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Just teach my kid the math (solipsys.co.uk)
136 points by ColinWright on Feb 18, 2018 | hide | past | favorite | 113 comments

When calculating in my head or on paper, I value quick estimations moving to greater refinement. Common Core math seems somewhat similar to this.

I never subtract, only add up from low to high. If the two numbers have the same number of differing digits, I move from most significant digit to least: 73-57=10+6. If they cross a round number threshold greater than 10s, I get to a convenient base first: 2018-1995=5+18; 173-57=43+73=110+6.

For multiplication, I do something similar: 6x57=300+42; 42x23=(42x20=800+40)+(42x3=120+6).

For division in my head, I use a kind of binary search to get a quick estimate and then refine it. 846/42=42x10? 42x20+6. 20r6. 6/42=1/7. 0.15x7>1, 0.14x7<1. 20.14.

From my understanding, the Common Core approach to math is much closer to this approach I developed intuitively than the traditional methods. I'm sure many others developed similar approaches. If I want an answer accurate to more than two decimal places, or if the math gets too complicated to keep in my head, I just use a calculator.

>From my understanding, the Common Core approach to math is much closer to this approach I developed intuitively than the traditional methods.

It is. I recently had to help my daughter with common core math and I saw the similarities.

I think the biggest problem people have with it is at first it's impossible to help your kids with homework which is frustrating for both the parent and the kid. I had to learn it before I could help her with it.

Now all we need to do is finally covert to the metric system.

I grew up being taught ‘borrow and carry’ methods. They are really methods of ‘apply the algorithm and success’. Common Core can be a better way of getting an intuitive sense of the numbers and quantities.

Some examples are over the top, like the 32-20 in the article. But one of the common criticisms my parents and grandparents used to level at people my age (when I was a teenager) was that we couldn’t run a cash register—-we didn’t know how to make change.

Kids won’t start out adding and subtracting as quickly. The ramp-up to learning may be a little slower but they’ll come out with a more intuitive sense of numbers.

All of this is just my opinion. I studied Math and have a couple of young kids going through these methods in school now.

This is why I'm not opposed to the Common Core approach. It seems that they really emphasize math conceptualization and chunking instead of a formulaic process list that was used when I was a kid. Given the poor math scores in the US compared to other advanced countries, I really hope that this helps more younger people understand math.

These things all go in cycles though. People were complaining about the "new math" back in the 1960s. As Tom Lehrer put it "Understanding what you're doing is the important thing, rather than to get the right answer."



Well, it is super useful do be able to do exact arithmetic. For example, to not get scammed when you buy anything, or share a bill. I've seen American university students who don't know how to compute a percentage of a given amount - so I'm not very impressed by common core math to be honest.

That being said, it's probably a cultural thing too. I noticed that in the USA (at least in California?) it is customary to exclude the taxes from the price that is displayed on the price tag. Since the tax rate is different for different products, it's pretty hard to compute the actual price that you pay, and in practice you just take whatever you need and don't have an idea what you pay. In addition, it is not very usual to split the bill (I'm not talking about simply dividing the price by the number of people, but about actually computing the price of the stuff you ate, and not paying more than that).

When I go to the grocery store, I usually keep a sum of the price in my head while I pick products. That way I'll know if something is off at the counter.

> When I go to the grocery store, I usually keep a sum of the price in my head while I pick products. That way I'll know if something is off at the counter.

I do the same thing, but it does depend on how much I'm shopping. If its big for the whole family I keep a rounded tally in my head, and use that sum to determine whether or not I look at the actual receipt to see if something was wrong. If on the other hand I'm buying say less than 15 items then I'll keep up with the actual to the cent sums when I'm paying in cash.

You get a weird look sometimes when you get to the counter, they say 23.45 and you hand them the money immediately.

I've done this ever since I can remember, but growing up in the USA you just had the added step that you needed to know what the sales tax rate was for the area you were currently in, and if you didn't know it then I'd just use a default rate of 8%.

The mathematics standards for Common Core were released in June 2010. So college freshmen this year would have been in 5th grade (seniors in 8th grade) at the earliest when introduced to Common Core style mathematics. Probably later, since schools needed time to adapt to the new curriculum and train teachers in the new methods. Seeing as how Common Core places a heavy emphasis on developing skills in grades 1-8, I don’t think that current college students are a good benchmark for the Common Core standards. Talk to some college students in 5-10 years to get a better idea of how things are progressing.

Also I would say that early CA Common Core for math basically just pushes up the current curriculum up about half a year, a mild improvement; more importantly it's a cleaner spec for outcomes.

It doesn't really re-order the curriculum, it uses a few new terms, there's mildly more emphasis on word problems, and there's earlier introduction to linear sequences. It's just a few touch-ups here and there, presumably to prepare kids better for Algebra 1. The whole reform is a modest start.

