Discussing my educational foundation in mathematics, I think, is probably very similar to many average people. But first, I think I have to provide a little background to begin this first post—my own background. What is my educational background in mathematics?
I really have no advanced mathematical education. I have no degree. I have no CV (curriculum vitae). My mathematical education is simply from the American educational system, grades 1–12, from the 1960s and 1970s. So my blog posts here will be limited to that, mostly basic stuff, basic algebra. I have forgotten most of the geometry, trigonometry, probabilities and statistics, and pre-calculus. I was a good student, but I did not take it any further than grades 1–12.
That being said, how did I learn about numbers?
I think like most people, I started learning about numbers either in pre-school or first grade by learning to count: 1 2 3 4 5 6... What are the counting numbers? They are the natural numbers, the positive integers, the positive integers greater than 0, etc.
In those early formative years, whether it was the first grade or earlier, we learn about these numbers. We learn how to count. Then, over time, and as we move on to the next grade level in school, we learn how to add them, how to subtract them, and then multiplication and division. Eventually we learn the properties: commutative property, associative property, distributive property, and identity property. Moving on, we learn about fractions (ratios) and how to add, subtract, multiply, and divide them. We learn about exponentiation (powers).
Where is this all going? Well, I will try to speed it up a bit. Eventually I want to point out some shortcomings in this educational process—the shortcuts and assumptions I spoke of in my 'Introduction' post. So let's just make a list here of things I learned through grades 1–12. A brief list, not necessarily complete or in perfect order, but an approximation, and then I will speak of those shortcomings, the shortcuts and built-in assumptions that, in later posts to follow, will become very important.
The point is, all of this is based on counting numbers, learning to count 1, 2, 3, 4, 5,...
If you continue on with your mathematical education, mathematics is so vast and so complicated that, in comparison to the simple education I have talked about above, this simple education is maybe one quadrillionth of one percent of the vast knowledge produced in the field of mathematics. It is probably even way, way less than that. But still, everyone started with counting 1, 2, 3, 4, 5,...
So now I will point out what I feel are some of the built-in assumptions and/or shortcuts in that grade 1–12 education that I think affect everyone in their journey of the math world.
The counting numbers, natural numbers or positive integers. {1,2,3,4,5,6,7,...} n+1. The set of natural numbers has countably infinite members. Each element of the countably infinite set of natural numbers, however, is finite.
The inverse natural numbers. {1,1/2,1/3,1/4,1/5,1/6,1/7,...} 1/n+1. The set of inverse natural numbers has countably infinite members. Each element of the countably infinite set of inverse natural numbers, however, is finite.
Because we begin to learn with counting numbers, we do not learn immediately that they are also a ratio, a fraction, 2 = 2/1, 3 = 3/1, 4 = 4/1, 5 = 5/1,... We do learn this eventually, and everyone knows this, but it seems silly to write them as a fraction/ratio because any number divided by 1 is that number, understandable, but it is a shortcut.
There is no shortcut for writing the inverse naturals. We are forced to write it as a ratio/fraction, there is no other way, we have to write 1 divided by n. This is entirely understandable, as I cannot imagine learning how to count where you are taught from the get-go 1/1, 2/1, 3/1, 4/1, 5/1,... You are imposing the notion of fractions/ratios and division on a child just learning how to count, and the first thing they are going to ask is why all the counting numbers are written so strangely. None the less, these shortcuts exist.
Another example is when we learn exponentiation. Generally we write 7, instead of , but to write seven to the inverse first power we have to write .
Once again, not a big deal, but once again a shortcut.
Now this next shortcut I am going to discuss I think is more of a big deal. Maybe even catastrophic. It is not that we do not know they are shortcuts or that we were never taught the long way of writing these expressions, but after years and years of reinforcing the use of these shortcuts over and over, we tend to build reinforcing circuits in our brains and we tend to forget the other ways or the not shortcut ways of expression, and I think in at least one case that is very detrimental.
Factorization of natural numbers and inverse natural numbers
At some point in our early mathematical education, we learn factorization. The prime numbers, the 'Fundamental Theorem of Arithmetic', also known as the 'Unique Factorization Theorem'. This basic theorem goes all the way back to the third century BCE and Euclid's Elements. It is foundational.
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