I asked my student today how she had done in her exams. She said she did well and after a moment added that she had perhaps not done so well in Mathematics. My heart sank.
Literacy and Numeracy are considered the two main pillars of education in India. When children start kindergarten (or even playschool) they are first taught skills to build strong foundations in Language and Mathematics. Subjects like Science and Social Studies are introduced in class three when the child is around 8 years old.
Children at a very young age learn anything provided it is presented to them in a concrete fashion. When a child sees a dog and calls it ‘doggie’ an abstract sound d-o-g-g-i-e becomes concrete. A child is able to form a picture of a four legged animal and learning takes place. However, the goal of mathematics education as mentioned in the position paper of teaching Mathematics of NCERT is that the child should be able to develop problem solving skills and also handle abstractions. How does one teach a subject that is meant to be abstract to children who are not developmentally ready to acquire it in the first place? Piaget’s Theory of Cognitive Development suggests that children from 7-11 years old need to use manipulatives to make the abstract concepts concrete. This stage is aptly called ‘concrete operations stage’. Hands on activities like using chips, counters and little objects to add, subtract, divide and so forth will accelerate learning. 7 to 11 years means class two to class six. Have you seen a class six student use objects or images to learn whole numbers or tests of divisibility?
The class six syllabus of Mathematics of the SSC Board (Maharashtra) includes topics such as ‘Use of Letters in Place of Numbers’, ‘Divisibility’, ‘Order of Operations and Use of Brackets’ which imply that the topics are indeed abstract. The aforementioned position paper suggests that there are several problems in successful mathematical instruction in India ranging from lack of quality teachers, inadequate training of teachers, outdated assessment techniques and in general “a sense of fear and failure regarding mathematics among a majority of children”. Are children afraid of learning Math before they fail or after, is left unstated. If kids are naturally inclined to learn, I am assuming the fear of learning Math takes place only after one has failed. Failure in my opinion takes place only when students are taught inappropriately.
If Piaget’s Theory was to be implemented in teaching in India (like UK did in 1966) the fourth stage of cognitive development- the formal operations stage (from age 11 years onwards) would mark the beginning of the child’s ability to understand abstractions. At this stage children are ready to understand when a letter (abstract) stands for an entity (concrete). Piaget’s theory of Cognitive Development has been criticized for being very tightly compartmentalized and that learning does not really progress from one stage to another.
The theory of Bruner similarly states that there are three stages of learning, the first stage- the enactive stage which is teaching and learning through the use of concrete items, the iconic stage where learning takes places through icons or images (hence instead of using five chips to add, a child might draw five small circles) and then moves to the symbolic stage where language and symbols are used (e.g. 5 + 5). According to Bruner, for learning to take place, these stages must occur in sequence not necessarily according to age.
I have struggled as a student in Mathematics (I studied in a convent school of the 90s in Mumbai) I hated it since the time I was in class one. I clearly remember a teacher teaching us subtraction making us colour squares in our square-line math notebooks. Iconic when we were supposed to be taught enactive?
Going by Bruner’s Theory, learning takes place when every child masters every stage. Those students who find it hard to grasp a topic at the symbolic stage need to be taught at the iconic even if it means it is for just one student in the class.
Well, it is hard but that’s why there needs to be a better method and thought put into teaching Mathematics if not at the systemic level, then at least at the class level. If we have to implement both the theories, teachers need to make Math concrete for students even if they are at a different stage according to Piaget. Because according to Bruner, what if students have skipped a stage in the progression?
Hence, dear teacher in secondary school- carry to class a box of marbles. Someone might need it to count. And know that it is ok.
Literacy and Numeracy are considered the two main pillars of education in India. When children start kindergarten (or even playschool) they are first taught skills to build strong foundations in Language and Mathematics. Subjects like Science and Social Studies are introduced in class three when the child is around 8 years old.
Children at a very young age learn anything provided it is presented to them in a concrete fashion. When a child sees a dog and calls it ‘doggie’ an abstract sound d-o-g-g-i-e becomes concrete. A child is able to form a picture of a four legged animal and learning takes place. However, the goal of mathematics education as mentioned in the position paper of teaching Mathematics of NCERT is that the child should be able to develop problem solving skills and also handle abstractions. How does one teach a subject that is meant to be abstract to children who are not developmentally ready to acquire it in the first place? Piaget’s Theory of Cognitive Development suggests that children from 7-11 years old need to use manipulatives to make the abstract concepts concrete. This stage is aptly called ‘concrete operations stage’. Hands on activities like using chips, counters and little objects to add, subtract, divide and so forth will accelerate learning. 7 to 11 years means class two to class six. Have you seen a class six student use objects or images to learn whole numbers or tests of divisibility?
The class six syllabus of Mathematics of the SSC Board (Maharashtra) includes topics such as ‘Use of Letters in Place of Numbers’, ‘Divisibility’, ‘Order of Operations and Use of Brackets’ which imply that the topics are indeed abstract. The aforementioned position paper suggests that there are several problems in successful mathematical instruction in India ranging from lack of quality teachers, inadequate training of teachers, outdated assessment techniques and in general “a sense of fear and failure regarding mathematics among a majority of children”. Are children afraid of learning Math before they fail or after, is left unstated. If kids are naturally inclined to learn, I am assuming the fear of learning Math takes place only after one has failed. Failure in my opinion takes place only when students are taught inappropriately.
If Piaget’s Theory was to be implemented in teaching in India (like UK did in 1966) the fourth stage of cognitive development- the formal operations stage (from age 11 years onwards) would mark the beginning of the child’s ability to understand abstractions. At this stage children are ready to understand when a letter (abstract) stands for an entity (concrete). Piaget’s theory of Cognitive Development has been criticized for being very tightly compartmentalized and that learning does not really progress from one stage to another.
The theory of Bruner similarly states that there are three stages of learning, the first stage- the enactive stage which is teaching and learning through the use of concrete items, the iconic stage where learning takes places through icons or images (hence instead of using five chips to add, a child might draw five small circles) and then moves to the symbolic stage where language and symbols are used (e.g. 5 + 5). According to Bruner, for learning to take place, these stages must occur in sequence not necessarily according to age.
I have struggled as a student in Mathematics (I studied in a convent school of the 90s in Mumbai) I hated it since the time I was in class one. I clearly remember a teacher teaching us subtraction making us colour squares in our square-line math notebooks. Iconic when we were supposed to be taught enactive?
Going by Bruner’s Theory, learning takes place when every child masters every stage. Those students who find it hard to grasp a topic at the symbolic stage need to be taught at the iconic even if it means it is for just one student in the class.
Well, it is hard but that’s why there needs to be a better method and thought put into teaching Mathematics if not at the systemic level, then at least at the class level. If we have to implement both the theories, teachers need to make Math concrete for students even if they are at a different stage according to Piaget. Because according to Bruner, what if students have skipped a stage in the progression?
Hence, dear teacher in secondary school- carry to class a box of marbles. Someone might need it to count. And know that it is ok.
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