> When I go to the grocery store, I usually keep a sum of the price in my head while I pick products. That way I'll know if something is off at the counter.

I wouldn't be able to CBA w/that during grocery shopping, I got way more to do whilst grocery shopping. Adhering my shopping list whilst checking for deals, and getting the correct items and correct amount. Figuring out in my head what combines with what recipe-wise.

For grocery shopping I ask for the receipt and then verify afterwards while discussing with my partner. Sometimes something's off in our advantage, sometimes in our disadvantage. Since its grocery shopping, we're talking about a few EUR here and there. If its non-grocery shopping, then I take quicker note.

As a rule of thumb the following rule appears to apply for me: the more expensive the item(s) the more cautious I am (denotes quality), but the more items the less cautious I am as well (denotes quantity). Why? Because its easier to do the former, and the latter becomes more difficult till we get to the CBA point.

Instead, I go for some deals, and then allow myself some treats as well, but never too much of either. The result is I mostly follow the grocery shopping list but I never know beforehand how much I exactly gotta pay, but I pay electronic either way; not with cash. I noticed when I've been in Germany and paid with cash, I took far better note of prices. Incidentally, groceries are generally cheaper in Germany as well...

> 2018 - 1995 = 8 + 15.

Now I too start with the addition, but my brain very much thinks in the vertical stacking of the numbers, moving right to left. How do we get from 5 to 8? Add 3, how do we get from 9 to 1 when adding: plus 2. How do we get from 9 to 0: plus 1, which we already have from the 9 to 1 case, so we pass the 1 here to the left. 1 to 2, plus 1, again using the value passed to us.

Little surprise I prefer functional languages.

I did A-level maths, I've programmed computers for 30 years since I was a kid.

Ive implanted algorithms to calculate solar irradiance varying by orientation, inclination and latitude.

I don't understand what my 7yo stepsons maths homework is asking him to do/learn.

It's just weird and I'm not sure he always does either, it teaches the how (weirdly) but never the why, I've started teaching him how to understand the practical uses for maths.

We play games now like "how many bricks in the wall" when we are out and about.

I struggled with maths as a child because I never saw what it was for, once you realise that it's the language of the universe things click why it is important, I'm not a mathematician nor do I need to be but I still appreciate those who are.

My (very limited) experience of more recent homework is that it makes perfect sense in the context of what the teachers are trying to teach, but no sense out of context to people who learned their maths decades ago.

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.

And techers are trying to teach how to make change?

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?

> And teachers are trying to teach how to make change?

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 a parent and I object not because I don’t understand it — it’s not too difficult to pick up — but because it completely misses the why of math. Being able to mentally estimate numbers or do simple arithmetic is a good skill, but having universal algorithms for solving problems and describing the universe is really what math is more about. I can trace my trajectory into computer science all the way back to practicing multiplication tables and discovering that I could multiply numbers in long form too large to fit in my head and yet get the exact result. It is the power of the algorithm. My kids are entirely missing that in their common core education...

> having universal algorithms for solving problems and describing the universe is really what math is more about.

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.

Traditional elementary math teaches you algorithms without discussing why they work or the cases where they break down. They seemed arbitrary and untrustworthy. I'm glad that worked out for you, but I don't think many students get the same value.

Part of the problem with this is that I feel elementary school teachers often lack understanding of why the math works. I remember hearing horror stories of people taking the Praxis and all the elementary teachers freaking out about the math -- one even said she had failed that section three times! It doesn't bode well that these are the people teaching our kids how basic math works. And it does show when they get to secondary schools.

> It is the power of the algorithm. My kids are entirely missing that in their common core education...

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.

Part of the problem though is that not everyone thinks about these things the same way. This is called Pedagogical Content Knowledge -- the meta-knowledge about teaching skills.

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.

Common core has an explicit goal of trying to foster an appreciation for different approaches, in complete recognition of what you said - that different people prefer different approaches, and also in complete opposition to what you and I grew up with. Schools used to teach one and only one way. I'm super curious if you have a problem with Common Core as implemented, or if like so many other parents who are scared of it, you haven't fully investigated what Common Core actually is. I know I might accidentally sound insulting or condescending, I am totally not intending that! I myself was scared of Common Core at first, and I heard and read lots of bad things from other parents who were scared of it, but hadn't looked into it carefully. My wife and I spent some time investigating the curriculum and the reasons behind it, and I'm a total convert, I now think what they're trying to do with Common Core is a great idea, and probably better than what you and I were taught when we were kids.

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.

Well it does come off as insulting and condescending and I don't really feel like pursuing this further.

Super sorry I made you feel bad. My apologies. Sometimes I argue instead of empathize, and undermine my own comments when I get excited and irritate the person I'm talking to. I honestly thought I might be helping someone who's going through something I've been through myself. I wish you and your kids success with their education.

