Bruner's Concept formation and Mathematics

1. Jerome
Bruner
C O N S T R U C T I V I S T T H E O R Y-
C O N C E P T F O R M AT I O N

2. Biography
Jerome Seymour Bruner (October 1, 1915 – June 5,
2016) was an American psychologist who made
significant contributions to human cognitive
psychology and cognitive learning theory in
educational psychology. Bruner was a senior
research fellow at the New York University School
of Law. He received a BA in 1937 from Duke
University and a PhD from Harvard University in
1941. He taught and did research at Harvard
University, the University of Oxford, and New York
University. A Review of General Psychology survey,
published in 2002, ranked Bruner as the 28th most
cited psychologist of the 20th century.

3. BRUNER’S CONSTRUCTVIST
THEORY.
Cognitive Structure
Learning Theory
Bruner believes that the essence of
learning is that one connects the
similar things and organizes them
into meaningful structures, and
learning is the organization and
reorganization of cognitive
structures. Knowledge learning is
to form the knowledge structure of
all subjects in the minds of
students.
Bruner is one of the pioneers of
cognitive psychology in the United
States, which began through his
own early research on sensation
and perception as being active,
rather than passive processes.
Like Ausubel (and other cognitive
psychologists), Bruner sees the
learner as an active agent;
emphasizing the importance of
existing schemata in guiding
learning.

4. Bruner’s Theory Continued
Bruner argues that students should discern for themselves the structure of subject content –
discovering the links and relationships between different facts, concepts and theories (rather than
the teacher simply telling them).
Bruner (1966) hypothesized that the usual course of intellectual development moves through three
stages: enactive, iconic, and symbolic, in that order. However, unlike Piaget’s stages, Bruner did
not contend that these stages were necessarily age-dependent, or invariant.
Piaget and, to an extent, Ausubel, contended that the child must be ready, or made ready, for the
subject matter. But Bruner contends just the opposite. According to his theory, the fundamental
principles of any subject can be taught at any age, provided the material is converted to a form
(and stage) appropriate to the child.

5. Principles of Bruner’s Theory
• Instruction must be concerned with the experiences and contexts that make the student willing and
able to learn (readiness).
• Instruction must be structured so that it can be easily grasped by the student (spiral organization).
• Instruction should be designed to facilitate extrapolation and or fill in the gaps (going beyond the
information given).

6. For Example:
• The concept of prime numbers appears to be more readily grasped when the child, through
construction, discovers that certain handfuls of beans cannot be laid out in completed rows and
columns. Such quantities have either to be laid out in a single file or in an incomplete row-column
design in which there is always one extra or one too few to fill the pattern. These patterns, the child
learns, happen to be called prime. It is easy for the child to go from this step to the recognition
that a multiple table , so called, is a record sheet of quantities in completed multiple rows and
columns. Here is factoring, multiplication and primes in a construction that can be visualized.

7. Spiraling According to Bruner
• The spiral approach to curriculum has three key principles that sum up the approach nicely. The
three principles are:
1. Cyclical: Students should return to the same topic several times throughout their school career;
2. Increasing Depth: Each time a student returns to the topic it should be learned at a deeper level
and explore more complexity;
3. Prior Knowledge: A student’s prior knowledge should be utilized when a topic is returned to so
that they build from their foundations rather than starting anew.

8. Spiraling Used In Mathematics
• In mathematics, we often return to the same content repeatedly but add complexity each time.
• For example, your teacher may first cover simple fractions, then more complex fractions, and then
start getting you to add and subtract fractions.
• Rather than focusing on fractions for an entire year, your school will spread fraction classes out
over the course of many years.
• Each time you return to fractions, your teacher will assess how well you retained previous
information, and then help you build upon that prior knowledge.

