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Be in Good Company - Life Lessons from My Parents Applied in School Mathematics


Be in Good Company

This article is based on some of the advises given by my father during my school days. The motivation to write this came while seeing a 50 years old photo captured in his camera. Due to privacy reason, I have modified that photo using AI and shared it here. Happy faces in that photo seem to be in good company and enjoying their moments. It can be felt that the kind of positive energy it generates can make challenging tasks look easy.


When it comes to study, being in good company doesn't mean about school, teachers and friends only, but it also includes books, other learning resources and the questions that one practices.

You may not have much choices while selecting school, teachers or friends but selection of books, other learning resources and questions for practice are completely in your control. A good company makes study enjoying, helps in setting and achieving right goals, facilitates deep and long lasting learning and leads to a satisfying life. 
A bad company ruins your study and goals. It may temporarily provide some escape but ultimately spoils the future.

Example of a bad book company is something like depending on all-in-one guide to examination that claims to contain all the answers as per the marking pattern of the examination and comforts you to make you prepare for the examination easily. Market is flooded with such guides and handbooks. Students feel that by studying directly for the examination through such books they are studying smartly and saving themselves from lots of hard work. But it is a disaster as it doesn't allow you to study the subject. It just prepares you on a set of questions. It becomes difficult to answer any other question as concepts are not built and just learning answers of several questions without getting into the concepts makes studying very hard and boring. Students suffer badly in this process and also become very weak in studies. They just struggle to manage the examinations somehow that too with low performance level. It's not study but it's just destroying one's study.

It's always better to study the subjects by understanding the concepts instead of merely preparing for an examination. It's ridiculous to plan separate study of the same subject for different examinations. If the concepts are understood well, that helps in all the examinations related to that subject. Hence, a good company in terms of books are all the text books published by reputed publishers and written by subject experts. Remember that studying any subject is not same as preparing for an examination on that subject. A good book focuses on building the concepts, provides good examples and exercises, makes students apply their learning through exercises and helps in generating interest in the subject. If you use good books and work a bit on your learning strategy, you will enjoy your studies.

Similarly, a good question makes you think, revise and expand your learning. You will enjoy solving such questions. You may not be able to solve some of them in your first attempt but still you will enjoy the exercise through which your brain goes while solving that. Each attempt will make you learn more of the concepts which otherwise may not be possible.  In fact it is possible to have a small set of such questions that can make you revise all the topics in effective way when you solve them. This is much better and enjoying way of revising the course than actually reading the texts again.

Bad questions are mostly trick based, require lengthy complex calculations, make you follow a set of routine steps and doesn't provide any quality exercise to brain. It may just waste your time without any learning. Learning tricks with serious deviation from logic and concepts is not good in long run.

Let us take few examples.

Consider this question;
Solve the following system of equations.
36x + 4y + 4z = 60
22x + 3y + 4z = 30
21x - y - 5z = 2
There are three equations and three unknowns. So, at first glance it seems to be a question that can be solved. This kind of question may be given to you as an example of tough question. You may be suggested that practicing with such tough questions is essential to be ready for tough examinations like IIT entrance examination. Give this question a try. You will soon realize that it offers practically no additional learning and consumes a lot of your valuable study time in arithmetic calculations. This is an example of a bad question. If you have to deal with such questions due to any compulsion, experiment with them. Here you can change one or more coefficients in the equations to get some special situations. You can watch the behavior of this set of equations by changing the coefficients and constants. This kind of experiments can help you in sharpening your concepts on the related topics.

Consider following question:

John walks at a speed of 4 km/hr whereas Mohan walks at a speed of 6 km/hr. Both starts walking on a circular track of length 2 km in the same direction at the same time. When will they meet again? What happens if they walk in opposite directions?

When you work on this question, you apply various concepts. You may apply first principle and think in an intuitive way. Walking with different speeds means the gap between them will increase with time. The track is circular. So, they will be at the same point on the track whenever the gap (difference or sum of distance traveled by them as the case may be) between them becomes equal to any multiple of the length of the walking track. You can visualize that the gap between them increases at the rate of difference between their speeds when they are walking in same direction. This rate will be equal to the sum of their speeds when they walk in opposite directions. At this rate the gap needs to become equal to the length of the track to meet them again. So, when they walk in the same direction, the gap between them increases at a rate of 6-4 = 2 km per hour. Length of the circular track is 2 km. So, they will meet again after one hour. 

Someone knowing LCM may visualize that it is a question of LCM. He will find that John makes one round of the 2 km long circular track in 30 minutes as he walks at a speed of 4 km per hour. Mohan takes 20 minutes to make one round as his speed is 6 km per hour. Lowest Common Multiple of both the times, i.e, 20 minutes and 30 minutes give the time when both have completed their rounds and are at the start point again. LCM of 20 and 30 is 60. That means both meet after 60 minutes or 1 hour of starting their walk. But wait. When you use LCM like this, you assume that they make some number of full rounds. Here, John makes 2 rounds of the track in 60 minutes and is at the start point whereas Mohan makes 3 rounds in this much time and he too is at the start point. So, both meet again at the start point after one hour. Will it be different if they move in opposite directions? Can they meet at some other points too when they are moving in the same direction? Try answering these questions with following data:

John's speed is 5 km/hr whereas Mohan's speed is 15 km/hr. Length of the circular track is 10 km. Find their meeting time in four cases as asked above i.e meeting at start point or any other point when they move in same direction or in opposite directions.

Such questions give you opportunities to apply your learning and even learn new concepts in organic way. Learning becomes meaningful. This is an example of a good question.

Use of technology and social media have added few new dimensions in learning experiences. You can connect and learn with people with similar interest from all across the world. There are discussion groups, blogs, online courses, social media pages etc. for almost all the topics. Collaborative learning platforms with intelligent analytics can make learning effective and fun too. These places are good to be if people are genuine, taking part in original discussions and help each other to stay motivated. But, they can be bad if they are filled with fake people profiles, create sensational discussions, have biased groups and can be demotivating for new members. 

So, be in good company.