Geometry was a great case in point for me. When working in the abstract, I had trouble with the topic. When I entered manufacturing, suddenly those things made great sense. Another set of realizations came with parametric modeling systems. Suddenly, I could "solve" lots of things geometrically, and or express them in various ways depending on what the givens were. Often, non-optimal measuring tools are available. Re-expressing something in terms of what can be measured is very powerful and useful.
In particular, proving something made a lot of sense then. And, parametric systems solve various geometric things quickly and easily, allowing somebody to express things in mostly geometrical terms instead of lots of numbers. In this way, geometry is very powerful. It can sharply diminish the amount of information that needs to be checked and or specified. It's kind of interesting to remodel things drawn pre-computer. There are lots of basic mistakes, and or inaccuracies and or contradictions between the dimensions. A stronger geometric approach eliminates those things, however it's laborious to compute. The concepts are invaluable though.
I personally think math should be more interactive. We've got the resources now to do that. Classic abstract math has it's place, and for those with the analytic skill and visual skill to master it, they should do so. Others can still benefit from strong math understanding, even if it's not directly computable, the concepts are valuable, if anything to understand the powerful software they could be applying to a task at hand.
And it continues up through your years of maths studies.
As you probably know one of my favourite things in recent times is the Fast Fourier Transform. We went through the Fourier Transform back in the day. There is a nice definition involving a big integral sign. We learned how to transform all kinds of functions and do whatever and at the end answer questions about it in an exam. If you look at the Fourier Transform from a software engineers view you see the discreet version operating on sampled data and it has a big summation sign which you can convince yourself is probably doing what the integral did.
But looking back I wonder if I really got the idea of the Fourier Transform when we were covering it back in uni.
For sure it was some time later I started to get pictures in my mind of what is going on and why it works. With the right pictures you can pretty much write down the formula without having to remember some arbitrary looking definition. Without those pictures I would never have been happy that I understood the Fast Fourier Transform at all.
Could it be that we have ended up with a bunch of maths teachers most of whom don't really have a passion for maths and don't actually understand it so well. That is to say they don't have those deep pictures in their minds. All they can do is regurgitate that blinding wall of symbols and definitions and hope you remember it long enough to pass a test. Thus breeding the next generation of maths teachers who don't have a passion for maths and don't understand it so well. Take away their text books and study guides and they are lost.
I was lucky. The guy teaching us calculus and such for two years knew his stuff inside out, would never move on until everyone got the idea. He would put in asides about how he used this and that technique when he was involved aircraft design.
When I was in high school I worked after school and summers in the back shop of the local newspaper and had helped one of the guys build a garage/shop at his house. One day when I came in he wanted to show me something. He drew the standard 3-4-5 diagram of the Pythagoran theorem. I said, yeah, that's the Pythagoran theorem, we learned about that in math. He was a little upset that I hadn't shared that with him. He explained that he used the theorem to check the 'square' on the garage foundation that we poured, and it was off by about an inch. I learned the theorem, but had never been shown any practical application for it.
In Calculus II class, I had an instructor that would give problems like: "there is a window that is 3 feet tall, and is on the second floor of the girl's dormitory, with the base of the window 15 feet above ground level. How far from the base of the building do you have to stand to maximize your view into the window?". Gave you some real-world practical examples.
I often hear parents (and teachers) say that many kids can't do word problems. And that is because they are only taught the "steps" required to solve a math problem and not the "reason" why you do these steps.
Like little robots students are taught to get good marks on standardized tests, but have know idea how to apply anything they are taught.
I bet most high school students couldn't solve this simple "rate" word problem
Pump "A" can fill a tank in 2 hours. Pump "B" can fill the tank in 3 hours. How long would it take to fill the tank if both pumps are used at the same time ?
I made a bad career choice and went to a university with no engineering school. So I ended up studying Art and Architecture.
