> Society exists because we pass on knowledge from one generation to the next.
Yes, but not all the knowledge, for then there would be no time for anything new.
> This means teaching young people how problems were solved, not just handing them the solution.
No, because this robs them of the time to build on that solution and do something new. This extends to all domains -- and all the way down to our most primitive survival needs. Most people have no idea how to build a home, they instead are typically 'handed' one (usually in exchange for money). Few can farm, hunt, and process food to the degree needed to ensure their own survival, but we eat the outputs from those who do, who are anyway also skipping steps by not having to learn and perform a solution that was maybe required hundreds of years ago when technology was less powerful. And taking advantage of all this knowledge we don't have to work for, then we can work for other, new things. This is not only fine but necessary for society to advance.
> Do we want a world where everyone is dependent on technology but nobody knows how it works?
We already live in such a world. Or rather, there exist critical things that no single mind fully comprehends.
So again, if you at least agree that not everything needs to be transmitted, then where are you going to draw the line? Do you think kids should learn to calculate cube roots by hand and prove that with a no-calculator exam? Why? How about logarithms by hand? (Though even before computers, pretty much no one did that either, instead they let a small number of people work really hard to produce many tables up to a few significant figures so they could use those results for bigger things.)
Don't you think the insistence on learning (and testing to demonstrate something was learned) so much "circus math" (math done for show, because there's vanishingly little other reason to do it by hand in current year) might have something to do with how so few students ever even get a taste of real linear algebra, statistics, calculus, and let alone more advanced topics?
I consider it a failure mostly of the education system that I didn't learn about rotation matrices until my first semester of college. They could have been introduced in 9th grade or perhaps earlier. But that would require cutting a lot of other stuff. (I'm still rather happy my 9th grade teacher taught us row reduction, made us do a few by hand and one big 3x4, but thereafter said "using rref() is fine". Meanwhile a Canadian friend reported to me that his "linear algebra" class in university consisted almost solely of doing row reductions of various dimensions by hand, all semester. Useless. But my same 9th grade teacher got very insulted when I wrote back some suggestions for topics he could introduce after taking my linear algebra class, even something as simple as the concept of the codomain, oh well.)
We already live in such a world. Or rather, there exist critical things that no single mind fully comprehends.
That is not the same thing. I am talking about technology which literally nobody understands. That's the premise in a lot of science fiction: a distant future dystopia where technology is slowly decaying and nobody understands it well enough to reproduce it. At best, people are able to scavenge parts from one broken artifact and use them to fix another.
And this is not a far-fetched idea. It has already happened. The fall of the Roman Empire led to a lot of technology being forgotten for centuries. I have also heard that, for example, CRT displays cannot be reproduced (because there are no more factories) and the knowledge to build a manufacturing process for high-quality CRTs would essentially have to be rediscovered.
Don't you think the insistence on learning (and testing to demonstrate something was learned) so much "circus math" (math done for show, because there's vanishingly little other reason to do it by hand in current year) might have something to do with how so few students ever even get a taste of real linear algebra, statistics, calculus, and let alone more advanced topics?
My original reply in this thread was about the use of calculators being bad, particularly in elementary school. I wouldn't exactly call the ability to quickly multiply 5 * 6 in your head "circus math." It's more like a bare minimum basic skill.
It's like learning how to type properly when you want to become a programmer. Yes, it's not strictly necessary to be able to type as a programmer. However, if you are in full "hunt and peck" mode, you are going to be very slow at getting your thoughts onto the screen. It's going to take so much mental effort just to input a line of code that your concentration on the deeper aspects of the problem will be adversely affected.
The same goes for math.
I'm still rather happy my 9th grade teacher taught us row reduction, made us do a few by hand and one big 3x4
If you can't do basic arithmetic, you aren't even going to be able to do one by hand. I would even go so far as to say you can't even understand what's going on in the process or what the bigger picture means (e.g. that matrix-vector multiplication is a linear combination of column vectors).
Are you going to be able to understand the Gram-Schmidt procedure or QR-decomposition or singular value decomposition without being able to do them by hand? I don't know, but I somehow doubt it. All of these processes lean heavily on basic high school algebra. If someone needs a computer to solve a quadratic equation (with integer roots) then they're not going to have much fun learning how to diagonalize a matrix.
Meanwhile a Canadian friend reported to me that his "linear algebra" class in university consisted almost solely of doing row reductions of various dimensions by hand, all semester.
That's really sad. My linear algebra courses taught me about the four fundamental subspaces of a matrix (and their relationships to one another), various matrix factorizations ([orthogonal/unitary] diagonalization, QR decomposition, singular value decomposition), the Gram-Schmidt procedure, the Cayley-Hamilton theorem, and many many more interesting things, as well as how to prove a lot of facts about this stuff. We also worked in abstract vector spaces and polynomial vector spaces and learned about quadratic forms and the normal equation which will give you any polynomial regression you want.
Anyway, the whole point of all of this is not to say that every single person needs to know all of this math. Only that some do. And I believe that if you create a math education system that leans heavily on technology without any manual work, you will not raise any students who are truly competent enough to do it. I have first-hand experience as a math tutor (for elementary and high school students) for the past 4 years. I have seen upper-elementary and high school students, unable to do basic arithmetic without a calculator, absolutely struggle to learn more advanced concepts. The reliance on the tool is a huge barrier to deep thought.
