Monday, June 27, 2011

From Real World Math Applications to Pure Math and Back

Can you go from pure math to real world math applications? The answer is no. You can not start with math only and then “apply” to real world scenario. It is, however, possible to use math in real world (even in fictional world, like in Harry Potter movies), practical applications, but the path and direction are different and needs to be clarified.

The reason you can not go from pure math to real world math application is in the fact and in the nature, definition (in a sense) of a number. A number is obtained as an abstraction, a common property for many objects. By this very definition, because it is abstracted from counted objects, because it is, now, a separate concept, representing a pure count, without any object associated to it, you can not tell, by looking at the number only, where it came from, what and if anything has been counted to obtain that number. In other words, a number does not carry any information about any object extraneous to mathematics! Hence, you can not say, just looking at the number only, or at the sequence of math operations on numbers, what its or their application, in real world, may be. Newton did not learn calculus first, then applied it to the gravitational problems! Quite the opposite happened. Newton was dealing with non mathematical objects and relationships like apple falling from the tree, Moon orbiting Earth, and other body motions. Unless they are quantified, these are not mathematical objects nor relationships. If they were, then you would see theorems in math books proved by apples, Moon, speed, etc. but it is not so. Math theorems are stipulated and proved using only mathematical objects, like numbers, sets, set of numbers, or other mathematical theorems and axioms. So, let make that clear, Newton first dealt with physical objects and only then he invented calculus. So, when someone tells you you will learn math then apply it, it is not quite true.

When you deal with a number, you deal with an abstracted common property, a separated concept abstracted from all the objects whose count it represents. The very moment you start adding 2 apples and 3 apples you are doing two distinct steps. First one is recognizing that the object to be counted is an apple. That recognition process, a categorization, that you are looking or holding an apple is outside mathematics, since you can be counting apples, pears, cars, books, cups. This recognition is a focus of research of cognitive science, psychology, biology, color research, even socal sciences. The second step (which is, actually, common to counting all those objects) is dealing with numbers 2 and 3. Even if you may not notice, when dealing with 2 and 3, you are dealing with counts that can represent not only apples, but millions of other objects that can be counted, or put in sets, then counted, to obtain 2 and 3. Hence, the result you obtained for 2 and 3 apples, i.e. 2 + 3 = 5, can be used in ANY other situation where you have 2 objects and 3 objects and you want to add their counts. You right away invented and used "pure" mathematics when counted these apples. It is this universality, the common numeric property of counted objects, that gives mathematics ability to be a separate discipline, to deal only with numbers. While to us, and to the field that uses quantification, is very important what, when, why and where something is counted or measured, to investigate only the counts' properties is the task of pure math. Pure math does not care where the numbers or counts are coming from. It is very similar when we create a set of any objects (of interest), but we are only interested how many elements are in the set and not which objects are part of the set (that information, which objects are elements of the set, we keep track of on a separate sheet of paper) ! Math knows and should know only about numbers and sets of numbers. Notice how math may be "motivated" by counting apples, but, the result obtained, i.e. 2 + 3 = 5, can be used when counting any other objects!

Also, math can't tell real world from fictional one! Look! If Harry Potter flies 10 m/s how many meters he will advance after flying 5 seconds?

Mathematics deals with numbers and with numbers only. It does not care where the numbers are coming from (but, in the field of applied math, we do care where the numbers are coming from). Now, you may ask, how we can apply mathematics at all, if the trace what is counted is lost in this abstraction, in this definition of pure number? Well, here is how.