> 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.

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.)

I think the problem with how these new concepts are presented, is that they are taking these ad hoc methods of doing arithmetic and trying to turn them into algorithms. My guess is that this is driven by the need children have to know how to get the “the right answer.”

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.

There are systems for doing rapid mental arithmetic. The Trachtenberg system is one that I played around with at one time. [1] I'm not sure how useful they are though outside of needing/wanting to do more rapid calculations in your head.

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

It's a bit weird when your mathematics education enables you to understand a book written several millennia ago, but not homework written several decades later.

... 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.

"Homework" is supposed to be practising skills they have already learned so they can make it more fluent, and have to concentrate on it less when they come to the next thing to learn. Learning things is done with the teacher there to help anticipate, identify, and correct misconceptions.

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.

As someone not from the U.S. I wonder, what does "outdated" mean? Can you give a simple example - e.g. comparison - what would be "math back then" and "math nowadays". Because to me math was always just math especially if it comes to numbers.

Here is one potential example from a number of years back. Maybe it counts as "back then." My daughter's algebra teacher was teaching the laws of exponents. When faced with "what is x^0," the teacher gave axiomatic reasoning of "x^0 is always 1. Nobody knows why, you just have to memorize it." When I confronted him about it, his response was along the lines of "we don't have time to get into concepts and teach reasons; we need students to memorize and move on to the next thing for the state tests."

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?]

>Nobody knows why, you just have to memorize it.

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.

Here's a simple example: Is sqrt(2)/2 or 1/sqrt(2) in "simplest form"?

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.

I wanted to echo the parent poster. I'm in my 30s and math made no sense as a kid - precisely because the problems were all artificial and uninteresting. I had to fight major math phobia in undergrad. For me personally, the most interesting application of math was figuring out orbits/positions of stars, etc. I have a 2 year old so if you have another favorite example like this, I'd love to hear about it.

> the most interesting application of math was figuring out orbits/positions of stars.

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.

You can teach multiplication as "here memorize that" or as "here is set of trics how to derive result" or even entirely diffreently.

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.

I don't mind this new way of teaching math, but on this note, I liked what Sal Khan had to say about flipping the process for school and homework.

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 a math teacher (Algebra II), I think this would be great. The problem is that most of my students won't go home and watch the videos. Either because they can't (lack of internet access, have to work to help support family, etc.) or are just lazy and won't. I've had to quit giving homework, even to the upper level classes, because nobody would do it. They'd just come in the next day and make excuses and have nothing done. This includes when they're homework was just to watch videos explaining problems they missed on a standardized test they took. Most just refused to watch the videos at home. It'd be great if they would watch the lectures and then be able to practice in the hour I see them daily, asking questions to clarify and such. But I can guarantee you it won't happen without a huge shift in the culture of education.

As a former inner city math teacher, I feel you. Getting students to do _anything_ outside the classroom was practically impossible. What worked for me for homework for a little while was having the students do 5 push ups for each day they missed homework in a row. I suddenly was getting 90%+ participation on students completing their homework! It was fantastic until someone told the VP, and the VP asked me if I was actually doing that, whereby they informed me that it was corporal punishment and absolutely against the rules. I could rant about that for a bit, but I'm sure you can guess what happened after the pushups stopped: homework stopped.

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.

> It was fantastic until someone told the VP, and the VP asked me if I was actually doing that, whereby they informed me that it was corporal punishment and absolutely against the rules

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

Exactly! I'm not in an inner-city area, but a rural one and there's a lot of the same problems. The poverty level of the county I teach in is 20%, and a lot of the students have to go home, and often get no support at home. And it's a clear night and day between the ones who do get support and encouragement and those who don'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.

I was at local selective school with students who overall did very well and ... only very few did homework or would watch videos unless it was graded or there was test.

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.

As a former math teacher, I would absolutely love this. I've been a fan of flipping the model ever since I first heard about it years ago from something on Khan Academy.

So, I know a lot less about math than you do. However, I have been studying lately myself, and reading a lot of Martin Gardner and Serge Lang; Both of those (and many of the other math types I read) seem to hold that making math practical can be useful... but it's not the point of learning math any more than learning how to spell correctly is the point of studying literature.

I think the key thing for making math worth knowing for a kid is making him realize that world is full of numbers.

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.

Most teachers are happy if you want to make an appointment to talk to them.

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



Understanding why has always been my primary way to learn things. Do you have any suggestions for books or courses I could dig into to better understand this approach.

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.

Everyone has a balance of these things, but in my experience there are two extrema to approaching maths. Some people are happy to be solving the puzzles, and simply enjoy the thrill of the chase. Others insist on knowing why they should be learning this, and what it's good for.

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.

