How wonderful would be the class where kids are talking about models for addition, subtraction, multiplication & division. Mathematical Modeling being a very fundamental necessity, though lately realized, should be practiced and taught right since primary education.
More than actual modeling at such an early stage, it would be highly encouraging to have students thinking on these lines. At a later stage when they actually have to handle complex problems, it would be on their finger tips on how to analyze the inputs, outputs and parameters affecting the inputs and outputs.
In this article, I have focused on how to teach unconventional methods along with conventional learning methods to trigger the learners’ mind towards understanding and exploration of Mathematical Modeling.
Starting with an example, once addition is taught in a standard way, there are ways of thinking addition symbol “+”as a Mathematical Model for adding the two numbers.
Now, lets say “addition” is being introduced to a kid who is learning it for the first time.
Since he/she would be knowing counting (assumed), lets do it this way, say we want to add two numbers 3 and 4. Addition is a box, in which we put these two numbers and it gives another number which is sum the two numbers fed to the box.
The question is what this box does ? ? ? Well that’s where we build what do we have inside the model.
Lets say, it takes the first number, it draws those many number of lines. Then it takes another number, draws those many number of lines, and then add all lines (because this addition model assumes that the person who is building it, knows counting) !!! . And you give the final count, and name it as output. Can you believe introducing a kid to “inputs” and “outputs” when he/she has just started learning addition.
Now, say you want to introduce addition of two digit numbers. For example 23 and 36, same procedure, define our old box, but yes this is an upgraded model now, because if this model is calculating like the “kid’s addition” model then, perhaps its going to take a lot of time.
So, here we expand the horizon, for example:
Standard Addition model: We put the standard convention method that is taught in schools into our this model. We add the digits at units place, (using the model for single digit addition), see if anything is carried, nothing then add the digit on tens place, and we are done. The output is thrown out of the box.
Split Addition model: We split the numbers themselves. 23 = 20+3, and 36 = 30+6, and now we can use our model of single number digit addition.
The idea is at this stage you know how to use a model inside another model. Vow, not bad !! Ok, at this stage, we can introduce the concept of verification of a mathematical model. So, for verification we first need to have the result which is right and then compare our models’ output to it. So, lets consider the conventional method taught by our teachers is the right one and produces accurate results. So, we use our model and compare the results. We do have many complex models for addition, wherein in my methodology, I believe Vedic addition is also a model for addition “+”. So, we expand the horizon of learning at this stage itself.
Similarly, after teaching models of subtraction, we can introduce verification models. For example “Digit Sums” is a good verification for the answer.
So, lets switch over to an example of multiplication model.
Now assuming that the learner knows counting, lets say our “kid” model, where in say 5 X 6 is read 5 times 6, that is the model draws 6 lines, 5 times, and count all the lines and gives the output.
At this stage itself, we can have some very interesting models, one I read it in “Left Hand Brain Mathematics”, I call them drawing models of multiplication. We know that kids are very crazy about drawing boxes, circles and what not. So, why not exploit their talent. So, lets make this circle:
Now, if we start with our pencil on “3”, and start the table of “3”, we start with 3 times 2 = 6, stop at 6, then 3 times 3 = 9, drag to 9, then 3 times 4 = 12, so here since we do not have “12” on our circle so drag to 2 and write 1 infront of “2”, making it look “12”, similarly complete the table.
That’s wonderful actually because of Maths has a beauty and creativity inside it. We need to give a spark of the Mathematical Modeling ideas, and they start growing in child’s mind on its own.
Just to conclude, I would like to describe an experiment I performed in a primary school. I focused on Mathematical modeling ideas for a week to a group of students, and then I had a group of students of the same class from another school. I just asked them one question, and asked for an answer next day. Being an aeronautical engineer, I am lopsided in my questions J. The question was: There is an aeroplane on the Runway, how can you find its weight? J
Next day, I checked everyone’s sheet, now-a-days kids are too strong at google, I was amazed to have a good data of weight of different aircrafts. Except for one, who had this on her answer sheet:
I was first shocked to see it, what did it really mean? On being asked to explain to everyone, the answer was “This is an aeroplane on the runway, and when it will pass through my box, it will find out its weight”. It was in fact fascinating as out of a group of 5 students, I was successful in imbibing a spark of mathematical modeling.
I strongly believe, that its just a thinking strategy needed right from the beginning to have a grasp of the mathematical modeling.