After High school I took a break for a few years before going to college for real. Once I had it in my mind to go to college, I signed up at my local community college and retook all of my high school maths. When I made it to Calculus I started signing up for other classes. I drilled, drilled, drilled the problems in every subject until my hands ached from writing. Slowly some patterns emerged and I started understanding more fundamental concepts about math, started to think symbolically. Working problems became an exercise in very careful symbol manipulation and not just arithmetic on steroids. It trained me in the kind of very careful mental discipline needed for a STEM degree.
I never really used much of the math I learned while studying Computer Science (outside of a couple small subjects), but that discipline, the intolerance for errors, and solving problems did have a huge impact. I also use very little of those maths in my day-to-day, but it's definitely left me with a rigor I bring to the job.
It turns out that what I really enjoyed was logic and set theory that you just have to learn when you start out in CS. I wish that I had learned that first instead of arithmetic tables or pointless long division. I use logic and set theory all the time in critical thinking and daily reasoning...it's so useful that it's practically automatic and subconscious at this point.
I think the real problem, and the one the original question is pointing out is, math doesn't mean anything to a youngster. There's literally no application for it in their life beyond very simple addition and subtraction, skills usually learned by 3rd or 4th grade. After that it's years and years of absolutely pointless busy work (to them). I also think Maths education should include more reading and math history.
Here's an alternative k-12 maths education route that I think I would have taken to much more readily, since I could have started to apply it as a reasoning and critical thinking skill immediately:
- K, True and False, Counting numbers - reinforces concepts of True and False they're already learning, teaches necessary basic numeracy (even Kindergarden aged kids get why counting is important)
- 1st and 2nd grade, T and F, AND, OR and NOT. More counting numbers, to 1000 and by 5s and 10s.
- 3rd grade, More complex boolean equations, basic boolean algebra, truth tables, implies -> operator. Negative numbers, count from -100,000 to 100,000 by 1s, 2s, 5s, 10s, 100s, 500s and 1000s.
- 4th grade, more complex boolean algebra, binary arithmetic (using +, -, * etc., it's the same thing but with new symbols!) Simple, boolean word problems (teach rational reasoning! [1]), boolean laws (commutative, Associative, etc.). Basic Set Theory.
[1] - The moon is in the sky and the moon is made of cheese. Is this statemen true or false?
- 5th grade, More Set Theory, boolean equivalency (T /\ F = F == F /\ (T \/ F)), base 10 arithmetic, more boolean word problems, set theory word problems, critical thinking and reasoning. Read simple articles and determine if they are true or false. Basic prepositional calculus.
- 6th grade, various elements of digital circuit design (diagramming, equivalency, etc.), more base 10 arithmetic, more set theory, more binary math, half and full adder truth tables and diagrams, algebra
- 7th grade, more basic algebra, fractions, decimals, digital circuit labs (woah, application!, make a binary counter, and maybe a half and full adder) maybe Karnaugh maps, more critical reading, deeper set theory (Jaccard coefficients), base conversion
- 8th grade, geometry, functions, more complex circuits, non-binary logic and reasoning, basic statistics, maybe basic probability, logical fallacies,
- 9th grade, more geometry and functions, more complex probability and stats (non-calculus), deeper logic and reasoning topics, inductive and deductive arguments, basic proofs, etc., end of year logic project, intro to abstract mathematics, predicate logic
- 10th grade, basic calculus, basic trig, more proofs and techniques, complex inductive and deductive reasoning, complex set theory, end of year logic project, more abstract mathematics, simple physics equations and labs!
- 11th grade, more calculus, more Prob&Stats, vector algebra intro, mid-year and end of year logic project, more proofs, more abstract mathematics, physics equations and labs!
- 12th grade, discrete math, vector algebra & calc, imaginary numbers, abstract mathematics topics, etc. simple calculus physics and labs!
I'm of the opinion that most children can learn the mechanics of basic calculus pretty simply. If you can do algebra, you can do basic calculus. You might not understand the immediate application, but physics and physics labs can be incredibly fun, calculate everything out and run an experiment.
I think logic should be taught before regular math because quite simply, the student isn't getting hung up on all these values and can focus on learning what an operation is. Young children also are learning topics like "truth and lies" anyways and this helps reinforce this and will come more easily to them.
Digital circuit breadboarding is fun and gives a nice application for logic, it will engage tactile learners who tend to struggle with math subjects.
Reading and critical thinking discussions should be central in the curriculum. It reinforces the need to read, teaches rational thought and gives application to the logic. The scientific method is covered as an application of math in simple physics labs.
Also, by not focusing an entire year on a single subject (say a year on Geometry), the student can flex as some topics will come more naturally than others. So if they're great at probability, but terrible at geometry, they'll get the slack in the year to get up to speed on their geometry.
Side benefits, a better understanding of critical thinking skills and rational thought, the fact that there are different kinds of number systems, a good transition from math into physics (and science), hands on with computer stuff, they'll be set up to understand things like bayes theory and proofs, it doesn't treat relatively simple subjects as "scary things only math priests do" like calculus, word problems and logic go hand in hand, it eases them into word problems and base 10 math as a matter of course instead of as a "special" subject. Geometry just becomes another piece of the puzzle instead of a different course, leading nicely into calculus. Probability and stats open up lots of possibilities for practical labs and long term assignments and non-binary logical reasoning. etc. etc.
At every grade, there's some way of spinning some of the subjects out into a practical, hands on lab...teach and reinforcing the applicability of math. Which of course is the entire problem all along. Reading assignments will bring along kids who are readers, I found math history unbelievably fascinating, you can go from ancient history to the computer age easily.
It shows future STEM jobs as part of the course work.
I'm sure I'm missing some topics and in real-life things might be shuffled around a bit, but I think I'm getting the basic gist across. There was no reason for me to struggle in school except that I could understand the application, and all of the adults around me seemed to get along fine with a calculator and basic arithmetic. But with this kind of curriculum, I would have been aware of a few dozen possible career paths in STEM (and elsewhere) that I never even conceived of. It would have kept me interested and focused.