I personally find Serge Lang's "Basic Mathematics" to be super useful in explaining the "why" of a particular formula in pre-college level math, as most books at my current level just seem to say "memorize this" He has an explanation of why x^0 is 1 and not 0 that doesn't involve calculus, for instance.

What would be an explanation that involves calculus? The only one I know is that x^(a+b) = x^a * x^b holds for positive a and b, and to uphold that pattern, it's necessary that x^0 = 1, x^-1 = 1/x and so on. For me, all of that was covered long before calculus.

Interesting. When I took high school math as a kid, there weren't many explanations, (and for some reason, I wasn't up for looking them up/figuring them out myself- weird, 'cause I totally was up for that sort of thing in some other classes) - just x^(a+b) = x^a * x^b and x^0=1 -- memorize these facts, with the emphasis on memorization rather than on the connection between those facts. Most of what I remember is a lot of guess and check and factorization; something I automated using the rom basic on my Tandy model 100. These were the mid '90s, and looking back with a less resistant mind, I think they really were trying to get us to develop a mathematical intuition, and that if I had read the math text book the way I read the history or English lit textbooks (they were full of stories!) I would have done just fine. But I still can't read math books the way I read literature, and I haven't figured out why yet. I know my first decade of reading, I skipped a lot, like if I didn't understand something, I moved on and filled out the blanks from what I did understand as best I could. (These days when I'm reading literature I get all excited when I see a word or phrase I don't understand, and I immediately stop and research. (the world is awesome for that today. Once I was blocked on my way home, waiting for a family of skunks to move off of the trail. I was introduced to rimbaud by miller, so I bought and downloaded a bunch of his stuff to my kindle. By the time I was done reading, it was dark and the skunks started to approach me, so I took off (without incident) but I wish I could do that with math, and I'm not quite sure why I cant. I mean, reading things I half understand, and halfway getting something out of it.)

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.

I actually referred to a different pattern to help knowing what x^0 is above. Not from calc, but just a different way than yours. If you start with x^3, x^2, x^1, x^0, x^-1, x^-2, and expand the forms, you get xxx, xx, x, ... where the pattern is each step divides by x. So x/x = 1 is the next, and 1/x is the next, and 1/xx is the next, ...

I don't know if this is useful but Richard Feynman used to play a game with his son where they would try to guess and compare the size differences between things as a way of learning units.

I’ve watched my daughter do her 3rd grade math homework both ways. She’s an ardent rule-follower at her age (and can’t quite understand why the boys in her class can’t stay out of trouble) and really enjoys doing her math homework. Watching her work on the extra credit math assignments and her get so exited when the multiplication problems extend into 8 digits is a lot of fun.

Often she will do the problem the “new” way taking up maybe 10 minutes and a whole sheet of paper which looks more like an art project, and then when she’s done check her work the “old” way in a few seconds, and then go back and find the mistake in the drawings.

She’s happy to do it “because that’s the way [my teacher] wants us to” but she’s faster and better at doing it the traditional way.

My personal opinion is that CC is perhaps better at teaching basic math skills to the bottom 50% and is less likely to lose students along the way, but at the cost of holding back the top 50% of students approximately 1 full grade-level by the time they graduate high school. From my limited sample sized poll students see the new methods as being just as ridiculous as parents see them.

And parents are generally pretty frustrated about this whole fiasco: https://youtu.be/wZEGijN_8R0

> My personal opinion is that CC is perhaps better at teaching basic math skills to the bottom 50% and is less likely to lose students along the way, but at the cost of holding back the top 50% of students approximately 1 full grade-level by the time they graduate high school. From my limited sample sized poll students see the new methods as being just as ridiculous as parents see them.

This is an issue that would occur regardless of CC, I believe. Unless you can differentiate them into separate classes fairly early, and are constantly checking to see if everyone is in the correct group (for instance, I have several students who are labeled as gifted in math who can't do simple addition without a calculator... they were listed as gifted in third grade, and then never tested again), one of the two is always going to suffer. Either you help the higher level kids at the expense of the lower, or the lower at the expense of the higher.

As for 'students see the new methods as being...' it's because they just mimic how their parents see them. Now, I don't teach elementary school so I don't know what they do, but reading comments here it seems they basically teach students how numbers work. And that's a good thing. I have high school students who didn't know that to multiply a number by 15, you could multiply it by 10 and by 5 then add the results. Along the same lines, they didn't know you could multiply by 10 then divide by 2 (or divide by 2 then multiply by 10) to get multiplication by 5. And this impacts them a lot more when we discuss variables and polynomials (which is mostly what Algebra II should be). They have no concept of how numbers work, so they don't understand why x^3/x = x^2.

The amount of time and space needed for her to do the problem on paper are irrelevant. The only important thing is that she's understanding.

It’s more the incredible amount of time spent drawing it out and ultimately often arriving at the wrong answer due to drawings becoming so overly complex.