It is true that pure mathematics deals with numbers only and, of course with mathematical operations on them. We have abstracted, separated a concept of number from all possible real world objects that might have been counted. Thus, when you say 5 + 3, you right away know that the answer will be 8. No real world objects are mentioned nor even thought of when we did this addition. We just selected two numbers, and decided to do addition (we could also decide to do subtraction or multiplication). Now, how then we can apply math to real world if we don’t have a trace of what is counted? There is a way! When we want to “apply” or rather, use, math in real world, we will drag the names of objects counted into the math! We will keep track of numbers obtained, to know where they come from, which objects’ counts they represent. How we do that? We will add  a small letter, or abbreviation, or a word, name, just beside the count to tell us what we have counted. For instance, we can write numbers, 3, 5, 7, 10 after we counted (or measured) something. In order to keep track what we have counted we will add small letters right beside the numbers, like 3m, 5m, 7seconds, 10apples. Now, very important thing. These added letters do not represent math. They are for us to keep track what is counted. Unfortunately, frequently, this is all mixed up and students are often told they are doing math even when they describe what are they counted, why (to give 10 apples), where (apples from the basket, he went 3m downhill, then 5 meters uphill,), when (7seconds ago, not after). All this reasoning, descriptions, units, abbreviations, meters, seconds, apples, ago, before, after, downhill, uphill,  DO NOT BELONG TO MATH. Why is that? Because, if you look in any pure mathematical textbook or book, you will clearly see that no theorems are stated or proved by mentioning apples, meters, seconds, pears, etc. All the theorems are proved strictly in terms of mathematical objects, numbers, sets, set of numbers, using other theorems and axioms. No outside objects or descriptions, like apples, cars, downhill, uphill, will ever enter a mathematical theorem or its proof.

Now, when we distinguished what is pure math and what is applied part of it, we can make more interesting and significant conclusions. Pure mathematics deals with numbers and numbers only. Since number 3, say, can represent an abstracted count of so many, many objects, wouldn’t be interesting to have its properties investigated? It looks like there is some value in the fact that one concept, a number, stands for counts for so many objects. We can compare number 3 with other numbers. We can say which numbers are greater than or less than other numbers. We can multiply them and see what numbers we are getting. And all the time we deal only with numbers. The value is, if we get some interesting result for a certain number or numbers, from our “pure” number investigation, we can use that result for all those examples in real world. That’s the value of applied math. But, in order to use it, say, in order to use 5 + 3 = 8, we have to make a match between pure math numbers and real world counts. Hence, IF we count say, CDs and IF we get 5CDs, and, again we count another set of CDs just mentioned. This is a simple example, but there are more complex mathematical results where we don’t need to reinvent the wheel each time we get the real world count, but instead we take advantage of ready to use mathematical result, procedure, theorem, solution.

Mathematics does not see the reasoning from other worlds. Math will see number 6 (given, or picked), math will see number 10 (given, or picked), and math will see the selected, required operation, addition (it could be subtraction or division too). Does it mean that these numbers, 10 and 6 came from thin air to mathematics? They came from counting apples, but, where are the apples then? The point is, remember, when we said that numbers, in pure math, are abstraction for all the objects they can represent count of. They are not from thin air, they exist in our math as our starting point. Our pure math has already numbers available for our use, 1, 2, 3,…10, …, etc.. How? We did not even need to find objects to count to obtain different numbers. We can start with 1, then add 1 to get 2, then add 1 to get 3, etc. That’s why we have numbers already available for us. We only pick them, and do the operations. Math, for that matter, doesn’t need to know if we counted apples, or pears, of chewing gums. It is enough for math to tell there is number 10 and number 6 and that we want to add them. It is us who will keep track why we counted (because Peter was hungry),  what we counted (apples, they are edible), when we counted (in the evening, when was the time for dinner).

How we are allowed, at all, to go from non-axiomatic worlds, physics, economics, finance, to, so strictly defined, axiomatic world of mathematics? Apparently, mathematics does not care whether the problems come from axiomatized system or not! And that tells us that mathematics can not correct logical steps or see the flaws in the system it proudly claims it models. Assumptions coming from non-axiomatized fields, like physics, economics, finance, and into a strict axiomatic system, like mathematics, can produce results that can wreak havoc back in the field where the mathematics is applied.

You may ask, at this point, how we can mix these two logic worlds. One world appear to be very fluid, the real world, with objects selected, of any type, and any kind of relationships. On the other hand we have mathematical logic world, where we only deal with numbers, or sets of numbers only, and with what appears to be quite precise rules, axioms, logic, and well defined sequences of math operations (of addition, subtraction, multiplication, division). Is there a logic that will merge and connect these two worlds? YES! You can mix these two worlds, but you have to be very careful with the World # 1, the real world's objects and scenarios. Your logic there has to be correct. Then, you can use logic used to link these two worlds, and it's  is the logic already familiar to you, but with new domains of application. How it is done? Here is how. Let’s say that logical statement from the first, real world, is labelled as “p” ,  and logical statement from the second, mathematical logic world is labelled as “q”, then we can create a new, logical statement IF “p” AND “q” THEN “s”, or in shorter notation, p ^ q => s. I am sure you are familiar with this logical statement. Here is example. Let p = “Peter is hungry” and let q = “there are 10 apples”. Then we can form a statement “IF Peter is hungry and IF there are 10 apples, THEN he will get 6 apples”.

Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define matheatics axioms and to define proofs of mathematical theorems. 

[ applied math, applied mathematics, learn math, math applications, math concepts ]

Friday, June 17, 2011

How to Teach Your Kids and Yourself to Think More Freely About Math and Real World Math Applications

When I was a kid, I remember, I was often asked, during a math chat, “How many apples are there?” or “If you subtract 3 apples from 10 apples how many apples are left?”. Or, "Peter has 5 apples. How many apples he will have if he gives Susan 3 apples"? Then, there were many more examples of counting objects.

I am sure that many of us have had this kind of first encounters with mathematics.

When I was 4th grade, an interesting idea came to my mind. Why do we need to know concept of an apple in order to subtract 3 from 10? How and when we become aware that an object is an apple, and not, say pear? And what if Peter doesn't own an apple? Isn't the ownership a legal concept? Does Peter really own an apple? Then, if Susan is really hungry and Stephen is not, then giving Susan 3 apples is way more important than actually counting apples. Also, turning left and drive 2km is not the same as turning right and drive 2km. Yet, left and right were not part of math. They are not quantities, yet they are so important. Those were my thoughts when I was in 4th grade..

After a while, and after many examples involving apples, cherries, cars, marbles I have been shown, it became clear to me that 3 and 10 are separate, abstracted concepts of numbers, and hence they can be dealt with independently of any apples, oranges, pears, and other objects used as counting example. Teachers are constantly talking about universal nature of math and even about pure math. Yet, somehow, teachers were always sticking to real world examples when explaining math concepts. At the same time they were talking where the math can be applied, but miss to inform us what is it we are actually applying, i.e. pure math. Fortunately, I have realized that math can be independent discipline and that the math tasks and problems can be tied to the numbers only, without any real world examples. But, that was a trouble, when I looked how the math is taught. I  did not talk to teachers about that, they were very strict, and I was afraid to raise any issue. Yet, I was the best student in primary school, in my class, and one of the best in school, in math. And the issue was why would you expect from kids to know non mathematical concepts in order to add two numbers? Why kids should know social structures and social relationships, legal concepts and then learn math from those examples? Math can be taught without them. The best approach is to say that math can be motivated by real world affairs and also it can be developed from math axioms.

Then, there was another puzzle. The opposite (of knowing non mathematical concepts) was also true. As a kid, I realized that I didn't know how the automobile engine works, yet I can count engines. Then, in 7th grade when physics kicked in, we have been counting objects that we only vaguely knew what they are, like, electrons, atoms, waves, particles. To me, if math says that definition what actually belongs to a set, in order to be counted, must be clear and non ambiguous, it should be followed. Yet, we do not know what is electron, but it was apparently sufficient to stick to "it looks like it's a particle, but, it can be a wave too, and it may appear and disappear time after time". So, apparently we can define set "of objects we don't know exactly what they are" and have that count too.

Throughout “schooling” teachers (not all), most likely, failed to educate us what math really is. What is left from so called math education, for many of us, is a bitter, frustrating, scary feeling, tight test schedules, unresolved problems, and many confusion about math concepts. Moreover, teachers were running through material without giving any deeper insights.

And that view I questioned through primary school. Later, it became clear that math can be developed as a discipline without waiting for real world examples. However, it is true that certain directions in math were motivated by physics problems, like calculus, but still, calculus can be developed from scratch without physics input. You can not expect from kids to fully know concept of force, energy, electrical field, in order to introduce calculus. Calculus can be introduced way earlier, right after irrational numbers.

Well, I did have my personal “fight” with math, and I think I won! And this happened relatively early, around 4th grade, when I realized that you can work with numbers independently from any real world examples, but I did question how we define sets, and can we really always define clearly what belongs to a set. Later, that proved to be related to Russell's paradox. That was, at least for me, key point when I started to like math, to appreciate its elegance, its independence as a discipline. After primary school I enrolled in Mathematical Gymnasium, then later, finished my Bachelor's Degree in Electrical Engineering (later redefined as Masters Degree, Bologna, around 2000), studied Astronomy along the way, and have exciting career in these fields.

