Fostering Creativity in Mathematics

Creativity is the ability to create something new or think about an original idea using imagination. It helps to find out sustainable solutions to the problem, to discover new ideas, to completely revamp or to design something unique. Our future totally depends upon creative ways to solve the problems that we are not aware of yet!

'Mathematics involves problem solving, logical reasoning, applying rules, putting restrictions and certainly does not include creativity!'

The above statement makes me inquisitive...and leads to the question, can we foster creativity in mathematics? 

I have been doing the activity of sunrise in my sessions (usually for set induction). The main objective is to check the creativity amongst participants and also it acts as an icebreaking activity. It starts with the prompt to everyone as draw sunrise in one minute! I set the timer and go around the class to see what everyone is doing. Infact I also ask everyone to do their own, not to peep into what others are doing. After one minute I ask them to show what they have drawn and look around what their peers have done, and there is a smile on everyone's face:) Why? Because most of them have drawn the sunrise as below

I also show them the similar drawing on my slide, and everyone is surprised as how I knew what they are going to draw? What is the reason, why most of the group do the similar drawing though they are from different regions, went to different schools and also have studied different subjects. Even they don't see such sunrise daily in their lives. So, after investigation we found the answer lies in the Sunrise drawing activity that they learnt during their foundation years, in primary school. This impression of sunrise drawing is so prominent that they do it unknowingly. 

There could be so many wonderful versions of sunrise (check out the image of sunrise between tall trees), but in actual drawing, the same picture is replicated. After this exercise, most of the teachers replicate the same in their classrooms too and found similar experiences!

So, the question arises, Do our school systems train the students to think alike and arrive at same solutions every time? Its alarming time that we all think about creative ways of thinking and how can we foster creativity?  that too in Mathematics!

Let's explore the mathematics where creativity can be medium of expression! Creative thinking involves divergent thinking, thinking about various ways the problem can be solved, and most often it is called as:

Out of box thinking

The example of joining 9 dots with 4 lines without lifting your pen/pencil, the condition is that all the dots should be covered, and lines should not overlap each other.

I have observed that in my sessions the participants try different ways of covering all the 9 dots with four lines. However, it is very challenging for them. Some of their versions are shown in the adjacent figure. They try paper folding and sometimes even tricking me! 

Finally, when prompted to think out of box, one or two of them try to think and practice something different. I then share this way where we need to go out of the box of 9 dots, make a triangle and draw a diagonal line from the base angle. This can be done without lifting the pen/ pencil. There could be other ways also! It is just matter of thinking outside the box to find out other version of doing this challenge! This example opens the participants up to think and now if I give some more example on similar lines, they are able solve it!

Divergent Thinking

The example of dividing the rectangle in 4 equal parts in 10 different ways and in just 1 min. 

Doing this activity with more than 500 teachers have given us lot of insights into their thinking process.

I have observed that most of the teachers can draw in these four ways and then they stop!

They keep of thinking and the time is also short so it is quite challenging for them to think any other way it can be done! 

There is a huge mental block! It is also observed that a very few of them would go to draw in some more ways, but it never goes beyond 6 or at the maximum 8 ways. So as if 10 is the limit!

How can a teacher explore more ways? The answer lies in the pattern how we divide the square! Whenever we start dividing a square in 4 equal parts, we first take half of entire square and then half it further. This process of dividing into halves can give us so many other possibilities. as shown in the fig in the left. We can further think about dividing diagonally and then start rotating, we will find so many possibilities of dividing the square in 4 equal parts!

And if we introduce an element of curve the possibilities will be numerous!

Thus, the most important factor here is to broaden thinking and explore different possible ways the same routine problem can be solved. This can be applicable to different problems or scenarios and creative ways can be discovered! 

Analytical thinking

This is one of my favorite examples which can involve all the participants. This is an excellent activity when you want to introduce how breaking down the problem into smaller chunks is essential.

If everyone in the group of 50 participants shake hands with everyone else, once only. How many handshakes would have taken place?

What I often notice is participants starts responding to oh I know the answer! This is a permutation/ combination example! The formula is 1/2 X n X (n-1)! or some other formula they start discussing. But when I ask them how they arrive at the formula? They are mostly clueless. The most common answer is oh this is what my teacher taught me! or I have solved other examples based on the formula.

The beauty of this example is in physically enacting it! 

But how? Doing it with so many people is tedious and time consuming! So, what should be done!

Yes, that's right! let's follow analytical approach! Breaking down the problem into smaller chunks and then arriving at conclusion.

Let's do this for 5 participants, how many handshakes with 5 people? Demonstrate it by calling them in front of the class. Let them count themselves! It is interesting to note what they are doing?

Some hypotheses can be built up for the numbers they start observing when handshakes are happening physically! Let us start with one person how many handshakes? None, how about two? one and what about 3? Immediate answer looking at the trend is 2. However, when they do it physically it actually turns out to be three! This is just like an eye opener moment. Thus, doing physically truly helps, that too in smaller chunks. But Mathematics demands symbolism, and this has to go towards generalization, hence let's denote the handshakes with number of dots! Let's see what happens!

As you can see here how the pattern is arising as we go on putting the handshakes with dots. This is giving rise to triangle numbers. How fascinating! A handshake problem entering into different concepts of mathematics!

This can be further elaborated with asking the participants to think about a connection in all these numbers. A relationship can be derived with the help of participants!

As you can see why there is 1/2 in the formula! This derivation of the formula has a long-lasting effect and also translational effect. i.e., it can be used in different examples or scenarios.

It is also very interesting to find out different ways of handshaking! I love the following ted talk by Karl and Eric which talks about mathematics and dance! 

One of the main aims of teaching mathematics is mathematizing student's thoughts, and if this needs to be achieved we need to include creative elements in the way we plan and present our content, model good practice and above all encourage our students to be creative!