... makes perfect sense in the context of what the teachers are trying to teach ...
What's the point with homework then?
School takes the best part of a childs day, and when they don't make it through, the parents can finish with a tired child.
What's the point with homework then?
School takes the best part of a childs day, and when they don't make it through, the parents can finish with a tired child.
If you want the parent to help with the homework, the parent will have to know the material they want to help with. In maths, that's rarely the case unless it's the same thing they learned. And that's now out-of-date.
Parents say they want their students to learn new, modern maths, and then insist that they are taught exactly what they were taught. And hated.
Having said that, I'm really not going to defend anything about the existing setup and system. This is a huge mess, and while I have my own ideas, there's no way a discussion here will change anything, and it's likely that things I do say will be taken out of context, twisted, and mis-understood.
For me math started to click when I started writing simple games for the spectrum, I realized quite quickly that what I needed was trigonometry and once I knew the name I could go to the library and get out books.
I was implementing a text based destroyer game (similar to mtrek/jtre though I didn't know those existed) and I had to implement handling bearings from the ship to the enemy ships, handle firing guns and missiles all that stuff, I made lots of horrible simplifications (shells experienced no air resistance so flew in perfect parabola's, the surface of the sea was a perfectly flat plane, no curvature (non-euclidean geometry would have been a step too far at 10-11) etc) but fundamentally it worked and was playable as a game, you could issue orders, fire shells (and later missiles though that made the game a lot more brutal and short) with bracketing and taking into time of flight - I was a strange child.
I got the original idea from a ZX81 kids book on game programming that implemented a simple side projection game where you set the elevation and velocity of your shot and aimed for the opponents ship, it was simple (couple of pages of code) and I wanted to do something a bit more realistic - as a side benefit I ended up reading up on naval warfare in quite a lot of depth, again a strange child.
Likewise with addition. You can mostly memorize that or you can tech them how to simplifie calculations. Thing like "17+5" is actually "17+3+2" which is easier to calculate.
More and more, educators in the US are realizing that memorizing things just wont work. You get students who can't recall if x^0 is 0, 1, or x, and they lack the mental model to figure out the answer. Math nowadays strives to give students the mental models to derive answers. I would hope today's students to go "I know x^3 is xxx, and x^2 is xx, so x^1 is x, so the pattern is divide by x, so x^0 must by x/x or 1".
This applies to even more common things. Many of us who did well in math did so because we saw patterns and relationships. Now educators are striving to help students see those patterns and relationships.
In similar fashion, and to address a concern a few comments up, many older US students (and adults for that matter) can't figure out percents. They were taught a formula of [base * percent / 100] and then they have to unpack the term "percent," where they remember there was something divided by 100, and then they find themselves all mixed up because of failed memorization methods. I can recall in grade school and high school, many kids did not realize they could find 20% of something by multiplying it by 0.2. They first would multiply by 20, then divide by 100, and when I told them they could just combine those two steps, their mental model would break. It got even worse if you tried to tell them "what is 1/10th of the item? now double that, and you'll have 20%".
To summarize, "outdated" vs "nowadays," the students would try to add algorithmic steps to their mental tool box that work in very specific situations, and now, many educators are trying to help children internalize concepts.
[edit, formatting. I wanted the xxx and xx to originally have an asterisk between them to show multiplication, but then everything went italic. Why does formatting have to be such a pain?]
While that's clearly a bad answer, it's not clear that an answer to the effect of: "That's a great question but answering it requires getting into a number of somewhat advanced topics. But if you're interested, here's a good place to start." would be.
Personally, I actually could not tell you the answer myself--although I could look it up.
I agree that, where practical, understanding "why" is preferable. But, even in more advanced subjects, you sometimes just don't have the basics to derive everything from first principles. High school physics is an obvious case when most students haven't had any calculus yet. But even intro courses at a lot of good colleges often ask you to accept certain things as true because understanding why they're true requires significantly more background in the subject.
The answer depends on whether you have a calculator handy. Because computing sqrt(2)/2 from a square root table and the long division algorithm is a lot easier than doing 1/sqrt(2). But on a calculator, 1/sqrt(2) is just as easy and it's simpler to write.
But at least as of the last 90s we were definitely teaching kids to put their radicals in rational-denominator form for basically "magic" reasons: the students had no idea why, and I suspect neither did most of the teachers.
He proposed that students should watch lectures at home with a few problems for familiarity, and that classroom time should be spent working through problems in a guided environment. This allows students to pause, rewind, rewatch, etc. the lectures and move at their own pace, while getting expert guidance on the problems in person (and not relying on parents who may not have even looked at the source material in decades).
As for the shift needed in the culture of education, I totally agree. There needs to be a shift in accountability for the students, their families, and for schools. While there were parents who hoped their kids would do well at the school where I taught, there was a pervasive home attitude of school will get you nowhere. Many of the kids had no role models in their life where they could see any value to education. One VP I knew, in a neighboring district, married into a poorer latino family. They valued hard work, but only really understood hard manual labor type efforts. He was constantly hounded as he was going through schooling by them. "Why are you wasting this time at school? Why are you not doing more to provide for my daughter? Go get a real job." After finally becoming a teacher and then a VP and making more money than any other member of the family he married into, they started to see how education could be an investment. I don't know how you do that on a scale where the majority of socio-economically depressed families and communities can be exposed to the value of education.
Sadly, the biggest issue I see is that they want school to be, basically, training for a job and real life. So, since you'll never solve a system of equation on a job (or so they believe), they don't see why it should be taught. But, yeah, that's not getting into accountability issues with students and their families. It's a frustrating cycle, and is likely going to push me from teaching before too much longer.
Basically, grades were instead of pushups. We did good overall, but when it comes to homework with no punishment for not being done, people skipped it.
Funny how being told to perform physical activity is seen as punishment, but having to perform complex mental activities isn't.