Here, I would try to keep examples simple, intuitive, and I would always emphasize that particular mathematical concept can be defined and derived without real world examples, but can be motivated by them.

In this article I want to explain a method how to teach kids, or primary school students, or anyone who is interested to understand math better, to think more freely about math, and to see that math is a game, puzzle game sometimes, whose variations can be created by you as well. You do not need to wait for someone else to give you mathematical problems and tasks. I hope that, after you read this post, you will have an unstoppable desire to revisit other parts of math, and to go on your own journey which will also include YOU creating new mathematical problems to solve.

Let’s start. Counting objects is the core of mathematics. Let’s say we have two kids, Peter and Stephen, who wants to learn math. Usually the learning starts with showing some objects, say apples, and asking kids to count them. Then, it comes addition of apples, subtraction, and perhaps later multiplication and division. Now, after, say one hour or so, or after even a few days, Peter and Stephen will always expect to see objects then to count them. That’s where the word “given” in math comes from, or even word “suppose”. Peter or Stephen will always expect to have a problem in front of him, to be “given” that he needs to solve. Needless to say, kids may be, at this point, scared to asked where these problems come from? Who “gives” them? Here is the answer.

It would be nice to show to kids that 2 + 3 = 5 is independent of any objects counted. This can be done by going through a number of examples with apples, oranges, cars, CDs, pencils, even time, like hours and seconds, and then show separately that 2 + 3 equals 5 no matter what are you counting. You can reuse this numerical answer for any objects you may be counting in future. At this point kids should be aware of numbers as separate concepts. They have to be aware that numbers can exist without any real world examples. They have to be aware that they can do two things with numbers. Deal with them as pure numbers, like 2 + 3 = 5, and, obtain these numbers by counting objects. It is very important to explain this fact to kids.

And, now, here is the main step!

Get a bunch of objects together, say 30 apples, in a basket. Ask Stephen to go outside of room. Then, tell Peter to specify how many apples he wants to be taken of the basket and put on table. Suggest to Peter that the number he specified is completely arbitrary, it’s up to him to tell the number. Or, he may have his own logic to do that. Let’s say Peter chose number 5, for whatever reason. Then 5 apples will be taken from the basket and put on table. Ask Peter again to chose another number. Peter says 3 and he takes 3 apples from the basket to table, in another group.

Now, explain to Peter how his arbitrary, starting selection, a starting number of apples dictated how many pencils will be counted. It also dictated the two numbers Stephen will be dealing with. Also, explain to Peter that it is him who specified how many apples were to be counted. There was no pressure from anyone else, no “given” number of apples.  It was Peter who “gave” the numbers.

So, in mathematics, not only that you can count some objects and tell how many of those objects are there, but also you can use a pure number to start with, and tell how many of things is to be counted. This count, specified by you, can be a “given” starting point for someone else’s calculation! Please feel free to select any “starting” number again and play with mathematical operations, addition, subtraction etc, using other numbers by your choice. The numerical results you get is important knowledge, and can be reused in real world applications.

Let’s say, after a while, you have a few pages with numbers chosen by you, and a number of results obtained by multiplication, addition, etc. You may want to sort them, as you wish, maybe by magnitude, smaller numbers first, bigger numbers later. You may ask yourself, how this can be applied in real life? But you can not go from math back to real objects because, as we have seen, when you write 2 + 3 you do not know what is counted. Can be anything. However, there is a way for real world application! If you start with “IF I counted CARS, and get 2, and then IF 3 more cars came by” what is the result? Hence, once you start keeping track WHAT you have counted, and WHY, you can use your mathematical results to match the numbers. Note here that math does not see where the numbers 3 and two came from or why. You can count cars passing by, you can count cars on parking lot, or these can be numbers you tell car dealer to move cars in the garage. Math can’t see that. Math only see you provided to it number 2 and 3 to add. It is you that you have to carefully keep track what did you count and why and where the result will go.

Back to Peter and Stephen. Now, it is time to ask Stephen to specify how many apples will be taken out of the basket while Peter is out of the room. Now Peter will have “given” numbers to add, and Stephen will become aware that it is him who can specify these initial conditions for counting.

