PSYCHOLOGY OF LEARNING MATHEMATICS
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Two trains of thought
 Behaviorism
(Skinner)
 Constructivism (Piaget)
Behaviorism – One can affect learning by changing
behavior
Constructivism – One affects “understanding” by
constructing his/her own knowledge
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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The prevailing theory in math education is
constructivism
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Constructivism is seen to entail the following
six items by most experts.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
“The particular problem of mathematics lies in
its great abstractness and generality, achieved
by successive generations of particularly
intelligent individuals each whom has been
abstracting from, or generalizing, concepts of
earlier generations.” R. Skemp
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
“The present day learner has to process not
raw data but data-processing systems of
existing mathematics.” R. Skemp
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Mathematics cannot be learnt directly from the
environment, but only indirectly from other
mathematicians, in conjunction with ones own
reflective intelligence.
It makes students dependent on teachers
(including textbook authors).
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
It can expose one to acquiring a lifelong dislike of mathematics.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS

First two principles to learning mathematics
 Concepts
of a higher order than those which a
person already have cannot be communicated to
him/her by a definition, but only by arranging for
him/her to encounter a suitable collection of
examples.
 Since in mathematics these examples are almost
invariably other concepts, it must first be ensured
that these are already formed in the mind of the
learner.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Rule 1 is broken by most textbooks and many
teachers.
Good teachers give ample number of examples
for each definition.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS

Pythagorean Theorem
 Given
a right triangle with right angle C then
c2=a2+b2
where c is the side opposite angle C.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Concept is widely used but is hard to define.
Mathematical concepts are among the most
abstract.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Concept development starts with classifying.
Pre-verbal examples.

Baby finishes drinking his bottle and places it
with empty bottles that he sees set in the
corner.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Boy sees a baby for the first time and it is
crawling. He pets it on the head like he would a
dog.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Low Level
Start with one object experienced from multiple
perspectives until it is classified.
c1
c2
c
c3
c4
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Similar objects are then grouped or classified
with the original object.
c
c’
c’’
c’’’
c’’’’
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
From this group we abstract invariant
properties by which we recognize object ch as a
member of the class {c, c’, c’’, c’’’, c’’’’}.
Classifying is collecting together experiences on
the basis of similarities.
Abstracting is an activity that makes us aware
of similarities.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Naming an object helps to classify it.
However, naming a concept can be limiting.
A name and the concept can often be confused.
4
vs.
3
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Two Types of Concepts
 Primary
– Those derived from sensory and motor
experiences
 Secondary – Those abstracted from other concepts
Defining a concept can short circuit the
intended process of defining it.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS

Two uses of definitions
 Specifies
the limits, boundaries of a larger class of
objects
 Provides means of describing new concepts of
lower order from known concepts
Example: Magenta can be described as between
red and blue
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Beginning examples should have little “noise”
More advanced examples should become
noisier
Collections of examples require inventiveness
and a clear knowledge of the
concepts/definitions they are being used for
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
If a particular level of abstraction is imperfectly
understood, all subsequent levels are in peril.
Contributory concepts necessary for each new
stage of abstraction must be available when
needed.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Pairing concepts together with a connecting
idea is known as a relation.
A transformation is applied to a relation to
change it or combine it with another relation.
The entire process of transforming relations
creates complex structures.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
The study of structures is an important part of
mathematics, and the study of the ways in
which they are built up and function is at the
very core of the psychology of learning
mathematics.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Conceptual structures are known as schema.
One of the most basic mathematical schema is
the set of Natural numbers with the operations
of addition and multiplication.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
One of the most misunderstood mathematical
schema is the set of positive fractions with the
operations of addition and multiplication.
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
Understanding – to assimilate into an
appropriate schema
not an all-or-nothing state
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Three failures to understand
 Wrong
schema is used
 Gap between new idea and the (appropriate)
existing schema
 The existing schema is not capable of assimilating
the new idea without undergoing expansion or
restructuring
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS

Five Dimensions of Understanding
 Nature
of task
 Role of the teacher
 Social culture of classroom
 Mathematical tools support learning
 Equity and accessibility
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Nature of the Task
 Students
develop mathematical understanding as
they invent and examine methods of solving
problems
 Paper-and-pencil worksheets – faster execution of
skills
 Watching a teacher at the board – imitation
 Doing and reflecting – UNDERSTANDING
 Role of the teacher
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
 Problematic
– a task is problematic if it promotes
new understanding. The task is an exercise if it
involves only previously learned material and skills.
 Problematic tasks promote reflection and
communication
 Problematic tasks require the use of tools
 Allow students time to explore new tools
 Use tools only when needed
 Make sure the tool is suitable to the task
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
 Problematic
tasks should leave behind a residue
 Outcomes of a task: (1) insights into structure, and
(2) strategies for solving problems

 Task
should connect with previously learned
knowledge (in and out of math class)
 Task should be significant
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Role of Teacher
Facilitate conceptual understanding
 Make sure task fits goal and is a genuine mathematical
problem
 Don’t be a professor of truth and correctness
 Select appropriate problems and sequencing to
promote learning
 Don’t intervene too much or too deeply – stems
initiative and creativity
 Provide essential information when it is needed
 Control the social culture of the classroom
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UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Social Culture of the Classroom
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INTERACTING IS ESSENTIAL
Reflection and communication depends on the social
culture of the classroom
Ideas are currency
Autonomy of students with respect to methods
Appreciate mistakes
Authority for reasonability and correctness lies in the logic
and structure of the subject
Students choose and share methods
Mistakes are seen as opportunities
Correctness based on arguments
UNDERSTAND THE PSYCHOLOGY OF LEARNING MATHEMATICS
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Mathematical tools support learning
 Tools
are not necessarily calculators and computers
 Meaning does not reside in the tool. It is formed by
using the tool
 Tools are used only to accomplish a task
 Tools should be versatile
 Tools should help to communicate effectively
 Tools should aid thinking