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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. |
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?