[opsunit]

> We start by finding a number that forms a perfect square that is close to 33. Here, let’s pick g=6, since 6^2=36.

As somebody who is numerically essentially blind I'm not only incredibly jealous and in awe of most of the comments in this thread but also utterly perplexed by the above. How does that even come to mind that 6 is a good starting value? Do you people just intuit this stuff or do you rapidly run through the options? If you're asking me to come up with something that when multiplied by itself is something near 33 then you're in for a long wait.

There's another comment in this thread:

>Everyone generally knows the perfect squares up to at least 12, and then for bigger values, you can use even powers of 2, which I assume people also know.

Do they? Jeez.

[MobiusHorizons]

In order to do math in your head, you have to have certain basics memorized, and you combine these to come to a solution. It’s pretty common to learn the perfect squares up to 12 or so

1x1 = 1 2x2 = 4 3x3 = 9 4x4 = 16 5x5 = 25 6x6 = 36 7x7 = 49 8x8 = 64 9x9 = 81 10x10 = 100 11x11 = 121 12x12 = 144

If you have that readily accessible in your head, you can simply do a linear scan until you find which two numbers the square is between. In practice you will probably skip a few or just recognize it as being close after some practice, but that’s the general idea.

As an aside the perfect squares are just a portion of the single digit times table that you would want in your head to do multiplication or long division.

[zdyn5]

Knowing the squares up to 12 isn’t that difficult to memorize and comes in handy… I imagine those who have them memorized did so in K-12 and just never forgot it.

[gamegoblin]

In every elementary school I have encountered in the US, one learns the multiplication tables up to 10 in 3rd grade (age ~8). Some schools go up to 12 (presumably because imperial units are somewhat base 12?).

Edit: On a lark, I just tested this on my wife who is not mathy whatsoever, and she was able to remember them up to n=12 (with a slight hesitation for 7^2 == 49).

[wrycoder]

For another lark, go to Kahn Academy and check out how he teaches arithmetic up to third grade.

You have to look really hard to find addition and multiplication tables. They’re there, with the rather lame comment that they’re good to know.

But, they are not in his main stream, which depends on gimmicks to figure out the answer, instead of just rote memorization of the tables (which is good for the brain, as well).

As an example, I know my multiplication table just fine through 12. But, something went wrong in the second grade, and my addition was terrible. I depended for years on tricks: to add three, count up very quickly on the three points of the numeral 3. Same for four. For five, count one point twice.

I was in trouble at adding seven, because I couldn’t do something like add three twice!

I was shocked that Kahn uses tricks like that.

[knaik94]

It's not worth feeling shocked over. When you are approaching fundamentals, it doesn't matter what method you use. The education system failed you. From my experience teaching kids with math anxiety, I learned it's impossible for people to accurately judge where their weakness at the core. But a pattern I found was that they were scared of doing basic things the "wrong" way. Short-sighted teachers worry about the "wrong" way because they feel like it will slow kids. The proper approach is to let kids use whatever way is fastest for them, and they'll learn other methods on their own as they continue practicing problems. Teachers who force a specific method for arithmetic make the problem worse because the kids end up more worried about doing it the right way rather than worrying about getting the answer right.

We live in an age with calculators, no one cares if you use your fingers to count. I have ADHD and feel limited by my working memory often, using fingers or repeating a number I want to remember over and over feels like having extra RAM. Even the way kids are taught to count is different depending on where you live. Studies show that kids who use fingers are stronger in quantitative reasoning. But growing up, I knew teachers who made fun of students for using fingers to count.

Imagining numbers as dots and counting or breaking a number into smaller numbers to add is not a "trick" it's an algorithm that is as valid as any other. It's counterproductive to associate the word "trick" with "wrong".

For a while I wrote my own system of dots to correspond with numbers, 1, 2, 3 I focus on the end points, 4 (I wrote it open) makes a square with four corners if you ignore the extensions, 5 I count when I change directions and the end points, 6 I imagine dots of a domino tile, 7 is basically two layers a four and then the end points of the character, 8 is similar to six but I count the two circle, and 9 is similar to six but I count circle and then both sides of the bottom curve (a 3x3) grid.

Even if my brain gets tired or distracted, I know I can still add by dots because it's so procedural and I don't need to "think", I just remember the starting digit and then count up as I follow the dots. I use saying the word out like as a form of RAM to this day. Repeating a word, to me, uses a completely different part of my mind, so I free up 100% of working memory and cognition. I have "forgotten" numbers while doing mental math and have reminded myself from hearing myself say it. Describing these techniques, I recognize I sound like a literal computer and almost not human, but it's struggle I learned to work past. It works, I can do relatively more advanced mental arithmetic compared to peer even.

For multiplication, I would recommend Anki. This kind of memorizing is what that entire system excels in.

https://www.theguardian.com/science/blog/2012/jun/26/count-f...

[opsunit]

I do the dots thing too. I've never heard anybody else describe it and, to be honest, it's quite comforting to hear I'm not the only one. In fact I don't just imagine dots, I imagine die faces. This obviously gets problematic after six. I too feel limited by my working memory: it makes mental arithmetic of numbers with two digits very very hard without an external store like fingers.

[wrycoder]

So, our methods for addition are nearly the same. I like your seven. Eight is just add ten and subtract two, and nine is subtract one.

But the thing is, doing addition and subtraction should be as fluid as multiplication. Your dots and my points are slower than having the table memorized, where the answer just appears in your head instantly. And so I made an effort to learn the table later in life. But, my memory isn’t great for that stuff anymore, so it wasn’t easy.

