So, as project for the day, I decided I would teach my second grader, 7-year-old son about functions (the math kind). He knows all the addition, subtraction and multiplication tables by heart and he knows how to solve math word problems and solve for unknowns and other "advanced" grade-school level math judo (for instance, he knows how to solve basic algebraic equations, he just doesn't know they are called that or any of the lingo involved). Even though he has the needed "machinery" to do advanced math (for his age and level), I forewarned him that he might not understand the topic and reassured him it was perfectly OK if didn't get a good grasp by the end of the class. Further, I emphasized that he was not going to be needing any of this knowledge for a few years, but I just wanted him to be acquainted with the basic concepts surrounding functions.
So, the first thing we did was define what functions are (in a language he could understand). I diluted the definition to something to the effect of: a function is like a machine that you feed it a number, it does stuff with that number and the gives back another number. He looked a bit puzzled, so I went ahead and wrote down a concrete example: f(x) := x, then together we walked through a table like this:
-------------- x | f(x) -------------- 0 0 1 1 2 2 3 3 4 4
at this point I could tell he was trying to make sense of this stuff. So, instead of explaining more material, I gave yet another concrete example f(x) := x + 1, just as we had done with the previous example, we walked through filling a table with inputs and outputs. He still looked a bit confused, so we did yet another example, this time using multiplication f(x) := 2 * x, as we walked through the input and output table, I saw the light bulb lighting up on top of his head, so to speak. Shortly thereafter, he interrupted me as I was slowly going through the table and said something to the effect so the 'x' gets switched out by whatever number you have 'f' thing. I told him that it was very clever of him to see the pattern, but there was more to it than that (you see, positive re-enforcement it's a bit of a balancing act with him because if I would've told him that he was exactly right, he would've left thinking he was a math functions expert). So, I was pleased that he grasped the concept that here we have this math construct/machinery that you feed it a number, does something with that number and then spits out the result. I was also a bit surprised that he saw the pattern that the input was replacing the mysterious variable 'x' in a mere three examples. So, I was confident that I could introduce more material and not leave him in the dust. I taught him the concept of x and y- axis, again on terms he could understand. So, I drew a 10-line by 10-line chart in a piece of graph paper. Together we labeled the axes and the origin (which I had just explained) and added the line ticks. Then I had him read me the inputs and outputs for the f(x) := x + 1 from the table we had previously created. I plotted point by point and then had him draw the lines from dot to dot. I could see by his smile that he was understanding the material thus far. So with basic concepts, definitions, examples and charts under our belt I decided to see if it was true that he understood the material (as opposed to my biased opinion). I wrote a short quiz consisting of three functions f(x) := 5 + x, f(x) := x * x and f(x) := x - 1 and he was supposed to fill the tables from zero to five for each of them. A little over fifteen minutes later, he told me he was done. He got all the answers correct with the exception of two. The first one he wrote f (0) = 0 even though it should've been 5 and the other he left blank f(0) for the last function even though it should've been -1. However, my wife (and his full-time teacher) later told me they haven't done negative numbers yet. Of course, I was (secretly) super excited that he had just done so well with a brand-new topic and one that most of his peers won't see for a few years. At the same time, I was amazed at his learning abilities and given that he is not a genius or gifted (to be objective), this probably is a potential that many 7-year-olds have but due to the nature of mass education goes unused, or perhaps it just means I'm crazy for trying to tech things to my kid that are out of his league.
Either way, before the Math Gestapo rears its ugly head, I will say that I'm well aware that there are many other facets and nuance to functions that were purposely left out of this lesson that I will try to introduce in time.
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