A good math education in arithmetic, algebra, and calculus is a combination of concepts and drill. This was a concept that I rejected back in school. Now I'm teaching-- not mathematics, but I have tutored and helped students catch up who have had problems.
There's a fair number of students struggling in pre-calc who have all the concepts just fine, from the bottom to the top. But when they're dealing with lots of terms and keeping a higher level goal in mind, their performance on a few simpler things, like fractions and quotient properties, falls apart. Maybe they missed a week or two in 6th grade when this was really solidified and practiced.
The terrible thing that tends to happen, once you stumble in math: the amount of concept content you have falls. The focus moves even more to rote, but focused on the "more difficult" stuff--- leaving whatever core deficit there is intact. There's solid reasons for this, but the outcomes are not great.
I also have a kid who is going into AP Calculus pretty young. I'm kinda nervous that he just has not had enough reps of practice, even though he scores very high on placement tests.
Our school does the much-lauded Singapore Math in elementary, which definitely tries to build intuition and looks at many approaches, and supplements with drills.
And I teach a competitive math class which definitely is all about finding different ways around problems and comparing and contrasting.
I do think most programs implementing it end up taking out a little too much rote and algorithms.
My little cousin goes to the same elementary school as I did. When I looked over his homework, I saw none of what I was taught. A glance at his textbook showed they had switched back to the old method of teaching by rote. Apparently school board members who had run in part based on their skepticism of the new techniques had voted in a change in curriculum. One funeral at a time indeed.
COMMON CORE DOES NOT SPECIFY PEDAGOGICAL METHODS!
It is, essentially. a list of topics that should be mastered at each grade level.
At some point the idea came about that for things like addition, subtraction or multiplication, we should teach multiple different ways of viewing the same concept (like addition) in the hopes that even if students don't really understand the classic approach one of the other approaches makes sense. So perhaps the classic explanation of borrowing in subtraction does not make sense to some students, but one of a few other equivalent ways of handling it does make sense.
But then it got turned into a system where all students need to learn all these different different methods, and apply them in both homework and on tests. Which totally defeats the point. The who idea was that some students may find some methods useful and intuitive, and others find different methods useful and intuitive, and as long as every student finds some method can can work with we are better off than only teaching the classic method of the concept.
I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”.
Which was such a thoroughly disappointing response and I hated trigonometry since it was mostly about memorisation rather than applying logic from understanding.
About 10 years later I did a math course through the UK’s Open University and the text book taught trigonometry through explaining the unit circle.
It made me so happy to finally understand and see the actual logic as opposed to punching numbers into a calculator, but I was also sad and a bit upset the unit circle was not taught in high school.
This is a bit of simple knowledge which is sadly unbeknownst to even most math majors and educators. Often, students are taught in calculus that cosines (and hence other trig functions) are computed with Taylor series, which is not really correct. In fact they use CORDIC, a highly optimized algorithm.
But CORDIC is based on repeated use of the sum formula, and a simple version of this can be taught without calculus. Just notice that:
cos(2x) = 2 cos(x)^2 - 1
Then for small enough x, you have (can be shown by drawing, but kind of annoying):
cos(x) ≈ 1 - x^2 / 2
Or rigorously: 1 - x^2 < cos(x)^2 < 1 / (x^2 + 1)
So divide x by 2 until you get a small number (x < 0.1 is usually good enough), use the quadratic approximation, and repeatedly apply the doubling formula. CORDIC uses a similar iteration based on the sum formula. There is no need to wait for Taylor series, limits, convergence tests, etc. You can even bound the error term!
I'm not entirely sure this would work (i.e. improve understanding), though — I have not taught high schoolers before. And there is a decent bit of work required.
I never tried a random angle like 13° in my head, but I figured it was kind of on a scale between the “round” numbers which I knew, which would give a good sense of the actual number, even if not accurate enough.
The point is, the unit circle explained the figures to me instead of them being just some magic numbers.
Likewise, there was an actual explanation of concepts like cos and sin in that they are the names for relationships between angles/sides (again, I can’t remember now, but it made sense), rather than them just being tools you’re told to use when.
