If you'd care to send me a photo of an example of the homework then I can try to give you some context and understanding. My contact details can be derived from my HN profile. I might be a little slow to reply at the moment as I'm in the middle of other things and only surfacing occasionally.

I understand “anchoring to fives and tens” but I don’t understand why anyone thinks it’s a good way to do anything except confuse kids.

Does independent research prove these approaches increase numeracy?

No, the "making change" comment is an example of how these techniques of thinking about numbers do turn up in real life.

> I understand “anchoring to fives and tens” but I don’t understand why anyone thinks it’s a good way to do anything except confuse kids.

It's a part of a broader strategy to talk about the structure of numbers and the operations on them.

> Does independent research prove these approaches increase numeracy?

Yes. I don't have access to it and can't give you any links, but there has been a lot of research about how to move away from the rote learning of algorithms and move towards understanding what numbers are and how they work. I've seen some of the research as a peripheral part of some of the projects I've been involved with, and it seems that these sorts of exercises really do serve to help students see more about what's really going on.

The problem is the parents pushing back because they don't understand it, and don't believe it can be useful or helpful. For me, it seems obvious that this is a better way to think about things.

I'm not sure I understand what you mean. What universal algorithms are there currently for solving problems and describing the universe? And why are these universal algorithms more important than estimation skill in practice?

Are you suggesting that there's always only one true way to do things in math, and that the quest for the one true way is the only noble pursuit?

FWIW, to me that sentence somewhat reads like a pre-conceived idea about math that could possibly impede your appreciation for valid alternative approaches.

It's worth considering that in computing, the algorithms that have been used for math in processors and standard libraries have been changing dramatically ever since the first transistor. And it's not because the newer algorithms are more universal, it's often because we're simply making different tradeoffs over time. Sometimes it's because better algorithms are being invented, because the algorithms we thought were great turn out to be not universal. Sometimes it's because a new fast approximation formula is discovered that is accurate to the precision of the machine.

There's very little in computing today that is "universal", despite the pedagogical emphasis on this abstract idea. There are, however, lots of algorithms that are canonized. Math constructs that seem universal because everyone knows them and they're popular, and alternatives aren't well known, enough that it becomes hard to imagine the alternatives.

> My kids are entirely missing that in their common core education...

Are you absolutely certain that's true? Common core, as the author pointed out, is still required to teach the old algorithms. Are you sure you're not having an emotional reaction to not quite understanding why some of the new methods are there and/or being unfamiliar with them? Your reaction here is exactly what this entire post was addressing.

From the article, "the Common Core State Standards in the U.S. still requires student fluency with these algorithms"

> Being able to mentally estimate numbers or do simple arithmetic is a good skill

It's not just about the ability to estimate or do simple mental arithmetic. Look at this example from the article: "Describe three ways to see why 35 x 12 is 420". Knowledge of these 'three ways' provides a deeper conceptual understanding of multiplication. I want my students to have a shot at deriving the algorithm for multiplication, not just implementing it.

One of the reasons common core math is so controversial is because it's one of these "shared experience" topics for which everyone has personal experience and considers themselves an expert, yet most don't realize that their own experience is probably quite different from nearly everyone else's. And the reason theirs is different is in large part because their mind works differently and they naturally fell into using skills matched to their own strengths and weaknesses. Take 20 people and pair them off into 10 pairs, and have one person in each pair walk the other through how they calculate 876 - 381 in their head. You will 10 different results[1]. The problem with common core is that it teaches a handful of these approaches rather than letting the student develop their own customized approach.

[1] I have actually done this experiment. Of the 8 pairings, I think we had 7 different algorithms. Only two people did something close enough to be called the same thing.

The implementation and communication about what Common Core is doing needs a little work. The author mentioned this, and I think he's absolutely right. The fact that math-educated parents like you are reacting negatively to Common Core is a symptom of the plan not being very well explained to parents, even though once people see it clearly, they often agree that it looks like a superior way to teach math.

I guess I'll just add a side note that fostering this appreciation for different approaches might actually be getting more directly at the universality you mentioned above. There's a deeper truth behind the meaning of subtracting two numbers than any one algorithm holds, and perhaps the only way to see it is to be familiar with multiple algorithms.

I found it a huge boost to my ability to do computations without paper or other tools when I learned it in elementary school; I think it's valuable, but I'm not certain that the manner in which it is often being taught now is helpful (which, AFAICT, is not so much a matter of the Common Core standards as poor curriculum attempting to implement the standards.)

The traditional algorithms aren’t very intuitive, but they work. Trying to make an algorithm out of the various ways you can break up a problem, ad hoc, seems bananas.

[1] https://en.wikipedia.org/wiki/Trachtenberg_system

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.

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.

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.

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.

Funny how being told to perform physical activity is seen as punishment, but having to perform complex mental activities isn't.

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.

One way I intend to do that is providing kids with a lot of measuring devices as toys. Rulers, calipers, pirometers, timers, you name it. If it can give you number for something real then it's a good toy.

Or the book "Maths for mums and dads" is useful.

https://www.amazon.co.uk/Maths-Mums-Dads-Mike-Askew/dp/02240...

https://www.amazon.co.uk/More-Maths-Mums-Dads-Askew/dp/02240...

My oldest who is 8 is quite good at math and on the math team and his school does do a lot to teach them but I want to help him how to think.

In almost every other field I can but I am not good enough at math to understand which "games" to play.

Both are valid, but if there is A and B and you're naturally good at A, the natural tendency is to ignore or avoid B. But that then limits your box of tricks, and makes you one-dimensional. By all means take advantage of what you're good at, but work hard on the other aspects to enrich your abilities. Also, they cross-fertilise, so you get more than you might expect.

With regards your oldest, it's worth trying a bunch of stuff to see if you can work out what they prefer. Then do both, but be prepared for having to work harder on one aspect than the other. Your oldest might like puzzles for their own sake, or might insist on knowing what it's for. Explore both. Find maths in strange places, work through puzzles just for their own sake.

The critical key is to be doing stuff all the time.

Caveat: I don't have children, I have no formal training, but I do a lot of outreach, enrichment, and enhancement.

the calculus explanation I heard from a friend that I was talking about this about was extrapolating from x^.5 x^.4, etc, etc, and as the exponent approaches 0 the answer approaches 1 for all non zero values of x - but this was maybe two years back and the friend in question was educated overseas, and I still am not educated enough to use calculus notation to write that down (or even really to understand if I understood him correctly.) so it could be garbled, or only tradition on the continent, or something like that.