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Constructivism values individual thinking
strategies: In Mathematics, there can be no
one fixed method to solve a given question.
Sadly, teachers insist on a particular method,
answer keys supplied to examiners allot marks
for a set pattern of steps, and we end up with
stereotyped answers. “Why can’t I solve it
using my method?” is an often heard query.
Following a set of steps may be beneficial as
it brings in some kind of standardization and
facilitates the teacher’s task, but in insisting
on ‘following a fixed method’ we fail to nurture
individual thinking strategies, we fail to allow
creativity and this is the first stumbling block
to constructivist thinking.
Constructivism involves sensory input:
Mathematics teaching is often considered
challenging as much of the content is ab- stract. We may not have a plausible hands- on activity for every concept in Mathematics,
but wherever possible a multimodal approach
(using both cognitive and psycho-motor do- mains) should be used. To learn more about multi
modal learning in Mathematics, I recommend
Rashmi Kathuria’s work which can be accessed
on http://mykhmsmathclass.blogspot.in/,
http://mathematicslearning.blogspot.in/,
http://mathematicsprojects. blogspot. in/.
Constructivism uses dovetailing, scaffold- ing and extrapolation: Mathematics involves
connections. An analytical teacher takes into
account the previous content that needs to
be dovetailed into the present content being
explored. One needs to provide the minimum
support that is adequate to the learner and
thus provide leverage to further learning. One
has to help the learner extrapolate
what is presently being learned
to what will be learned in the
future. When my daughter
learned formally about odd
and even numbers,
explo ring
“What did you do at school?” is a routine question that most mothers ask their children when they
return home. My daughter Priya was in Class One and when I asked her about her school day, she
led me to the tamarind tree in our courtyard, picked a little leaf and said “I learned this.” Elaborating
further as she pointed to every pair of leaflets, she said “Look, this is two ones are two, two twos
are four, ...” “Is this what teacher used in class today?” I enquired. “No, no. Teacher wrote this on
the blackboard and made us say ‘two ones are two’. But mamma, I had seen this ‘two ones are two’
leaf while playing.” I loved the new name that the tamarind leaf had got. A ‘two ones are two’ leaf!
Mathematics abounds all around. My daughter taught me this is through many more examples.
Here is another interesting one.
Priya was about three when she was helping me arrange eggs in the refrigerator. As we placed the
eggs (in twos) in the egg-rack, she commented “5 is not a partner number, 6 is a partner number, 3
is not a partner number, and 4 is a partner number.” Amused that my daughter was hinting at odd
and even numbers, I asked her to explain. “With 3 eggs, we can place 2 eggs next to each other, but
1 is left behind. When there are 4 eggs, all get partners, none are left behind.” I told her that ‘not a
partner number’ is the same as odd and ‘partner number’ is the same as even, but she seemed hap- pier with her terminology; it made more sense than ‘odd’ and ‘even’. Later, when she learned about
the same in school, I reminded her of this incident. I had read about constructivism and designed
constructivist activities, but it was this experience that gave me a chance to explore constructiv- ism. Based on this and my experiences as a teacher, I share a few thoughts about constructivism.
Image courtsey: Mukesh Malviya Govt School
Teacher, Pahawadi, Shahpur, MP
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I reminded her of the ‘eggs and partner numbers’ in- cident. Next I took a number of small circles and we
arranged them in pairs. So if we took seven pairs, we
had 14 circles. If we had 20 circles, we had ten pairs.
The next step was to try and arrange circles in dif- ferent combinations, not just pairs. For example, 16
could be arranged as a pair pattern (2 x 8) but going
beyond pairs, we could arrange 16 as (4 x 4); 15 could
be arranged as a 3 x 5 pattern; 18 could be arranged
as either 3 x 6 or 2 x 9 pattern. This was dovetail- ing what Priya knew about even numbers into fac- tors of a number, which was something she did not
yet know. I had to provide help for one example. The
remaining examples were like a game. This step of
providing minimal support is scaffolding. Construc- tivism also takes into account extrapolation. This ac- tivity of arranging circles was now made challeng- ing by giving 31 circles or 19 circles to arrange in a
pattern. This helped introduce the concept of prime
numbers. Meaningful connection between what is
known and what needs to be known is the crux of
constructivism.
Constructivism encourages queries: A healthy
learning environment welcomes questions. I once
had a question from my daughter: “When we add,
subtract and multiply, we begin with the unit’s place.
Why do we follow a reverse order when we divide?”
Such questions indicate that the learner is looking
for meaning, and this is the corner stone of con- structivist learning.
Constructivism is contagious: Learners who in- dulge in constructivist learning apply it to all forms
of learning. They tend to use it for all subjects. They
tend to experiment, to interact with content. They
look out for alternative ways to arrive at knowl- edge gaining. Most important, they see application
of what they learn to real life. When Priya learned
that metals expand on heating, she had this expe- rience to share with me. She said “When we leave
for school each morning, the two panels of the iron
gate of our housing complex slide open easily. But
when we come home in the afternoon, they are hot
and have expanded. So the iron panels are touching
each other and we have to apply force or sometimes
kick the gate to open it.” I had experienced the same
phenomenon but my adult mind had not made the
connection. Allowing learners to see the application
of what they learn and encouraging them to quote
examples beyond the textbook should be a prime fo- cus in constructivist learning.
Learn from and with your learners: All teachers
need to learn from and with their learners. Learners
could be forming connections based on misconcep- tions, and this will mean learning something errone- ous. My daughter is now thirteen and learning about
tests of congruence of triangles. Recently she told
me that when she was small and had seen figures
of triangles, she thought that segments with one
stroke across them were smaller than those with
two strokes! Thankfully this misconception was
corrected. Thus constructivism has a lot to do with
the ideas that the learner forms about content and
here vigilance on part of the mentor is required. Else
such misconceptions affect further learning. Teach- ers need to be vigilant about how learners learn, how
they think and what they think.
AGNES D’COSTA is an Assistant Professor at Pushpanjali College of Education, Vasai, Maharashtra. She
holds a Ph.D in Education in the topic ‘A Study of the Relationship Between Multiple Intelligences and
Teacher Effectiveness of Secondary School Teachers.’ She was Project Coordinator for the project ‘Open
Educational Resources in Teacher Education’. This was recognized as an innovation in Teacher Education
by NCERT, June 2012. She is an avid contributor to Open Educational Resources. Her work can be accessed
at www. wikieducator. org/User:Agnes. She may be contacted at c.dcosta@rediffmail.com
My experience with my daughter has taught me how learners think. Once a teacher is in sync with how the
learner thinks, the strategies used to stimulate learning can be aligned to the learner’s thinking strategies.
Constructivism fosters a ‘learning to learn’ attitude, an asset in today’s era. As educators let us learn how
students learn, so that learning is enriching and enjoyable. And before I end, thank you, Priya, for being my
teacher!