These exercises should be continued with measuring and specifying length. First Peter will be the one who will specify the distance between two objects in the room, by moving those objects apart. Then, Stephen will come in the room and measure the distance. Second exercise will be for Stephen and Peter, to specify the distance and then measure and move two objects to match that distance. It is of fundamental importance here to make Peter and Stephen aware that it’s them who specify the initial numbers, counts, and then the change is done in real world by their own specification! During these exercises kids will start thinking more freely about mathematics, and starting creating their own mathematical problems, instead of only dealing with what is “given” to them.

In next post I will write how we separate and connect two logical worlds. First one is how we create rules and keep track what and why is counted and the second world is the world of mathematics that will deal with these numbers. With the second world we are already familiar at this post.

Real World Math Applications

Peter may want to ask “Ok, I have done all this math calculations, and what I get is a number as the result. Where is the real world application?”

Here is the answer. Let’s say Peter obtained number 5 as counting pencils. What now? What possible application can be created using number 5 and pencils?

Here is how the real world examples can be created. Make this sentence:

“If I have 5 pencils AND IF 'something else A' THEN 'something else B .”

Note first “AND IF”! Following that it can be a true statement about anything you can imagine that can depend on the number of pencils counted. You have a true statement that you counted and obtained 5 pencils. Then, you may say, IF I have 5 pencils AND IF it’s sunny outside THEN I will decide to go out and draw landscapes. Note how the number of pencils influenced your decision to go outside and draw landscapes. Please note how open ended logical connective “AND IF can be related to any specified fact YOU have in mind! The very true and accurate calculation of 5 pencils can lead to an action that you can attach to this number! It is really up to you or up to your analysis what will depend on number five. It can be your creative new statement, new initiative or new decision that will start once you have 5 (and not 3 or 6) pencils in. That connective on the other side of "AND IF" can be another count, obtained from counting other objects too.

Now, teachers will rarely ask you to do this! Teachers will just stop after you calculated 5 (pencils or other objects) and leave you hanging there asking “what for?” and “what now?” I want to change that. I want to show you how the math calculations are actually applied, used, and utilized in the real world.

[ applied math, applied mathematics, learn math, math, math and real life, math concepts, math education, math tutoring, mathematics ]

Tuesday, June 14, 2011

The link between pure and applied math, physics formula and their moral, ethical, emotional processing in our brains

Newton Second Law, F = ma. 

In addition to defining force, and showing its relationship with mass and acceleration, the formula is interesting because of few other aspects. It is a given way of thinking, a specific direction of thinking we can use in solving some important questions in real world. This value of directional thinking has its merit even if you can’t tell where the formula comes from. Given formula like this is comparable to an invention, it’s a postulate, an affirmation, a statement, a premise, given a priori, for your use.

So, let’s see what is the additional significance of this formula. 

Since you can count almost anything, and conversely, can specify count of anything, finding relationships between counts can be very interesting. For instance you can initially specify that mass count will be 5 kg, i.e. m = 5kg. You specify that it is the mass we want to count or measure (measure is counting how many units of certain amount is in target quantity) and then you specify actual number, in this case 5. Note how you specify the starting number. 

Then, acceleration. Pick any value, set any value, say 3 m/s^2. 

Now it’s the time to find out how much of force there will be. But, look, the count of force, the amount of force is determined by the counts relationship between the counts of mass and counts of acceleration. Of course, if this relationship is not specified we could give force either arbitrary value or the value implied by specific requirement or application. In both cases we will give the starting number for force. However, since the quantity of force is bound by the relationship between mass and acceleration, we will use these two to obtain the amount of force. The magnitude of force is linked to the counts obtained from mass and acceleration. This is very elegant point! We are allowed to count objects. We are allowed to specify initial, starting quantitative value for objects. We manipulate counts obtained from two objects, and then use that count to obtain amount of the third object. We are allowed to specify a starting number for mass and acceleration apparently from thin air. But, also, if the physical law, which I would call law of quantitative relationships of quantifiable concepts, objects, or processes, is specified, then, instead of us to give the count for force, the multiplication of counts for mass and acceleration will dictate and give us that number! In this case it will be 15 N. Note how, before we specified mass and acceleration values, we did not know at all what the value for force will be. Also, note how counts of mass and acceleration dictate how much force we will count, then possibly apply. Of course, the genius of Newton is to know which quantities to measure and put in quantitative relationship. What to measure and what quantitative relationship to establish between physical objects is the major task of Physics.