Despite her spending tremendous care at trying to neatly diagram exactly as she was taught, the methods apparently do not scale well past 3 significant digits.

It becomes an exercise in frustration which can take up to 10 minutes trying to solve a multiplication problem which she knows how to do properly in the blink of an eye.

There’s lots of graphite going on the page, but I’m highly skeptical of any learning. Certainly any love of math quickly goes out the window during the process.

The woman in your video is just plain wrong. She's as wrong as if she claimed the sky is green or 1 + 1 = 14.

The Common Core doesn't specify curriculum. Period. Anyone who cites a specific problem or worksheet as evidence the Common Core is bad doesn't know what they're talking about. It's just evidence of an incompetent teacher.

What seems to happen in practice is the parents end up asking the teachers why they're teaching that way, and the teachers can't actually explain it. Obviously, the parents don't walk away impressed.

When I got my teaching credential for high school math, it became apparent and was openly recognized that many grade school level teachers choose that level of certification because they are, themselves, afraid of "math expected at the high school level." We expect many teachers who are math-phobic to lay down the foundational layers of mathematics with our kids. They cling to whatever the pre-made material and lessons are in the texts and often don't know why things are to be taught the way the texts want. Scary stuff.

I don’t have any problem with common core or “new math”. I was a math major myself and have two kids in public school in SFUSD, and I think it’s valuable to develop ease and fluency in problem solving. An example in another thread, 2015 - 1985, is a good example. I like the idea that kids learning arithmetic would see how to solve this without a plug and chug approach.

Here’s why I don’t find it promising: this is mimicry. The US found that other countries that teach math well don’t rely excessively on plug and chug. But these countries draw math teachers from the top math students. These teachers teach this way not because there is a new set of standards to adhere to but because this is how people who are good at math do math.

Until the US draws math teachers from top math students, the new math will just be a new set of plug and chug steps. This is why the parent in the blog entry is frustrated- to them, new path is just more steps than old math. It may feel that way to the teachers as well, if they aren’t understanding what they’re doing.

If we want to mimic Finland, we need to start by drawing teachers from the upper echelons of math students. And once you do this, much of this “new math” will happen organically. That may in fact be the only way it can happen.

The problem comes with how elementary schools are structured here (at least the ones in the district where I teach). If you even have enough teachers for each teacher to only have one grade (i.e. no split classes), they've got one teacher teaching each class until 4th or 5th grade. So you have the same teacher teaching history, science, math and language arts. This person probably struggled with math and often shuddered at the thought of the math portion of the Praxis test to become certified. They don't know how to do the math, but, because of how elementary schools are taught, you can't just hire a math teacher to teach math to all the students.

Personally, I could easily see the benefits of shifting the students to a schedule where you went to a math class, taught by someone who knows math (i.e. the best math students), at the very beginning. It would also help because you could then make it where someone who is good at teaching, say, history, doesn't have to worry about passing a math content exam. Or at least one as in-depth as the math teacher.

But, you also have to attract people to teaching. It's a thankless job, and, in my state, they're currently under attack on pension reform and pay and such. Our governor wants to remove mandatory sick days, for instance. And he wants to guy the pension. Or, he'll keep it all and gut all these other programs and then blame it on the teachers. Public education in the US as a whole is under attack, and that's part of the problem of getting the best math teachers. I'd be willing to bet Finland doesn't have to deal with the issue of charter schools like is currently happening in the US, either.

> We need to communicate the true goal of given exercises to parents.

This is spot-on, and can't be emphasized enough. The goals also need to be communicated multiple times, more clearly to the students, in writing in the homework. So much of the homework asks what seems like trick questions when I look at it. Whenever they weren't paying 100% attention in class, it seems like trick questions to them too. The exercises feel like they intentionally withhold stating the goals clearly for fear that it would give away the answer or fail to make the students struggle enough on their own.

> Make showing/explaining your work interesting.

This is true, but tough. Not every exercise should be an essay answer or story problem. The homework my kids get has lots of variations on exactly the example the author gave, and it doesn't feel like it's helping generate any appreciation for showing the work or for coming up with alternatives. It usually just feels like forced busywork.

My real appreciation for showing my work only came in college physics when I finally realized that I make too many mistakes when I try to think my way through steps. Writing down every minor step was finally more efficient and accurate than not doing it.

This is a really good essay on what's wrong with how we teach mathematics to children: A Mathematicians Lament https://www.maa.org/external_archive/devlin/LockhartsLament....

It's rather long, at 25 pages, but it's a very good and entertaining read. It was written in 2002 but it still holds water today.

My problem with the "new math" is the lack of volume. Developing a sense for the numbers is best done by working more with numbers; in the "old math" kids had to do thousands of additions in the first and second grade, and developed ten times the sense of numbers compared to kids today.