The tricks don’t work very well when you are asked to count backwards from one hundred by sevens. (Which used to be on a standard cognitive test!)

I don’t think the system failed me - it was a good school - so, I think it was a joint effort. I didn’t learn the table properly for an unknown reason, and the school didn’t realize that, because my methods worked and were never tested in competition.

I don’t think the tricks are “wrong”, just suboptimal. That’s why I think Kahn should teach tables, and only fall back on other methods when necessary.

[knaik94]

I was brought up in a academically competitive environment, both in school and at home. I also have a natrual affinity towards mathematics and puzzles, but none of that makes a difference. This kind of arithmetic math inutition is nothing more than practice. You might be a little removed from the last time you had to do this level of math, so it's not fresh. I have tutored high school math for the last 10 years, so I built that intution over time checking students' work. But that only makes a difference in speed.

The reason the blog post and I made the assumption about perfect squares up to 12 is because it's a consequence of just knowing multiplication tables up to 12x12. The squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. The most basic approximation of a square root would be rounding to the nearest perfect square. The author didn't pick g=6 because they had some hidden intuition that 6 would be close, they picked 36 instead of 25 and knew that'll be 6. If I wanted to know the square root of 72, that's between 64 and 81. You'd know it's closer to 64 because the difference between 64 and 72 is smaller than 81 and 72. The actual numerical differences aren't that useful, you just need an understanding of what it's close to.

The reason why I assumed most people would know even powers of 2 is because of CS. It's just comes up so often, for example looking at algorithm complexity in relation to simplying with respect to log base 2 or binary representation of integers. The numbers also come up when thinking about primitives conversion such char to integer or how floating points work. As well as understanding amortized memory allocating algorithms, like how much bigger to make a dynamic array when it's filled. Even if you don't explicitly know why, numbers like 4, 16, 64, 256, 1024, 4096 are familiar, which are 2^2, 4^2, 8^2, 16^2, 32^2, 64^2. They are all also powers of 2 and you could write them as 2^2, 2^4, 2^6, 2^8, 2^10, and 2^12. I felt like it's a fair assumption on HN.

When I mentioned visualizing the graph, I just meant the non linear mapping between numbers and their squares. That is more raw intution, but it's not numerical in any way, it's knowing how the graph looks. And that comes from remembering the relationship between algorithm complexities, like log(x) vs sqrt(x) vs x vs x^2 vs 2^x.

An an educator, I strongly disagree with the idea of anyone being numerically "blind". If you struggle with rapidly finding that option, my only advice is brush up on some multiplication tables, and to give yourself time to speed up. Math anxiety is a problem for some, but that's a problem related to fear of failure and not cognitive ability.

[opsunit]

I've often wondered how others get the multiplication tables to stick. When I was in school and tasked with learning and being tested on them I remembered them - just about - for long enough to be tested (primarily from a quick reminder on the way to school) and then they were gone.

My overall experience was that they were a very boring song with terrible lyrics. I can't ever say that any meaning clicked especially. The teacher called out the first part of the verse and I "sang" it internally and hopefully got it right. Having gotten through that it was gone by lunchtime in time for a different set of words to the same song next week.

I tried again about 30 years later as a adult and had much the same experience. You might as well have been asking me to remember items on a tray.

To me it's like this:

chicken x tree = rock

brick x kangaroo = Susan

boat x walnut = dinosaur

Now imagine you have to remember 288 of those (because you might be asked to produce either side of the equals sign) and somebody asking you to recall one arbitrarily.

How do you get your students to get them to stick?

[knaik94]

https://www.mathmammoth.com/practice/multiplication

https://www.mathmammoth.com/practice/multiplication-table

It's all about practice. For some, it's easier to create something competitive from it. Some of my students really enjoy challenging me, because they are at that age in life where they want to stick it to "the man". So we do 60 second times competition to see who can finish more.

At its core, learning this is boring. Having a benchmark that you work to improve, gamifying it, is the fastest way to improve. It's like typing or exercise, the fundamentals aren't cognitively taxing. I think it's a little harder to practice after you leave school, because there's no math or science class where you would immediately apply and practice what you learned.

There's no general solution that would work for every student, and not every student has the same weakness. The website I linked is genuinely one of my favorites, it's like duolingo for math, for free with no account needed. I think the second link will help build intuition behind the relationship between the numbers. All of the quizzes/games on this site are fantastic.

https://www.mathmammoth.com/practice/sieve-of-eratosthenes

This is fantastic to understand numbers and primes. Start with multiplication/division. Then work with GCF, which is greatest common factor between two or more numbers, for example 24 and 36, 6 * 4 or 12 * 2 == 24 ; 6 * 6 = 36, 12 * 3; so the GCF of 24 and 36 is 12. GCF is a way to practice multiplication/division naturally.

Arithmetic puzzles are also a good way to get some practice where it's not just boring. Fermi questions are as well. There's more advanced algorithms for multiplying things like large numbers in your head, but in my opinion, an estimate is good enough for 99% of real life situations. Fermi questions help build an intuitive math sense that will be applicable to real life situation. For memorizing, using "tricks" is fine, like the 9 multiples all add up, looking at digits, to 9. 9 * 8 is 72 which is 7 plus 2. Multiples below 10 also start with 1 less than the digit that is not 9, so 8 - 1 = 7, and 7_ and the second digit adds up to 9, so 2, 72. I'll leave the rest as an exercise for reader. Good luck.