At this point role in analysis of mathematics and physics ends. They fulfilled their roles. Even more precisely, mathematics role ends when the multiplication is done. Math did not care whether you multiply 5 apples in 3 baskets, or, as in this case, 5 kg with 3 m/s^2. Math did its job by multiplying the numbers 5 and 3 and giving back the result to you. It is you who kept track of what is multiplied and to what the multiplication result refers to. And then at that point the role of physics ends, in this application. How? Once you obtained a count, amount of a certain quantity, a number, in this case force, the role of physics ends. Physics of course had a part in deciding what to count. The very selection of mass and acceleration, and decision that you are going to quantify, i.e. count them, defines physics. So, physics can be defined in specifying what to count and trying to establish numerical inter relationships between such obtained counts. Logical methods are, of course, used throughout physics and math, but initial assumptions are left to geniuses to discover and postulate. 

So, again, once you obtained the number, an amount, a count of certain property, in this case force, the role of Physics ends in this analysis. Everything else, what are you going to do with this force of 15 N, why, when, is matter of other sciences. Is it a force that has some value for human experience? Is it ethical to apply this force? How about moral value? Is it a force that will be apply to start a boat motor that can save lives later? Is it a tangential force on the bomber's engine pylon? All these questions are started after the physics did its job, in selecting force as a property of interest and specifying an amount for it. The interpretation of what that force is and can be is a matter of human experience, how our brain process and interpret the effect of the force applied, how biochemical triggers and neurological reactions process it, which neural path configurations in our brain, fuelled by oxygen and ATP are activated. What the force of 15N did can be subject of social analysis, cognition, psychology, maybe war tactics analysis, it may trigger our emotional, moral response, it can have legal consequences, economic consequences. And this is a boundary and a limitation of physics and mathematics. It will give a quantitative relationship inside certain system, in this case physical, but it stops there, because it can not define and measure our human experience, our human valuation of it. Is there a way to connect physics and our human experience? There is. The way is mapping, associating the physical actions with what do they mean to us from moral, ethical, emotional point of view (for instance, is physics used to develop nuclear and conventional weapons, are we going to build wind farms on the shore, are we going to construct a dam and possibly change ecosystems with the artificial lake). But, this map is not created by physics laws. It’s created by us.

[ applied math, applied mathematics, formula, math and physics, math tutoring, mathematics, physics, physics  formula, physics education, applied physics ]

Monday, June 6, 2011

Motivation to Use Graphs in Math, Physics and to Know Arbitrary Surface Area Calculations

Now, you may ask, are there any other reasons to learn calculus and integration, other than calculating the surface area of arbitrary surfaces of physical objects (see my previous post below). Yes, there are. The point is that mathematics uses graph as a tool a lot. Graphical representation of functions is the most common use. Also, mathematicians, physicists, economists like to represent a number, a count, as a length of a line in a coordinate system, where length matches the count in question. Then, since we use line to represent counts or numbers obtained, the product of two numbers will automatically be shown as surface, bounded by these lines. Then, volumes can come into picture too. It is just pure convenience, call it visual convenience, if you wish, that we have found lines and surfaces to represent our quantities. We are still interested in numbers which these graphical elements represent.

As you know you can count many, many things. Moreover, what you have also seen, you can play with numbers without knowing where they come from. For you know that, for instance, 2 + 3 is equal 5 no matter whether you counted apples, CDs, meters, kilograms, pears, cars, pencils, rockets, measure length etc. You can work with numbers, 2, 3, and 5 now independently of any objects from real world. And it is helpful to know in advance that 2 + 3 = 5, so you can use in any possible counting situation.

Now, let’s say you want to count how many cars are driving by each minute on the street. These can be the result of your measurements: 1st min 15 cars, 2nd minute 23 cars, 3 rd minute 10 cars, 4th minute 12 cars, 5th minute 22 cars. Of course you can draw a table with the columns showing minutes and associated count of cars.

