How to Think like a Mathematician?

[dropcap]H[/dropcap]ow do great ideas take form? How does one think and develop great concepts? And above all, how do some people are able to think in an abstract manner while others are not? We have a perception that one needs to be a genius, great scientist or a mathematician to develop great ideas. What we fail to realize is that these geniuses are born and nurtured among us just taking a different path in their growing years. They develop certain ability or a strategy that helps them master any concept. I feel any strategy in a right direction is fruitful given it is practiced for a long enough time. I will try to list here few of them that a learner can adopt and practice to be better at Mathematics. Though the context here is Mathematics but a learner can apply this in any field of understanding and knowledge discovery. .


When we start learning anything or try to grasp a concept, it is important to start using the first principle (The first principle approach is to start at the beginning of the concept. You clear your basics; understand it and build subsequent ideas upon it.) As soon as the bas concept is clear the next important thing in learning anything especially Mathematics is to doubt. You should not just accept anything as given rather doubt it to your level of understand by putting forth contrary examples or situations. Test it yourself approach is the best for clearing your own concepts and understanding. It is a two-way approach; one it tests the truth and falsity of the statement given and also helps you grasp the concept thoroughly. .


Thinking about the problem is necessary inorder to make new connections. Try to spend as much time as possible with the problem. As soon as you start to doubt; give your mind some time to absorb the content and ideas. When we think too hard on a problem and then give our mind some space to make connections among discreet ideas it results in a much better understanding of the problem at hand. .


When we reach conclusions, it is important to start writing. Many of us believe that we can do this or achieve that but fail to give it a concrete form. The best is to start writing down all the ideas and try to make a mind map. Once we have written all the ideas and conclusion; the next step is to organize it and try to make logical connections among the ideas that we have written down. You will realize that seemingly discreet ideas suddenly start making wonderful connections and relationship. This might not lead you to ultimate answer but will surely take you forward with the work. .


It is easy to reach conclusions but difficult to verify it and say in absolute terms. So test it out with examples and relate it with other subject areas. It will help you to understand a concept thoroughly not just broadly. You will be yourself surpirsed to discover the constituent of your own thories and conclusions. Such a process will help to keep the undesired results out and further improve our results. If your conclusions seem correct, you’ll move forward and using your own theory, you can build your own examples and problems. .


All the vague ideas and conclusions that have been verified; tested with examples and improved must be written in general ideas annd theories. It is important to build a theory so that a logical sequence to your understanding of the problem can be identified. It will also give other readers a clear path to follow which ultimately they should also doubt to keep up the spirit of mathematics. Once the theory is ready; the hardest and the most critical step is to formulate your theory and prove it.  It requires tremendous amount of effort and some prior mathematical knowledge to formulate a proof. Once you have formulated a proof the work is done and you can repeat the process to DOUBT. .



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