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C-I-Y (Check it yourself) -- Different Learning Environments

 Here, we take a question and create three learning environments. You can check yourself, which environment offers what benefits. We talk about a formal learning environment where student is a part of school/ college/ coaching/ tuition etc. and is helped by the teacher in learning various topics as per the prescribed syllabus. These learnings are immediately helpful in getting marks in the examination but are also highly required in various applications and competitive examinations at later stage. So, learning them right at the time they are being learnt is necessary to avoid various blockages that can restrict the growth later. Let us take 3 scenarios:

Scenario A:  Student asks a question. Teacher answers that. Student is now able to answer that question.

Scenario B: Student asks a question. Teacher also asks few questions to understand why the student had to ask it. He determines the area of difficulty for the student and helps student to answer the question by addressing the problem area. The student is now able to answer the question.

Scenario C: In addition to all that in Scenario B, the teacher also creates few variations in the question and makes the student think about them. He also encourages the student to create variations on his own and discusses about them.

You can think about what benefits these approaches can provide in learning.

Let us take a question and create these scenarios.

This question asks to find the radius of a bubble at the surface of the water while it is given that the same bubble has radius 1 cm when it is at the depth of 10.336 meter in the water. Temperature remaining unchanged.

In Scenario A, teacher uses the theorem that at constant temperature, volume of a gas (air bubble) is inversely proportional to pressure. He takes the pressure at the surface as atmospheric pressure and adds the pressure of water column on it to get the pressure at the bottom. He finds the volume of air bubble at the bottom using the given radius. Since, pressure and volume are inversely proportional, he takes the product of them at the bottom equal to the product of them at the top. This way he finds the volume of air bubble at the top and then uses the formula of volume of sphere to compute the radius of the bubble for this volume. Student gets the answer and can answer this question if asked in the examination.

In Scenario B, the teacher tries to find the reason behind the student asking the question. Student is considering water column (depth of water) as pressure. He gets that as 0 for the bubble on the surface. That disturbs his ratio of pressure at the bottom and the pressure at the surface. He gets volume of bubble at the surface as infinity and is confused. When asked to consider atmospheric pressure as well, he is not able to relate that this pressure too can be in terms of height of water; or units of atmospheric pressure and pressure due to water column can be made same. The teacher guides the student to answer this question by providing support wherever needed.

In Scenario C, the teacher does all that of Scenario B and goes further to make the student think about variations. He puts the student to few more questions involving density of the liquid, temperature etc. so that the student gets some practice to think about those concepts that were holding him from answering the question. 

The additional questions and the summarized answer are as follows:

"As the bubble moves up, the pressure around it changes. The question says that the temperature remains same. Volume of the air inside bubble is inversely proportional to pressure. 

Now, pressure at the bottom of the water = atmospheric pressure at the top + pressure of 10.336 m of water column. Atmospheric pressure in terms of water column is approximately equal to 10.335 m (you can convert 760mm of mercury to water column by multiplying by density of mercury or whatever way you can). So, pressure at the bottom of the water is 2*atmospheric pressure. This is because the height of the water column given in the question is same as 10.335 m.

Pressure at the top is equal to the atmospheric pressure only. So, the pressure at the top is half of the pressure at the bottom. Since Volume of the bubble is inversely proportional to the pressure, volume at the top will be double of the volume at the bottom. 

Assuming that bubble is spherical, the volume is directly proportional to cube of the radius. Hence, the volume of the bubble at the top will be equal to cube root of 2 times the volume of the bubble at the bottom. Cube root of 2 is 1.26. Volume of the bubble at the bottom is 1 cm. Hence, volume of the bubble at the top is 1.26 cm.

But, there is another option given as 1.28 which is very close. Some rounding off in finding cube root, or conversion of unit of pressure may result in getting this answer too. So close choices could have been avoided.

It will help if students also explore following:

1. Radius of the bubble  at the bottom of the water if half of the water is taken out. 

2. Let the question remain same but change the liquid. What should be the density of the liquid so that the second option becomes correct?

3. Temperature at the top is double the temperature at the bottom. What will be the radius of the bubble at the top in the original question?

You can create more variations and see the behavior of the bubble, air inside that etc."

You can check yourself about which learning environment provides what kind of benefits and try to put yourself in such learning environment that suits your goals best.