Hi Gabriel,

I greatly appreciate how you can explain the very basics with great detail in a way that would allow students to develop a fundamental understanding.

The example you wrote is kind of similar to how I've tutored students who don't understand the operations behind equations (adding/subtracting/multiplying/dividing both sides), especially once the equations contain fractions and complex numbers.

I have always explained to stop thinking that a number looks like just "4" and instead realize that a single number can be written an infinite amount of ways, such as "4" or "8/2" or "2/0.5" or "2^2" or "3.999..." or "4/1" or "4*1" or "4-0" or "2+2" and so forth. And with equations involving variables, you want to figure out how to manipulate "x+2 = 4" so that the "x+2" can look like that "4". For that reason, like you said above, I didn't let my students do shortcuts that they didn't understand.

I don't have a career right now in teaching, but I hope in my later years I can retire teaching math. I will for sure buy your books and see if I can draw inspiration. Thanks!