> Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring
That was me. I grew up believing I hated math. Struggled all the way through middle & high school to AP calc and just found it incredibly boring and tedious. Ended up opting out of doing engineering/science in undergrad because I just couldn't stand doing all the math.
Long story short, years later ended up going back to school for CS and took discrete math as one of my first courses, and remember being blown away by how cool it was. All this time thinking I hated math!
Hard to say exactly what the difference is. Partially I think my brain just groks discrete concepts more easily.
But also the class had a heavy emphasis on proofs, which I think was really important. At a certain level this type of problem-solving can start to resemble philosophy. Chugging through a proof, figuring out just the right way to construct it and slapping a triumphant "Q.E.D." at the end is an empowering experience, especially the first time. There's a world of difference between "you throw a ball, solve for its velocity at time x" and "prove that there must be a ball" (I'm embellishing of course). It's a difference between obtaining an answer for a specific instance of a situation, and shedding light on some fundamental/universal property of the world. To me that feels profound in a sense, which makes it exciting.
Proofs don't belong solely to the domain of discrete math, of course, so this probably isn't as much a testament to the subject as it is to the general problem-solving approach. It would be nice if students could get exposed to this a bit earlier, I think there are many folks like myself who would realize that they can love math too.
What you're describing is the basic shift between what lower-ed science/math is like and what "real" (college) science/math is like. The problem is that everything they teach in highschool and below needs to have an escape hatch for "what if they have anti-ADHD* but no clue what's going on?" That's why you were able to solve for v without obtaining any knowledge about the universe, and why taking discrete math did what they claimed geometry would do.
* It's kind of a stupid way to put it, but by anti-ADHD I mean sufficiently controlled behavior combined with the ability to focus on any rote task until it is completely learned (basically, whatever traits or life conditions you need to have the opposite of ADHD). No matter what, you're not allowed to fail that group.
awkward way to put it (although i dunno how else to describe them), but i think everyone knows the students you are talking about. these people are the reason why highschool math/science is hell, and you can't get away from them by taken honors or AP courses.
Don't worry, algebra, systems of linear equations, synthetic geometry, analytic geometry, infinite series, differential and integral calculus, etc. can also be made 'cool' and intellectually stimulating. Likewise discrete math can be made dull and lifeless. The difference has much more to do with quality of problems and teaching than with your brain.
Calculus is totally based on the teacher. I had an awesome Calculus teacher (He actually was a Physicians Assistant and had degrees from Yale and Harvard but volunteered at my small Christian School). He taught me first class why calculus was awesome by challenging use that everything else in math was fake numbers. Showed us the difference between 1/3 and 0.33333 and studying the speed of two trains word problem was always wrong. He than stated that with Calculus you could see the world as we see it. We than used functions all semester long that would eventually get "exact" and it was an awesome ride. To bad we had 3 students and the other 2 were total math geniuses and got perfect math scores on their SATs. They always made me feel like an idiot.
> It's a difference between obtaining an answer for a specific instance of a situation, and shedding light on some fundamental/universal property of the world.
They're also completely different questions in the sense of what you mean when you say "this result is correct."
The ball velocity is really a simplified model that gives you a correspondence truth (you verify by running the experiment and taking a measurement.) The exidtence proof gives you a coherence truth, i.e. there are no unknown factors and ur statement is absolutely true.
Just giving terminology to your intuition: coherence truth vs correspondence truth.
That was me. I grew up believing I hated math. Struggled all the way through middle & high school to AP calc and just found it incredibly boring and tedious. Ended up opting out of doing engineering/science in undergrad because I just couldn't stand doing all the math.
Long story short, years later ended up going back to school for CS and took discrete math as one of my first courses, and remember being blown away by how cool it was. All this time thinking I hated math!
Hard to say exactly what the difference is. Partially I think my brain just groks discrete concepts more easily.
But also the class had a heavy emphasis on proofs, which I think was really important. At a certain level this type of problem-solving can start to resemble philosophy. Chugging through a proof, figuring out just the right way to construct it and slapping a triumphant "Q.E.D." at the end is an empowering experience, especially the first time. There's a world of difference between "you throw a ball, solve for its velocity at time x" and "prove that there must be a ball" (I'm embellishing of course). It's a difference between obtaining an answer for a specific instance of a situation, and shedding light on some fundamental/universal property of the world. To me that feels profound in a sense, which makes it exciting.
Proofs don't belong solely to the domain of discrete math, of course, so this probably isn't as much a testament to the subject as it is to the general problem-solving approach. It would be nice if students could get exposed to this a bit earlier, I think there are many folks like myself who would realize that they can love math too.