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Learning algebra in my 60s

theguardian.com

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Re: Learning algebra in my 60s

#111

Post by deelowe »

ravi-delia wrote: post #85
jjice wrote: post #54

Earlier quoted context omitted.

> ...still ended up going down an execution rather than concept focused route. 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…

When I was learning math in public school, my family couldn't make heads or tails of my homework. Even in Kindergarden we learned that '2 = 1 + 1' was the same as '1 + 1 = 2', and we got started with '1 + ? = 3' even in first grade. A focus was put on mental math, teaching techniques to break two digit numbers up, before we did pencil-and-paper work. Whenever possible there would be more than one technique, and the c…

Common core. It's become a taboo term in certain circles.

Re: Learning algebra in my 60s

#112

Post by bslorence »

sfpotter wrote: post #83
2OEH8eoCRo0 wrote: post #13

Earlier quoted context omitted.

> “Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”

Many other comments here don't seem to have spent much time reflecting on why his niece said this. Speaking as an actual math teacher, this is a very important thing for students to try to come to grips with. Memorization and "learning without understanding" have a bad rep, but memorization is a tremendously valuable skill. Instead of thinking of "learning by memorization" as doing a disservice to learning math, cons…

Maybe "asking 'why'" isn't the right way to express it. Perhaps what the OP was really after here was a concrete illustration of the formula, rather than a rigorous derivation of the formula from first principles.

There are many concrete geometrical illustrations of fundamental concepts in algebra, but most people aren't even aware that they exist.

The niece's comment, oddly, both points to this and glosses over it. You don't teach a child multiplication by only forcing them to memorize times tables. You also show them stacks of coins, etc. But we do teach children algebra by only forcing them to memorize symbolic procedures. Meanwhile Book 2 of Euclid is free online, or very cheap from Dover.

Re: Learning algebra in my 60s

#113

Post by hgomersall »

sfpotter wrote: post #83
2OEH8eoCRo0 wrote: post #13

Earlier quoted context omitted.

> “Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”

Many other comments here don't seem to have spent much time reflecting on why his niece said this. Speaking as an actual math teacher, this is a very important thing for students to try to come to grips with. Memorization and "learning without understanding" have a bad rep, but memorization is a tremendously valuable skill. Instead of thinking of "learning by memorization" as doing a disservice to learning math, cons…

I can distinctly remember in my second or third year of engineering undergrad doing some stuff with the Radon Transform. At the time I had nothing to map it too. The maths to me was completely abstract. I just sat there, thought hard, and worked through problem after problem until it became intuitive. Now I have difficulty understanding what I found hard, because it all fits together with loads of other things into a coherent mass of concepts.

It was interesting to me because it was one of a few times as an adult where I realised the framework was completely missing and I would just have to start from scratch, which meant lots of rote effort and thinking hard. There was no shortcut.

Re: Learning algebra in my 60s

#114

Post by dhosek »

I’ve been tempted to pick up an adjunct section of algebra at a local community college and invert the usual style of teaching and start with word problems. A lot of people have an intuitive sense of how to figure out, say, how to scale a recipe but when it gets turned into symbolic math it becomes a challenge for them. I’d kind of like to take advantage of that and start from the word problem and then move to how to turn that into symbols so that instead of thinking y = 1.4x-2.8 we'd say think in terms of you pay $1.40 for each donut but the first two are free. If all the x² and x³ have particular meanings, you’re less like to think that x²+x³ = x⁵ but recognize that you can’t simplify the expression until you know what x is.

Re: Learning algebra in my 60s

#115

Post by ziddoap »

wawjgreen wrote: post #63

My litmus test of someone being intelligent or not--among other such tests--is what they say about Shakespeare. This might anger the Brits, but Shakespeare is garbage--and the guy says: "I read .... most of Shakespeare." One famous guy lambasting Shakespeare (Tolstoy) was only marginally smarter, because according to his "Confessions", he read it many times over (incl. in English) and STILL could not find any artisti…

>Jung was not as bright as is commonly believed as well. He once claimed that UFO's were all imaginary-- a sweeping generalization by someone claiming to be scientist-- whereas I for sure know that UFO's are real for I have seen one and to prove that I wasn't seeing things, I had a camcorder ready which recorded the space-ship going vertically up very slowly. I have lost the video though but it certainly happened 20 years ago and I still remember it.

Your evidence that Jung was not as smart as believed is that you recorded a UFO 20 years ago (but can't prove it) and he said UFOs don't exist? Forgive me, but that's the opposite of convincing.

Re: Learning algebra in my 60s

#116

Post by dhosek »

deelowe wrote: post #111
ravi-delia wrote: post #85

Earlier quoted context omitted.

When I was learning math in public school, my family couldn't make heads or tails of my homework. Even in Kindergarden we learned that '2 = 1 + 1' was the same as '1 + 1 = 2', and we got started with '1 + ? = 3' even in first grade. A focus was put on mental math, teaching techniques to break two digit numbers up, before we did pencil-and-paper work. Whenever possible there would be more than one technique, and the c…

Common core. It's become a taboo term in certain circles.

Arrrgggghhh! I was getting my math teaching credential when common core was getting its start and what I want to shout from the rooftops is:

COMMON CORE DOES NOT SPECIFY PEDAGOGICAL METHODS!

It is, essentially. a list of topics that should be mastered at each grade level.

Re: Learning algebra in my 60s

#117

Post by bowsamic »

mattwilsonn888 wrote: post #8

It's a shame he went to all the trouble of avoiding conventional education (for the wrong reasons), but then still ended up going down an execution rather than concept focused route. As soon as you start memorizing operations and treating math more like a narrow grind only approached through arbitrary problems solved primarily through computation or use of rote application of poorly understood technique, you lose the…

I disagree. My favourite lecturer for general relativity said the important thing is fluency, which comes from repetition. It’s great to know everything, but you should be able to do the basics, such as calculus or algebra, very quickly and without thinking. You will need that brain power for the new stuff. The best way to gain mathematical fluency is by lots of practise.

Re: Learning algebra in my 60s

#118

Post by jonathanlydall »

scythe wrote: post #102
jonathanlydall wrote: post #69

Earlier quoted context omitted.

When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer. 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…

>I asked the teacher where the numbers which were spat out by the calculator came from 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 repeate…

I can’t remember the details now, but my recollection was that the unit circle enlightened me that they were ratios. It was even possible to reason about “round” angles (like 90° and 45°), in your head without the need of a calculator.

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.

Re: Learning algebra in my 60s

#119

Post by guidorice »

victor106 wrote: post #32

My friends dad in his late 60's wanted to learn math and used the No Bullshit math and physics from https://minireference.com/ and loved it. Their Linear Algebra is quite good too.

I highly recommend the Ivan Savof books as well. A Statistics title is upcoming.

Re: Learning algebra in my 60s

#120

Post by dhosek »

jonathanlydall wrote: post #69
jjice wrote: post #54

Earlier quoted context omitted.

> ...still ended up going down an execution rather than concept focused route. 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…

When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer. 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…

There’s a nice diagram I created an EPS of (which I can’t find now) that shows the unit circle derivations for tan/cot and sec/csc.

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.

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