Theory is theory, but practice is practice. Just get over the addition and multiplication, please. That's not where the true math is. Don't spend so much time on finding the true meaning of addition by carry, you won't have time to teach kids logarithms and integrals later.

> the "old math" kids had to do thousands of additions in the first and second grade, and developed ten times the sense of numbers compared to kids today.

This must be variation from school to school or state to state. My kids are stuck with more homework than I had when I was a kid. I actually wish they had less homework, they're not getting enough exploratory time during their school days. They also have a better intuitive sense of numbers than I did.

> Just get over the addition and multiplication, please... you won't have time to teach kids logarithms and integrals later.

Generally speaking, I think calculus is being taught earlier now than when I was in grade school. My kids (grades 5&7) are being introduced to some interesting arithmetic concepts that I didn't get when I was a kid that I think will complement higher math.

As a parent, I'm not seeing any issue with lack of volume or of progression toward higher math concepts, FWIW...

> in the "old math" kids had to do thousands of additions in the first and second grade, and developed ten times the sense of numbers compared to kids today.

I don't think that's generally true; the kids that were particularly successful in math certainly developed more sense of numbers than mod kids today, but AFAICT very many developed almost no sense of numbers.

Learning by rote doesn’t teach you anything.

All humans learn their first language by rote. Much more of higher ed is rote learning than you think. If it wasn't rote, we would never have to see examples or practice something, but the majority of what we actually spend our time doing in school is seeing examples and practicing. If there was no rote learning in math, doing long division would be an order of magnitude more painful, and you would always derive your formulas from scratch.

I believe critical thinking is as important as anyone does, but there's no critical thinking without foundational skills and knowledge. Education simply cannot be 100% critical thinking, even if we wanted it do.

"Rote learning is widely used in the mastery of foundational knowledge."


Spending lots of time on the treadmill won't make you a great soccer player, but still trains a basic skill useful for the game. Similarly, doing lots of arithmetic by hand isn't enough to solve all kinds of real-world problems, but it does help develop an intuition for numbers, which then helps with the problem-solving part.

Rote arithmetic may help you develop an intuition for numbers. It's hit or miss though and there are more effective and efficient methods than just doing a large volume of arithmetic. I believe the current trend in Math education is intended to take advantage of the more effective and efficient methods.

Your treadmill is a good example actually. Much of the exercise world has realized that just jogging for a long period of time while effective is also not the most efficient way to exercise. Instead mixing periods of hi intensity workout with periods of lower pace workout can allow you to get more value with less time from you routine.

Are you sure that it doesn't teach you anything? Somehow the students of "old math" were able to put a man on the Moon.

Only a fraction of them. And those of their generation who were that good talked about math as something to be understood rather then memorized.

So you're advocating the return of the slide rule? As that's what was used for most calculating, the "old math" method being too cumbersome for such large problems.

As someone who actually learned to use a slide rule before pocket calculators became available, that's probably not the best example. Slide rules were good for getting "[EDITED: precise not accurate] enough" results in various situations where they could do so more quickly than solving by hand. You still needed to understand orders of magnitude, precision, and generally having a feel for numbers to use one.

Note also that a lot of calculations on the Apollo program were actually done in part by hand by human "computers." There was even a whole movie about it a couple years ago: https://en.wikipedia.org/wiki/Hidden_Figures

Those human computers used slide rules, as even with them the problems were so ridiculously complex they took months. Their were many different slide rules with varying degrees of precision


Really? In the case of math, it provides you with the ability to add/multiply/etc. Now, maybe you don't think that being able to do fast and reasonably accurate mental or paper-based arithmetic is important in a world in which calculators are ubiquitous but rote learning does help give you that ability even if you don't understand deeper mathematical concepts.

Asimov's short story "A Feeling of Power" is a fun read by the way.

Rote learning also has a role in things like language vocabulary but in that case there are certainly differing approaches to developing language skills. My understanding is that there's a lot of memorization in things like medicine as well.

Is there a common core method to teach arithmetic in different a different base, like 8 or 16?

The "old way" algorithm still works with different bases, you just change extend or truncate the rule for when to carry.

Easy to teach your kid hexadecimal arithmetic using old way, like how my parents taught me in like grade 4 or 5 - and now I program for a living, haha

If you ignore the "anchor to 5s" part that is ridiculous, the other parts work in other bases too.

  D3-A7 = (D3-B0) + (B0-A7) = (D3-B0) + 9
        = (D3-D0) + (D0-B0) + 9 = (D3-D0) + 20 + 9
        = (D3-D0) + 20 + 9 = 3 + 20 + 9 = 2C
(I swear that I didn't cheat to make the calculation and I made it in the "string" representation of the hexadecimal form, without looking at the decimal form.)

In hexadecimal can try to anchor to 8s, it should work. Perhaps anchoring to 4s may be better?

