11–20 of 102 posts
I'm sure there are things that I'm missing out on by having this gap in my knowledge but there's a lot of knowledge out there and I spend most days learning, I may get to it one day.
We had access to all of the symbolic algebra tools and were even expected to use them regularly for both courses. It was great!
I'm not sure how well this would extend to introductory courses though, especially if the standardized tests still expect integration by hand.
Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do" The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a…
You are spot on the money. I've seen "progressive" educators who believe that technology, including CAS is the answer to teaching calculus to high school and tertiary students. I think the tech is mostly useful _pedagogically_ after the student has a very firm grip on the content. Using it as a shortcut to "real world problems" leaves the student with no real understanding of the dynamics/functions involved, leading in turn to believing the machines are magic and always right, regardless of whether you've hit all the right buttons in the right order.
The most extreme example of this I've seen was in a classroom I was auditing. The students were learning about harmonic motion and trig functions by using a mass on a spring and an ultrasonic transducer. The students were told to "keep all the digits, because the machine gave them". Students were doing sums with 14 significant digits. When I pointed out that most of them were meaningless, being measures in the Angstrom range, the "teacher" said "the machine gives those figures".
Even when I pointed out that measuring 10^-14 m required a very high frequency, which meant high energy, all supposedly from AA batteries that last for months, he agreed that it did seem strange, but after all, the "machine gave the figures".
The man teaching (he was head of department AND author of the state mathematics syllabus) was unaware that only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.
Now as I said, for me maths is easy. It's not for most people. Most people won't grasp it all as "but that's obvious", so for these people having the skills to check the result the machine gives is vital if they are not to cause all sorts of fun as bridges collapse, cars explode and mortgages go into default. Those skills are developed by doing it by hand, and learning what numbers change in what ways.
TL;DR Modern trend to use CAS in mathematics education is well intentioned but misled, in this retired maths educators very informed opinion.
>Can Calculus be taught with differentiating or integrating by hand?
And then the answer is yes, as this is how it is done.
Now, why would be want to do otherwise? Why would we ever want people to learn less?
Yes. I regularly solve (mostly motion control) problems that require integration and differentiation and I absolutely cannot and have never been able to do those things by hand but I understand the concept and with a computer by my side that's plenty. I'm sure there are things that I'm missing out on by having this gap in my knowledge but there's a lot of knowledge out there and I spend most days learning, I may get…
Years ago, my college multi variable calculus and linear algebra courses were both taught primarily using course materials that were interactive Mathematica Notebooks. We had access to all of the symbolic algebra tools and were even expected to use them regularly for both courses. It was great! I'm not sure how well this would extend to introductory courses though, especially if the standardized tests still expect in…