My grandson Varun is in kindergarten class in Ashley Elementary School in Frisco in the US. During his last visit to India, his dad asked me to help his son in doing his Mathematics homework. He was attempting to solve elementary problems relating to subtractions.The first problem was ten minus one. He opened his ten fingers of his two hands, folded one finger and counted the remaining fingers and put the answer ‘nine’ in the blank block meant for posting the answer. The next problem was ten minus two. He solved the problem in the same manner and posted the answer correctly. The subsequent problems were ten minus three, ten minus four etc with gradual and increasing degree of difficulty. When he reached the stage of ten minus seven, I suggested a shortcut to get the answer. I advised him to count from seven onwards up to ten and fill the answer. My grandson looked at me quizzically and told me point blank, “Grandpa, you are trying to mislead me. What you are trying to teach me is additions, but what I am doing is subtractions.”

I learnt a lesson from my grandson. Cutting corners for arriving at an answer is taboo in the initial stages of learning. Following the given logic or procedure is more important. It is always proper to learn in a hard way following due methods of learning. Procedural sanctity is more important than the end result. Parents are often seen helping their children, and teens sometimes help their pre-teen siblings in their homework. This anecdote would perhaps help them in getting right insight in teaching.Meaning of Difficult Words and Phrases To cut corners-to do something in the easiest way -ADVICE: There is no substitute for hard work.

Learning is more important: “If you as parents cut corners, your children will too. If you lie, they will too. If you spend all your money on yourselves and there is no portion of it for charities, colleges, churches, synagogues, and civic causes, your children won’t either. And if parents snicker at racial and gender jokes, another generation will pass on the poison adults still have not had the courage to snuff out”- Marian Wright Edelman (quoted from a website) Conceptual Clarity Is More Important Than Leaning By Rote There is cutthroat competition in getting admission to kindergarten class. Neelima is a bright kid, but she is reticent by nature. In the interview, the interviewer asked her whether she knew about circle. She cogitated for a moment, took out a bangle from her left hand, put it on a piece of paper and picked up a pencil from the table and drew a pencil line along the periphery of the bangle. The interviewer asked no more questions to Neelima. She got the seat.

Learning by rote, without application of mind may be of some help in remembering tables etc. But gaining conceptual clarity is more important at every stage of learning, more so at the elementary level of education. Understanding Zero When I was a student, I could not properly comprehend the following two mathematical axioms. The first axiom is that any numeric value multiplied by zero gets reduced to zero and the second axiom is that any numeric value upon division by zero becomes infinite. After my school studies, I understood the concept of zero which I would like to share with you. Zero must be understood as a numerical with infinitesimally small value. Any numerical on multiplication with a fraction becomes less in magnitude. For the purpose of illustration, I take fractions in decimal form. 100x0.9=90 100x0.5=50 100x0.10=10 100x0.01=1.0 100x0.001=0.1 100x0.0001=0.01 100x0.00001=0.001 100x0.00000=0.0001 As the second numeric value becomes small and small in magnitude, the resultant of the multiplication becomes smaller and smaller. If the decimal numeric value tends to zero, the resultant of the multiplication also correspondingly tends to zero.

Now let us take up the second axiom or proposition. Any numerical value on division by a fraction or any numerical value less than unity becomes greater in magnitude. The following illustration makes the point very clear. 100÷0.50=200 100÷0.10=1000 100÷0.01=10,000 100÷0.001=100,000 100÷0.0001=1,000,000 When the decimal numerical figure tends to be zero, the resultant of the fraction also gets magnified tending to infinite. This is the QED of the second axiom. Practical Problems In Explaining Concepts We are explaining the concepts of division to Varun who has just joined Grade-2 in the Ashley elementary school in Frisco, through the following illustration. “Suppose you have eight pieces of pizza and four persons are there, then how much each would get?” “Two pieces each.” Explanation: (8÷4=2) “Suppose you have eight pieces of pizza and two persons are there, then how much each would get?” Explanation: (8÷2=4) “Each would get four pieces.” “Suppose you have eight pieces of pizza and only one person is there, then how much each would get?” “All the eight pieces.” Explanation: (8÷1=8) “If zero people are there,” Varun questioned?

We have stumbled on the logic we laboured to explain the concept of division. Anything divided by zero becomes infinity. The pizza cannot possibly become infinity. The question of division doesn’t arise in this case. Logic needs to be applied within the limits of rationality. Basic Quality of A Teacher To teach students, teacher must be a student first, keeping abreast with the current knowledge.When I was a student, I learnt that in the solar system there are nine planets. In the neighborhood park of Frisco city, in a model all the nine planets are displayed. To test the knowledge of my grandson, I put the following question to him.

“How many planets are there in the Solar System?” “There are eight in number.” “No. Varun there are nine in all.” “No grandpa, there are only eight planets.” “You see that in the model displayed here, nine planets are displayed.” “The model is incorrect grandpa. Pluto is too small to be counted as a planet.” Instead of arguing with my grandson, after we reached home, I asked Srikant, my son-in-law about the reply given by Varun, Srikant said that his son was absolutely right. NASA declassified Pluto as planet in accordance with a new definition of a planet. I consulted the Wikipedia (free encyclopedia online) which states as under. ‘-On August 24, 2006, the International Astronomical Union (IAU) defined what it means to be a “planet” within the Solar System. The definition excluded Pluto as a planet and added it as a member of the new category “dwarf planet” along with Eris and Ceres—-’ In another occasion, I asked Varun about the continents. I remembered to have read in my schooldays that there are six continents. He  replied that there were 7 or 8 depending upon the way we distinguish the continents. On consultation of Wikipedia, I found that my grandson was correct. Material knowledge we have is not complete. Discoveries of new facets of truth of material world, provides us new insights. He  horoughly exposed my belief that I could teach elementary school students as unfounded.

A few more things I leant from him. The alphabet ‘Z’ is not be pronounced as Zed, but as Jee and certain fundamentals about phonics. When I was a student, I was not aware about phonics.I used to learn spelling of words by heart. Accordingto the current modern methods of teaching, commit to memory mathematical tables is also taboo. Abstract And Concrete Knowledge Learning Mathematics in abstract terms causes difficulty for students. During my time, many students in Class X used to opt for science stream, instead of Math stream due to perceived fear in understanding Algebra. The problem is not with the subject, but the way teachers used to explain the subject to students. For instance, according to Pythagoras theorem, in a right angled triangle, sum of the squares of the sides is always equal to square  of the hypotenuse. To put it in Algebraic form,in a right angled triangle, if ‘a’ and ‘b’ are lengths of sides and if ‘c’ is the length of the hypotenuse, then a²+b²=c². A lot majority of students remember the formula and successfully solve problems relating to Pythagoras theorem without really understanding the concept.

Right angle means an angle making 90º. It is an angle a square or a rectangle makes at corners. Visualization of abstract concepts in graphic way helps in concretization of the knowledge. The theorem could be better understood , if we remember that a square cloth stitched with one side equal to hypotenuse is just sufficient to stitch two square cloths one equal to side ‘a’ and the other equal to side ‘b’. The cloth example imparts graphic clarity of the theorem. Conceptual clarity is more important than problem solving.(The author is presently working as Consultant to Central Power Research Institute at Bangalore)