[Disclaimer: I don't like it, but the method to do the calculation works in other bases.]

This is like the things you develop yourself as a child although broken down and over simplified.

Maybe this should just be for the less intelligent children who do not have the skills to learn their own more efficient method.

I can see this wasting too much of an able pupils time.

Do you have streaming in America where you filter off children by ability like in Europe?, If so then maybe this is good for the lower streams to gain them insight into what more able pupils do anyway. (Though maybe not so convoluted).

> For instance, our standard arithmetic algorithms are somewhat bizarre - they are the end result of a human process of codifying arithmetical thinking, designed with the extra goal of using as little of precious 17th-century ink as possible

I've long thought that this is partly why Mathematics are, paradoxically, at once terrifying and tedious for students. To learn a second language, you must immerse in it, using it as much as you can. Math has been attached to a language that is not used in daily life (except in a few situations, mostly very basic usage) and where the parallels to real usage feel contrived. If they weren't contrived, the question "what's the point of learning this" would get much clearer answers.

When I was a student, I found it curious how I had trouble expressing certain concepts in that language, but when working with the same concepts in a programming language, things were much clearer.

I can't offer a solution, but I think learning programming at a young age can help, though I'm personally against requiring it in classrooms, increasing the pressure on kids who are already exhausted with the rest of the curriculum.

> We need to be sure not to insist on one approach when analyzing a problem

It's encouraging to hear from educators out there who understand communicating the value of problem solving and comprehension is more important than rigid adherence to methodology and procedure.

I remember the most useful aspect of my secondary school education (12-16yr) was learning how much I liked problem solving. However I came to this realisation through solving various simple math homework problems without following procedure (procedures which I remember finding particularly irritating for not explaining themselves) only to be lectured on how I had not followed the rules in spite of arriving at the correct solution and understanding the problem (albeit not well enough to figure out how the procedures worked).

My teachers did exactly the opposite of what the above quote is saying - they insisted on one approach, without explanation, and it totally alienated me from math at the time. Thankfully I was never a very "good student" and it didn't dissuade me from enjoying the problem solving aspect, but it did leave a lasting impression of "i'm not good at math".

its my understanding that common core math is of the theology that its more important to learn how to think than to solve the question.

its also important to understand that if a kid uses 'the old way' to solve a question, they do not get points for it. remember, this is coming from an ideology that being taught how to think is priority number one, and priority number two is to not deviate from that thinking.

it was always frustrating in hs, especially calculus when I would get 0 points for questions i had right because i used a shortcut, trick, or simply solved it more efficiently. then I got to college and the professor encouraged that. on the first major exam I scored a 96, far above anyone else, and the professor asked to shake my hand, because no one ever solved equations so efficiently before.

i attack and do problems differently, sometimes out of necessity. dyslexia is an adversary of mine, and i find it better not to fight it on its home ground. i also read slow, and take much longer to absorb a problem into my head than other people. but once i do, i own it. unless people make me do it their way in the process. its like running 100m hurdles with 30 hurdles.

im sure you love how you solve problems. you can do it fine your way, i cant. once you start forcing everyone to think one way and evaluating them, your measuring each animal how well they climb a tree.

I just asked my stepdaughter "What is 33 minus 18?"

"Because ... 18 plus 18 is 36 ..... it's 18 minus 3 ... 15!"

Isn't it suppose to be:

  18 to 20 is 2
  20 to 30 is 10
  30 to 33 is 3
  2 + 10 + 3 = 15

Also, how does she know it's 18, plus an additional 18? Why not 18 + 20? What happens if it's 33 minus 21? 21 + 21 is 42...

Just curious - but at least she's got her own system that works!

"If it's 33 minus 21, I'd probably do 3 minus 1 and 30 minus 20, because it's easy numbers."

> The 21st-century is not looking for humans who serve as calculating machines, but instead it seeks problem-solvers and innovative thinkers.

I distinctly recall how one of my grade school teachers justified learning maths: "In the future, you aren't going to be walking around with calculators in your pocket!" Yet today I routinely check even simple calculations with the calculator in my pocket.

I've found that error-checking is the key to doing 'well' in maths, yet that's something humans are bad at and something machines are great at. Given this, I'd love to see a future where learning to do math strongly emphasized learning how to best leverage computers for error-checking. That's a skill I wish I had developed earlier...


“remember, base 8 is just like base 10, really, if you’re missing two fingers. Now, anyone who gets this right can stay after class and help clean the erasers...”

The funny thing is that New Math's considerable focus on non-base 10 arithmetic and boolean logic was an academic exercise largely divorced from practical application for the vast majority of people at the time. While that may still be true in terms of the overall population, of course they're highly relevant to both programming and digital circuits.

Indeed. I’m post new math, but I had Boolean logic in 10th grade and it was a very helpful step up.

