Email me if you're curious about class materials, especially if I was the instructor.
I have also been an instructor for I-STEM at Stony Brook University where I've introduced high school students to complex numbers, group theory, the pigeonhole principle, Hamiltonian mechanics, and graph theory with applications to error-correcting code.
Here is a note on what the dx means when integrating in a Calculus I, II, or III course.
Before getting into things, I want to point out a few resources. Bill Thurston's "On Proof and Progress in Mathematics" and Paul Lockhart's "A Mathematicians Lament" do a fine job of describing perceptions of math and the current state of US public math education. Also, my high school history teacher, Russell Brown, gave a TED talk on the current state of general education in the US. He taught me far more than just history and I am very grateful to him.
Now, to begin, as is often the case, "Calvin and Hobbes" speaks to the truth of the matter (all credit to Bill Watterson).
Many of my students are not so unlike Calvin who concludes that 3+8 = 6. If they ask themselves the simple question of, "If I start off with 8 and I add something positive to it like 3, could I end with something smaller than 8?", they'll find that 6 cannot be the correct answer. But unfortunately, it seems students are not taught to ask such questions, to actively reason through and understand their own though processes. Instead, they are told the steps to compute something, whether its 3+8 or the derivative of cos(3e^x), but not taught how to interpret what everything means nor to check whether their answers make sense. And who can blame them when the way math is taught in the US is akin to a piano teacher having a student only play the C major scale for weeks on end with nothing to look forward to except for the G major scale? (an example of Jacob Lurie). It's hard to feel motivated, less alone any passion for math.
Unfortunately, I'm not sure if much can be done towards igniting passion for mathematics once a student is at the college level. But at the high school level, there is more hope and that is why I get involved with I-STEM at Stony Brook to promote mathematics. My hope is like that of Paul Lockhart: let the students explore math in a way that feels like discovery. Instead of using supposedly "novel" ways of multiplying numbers by first drawing boxes with diagonals, I think it's more personal to tell stories instead of definitions. Then ask the students to offer their own definitions to compare to the "textbook" definition. Make critical thinking fun. Teach them to ask meaningful questions and to be inventive without fear of making mistakes like Hobbes below who stumbles upon imaginary numbers. Mistakes are great teachers.
