UNIT-3
The Link between Primary Mathematical Concepts and higher Mathematical Concepts
Now, when we come to the higher Mathematics concepts that comes from sixth class to 10th class, we encounter a variety of topics.
Let's take one topic and see how the primary mathematical concepts pave the way for a clarity about the higher concepts.
Let's take the first concept of percentages or discounts or compound interest.
For example, in percentage when you say we have a person who got 60 per cent, it is clear that a person didn't get exactly 60 out of 100, he might have got something out of something, let's say you got 360 marks out of 600, so PER CENT or per hundred is a concept clearly connected to fractions.
A person in the primary mathematical stage, having learnt fractions perfectly as illustrated in the last chapter would easily understand the concept of percentages, discounts and interests.
This is because once he gets the concept of fractions connected to real life, the concept of percentage or discount would be just merely an additional application than something entirely new, but notice that in most schools, this concept is not at all clear in the primary section, so when they come to the higher level, it becomes mechanical like you say, 50% means 50/100 directly, we don't know why we are saying, 50% is 50 by 100, we don't know why, but actually, the concept is very simple and this detachment from real life is needless
This concept is seemingly so difficult for most people simply because and merely because in the primary level, they did not have a concept of fractions connected to reality and real life and through games as we saw in the last chapter.
So if you don't do that, this problem will never get solved. The only way to solve the problem in higher mathematical concepts is the same. There's only one way to solve the problem of mathematics on the higher level and that is to make the foundation in the primary level extremely strong.
By strong here we mean connected to real life and real world through activities and games.
So that is the only way.
Now suppose we take geometry, even geometry is mainly about numbers, manipulation of numbers with additional concepts of cube, cone, cylinder, circles etc which are easily learnt.
When you take algebra the concept of the x and y will be very clear if you're clear about numbers as such. Suppose you say 2 + x = 5, a good student will clearly understand x is simply an unknown number, the student will look at x not with a phobia but he will simply look at that number which when added to 2 would give 5 and he'll be confident that it should be three. And thus all rules in algebra would seem simple and floeing from number concrpts only with X simply seen as an unknown number.
So as you can notice, once the primary basic mathematical concepts are crystal clear and connected to real life the higher level concepts actually become quite simple, and also the student has a power to understand the concepts in a powerful way. He becomes actually powerful when he goes on the higher level concepts like In trigonometry, we have ratios that relate to angled in a right angled triangle
Ratios are simply fractions and again a student who is comfortable with fractions would grasp trigonometry too which connects angles with lengths or ratios.
In calculus they talk about rate of change of one variable or quantity with respect to another, how one thing changes with respect to another every moment,
Here too it is like algebra and if we can arithmetise algebra then the concepts of differentiation, limits and integration would really be clear.
Same thing with other topics like statistics, orobsbilty , series, permutations , coordinate geometry etc.
In all these you have simply quantitive relationships between two or more measurable quantities and a clear concept in a certain situation.
As we go even on the highest level, the relationships and patterns become more.
All the people who finally become scientists are those people who have very solid foundational concepts in the primary level, because they can think mathematically, are comfortable in mathematical concepts even the higher level concepts, whivh in their mind are rooted to the basic concepts and are learned conceptually and are not detached from real life.
But what's happening in schools? If you notice in schools, all these higher level concepts are detached from reality, they are again, played as a game, whether it is trigonometry or coordinate geometry or geometry.
We know very well the students get marks, but they're not able to use this mathematics that they're learning because they don't have any idea why they are learning and what they're learning and there's a common experience that so many people ask, Why am I learning this? Like for example they keep on doing this derivations in trigonometry but they don't know why they're doing it. They don't understand that trigonometry is relating one thing with another, for example, in physics you have a concept like sin(i)/sine(r) is equal to refractive index where you are relating using the trigonometry for a certain real life application.
In the same same way you're using the sine wave, in physics where you are showing the relationship again of one variable with another.