Time (mins)

That’s an excellent way to do it. Now, let’s say you want to represent count of cars not only by numbers written on paper, or by table,  but with marbles. This means that for 10 cars you will have 10 marbles. For 22 cars you will have 22 marbles. Maybe you did not have a paper at hand, so you needed to use marbles. Then you can use the marbles to count minutes. One marble for one minute, two marbles for two minutes etc. You realize that you can use marbles to represent the count of anything, apples, pears, oranges, rockets, cars, cars, etc. But, marbles doesn’t seem to be practical too much. You have to carry a lot of them, and, although they carry exact information about the counts of things they represent, they are too heavy to carry around. What else can be used to represent conveniently numbers of anything? It was discovered that length can be used to represent counts of other things. You draw the line of certain length and say that it represents a count of some objects. In our example we will draw a line of 10 mm to represent 10 cars, then line of 23 mm to represent 23 cars. The similar things can be done for time measurements. We will draw a line of say 1 cm for one minute then 2 cm for 2 minutes etc. We are using length of line to show the numbers, counts we have obtained. Please note, selecting line as a tool is our choice, because of convenience since its length represents different counts, numbers, quantities. Why it is convenient? Look at the graph we can make with lines. You can immediately see when a quantity is bigger than the other, you can see the trend between quantities and their other relationship, like minimum, maximum, etc. Of course all this things can be seen from a table too, but you have to go row by row, and do a lot of comparison. Table is one of the ways to represent pairs of numbers as we did here. Moreover, we can use either table or graph! But note how graph is more convenient, because it visually shows relationship that can be harder to spot in the table. Let’s repeat again, we used length of a line  (as we first did with marbles, or as you used fingers to count objects) as a universal way to represent counts of all other objects we are interested in any particular measurements or calculation. You also can line up numbers along the line, matching the length of line with a corresponding number. Note that you use line, length of line for pure convenience! There is no law that tells you need to use line, length of line to represent numbers and counts. It just happen to be a convenient visual way to use line to represent numbers.
There is nothing wrong using marbles or fingers, it's just seem to be impractical to communicate some ideas, however, these two methods of representing numbers are completely correct too. For instance while you think you can represent only integers with you fingers, you can actually represent any number! How? Say, you want to show number 3/5 with your fingers. You will show first 5 fingers and say this is how many times I divide number one. Then you show 3 fingers and say this is how many times I will multiply the quantity I've just obtained by dividing number one into 5 parts. The result of these two simple operations gives exactly 3/5 and it's communicated with fingers only!

Graph of a function serves the purpose to represent and show all pairs of numbers. Line on a graph should be seen as a continuous set of PAIRED NUMBERS.

Here are some examples of lines in action! Of course these are called graphs in mathematics, physics, engineering, economics, or in any other discipline where we have to show some quantitative relationship.

Fig 1. Examples of graphs

We are, perhaps, used to look at the graphs and focus on their shape, or on, even, their aesthetics values or characteristics. But, aesthetic value is of far less importance, as graphs are concerned. No matter how attractive graph looks, the line of it represents the PAIRED NUMBERS which are the lengths from each coordinate. That's the main purpose of graph. The shape of graph tells us how these pairs of numbers differ from one to another by their magnitudes.

So, using graph is about calculating surface area that can represent not only physical objects, like Pirate Island, airplane vertical stabilizer, or ship’s sail, but any surface that can come from graphical representation of various amounts, measured, counted, obtained by any way, and their quantitative relationships. We came up to the point that surface will represent something because we agreed, at the first place, that LENGTH represents something, that length of line we draw in graph corresponds to a number, count, obtained by measurement (or by any other mean).

Perhaps, it is a good place to describe steps that we use in these scenarios and how we can use thus acquired knowledge.

  1. First we measure quantities of interest and note their relationships.
  2. We represent the counts of these different quantities by lengths of line. We draw a graph.
  3. Since we started dealing with lines, automatically then the surface bounded by these lines may represent a quantity of interest. Example is speed, v, and time, t. If we plot graph v(t), then automatically surface area under the graph will represent path taken.
  4. We use our “method of rectangles” to calculate surface area, if required!

Note how we first focus, without considering the usage of graph, on relationships and measurements, mathematical treatment, of different objects. Then, we can use numbers by themselves, or use  table, or graph, for our further analysis. In graph we represent all those counts and numbers by lines of certain length, to show us, visually, quantitative relationships between measured objects.

Not also that lines, curves generated can be analyzed in their separate worlds of geometry or analytical geometry for example.

Note that coded color scheme can be introduced as the third dimension on the two dimensional graph. Volume integration is possible even here, but first you have to have a map, an association between a color and the number it represents.

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