I have to teach myself Calculus for my engineering licensing exam. But I did get all A's on my math exams in university by just studying the text. Even though I go A's on all the exams, the Math Department required good attendance to get an A, so all my final grades were B's because I couldn't stand sitting through hours of listening to people confused about math.
I think the Schamm Outlines are excellent for self-study. And about 25 years ago, I was able to buy copies of nearly all the math outlines they published in a second hand store for 25 cents each, about 20 texts in excellent condition.
I also sold them all to a high school math teach for 25 cent each after I moved to Taiwan. He wanted to buy two or three, but I mentioned that he would never have another chance to get them all so cheap.
Comments
Geometry was a great case in point for me. When working in the abstract, I had trouble with the topic. When I entered manufacturing, suddenly those things made great sense. Another set of realizations came with parametric modeling systems. Suddenly, I could "solve" lots of things geometrically, and or express them in various ways depending on what the givens were. Often, non-optimal measuring tools are available. Re-expressing something in terms of what can be measured is very powerful and useful.
In particular, proving something made a lot of sense then. And, parametric systems solve various geometric things quickly and easily, allowing somebody to express things in mostly geometrical terms instead of lots of numbers. In this way, geometry is very powerful. It can sharply diminish the amount of information that needs to be checked and or specified. It's kind of interesting to remodel things drawn pre-computer. There are lots of basic mistakes, and or inaccuracies and or contradictions between the dimensions. A stronger geometric approach eliminates those things, however it's laborious to compute. The concepts are invaluable though.
I personally think math should be more interactive. We've got the resources now to do that. Classic abstract math has it's place, and for those with the analytic skill and visual skill to master it, they should do so. Others can still benefit from strong math understanding, even if it's not directly computable, the concepts are valuable, if anything to understand the powerful software they could be applying to a task at hand.
As you probably know one of my favourite things in recent times is the Fast Fourier Transform. We went through the Fourier Transform back in the day. There is a nice definition involving a big integral sign. We learned how to transform all kinds of functions and do whatever and at the end answer questions about it in an exam. If you look at the Fourier Transform from a software engineers view you see the discreet version operating on sampled data and it has a big summation sign which you can convince yourself is probably doing what the integral did.
But looking back I wonder if I really got the idea of the Fourier Transform when we were covering it back in uni.
For sure it was some time later I started to get pictures in my mind of what is going on and why it works. With the right pictures you can pretty much write down the formula without having to remember some arbitrary looking definition. Without those pictures I would never have been happy that I understood the Fast Fourier Transform at all.
I was lucky. The guy teaching us calculus and such for two years knew his stuff inside out, would never move on until everyone got the idea. He would put in asides about how he used this and that technique when he was involved aircraft design.
In Calculus II class, I had an instructor that would give problems like: "there is a window that is 3 feet tall, and is on the second floor of the girl's dormitory, with the base of the window 15 feet above ground level. How far from the base of the building do you have to stand to maximize your view into the window?". Gave you some real-world practical examples.
Like little robots students are taught to get good marks on standardized tests, but have know idea how to apply anything they are taught.
I bet most high school students couldn't solve this simple "rate" word problem
Pump "A" can fill a tank in 2 hours. Pump "B" can fill the tank in 3 hours. How long would it take to fill the tank if both pumps are used at the same time ?
Bean
I have to teach myself Calculus for my engineering licensing exam. But I did get all A's on my math exams in university by just studying the text. Even though I go A's on all the exams, the Math Department required good attendance to get an A, so all my final grades were B's because I couldn't stand sitting through hours of listening to people confused about math.
I think the Schamm Outlines are excellent for self-study. And about 25 years ago, I was able to buy copies of nearly all the math outlines they published in a second hand store for 25 cents each, about 20 texts in excellent condition.
I also sold them all to a high school math teach for 25 cent each after I moved to Taiwan. He wanted to buy two or three, but I mentioned that he would never have another chance to get them all so cheap.