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1. Although when I think about educational malpractice, I remember a program I was teaching in where the director asked me about why the numerator and denominator were called what they were and I explained the denominator identified what kind of fraction (and made the analogy to denominations of currency) and the numerator was how many (the “number”) of that kind of fraction. She then, that same day, gave a talk to the assembled teachers saying that she’d asked me and I’d said that’s just what they were called. I nearly stood up and screamed in anger. If I hadn’t desperately needed the money, I would have quit on the spot.
We were simply taught which equation to use for particular scenarios and how to use calculators with particular equations.
When you bought it up, I do recall the connection being used by Open University to explain things, which I also found a bit thrilling to learn.
Although only anecdotal evidence, I have asked a few South African colleagues over the years about the unit circle and it seems like none of them were taught it during school, so it seems like it's just not in the curriculums over here, which I feel is a tragedy.
Here’s how to create this. Draw a unit circle centered on the origin. Draw a ray from the origin where θ will be the angle measured from the positive x axis. You know sin and cos already: you draw a perpendicular from the x axis to where the ray intersects the circle. sinθ is the distance from the origin to where your perpendicular hits the x axis and cosθ is the distance from the x axis to where the perpendicular hits the circle.
Now, draw another line perpendicular from the x axis at (1,0) to your ray (tangent to the circle). Let’s call the point where this line hits your ray T. The distance from T to (1,0) is tanθ and the distance from the origin (0,0) to T is secθ. (As an added bonus, you can easily see that 1+tan²θ = sec²θ). We can do the same process, drawing a tangent line from (0,1) instead of (1,0) to get cotθ and cscθ. You can do some simple math with similar triangles to get familiar formulae like tanθ=sinθ/cosθ etc.
Explaining signs of the functions outside the first quadrant is left as an exercise to the reader.
I'm not a mathematician, but that sounds like poor teaching to me. Even if it's not part of the programme, learning the unit circle takes literally a few minutes and is an invaluable tool afterward. I don't know if it's possible to develop an intuition for trigonometry without learning the unit circle.
It also has a downside... The British Empire depended on robust trigonometrical education. In those days, you couldn't efficiently navigate the world's oceans without it and they needed a steady supply of ship's masters for the thousands of ships that made the empire work. Earth is the unit circle.
What exactly is the downside and for whom?
> The British Empire depended on robust trigonometrical education [...] and they needed a steady supply of ship's masters
The British Empire was already collapsing rapidly when the Open University was founded in 1969. I doubt that ensuring a supply of ship masters was a motivation in choosing to include trigonometry in the syllabus.
This is pretty easy to do for a polynomial term, but much harder for x raised to a non-integer power[1], or sin(x). There is no one way to "derive the formula for a derivative"; different functions have to be analyzed differently.
[1] Wikipedia suggests that the simplest way to derive this formula is to begin by establishing that exp(x) is its own derivative. But that's not an approach I'd be likely to take with students new to calculus.
This is something that made me really disappointed when I realized it in school. It was always a step by step process taught to me, and I loved math. I took my AP Calculus class in my senior year of high school and my teacher (fantastic man) showed us how the formula for a derivative is derived. I was blown away. We had just learned and it were told to memorize it the previous year and I had just accepted it.
After that, math changed for me. Everything had a reason that was connected to everything else. If you understood where it came from, you could do even more than if you were just given a formula to follow. How beautiful.
I realized that I had 5 years (assuming we only cared to do so after algebra) of match education that could have been better taught if done conceptually. I don't blame teachers. You need to get everyone to pass your class for state requirements, so you try to streamline it to get this year done, and not prepare for the future. Also, most math teachers I had did it as a job, not because they loved it, and there's nothing wrong with that. When I had that calculus class during my senior year though, I realized that that man loved math. Hard but fair. I learned so much and that teacher taught me how to develop a work ethic.
I wish meta-learning and a more concept based approach could be applied to high school courses (at least looking back). I understand why they aren't, but man would I love to see how it would play out.