Learning -to-learn---for Math

Learning to Learn Srinivasan Nenmeli-K

This article is about the educational approach or teaching methods [pedagogy] which could be taught to students at a very young age---that is ,elementary school or later at Middle school ,perhaps reinforced again when students attain greater maturity in different subjects....

While the methods could vary for

different subjects such as

literature,science or mathematics,some of the methods explored or suggessted here are common to several academic subjects/disciplines. [I am not including fine arts like music,drawing here.]

Mathematics To illustrate the methods of "learning to learn",math and physics would be easy to begin with.Take math,for instance. Four

methods to be learned [to learn faster and better]are:

1 Seeing the pattern 2 Building math /real-world models

3 Learn to formulate a problem before solving it. 4 learning from examples--generalising from particular cases [Of these three, we find attempts are made to teach the first at middle school level,and building math models at high school level in normal curriculum...How effective is this teaching?---this has to be discussed separately, and may depend on the teacher/school.] Perhaps the most important method is: 3. to learn to formulate a problem in mathematical terms, using equations/inequalities/graphs /figures before solving it.

1 Seeing the Pattern

Let us consider a simple means of teaching this: You ask the students: Can you see the pattern in this sequence of numbers: -2, 0, 2, 4, 6, 8,10..... The student should be able to recognise the pattern: there is an addition of +2 with each number to get the next one. To extend this further,students are asked to find a scheme or formula for a number far removed from any given number. Let the given number in the sequence be called 'the first number",say 'a'. In this sequence,take a= 4

What will be the third number from 4...it would be '8'. so from first number ,we get the third number say c = a +4 or c = 4 + 2 (difference between numbers) = 4+ 2x2 What about the fourth number: 'd" d= 4 + 3 (difference) = 4 + 3x2=10 The Fifth number: "e"

:

e=4 + 4x2= 12

Now ,the students are asked to 'see the pattern' in the expression: third

number:

c= 4 + 2x2

fourth number:

d= 4 + 3x2

fifth number:

e= 4 + 4 x2

The common factor in these expressions: the first number '4' and the difference "2" Let us write: nth number = first number + (n-1) difference or

a(n) = a(1) + (n-1) diff ---------(1)

Now we have a formula developed: a(n)=a(1) + (n-1)d Such examples can be illustrated as part of curriculum to learn how to see the pattern. 2 Building a model Here the students learn to build a model with two factors in mind: 1. MModels are simplified descriptions, and

2.models capture the essential feature or factors of the problem. As an illustration, you can teach the simple linear model of direct variation: For instance: the shipping cost depends on the distance from the supplier to the customer. The student learns:

shipping cost C = constant(distance)

For sending by road, the Amaze Co charges $0.15 for each 100 kilometers.The distance is given in kilometers. Then the shipping cost C= 0.15(distance/100) The same company charges $0.65 for each 100 kilometers by air-flight. Shipping cost C= 0.65 (distance/100) The the student is told that the packing cost does not depend on distance [logical!] and is a constant: it is $2 for road transport and $8 for airflight for weight less than 25 lbs. Total charge = packing cost + shipping cost C= 2 + 0.15(distance/100)

Road transport

C= 8 + 0.65 (distance/100)

Air transport

By this process, the student learns to decompose the problem into constituents [here into packing cost and shipping cost] and the shipping cost is directly proportional to the distance. Now you tell the students to plot c versus distance in a graph paper.

The student also learns that the resulting model is a straight line or linear model... This will enable him to learn 'fixed cost '[the intercept] and the rate or slope which is the variable cost. This illustration is a simple one but explains 'how to build models with simple steps' or how the model can

evolve with

greater complexity . For instance, you can state that the shipping cost This approach is different from the normal way of teaching that there is a linear model or equation connecting the cost and the distance. 3 Learning to formulate a problem: to ask the right question in a precise way!

At present, this is attempted by using word problems and some figures and tables. This should be improved by graduated steps to formulate the problem. Example:

To find the average cost of making bread for a bakery:

1. first formulate the cost equation: 2 develop an equation for making N breads in a day: Cost = fixed cost + variable cost for N bread. variable cost= material cost + labour cost + utility [water+power] cost

3 Develop equations for each cost item 4 Put them together:

total cost = fixed cost + N ( material +

labour+utility) Average cost = total cost/N 5 Analyse the results using graphs.

It is better to spend 30% of time in formulating problems in each session. One interesting way is to draw pictures or graphs and let students do them on the white boards. Mind-mapping I have not explored this;you can try this: Using "mind-mapping" to formulate math problems...One can explore various factors/variables involved and pick the essential ones.This approach can be used to develop models too.

4 Learning from examples A great way to learn theory is to examine a simple example .You first study an example,analyze that and then make a general deduction or

find the common principle.

To illustrate, first study this example: John buys x number of

cartons of wool ,each costing $4.

The shipping charge for the cartons is $8. The total bill amount

is $40. How many cartons of wool he bought? The cost of cartons of wool= totalbillamount - shipping cost = 40 - 8 =32 $ GIVEN:

Cost of one carton of wool=4 $ Number of cartons of wool bought= 32/4

Now we can generalize this as follows: Cost of items = total cost -shipping cost Number of units bought = cost of items/unit cost If unit cost = k, total cost cost = C,shipping cost s, let

N be number of units bought. N = (C- s)/k

Many times ,learning from examples is lot easier,because examples are more concrete and easy to understand than abstract formulation... This is an illustration of going from particular case to general formulation.

Another example:RATE PROBLEMS John traveled in his motor-bike at a speed of 80 miles per hour. he traveled for 4 hours.What is the distance travelled? Distance traveled = 80 x 4 = 320 miles. Speed is just the the distance traveled in one hour... So, the speed is also called 'RATE'.

Distance traveled = rate x time We can extend this: A gas pump delivers 5 gallons of gasoline per minute.How long it will take to fill a gas tank of 80 gallons in a truck? Amount pumped = rate of pumping x time 80 = 5 x time Time = 80/5=16 mins We have seen two examples....let us generalise: D = R x t

Summary: By the four methods given here, the students can learn to learn math in various units...The teachers can reinforce these four methods and give some clues for each type of problem..This way, learning math could be great fun and also pretty easy....The students should be encouraged to learn on their own,with limited help from the teachers....What is more,text books can be modified to suit this appraoch --emphasizing "learning to learn".