That's <adjective> Arithmetic not Mathematics

I thought Hacker News preferred original source material rather than reblogging.

Or is the original source not preferred in this case for some reason, for example being hosted on Medium?

Original source: https://medium.com/q-e-d/just-teach-my-kid-the-expletive-mat...

And just for completeness ... now reverted because there is a good reason for re-hosting the article - Medium is blocked in some countries, and the re-hosting is to make it more widely available.

The original source you linked to in the article is a lot easier to read. sorry/


That's fine - I'm hosting the copy (with permission) because medium is blocked from some locations.

Personally, I hate medium's default settings - I just feel like my face is being pushed through mush. But that's a personal thing, and I've certainly lost that battle.

<fx: shrug />

Thank you for that - Medium is blocked in Malaysia, for example, which I can bypass by using Google DNS on WiFi, but not on 4G...

Interesting. Is it blocked for content reasons (i.e. it hosts political material that is offensive to the censors of the local internet infrastructure) or techno-political reasons (Medium-the-hosting company is banned from providing content in the area due to the company itself)?

It has horrible always-there headers and footers that take a third of my 12" screen.

Web designers really need to get their act together. Nothing of such bad quality gets out of the door in any other real industry.

Thanks, we've updated the link.

As explained elsewhere, this is a copy, hosted at the request of the author, because Medium is blocked in certain countries. Your change means that people in those countries will, if they come to HN, no longer be able to read this article.

I'm not sure that's a good thing - did you take the other comments into account when you changed the link?

Alright, we've changed it back.

Thank you - I've added some text at the top of the re-hosting to make it clear why it's being hosted there in addition to Medium - I hope that makes it clearer.

Is there a way of avoiding this problem in the future, and possibly reducing your workload?

Is the issue that parents don’t understand because it’s different from how they were taught or the teaching method does not work?

There is a reason why most people hate math.

IMAO the current system of teaching math is ancient and above all deprecated. We should not punish our kids with it. I mean, what's the point of drilling stupid sums if you'll hardly ever need it in your life? I think it is totally stupid and irresponsible if anyone does calculations without a calculator for almost any application, not excluding: store checkout, health care, aviation, etc.. The current educational methods are way pre-computer era, it desperately needs an upgrade.

So why not start teaching kids concepts and let them use the technologies we create to solve problems that our minds are not wired for? Ever seen a teacher without a cheat sheet checking the answers? Guess why? Because their own calculations are too slow and above all not to be trusted! And should I not use a debugger because theoretically we can write proper code without it? Come on..

Please stop teaching our kids deprecated workflows. Teach them the best practice, the best way we know and can do now, because: "As the twig is bent the tree is inclined".

You want us to teach them best practice? Then we need to teach them maths, without relying on calculators. I teach secondary maths (specifically Algebra II), and they can't understand what I'm supposed to be teaching them because they have no clue how numbers work. Why don't they know this, if they're 15/16? Because they were given calculators at too young of an age and know "just push the button". What you're arguing is for a complete removal of teaching math, period. Also, why would you want to pull out a calculator every time when it's often quicker to do some mental calculations in your head? For instance, I've seen my students multiply by two in the calculator. Then be amazed when I can just say it.

Over-reliance on the calculators is already a huge issue, and we don't need to make it worse.

> What you're arguing is for a complete removal of teaching math, period.

No. I suggest focussing on explaining concepts, not drilling sums. And explain them why in ancient times the teacher used to use a cheat sheet because drilling didn't work because science found out our minds are not wired for that.

Besides, from over 7 billion people on Earth, how many need Algebra 2? Please try to be honest and give us here an estimate teacher. In your life Math is important apparently, but in most lives it's not. Teach it to those interested please and don't push it through the throat of those not interested.

> I suggest focussing on explaining concepts, not drilling sums.

The research I've seen suggests that having a bunch of stuff at your finger tips then makes it easier to have the concepts crystallise. Trying to start with just the concepts doesn't work.

So what's needed is both, significant practice to help embed the simple underlying facts, with extensive exploration of the links between the facts to help the concepts emerge, be identified, and then distilled.

To me, it seems that almost everyone goes full extreme: "Must do this", "Must do that", whereas, as in game theory, it seems obvious that a mixed strategy is likely the best option.

I mean, what's the point of drilling stupid sums if you'll hardly ever need it in your life?

"Hardly ever"? So when checking the restaurant bill, the change at the grocery shop, the tax return, a bank investment proposal, the costs of building/renovating a house, the time left for a mortgage, do you always use a calculator?

If I need to be sure? Yes!

Haha, I do not trust machines -- I check the calculator by doing the math by hand.

The idea is to give them "number sense." Kids trained on calculators don't bat an eye at 12+6=72. "The calculator said so." "What do you mean I pressed the wrong button?"

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