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.
And, now, here is the main step!
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.
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