This can help student to understand the process of mastering mathematics.
Learning is the most common word that is used in education. Though simple in comprehension, the scope of it is elaborate and comprehensive.
We take it first in the context of Mathematics, where learning means acquiring proficiency in the following areas;
· Conceptual Knowledge
· Procedural Knowledge
· Reasoning Skills
Conceptual Knowledge; conceptual grasp in Arithmetic, Algebra, and Geometry etc. This knowledge entails understanding the causal relationships in the domain and demands building of the correct “schemas” in the brain. The concept of place value in the base-ten arithmetic is one such simple example.
Procedural Knowledge; these are behavioral skills needed to carry out any set of operations. Say in the context of simple 2 digit addition, it would include placing the two numbers one below the other, adding digits and carrying over.
Reasoning Skill; having mastered the conceptual and procedural knowledge the ability of the mind to use that knowledge in new contexts and formats, more in the area of application. A simple example is the reasoning in extension of the group of natural numbers to whole numbers to integers to real numbers right up to complex numbers.
Our understanding of working with students and also with teachers tells that a large majority are very weak in conceptual and reasoning areas. At best most memorize the questions from the books and hope that the same set of questions and in the same format will be asked.
We have asked numerous students of grade 8th in schools across geographies and different socio-economic clusters the following question from grade 5 mathematics syllabus ;
“Give us the smallest and the largest fraction and explain your answer?”
There were 3 types of answers to the above question, namely;
· Small numerators (or denominator) make small fractions and bigger numerators bigger.
· Larger numbers in the numerator (or denominator) make smaller fractions.
· Fractions become larger when the numerator approaches the size of the denominator.
Analyzing the responses to the above gives interesting insights into the understanding of the concept of fractions.
Answer 1…..Size of fraction is incorrectly represented by two independent numbers
Answer 2…..Partial understating that fractions behave differently from single numbers-“in fractions, the smaller is somehow bigger”
Answer 3…..Partial understanding that the representation of fraction has to be seen as a relationship between two numbers.
The following construct has to be explained to the students if they have to fully comprehend the meaning of fractions;
· Understanding Multiple Meanings of Fractions
o Fraction as a part of a whole
o Fraction as a part of the collection
o Fraction as division
· Understanding the Equivalence of Fractions
o Basic concept of Equivalence
o Comparing Fraction Items
o Concept of a whole
· Fraction Number Sense
o Working with the fractional quantities
o Estimating fractional quantities
This puts into a perspective the Conceptual Knowledge Paradigm.
*This entire construct is given by NCERT.
As discussed earlier the Procedural Knowledge essentially means knowing how to do things. Unfortunately most people and even a dominant majority of academicians also believe that procedural knowledge is all about operations only.
It is very important and separate and very important set of learning in the brain. The entire construct of this knowledge is based on the following scientific generalizations;
· Learned best by doing and not by instructions
· Improves with repeated exposure
· Demands practice on a regular basis
· Often automatic and effortless
The critical thing is to understand that in procedural learning, repeated exposure leads to jumps in performance, until the skill becomes so practiced that becomes automatic.
There is a huge amount of neuroscience that helps both the academicians and the students to know how our brains get engaged in learning while being in practice. We use those insights to build strong “neural pathways” to help students do well in practice to help in their overall performance.
Learning a procedural skill is absolutely different from understanding the concept. Whereas conceptual learning is based on the theory of schemas, the procedural skill demands practice.
Take the case of Mental Mathematics. Students who do rigorous practice on operations and multiplications tables become very good on mental mathematics. But most of the students give up this after the primary school and by the time they are in middle and senior school struggle on basic operations of Multiplication and Divisions.
This is a very important skill that we will focus in building the proficiency in the mathematics.
So the important thing to understand is that drill or practice is not a substitute for conceptual learning. Both work in complimentary mode and reinforce each other.
The last and the most difficult is the skill of reasoning. With conceptual and procedural learning in place the next level of generalization is acquired and this is by creating trajectories of reasoning.
At a very simple level the entire Algebra is generalization of Arithmetic. If 20+2 is 22 we are in the domain of Arithmetic. The moment we say that there is a value x which when added to 20 gives 22, we shift into the area of Algebra.
In our content, it is important to focus on developing the reasoning capability of the students with multiple approaches.