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...
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.
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?