not uniform (this is why in their minds adults stop aging).
Projection of virtual relationships
The relationships between units and groups and the basic operations with numbers are
fundamental in early mathematical thinking. Where these relationships are projected,
particularly without manipulating objects, they can be referred to as virtual. Comparing
and projecting such relationships are perhaps the most critical consequences of allocentric
thought. For example: a math problem may state that Don is 3 years older than Elsa, or
that Jamal is younger than Jo by 4 years. Before a child can determine which numbers to
add the child must understand these relationships. Projecting virtual relationships permits
the child to prioritize, sequence, and identify patterns
Self-regulation (cognitive, behavioral and emotional)
Mathematical thinking, particularly as in problem solving, requires the ability to set goals
for the relatively long- and short-term, prioritize them, and estimate which would be easy
and which would be difficult to attain, and monitor and regulate progress. The quality of
such self-regulation is increasingly challenged by the mathematics curriculum.
It is now appropriate to consider for example the case of the development of children’s
sense of natural numbers.
The topological space can be thought of as a model of the environment constructed on a
rubber sheet. As the sheet is stretched in different directions, the relative distances and
directions between locations may change, but the basic layout and the sequence of places
along a route are preserved intact.
2
3
Piaget, J., & Inhelder, B. (1948) Op. Cit.
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Children’s Sense of Natural Numbers
Normally, children younger than 5 years tend to confuse the quantity of objects with the
amount of space these objects occupy. For example, they would point to a longer pattern
as the one with more objects even if it contains the lesser amount. This naive sense of
quantities changes as children learn to count using numbers. Numbers have two attributes,
ordinality and cardinality. Ordinality refers to the ordered set of numerical terms (“one”,
“two”, “three”…). Cardinality refers to quantity the number represents. The following
table defines five principles that children must learn to construct the mathematical
number sense:
The five principles of counting, their learning significance and cognitive challenge 4
Principles of counting:
Significance
Cognitive challenge
Ordinal principle I: Stable and
complete order of numeration (e.g.,
“twelve”, “thirteen”, “fourteen”,
etc…)
The language of a numeration
determines how well children
understand the base structure, place
value (units, tens, etc.), and later
also arithmetical operations.
Cultural (language) differences in
numeration systems correlate
positively with the types of errors
made by 3 and 4 year old children5.
These differences impact
children’s ability to count, their
memory span, and eventually their
cognitive representation of
numbers.
Ordinal principle II: One-to-one
correspondence between a number
and an item to be counted.
The analysis/categorisation of
sequence facilitates the
coordination of the concept of
number with the concept of groups.
Besides verbal fluency
(automaticity), and memory,
counting requires coordination that
preschool-children may still be
lacking even when they understand
the concepts involved6.
Cardinal principle I: The last
count represents the whole.
Counting is the trivial of addition
(+1).
The emphasis is on the need to
ensure that the whole group is
counted, each member only once:
A need for precision.
Cardinal principle II: Any order
of counting results in the same
The emphasis on the flexible
concept of group. This flexibility
The flexible mechanics of counting
requires systematic and organized
4
Adapted from Ben-Hur, M. (2004). Forming Early Concepts of Mathematics. IRI Inc.
Glencoe, IL
5
Miura, I. T., Okamoto, Y.,Kim, C. C., Stere, & Fayol. M., (1993). First graders’
cognitive representation of number and understanding of place value: Cross-national
comparisons: France, Japan, Korea, Sweden, and the US. Journal of Educational
Psychology, 85, 24-30.
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For example: Gleman, R., a& Meckin, S. (1983). Preschoolers’ counting: Principles
before skill. Cognition, 13, 343-359.
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number
forms the foundation for the two
basic properties (for addition and
multiplication) the associativity
and commutativity
input. It requires conservation of a
constancy (number) over changes
(in the configuration of the group)
The abstraction principle: Any
kind, or mixed kinds of objects can
be grouped and counted
Numbers are conceptualized as
versatile tools for problem solving
Numbers are projects of virtual
relationships
Counting helps children explore the relationships between numbers and establish the basis
the arithmetical operations. Learning to categorize what has been counted vs. what is yet
to be counted, teaches children to coordinate their perception and their actions.
Understanding that the strategy they choose for counting does not affect the result helps
them reinforce their conservation schema and enables them to develop the intuition for
the associative and commutative properties of the two basic operations (addition and
multiplication). Ultimately, counting becomes a useful tool they learn for solving
problems.
A cognitive approach to teaching mathematics requires that teachers not only
know the instructional challenges, but also that they know how to mediate them. For this
reason it is important that teachers of mathematics study the work of Reuven Feuerstein,
and the example of Feuerstein’s Instrumental Enrichment. This work provides effective
strategies and a particular program that are aimed exactly at the development of these
critical cognitive challenges.
meirbh@aol.com
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