Now, these are concepts to show relationships between two things. So once basic concepts of relationship of numbers of numbers and fractions, are absolutely reality oriented, once a student can observe patterns and grasp higher level concepts as connected to reality,
a student feels Mathematics to be simple and also he becomes adept in USING Mathematics in real life work.
Again, we see the power of grasping of all Mathematics concepts conceptually and the uselessness of detaching Mathematics from real life.
Chapter 4
The Relationship of Mathematics and other fields and subjects.
Maths is a very powerful subject and since it deals with measurements, pattern recognition and logical connections directly or indirectly, it is related to almost every subject you can think of.
Relationship of maths to physics.
It's a very common fact that in physics, you will find lots of equations. You'll find for example, trigonometry and calculus used directly because physics is about the relationship of one variable with another.
Lets take the equation of the Universal law of gravitation to see how Mathematics becomes very important in Physics.
The is
F = GM1M2/r2
This is actually a proportionality relationship.
F is directly proportional to M1, F is already proportional to M2 and F is inversely proportional to R square and there is a proportionality constant
This is a concept in sixth standard mathematics called Proportionality You have a direct proportionality or you have a inversely proportionality concept.
If a student doesn't understand these concepts, he will not understand this physics equation. It would not connect to reality. But if a student understands these concepts, then he will be able to SEE the physics concepts and the actual reality there!! He will see that one mass is attracting another mass and if you double the distance, then four times the force reduces, if you triple the distance, nine times of force reduces etc
That reality of the equation will be seen by him,
But largely this is NOT the case and most students see this equation like an abstract equation with obviously no connect to real life. That's the reason why they have problems in physics when they come to physics, F = GM1M2/r2 is a mechanical equation manipulated mechanically
If the maths were known, if that proportionality concepts were known that physics would have become very clear because mathematics, we repeat is a connection to reality of relationships of one variable to another. We're able to connect the whole of physics, to reality only when we connect the equations properly to reality.
Math and physics are deeply inherently interconnected.
In fact many of the maths concepts that came were invented by physicists down history to manage quantitaive relationships.
The relationship of maths with economics, politics, management etc is obvious. There's a lot of statistics, there's a lot of percentages. Obviously, there's a lot of mathematics involved with economics and social sciences, because when you talk about GDP, for example, you must have an exact understanding of the concept.
Nowadays there's lot of lot of propaganda done by manipulating numbers, but if you are good at understanding the concept of a number, concept of percentages, how to be analytical through numbers, you'll be in a better position to analyze data, huge data, whether it's statistical or percentages, or other concepts in economics. Even calculus is nowadays used, because things are changing and wherever there is change, wherever there is data, wherever there is a relationship of one variable with another maths comes in inevitably, as we said earlier
Maths is an inherent natural and intrinsic part of reality.
Again, Maths is obviously very very deeply connected with engineering, whether it's computer science or architecture or civil engineering. You have absolutely deep advanced applied mathematical concepts to handle complex situations.
For example, Google is nothing but a way by which hundreds of variables are manipulated, through very complex mathematics, but if you notice that at every step, it is measurements, it is relationships, it is patterns, it's logical connections quantitatively connected to each other.
So obviously in technology also maths comes.
Such connection of Mathematics to various fields are innumerable.
Also Mathematics being so precise enhances senses and directly ot indirectly comes in arts like painting, music, dance, sculpture, architecture etv
Also a student who masters Mathematics the way we have ecplsinef in this nook becomes very sharp in logical thinking too and very precise .
So, what IS the power of mathematics?
In conclusion, we can say that mathematics is that science by which we have concepts which handles data, which handles relationships, which handles patterns so that we can gain power in any subject by using quantitative measurements, relationships and patterns to HANDLE large data to come to certain creativity with respect to creating large scale things, like a bridge, skyscraper, going to the moon etc etc
So, Mathematics is a literal power. It expands your mind. It enables you to handle large amounts of data.
We close the whole book with one great quotation:
“Knowledge is power'
but we would like to add that knowledge is coded through mathematics. So it is mathematics which makes knowledge a great power.
THANK YOU!!