What are the other differences between learners that a teacher needs to keep in mind, while teaching?  Let us see an example in which a teacher took the pupil's background into account to help him learn.

Sumit, a fifth standard child in a rural school, was being introduced to the formal procedure of addition and subtraction. The teacher tried to gauge how much he knew, before-she taught him the formal method.

Teacher How much is 8 plus 11?

Sumit 19

Teacher How did you do it?

Sumit I rowed. I took 11 and added on 8.

Sumit had used the strategy of "counting on " from the larger number, and could even describe his method in words.

Teacher What about 22 plus 19?

Sumit (writing 22+19): 41?

Teacher: Did you count from 22 by ones?

Sumit : I took the 10 from 19 first, and that's 32, and then I took the 9, and that's 41.

This time Sumit had used "regrouping" to facilitate his addition.

Thereafter, he was given the written problem:

His answer was

How did he get this answer? He added the 8 and 5 in the units column correctly to get 13, put 1 below them and "carried over" the 3. Then he added 3 to 1 in the tens column to get 4. Hence, his answer!

He was quite convinced that his answer was right. The teacher decided to pose the question differently. She said, "If you had eighteen marbles and you got five more, how many would you have altogether?" Sumit counted on his fingers and said 23. Raven the teacher pointed out his written answer to him, he slowly agreed that it was wrong. Isn't it interesting that he was willing to accept his own intuitive method (the informal procedure) as right, rather than the formal written method?

In this example, Sumit demonstrates a well developed skill of using appropriate and efficient strategies to add numbers. However, he finds the formal manipulation of symbols difficult, perhaps due to various reasons. It could be that Sumit has yet to develop an understanding of 'place value'. It could also be that Sumit does not find the given task of addition of numbers with the algorithm meaningful. The moment the teacher posed the problem in a context and ' with reference to concrete objects (counting marbles), Sumit

was able to understand it, and hence solve it The example above clearly demonstrates that Sumit had evolved his own strategies of doing addition intuitively: 'counting on' and 'regrouping'. He was aware of patterns 41 in numbers, and hence was able to regroup to add some - large numbers with ease. The example also shows how the teacher tried to assess Sumit's background, and use this knowledge to make the problem comprehensible to him in two ways:

i) by giving it a relevant context, and

ii) by concretising it for him.