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Context behind innovative approaches
I walk down memory lane and arrive at 1998. I was working in a K-5 elementary school in a nice town in Southern India. One morning, one of the math teachers in the school gave some homework to his children. That evening, children were seen happily playing with broomsticks and empty matchboxes in their homes. The parents were very disappointed to see their kids playing and not `doing' any mathematics or using their textbook and notebooks. They complained to the school principal (me) about the teacher. The principal sent them back assuring them that nothing had gone wrong and asked them to meet him after a week if they saw any problem.
The principal told the teacher not to worry, assured him that everything would be fine, and asked him not to give up his experiment. After a week, the same parents came back to the school and reported that their kids loved doing math with an unabated passion never seen before! As the school principal, I was greatly relieved! My efforts to train my colleagues to use innovative methods worked!
I have witnessed many such instances since my first day as a teacher 30 years ago. I have come across several colleagues who began using activity based approaches but discontinued them soon, thinking that it was time consuming and not relevant to quantity driven syllabus requirements, a painful reality in much of the third world, particularly in Asia and Africa. One main reason for this is the fact that most textbooks don't encourage teachers to think `out of the box'.
Let me share two math lesson concepts, at the elementary school level, that I taught by using innovative approaches to teach math.
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Teaching Unit Conversions
Being familiar with unit conversions is a skill that will help students from elementary schools to post secondary schools. Here's an approach that works:
Teaching Unit Conversions
If we ask children to convert 100 centimeters into meters, they will usually answer correctly. But when such conversions are required at slightly more complex levels, it becomes important for children to acquire conceptual clarity at the very beginning. In most textbooks in the Third World, unit conversions are usually illustrated in a round about manner. This proves to be difficult for children. Way back in 1988, I tried a simple, out of the box method with my kids, as illustrated below:
Math question: 28 cm³ = __________ m³?
= 28 X cm X cm X cm
= 28 X 1 cm X 1 cm X 1 cm
= 28 X (1/100) m X (1/100) m X (1/100) m
= 28 X (1/1000000) X m X m X m
= (28/1000000) m³
= 28 X 10-6 m³
This step-by-step method not only makes understanding easy but also lays a firm foundation for learning math and science in middle school or even 9-12 physics; for instance, when handling Fundamental and Derived Units, at more complex level, as the following example would illustrate:
Textbook question: An object is measured to have a density of 120 grams/cm3. What would be its density expressed in Kilogram per liter?
Solution: = 120 grams/cm3
= 120 grams / cm X cm X cm
= 120 grams / 1 cm X 1 cm X 1cm
Thus the numerator is 120 grams and the denominator is 1 cm X 1 cm X 1 cm.
The next step is to convert the above quantities into kg and liter (cm3) respectively.
This can be done as follows:
As 1000 grams = 1 kilogram, 120 grams = (120/1000) kilogram = 0.12 kg
As 1000 cm3 = 1 liter, 1 cm3 = (1/1000) Liter
1 cm X 1 cm X 1 cm = 1 cm3 = (1/1000) Liter = 0.0001 L.
120 grams/cm3 = 0.12 kg / 0.0001 L = 1200 kg/L.
That is, the density of the object = 1200 kg/L.
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Teaching Simple Division
Twenty-nine years back, it was dawn in a remote village in eastern Bhutan. As usual, the aromatic cup of Assamese tea and the spectacular sight of a snow-clad landscape outside my window made me ecstatic! But inside me, there was something in contrast: Worry.
I was greatly worried about my previous day's mathematics class in Grade 2. I was trying to teach single digit division to my children but with no success, despite meticulous lesson planning and `proven' teaching methods. I realized that I had failed in my self-administered litmus test, one more time.
More recently, fourteen years back, I happened to work in a primary school catering to children of tribal folk in Orissa, India. After several years of handling crests and troughs of classroom interaction, here I was, undertaking classroom observation of a young teacher. She was teaching exactly the same lesson that I had taught in Bhutan 15 years back as illustrated above. The difficulty that this teacher faced was strikingly similar to the one that I had encountered. Children could not understand the concept. The teacher was teaching the math question: 122/3 = ?
I was glad that I was there to observe the class and witness the difficult circumstance, so that I could help the teacher. As the academic watchdog, I strongly felt responsible for the issue. That evening I got back to my hotel room with a slightly heavy heart and pondered over the issue. I was not able to sleep well. At around 11.30 PM, I eventually got the solution stemming from somewhere in my inner mind. The solution that flashed through my mind is documented below:
122/3 = (90+30+2)/3
= (90/3) + (30/3) + (2/3)
= 30 +10 + (2/3)
= 40 + (2/3)
= 40 -
I explained the steps to the teacher who in turn, tried my strategy in her class. She later told me that her children understood the concept very well.
Although the process used in the solution is longer than the commonly used ones, it has the following advantages:
- The algorithm offers `transparency' as they progress in solving the problem.
- Children apply their knowledge of `factors' to split the dividend.
- It presents the logic behind fractions [for instance as to how we get the fraction `2/3' in the above illustration]; children not only learn `what' but also `how'.
- It serves as an effective springboard to teach LCM (Least Common Multiple).
- It reflects the flexibility that numbers can offer.
I have always found that any lesson concept, however abstract it may be, can be taught in a child friendly manner. I think that how we teach is more important than how much we teach. I am sure you endorse my viewpoint.