I think I'm closer to understanding you now. For a professional, they need to know (that is, reproduce without looking up) about 70k ± 20k facts in their field to be effective (https://apps.dtic.mil/dtic/tr/fulltext/u2/a219064.pdf). Otherwise they're just too slow. Over time though, those facts change.
For some concepts, especially ones early on like arithmetic, sometimes memorization and mechanistic drilling is a prerequisite for solidifying the more general concepts at play which is what will really boost your skills (understanding what is actually happening when doing arithmetic). And sometimes you just need to do the simpler activity over and over before 'getting it' (like Feynman's counting with beans story he gave in order to justify his ability to explain fundamentals of QED without years of undergrad work first).
I agree that if you just show how to use the tool and never explain what's going on (and have no way to vet understanding that any explanation was sufficient) then it can cheat learning and further understanding... The most apt comparison is that of always asking your friend what the answer is, never sparing a single neuron for trying to figure it out yourself. Calculators can be like that to some people. Maybe the key is the subtle difference between "let me ask the calculator (or hey, Alexa)" and "let me use the calculator/Alexa". The second implies a belief of being able to do without, of seeing an alternative potential path towards the goal even if it takes more time.
And as one of the first bodies of math children will be exposed to with expectation to master, there's room in arithmetic for manual tedium to help solidify concepts that will be present in the future like copying things down in different places, value substitutions, learning and following (or programming) an algorithm, experiencing directly the tight feedback loop of seeing a problem or sub-problem and recalling a memorized answer instantly, parsing out numerical information from text or graphs, and so forth -- but eventually too, using a calculator or some device for almost all actual arithmetic. Even professional mathematicians will often outsource their brain for simple arithmetic, and anyone trying to multiply two 3 digit numbers in their head is likely just wasting the effort of trillions of firing synapses...
Anyway there's certainly room and need for some by-hand or in-head stuff, but I still maintain there's also a lot of wasted time (especially as subjects get more and more advanced) taken up by such activities and we'd do well to cast critical eyes on both traditional and newer computer-aware curricula. And then as a pipe dream, not holding everyone to the same pace.
From my experience tutoring (especially young children but also teenagers) I have discovered what seems to be a human universal: the brain is always looking for shortcuts. I have witnessed kids attempt to develop strategies for cold-reading me, the tutor, instead of learning how to solve the problem directly. They’ll make these educated guesses and then try to see how I react. I’ve had to develop a poker face just to get them to give up on that nonsense and turn their attention to the problem at hand.
I suspect that many tutors, parents, and even teachers never realize that kids are doing this. Then they’re shocked when a kid who seemed to be doing so well bombs the exam.
I feel the calculator is part of this phenomenon. In addition to immediately providing the answers so kids don’t have to figure it out themselves, it also creates the expectation that there are shortcuts in math and that teachers are just giving them busywork because that’s what adults do to kids.
That’s the heart of the problem, to me. There are no shortcuts to understanding.
Yes, but not all the knowledge, for then there would be no time for anything new.
> This means teaching young people how problems were solved, not just handing them the solution.
No, because this robs them of the time to build on that solution and do something new. This extends to all domains -- and all the way down to our most primitive survival needs. Most people have no idea how to build a home, they instead are typically 'handed' one (usually in exchange for money). Few can farm, hunt, and process food to the degree needed to ensure their own survival, but we eat the outputs from those who do, who are anyway also skipping steps by not having to learn and perform a solution that was maybe required hundreds of years ago when technology was less powerful. And taking advantage of all this knowledge we don't have to work for, then we can work for other, new things. This is not only fine but necessary for society to advance.
> Do we want a world where everyone is dependent on technology but nobody knows how it works?
We already live in such a world. Or rather, there exist critical things that no single mind fully comprehends.
So again, if you at least agree that not everything needs to be transmitted, then where are you going to draw the line? Do you think kids should learn to calculate cube roots by hand and prove that with a no-calculator exam? Why? How about logarithms by hand? (Though even before computers, pretty much no one did that either, instead they let a small number of people work really hard to produce many tables up to a few significant figures so they could use those results for bigger things.)
Don't you think the insistence on learning (and testing to demonstrate something was learned) so much "circus math" (math done for show, because there's vanishingly little other reason to do it by hand in current year) might have something to do with how so few students ever even get a taste of real linear algebra, statistics, calculus, and let alone more advanced topics?
I consider it a failure mostly of the education system that I didn't learn about rotation matrices until my first semester of college. They could have been introduced in 9th grade or perhaps earlier. But that would require cutting a lot of other stuff. (I'm still rather happy my 9th grade teacher taught us row reduction, made us do a few by hand and one big 3x4, but thereafter said "using rref() is fine". Meanwhile a Canadian friend reported to me that his "linear algebra" class in university consisted almost solely of doing row reductions of various dimensions by hand, all semester. Useless. But my same 9th grade teacher got very insulted when I wrote back some suggestions for topics he could introduce after taking my linear algebra class, even something as simple as the concept of the codomain, oh well.)