9. Three Stages
of Bruner’s
Theory
 ENACTIVE REPRESENTATION
(ACTION-BASED)
 ICONIC REPRESENTATION
(IMAGE-BASED)
 SYMBOLIC REPRESENTATION
(LANGUAGE-BASED)

10. Enactive
Representatio
n
(action-based)
I N T H E E N A C T I V E M O D E ,
K N O W L E D G E I S S T O R E D P R I M A R I L Y
I N T H E F O R M O F M O T O R
R E S P O N S E S . T H I S M O D E I S U S E D
W I T H I N T H E F I R S T Y E A R O F L I F E
( C O R R E S P O N D I N G W I T H P I A G E T ’ S
S E N S O R I M O T O R S T A G E )
T H I N K I N G I S B A S E D E N T I R E L Y
O N P H Y S I C A L A C T I O N S , A N D
I N F A N T S L E A R N B Y D O I N G , R A T H E R
T H A N B Y I N T E R N A L
R E P R E S E N T A T I O N ( O R T H I N K I N G ) .
I T I N V O L V E S E N C O D I N G P H Y S I C A L
A C T I O N - B A S E D I N F O R M A T I O N A N D
S T O R I N G I T I N O U R M E M O R Y . F O R
E X A M P L E , I N T H E F O R M O F
M O V E M E N T A S M U S C L E M E M O R Y , A
B A B Y M I G H T R E M E M B E R T H E
A C T I O N O F S H A K I N G A R A T T L E .

11. The Enactive
Stage In
Mathematics
 I n t h e e n a c t i v e m o d e , t h e c o n c e p t i s r e p r e s e n t e d t h r o u g h t h e l e a r n e r s
a c t i n g o n c o n c r e t e , p h y s i c a l o b j e c t s . F o r a l g e b r a , t h e c o n c e p t o f a n
u n k n o w n q u a n t i t y c a n b e r e p r e s e n t e d b y a b a g o r b o x o f a n u n k n o w n
n u m b e r o f c o u n t e r s .
 T h e t e a c h e r c a n a s k q u e s t i o n s s u c h a s “ W h a t h a p p e n s i f t h r e e
c o u n t e r s a r e p l a c e d i n t h e b a g ? W h a t i f o n e c o u n t e r i s t a k e n o u t ?
W h a t i f t h e r e a r e t e n b o x e s w i t h t h e s a m e n u m b e r o f c o u n t e r s i n
e a c h ? ”
 A v a r i a b l e c a n b e r e p r e s e n t e d a s a c h a n g i n g l e n g t h , f o r e x a m p l e ,
l e a r n e r s c a n m e a s u r e t h e l e n g t h o f a p e a s t r e e a s i t g r o w s .

12. Iconic
Representatio
n
(image-
based)
•I N F O R M AT I O N I S L E A R N E D
A S S E N S O R Y I M A G E S
( I C O N S ) , U S U A L LY V I S U A L
O N E S , L I K E P I C T U R E S I N
T H E M I N D . F O R S O M E , T H I S
I S C O N S C I O U S ; O T H E R S
S AY T H E Y D O N ’ T
E X P E R I E N C E I T.
•T H I S M AY E X P L A I N W H Y,
W H E N W E A R E L E A R N I N G A
N E W S U B J E C T, D I A G R A M S
O R I L L U S T R AT I O N S A R E
O F T E N H E L P F U L T O
A C C O M PA N Y T H E V E R B A L
I N F O R M AT I O N .
•T H I N K I N G I S A L S O B A S E D
O N U S I N G O T H E R M E N TA L
I M A G E S ( I C O N S ) , S U C H A S
H E A R I N G , S M E L L O R
T O U C H .

13. The Iconic
Mode in
Mathematic
s
IN TH E IC ON IC MOD E, A PIC TU R E
ILLU STR ATES TH E U N K N OW N
QU A N TITY TH E PIC TU R E TA K ES ITS
MEA N IN G FR OM TH E LEA R N ER ’S
PR EVIOU S EN A C TIVE EXPER IEN C ES.

