Learning is also a responsibility

In general, Maths needs lot more participation from students than any other subject. Understanding of a concept, even at a face-level, requires immediate involvement from the student. More involvement from a student will earn him even deeper insight into how maths works.

If students are not willing to think (as is often the case with maths) they are less likely to understand the working of the Maths. Understanding can happen when there is both, teaching efforts and learning efforts. While there is a lot of focus on teaching efforts today, we don't seem to pay much attention to the need of learning efforts.

I have observed that students who are unwilling to do the maths work are most likely to claim that they haven't understood the sum - when the time comes to check the sums. This is almost a sure sign of an excuse. This distracts teachers into thinking that better explanation would lead to the understanding. However for these students 'not understanding' is an excuse to avoid work and a way to put the ball in teacher's court.

This has resulted in commonly seen attitude amongst students (and parents as well) that teacher is whole and sole of education. We all understand that teaching is teacher's responsibility. But do we understand that learning is equally serious and demanding responsibility on the students ? Good understanding of a concept can happen only when both, teachers and students, are making efforts to meet their responsibility.

PS: I was made aware of this parity by a teacher friend of mine. I was making a feedback form for the course I had just taught. I asked my students, "What contributions did you make to make this course successful and useful to you ?" and got back equivalent of a blank-look saying, "us ? do we have to do something as well ?".

End of Innocence

These days we often talk about how smart the kids have become. They know how to manage cell-phones, they are up-to-date with the kid's movies, they know the prices of cars, they even know the cool places to eat in town. When kids talk, grown-ups often have a look on their face which is somewhere between amazement and admiration. Children are certainly smarter today if you compare with kids of same age from a generation back.

What is not talked about or realized is the early loss of certain innocence in these kids. What is not talked about enough is that kids are becoming wiser, as in cleverer, rather than being world-wise.

There is another draw-back to knowing lot more at an earlier age (even more dangerous is to know that you are admired for it). Children tend to think that just because they know things they understand things. The over exposure of information creates a barrier in many kids to learning. The smart-type kids have a mental block in learning something, which they know, from a new perspective. 

There is a certain benefit in letting children be children for some more years. These additional years expose children to diverse experiences. At a later age their ability to see the inter-relations and consequences is better developed. They can infer things with better judgement and sensitivity. Questions and concepts, especially those in the Social Sciences, can be posed, debated and answered with far greater depth.

This is not to say that delaying learning process is better - children should learn at all ages. But we should let children stay innocent till the age when they can compose the bigger picture themselves.

I can't give you no money...

Once you understand something, you tend to look at it in a completely different light. And you wipe out the memory of how it used to be before. I came across a curious case of this while teaching negative numbers.

The negative numbers carry sign of, - , in front. The sign's job is to suggest the value of the number. The value may be compared to zero, such as -30 or it could be a relative value such as -15 km from my house. The negative sign here is like an adjective.

Unfortunately, the exact same sign, -, is also there for the subtraction operation. Here it represents an operation between two numbers. Namely, the operation of taking difference between the values of two numbers. It is like a verb here. If you can't guess which role the, -, sign is playing then that leads to a great confusion.

Soon things get more complicated. The numbers whose difference is to be taken by subtraction could themselves be positive, +, or negative ,- . Many are unable to recognize the dual role played by, -. This may be one reason why so many children find concept of negative numbers baffling.

It is less confusing if its put in words, such as, "Subtract minus four from three". Here the two jobs played by our friend are clearer. Once you learn to recognize it as an operation or a value, then you see negative numbers in a more generic light.

PS: From another angle, subtraction operation is like finding the value of second number with respect to the first number. But that's even deeper to realize.

You can only learn it.

Recently a student of mine, who tried teaching maths to younger brother, narrated the frustrating experience to me - "its so obvious to me, but he doesn't get it. He just doesn't get it".

This is especially true of Maths. Maths is the ultimate compact way of saying things. Obviously a lot of thought is required to unpack all the thought that is hidden in a simple looking maths statement. A teacher can show what happens when different levers are pulled. But to understand how the Maths-machine works a student has to figure it out for himself/herself. Teacher may use various props to nudge a student to think about the inner working of maths. But for the student, think he must.

Maths demands a lot more "willingness to learn" from a student in that sense. The best kind of maths-teaching is when student does most of the thinking and teacher remains invisible.

Some things can not be taught, but can be only be learned.

Form versus Format

In today's schools many things are over-designed. There are formats for all kind of written works - notes, tests, letters, comprehension etc. Children have to follow these formats else they may lose marks.

No doubt this makes it easy for teachers to detect outliers and non-conformists. And indeed it is easy to catch sloppy children by strictly adhering to various formats and rules. However, this has far-reaching, unintended consequence.

Its become harder for a child to develop a sense of neatness in his or her work. There is a difference between a 'format' and a 'form'. Children quickly learn to mimic the format without developing internal sense of form and proportion. An elegant written work has a form - spacing, margins, tabs for paragraphs and justifications.

Sometimes, when kids ask me how they should write some text or solve a maths sum, I tell them to do it in a way that looks neat - no rules. And hope that they will learn to differentiate between neat work and sloppy work.

While it is easy to learn the format, it takes time and an eye to learn the form. Children wouldn't learn elegance of written work if we don't give space and time to explore.

Addicted to Eraser

They hold pencil in one hand and eraser in other hand. This seems a common style of writing these days. No sooner they make a mistake they want to erase and correct it. It may be a factual error, or a compute error, or even a case of bad hand-writing. Erasing your mistakes may seem right thing to do - after all children are conscious of their own mistakes, but it has unwanted consequences.

Firstly, by erasing their mistakes they forget about it and make similar mistakes down the line. Leaving a mistake on the page - there to see - helps in identifying it next time you do it. Secondly, most children are not careful in erasing, as a result their writing only becomes more smudgy. Some times the paper is torn as well. Why can't you simpley draw a clean line across your mistake and leave it there for the future ?

Secondly, this habit seem to undermine their confidence. The line between a 'mistake' and 'not getting it exactly correct' is very thin. For example, ask children to draw freely and they hesitate - unless they can use an eraser now and then. By relying on eraser so much, we seem to have closed the feed-back loop that allows a child to learn from their 'not exactly correct' wanderings.

In our efforts to improve educational quality, somehow we have sent a clear signal of what is correct and acceptable. Children are reacting to it by relying more on erasure. Will they ever learn to get things correct in first place, or draw lines confidently ?

Maths as a story

It has amazed me to see the extent to which children equate maths with numbers and formulae. They have strong conviction that maths is numbers and rules. They hardly think about maths as a story, drama or an event. As a result simple word-problems baffle them.

Well before we get down to actually doing the maths, we must understand what happened in this story. Say, Meena goes to market to get a dozen bananas, a dozen times. We need to play this drama in our mind, only then can we figure-out what to do. Is this a multiplication or a division sum. A fraction or a square-root sum. Unfortunately, well before children are hooked to stories of maths, they are made to learn the maths of the story.

One way out is to start teaching elementary maths only through stories. It doesn't matter if they don't know the generalized rules, as long as they know how to resolve the situation in the story. In Meena's case she could have made 12 trips and buy 12 bananas each time (hence 12 times 12, which is 144 bananas). Or, she could have made 12 trips to buy one banana each (hence 12 times, which is 12 bananas). These possibilities are there. The maths (as symbols and rules) hardly enters here.

It would be easy to teach children generic rules once they understand many maths-stories. However, if they know the many generic-rules, its not easy to figure-out which of these would apply to a specific story-sum. So treat maths more like a story than a formula.