14. The Symbolic
Representatio
n
• T H I S D E V E L O P S L A S T . I N T H E S Y M B O L I C
S T A G E , K N O W L E D G E I S S T O R E D P R I M A R I L Y A S
L A N G U A G E , M A T H E M A T I C A L S Y M B O L S , O R I N
O T H E R S Y M B O L S Y S T E M S .
• T H I S M O D E I S A C Q U I R E D A R O U N D S I X T O S E V E N
Y E A R S O L D ( C O R R E S P O N D I N G T O P I A G E T ’ S
C O N C R E T E O P E R A T I O N A L S T A G E ) .
• I N T H E S Y M B O L I C S T A G E , K N O W L E D G E I S
S T O R E D P R I M A R I L Y A S W O R D S , M A T H E M A T I C A L
S Y M B O L S , O R O T H E R S Y M B O L S Y S T E M S , S U C H A S
M U S I C .
• S Y M B O L S A R E F L E X I B L E I N T H A T T H E Y C A N B E
M A N I P U L A T E D , O R D E R E D , C L A S S I F I E D , E T C . S O
T H E U S E R I S N ’ T C O N S T R A I N E D B Y A C T I O N S O R
I M A G E S ( W H I C H H A V E A F I X E D R E L A T I O N T O
T H A T W H I C H T H E Y R E P R E S E N T ) .
• A C C O R D I N G T O B R U N E R ’ S T A X O N O M Y , T H E S E
D I F F E R F R O M I C O N S I N T H A T S Y M B O L S A R E
“ A R B I T R A R Y . ” F O R E X A M P L E , T H E W O R D
“ B E A U T Y ” I S A N A R B I T R A R Y D E S I G N A T I O N F O R
T H E I D E A O F B E A U T Y I N T H A T T H E W O R D
I T S E L F I S N O M O R E I N H E R E N T L Y B E A U T I F U L
T H A N A N Y O T H E R W O R D .

15. The Symbolic Mode in
Mathematics
• In the symbolic mode, the concept is represented in an abstract or a conventional way. Bruner
considers words (such as ‘five’, ‘add’ or ‘tin’) to be symbolic in the same way as conventional
letters and signs are (e.g. 5, + or t).
• FOR EXAMPLE
When students learn the knowledge of squares, they do not grasp the concept and nature of squares in isolation,
instead link the knowledge of squares to other geometric knowledge, such as quadrilaterals, parallelograms,
rectangles, diamonds, and so on. And they conclude the new knowledge into the original knowledge structure, so
as to form and perfect the quadrilateral knowledge structure diagram constantly in the mind.

16. The CPA Approach of Bruner’s
Theory
The CPA approach, that is the CONCRETE, PICTORAL and ABSTRACT approach lies at the
heart of Mathematics.
It enables a natural and supportive progression for learners as they develop their
understanding and skills in math. For example, in exploring equivalent fractions with
students you can begin with multi-link cubes, then progress to the pictorial problems in the
book and then on to questions with no visuals provided except numbers and the necessary
mathematical symbols. This will result in a fully embedded understanding of the concepts
throughout the classroom.

17. Concrete Stage
• Concrete is the “doing” stage. During this stage, students use concrete objects to model problems.
Unlike traditional maths teaching methods where teachers demonstrate how to solve a problem,
the CPA approach brings concepts to life by allowing children to experience and handle physical
(concrete) objects. With the CPA framework, every abstract concept is first introduced using
physical, interactive concrete materials.
• For example, if a problem involves adding pieces of fruit, children can first handle the actual fruit.
From there, they can progress to handling abstract counters or cubes which represent the fruit.

18. Pictorial (Representational)
Stage
• Pictorial is the “seeing” stage. Here, visual representations of concrete objects are used to model
problems. This stage encourages children to make a mental connection between the physical
object they just handled and the abstract pictures, diagrams or models that represent the objects
from the problem.
• Building or drawing a model makes it easier for children to grasp difficult abstract concepts (for
example, fractions). Simply put, it helps students visualize abstract problems and make them more
accessible.

19. Abstract Stage
• Abstract is the “symbolic” stage, where children use abstract symbols to model problems. Students
will not progress to this stage until they have demonstrated that they have a solid understanding of
the concrete and pictorial stages of the problem. The abstract stage involves the teacher
introducing abstract concepts (for example, mathematical symbols). Children are introduced to the
concept at a symbolic level, using only numbers, notation, and mathematical symbols (for example,
+, –, x, /) to indicate addition, multiplication, or division.

20. Figure 1. shows how to use CPA
approach