To illustrate what I have in mind… most people who have studied mathematics seriously, even teenagers, can quickly sum up all numbers in a sequence. For example, what is the sum of the numbers between 1 and 99. That sounds hard? So maybe you can look up a formula online. Maybe. But once you know the “trick”, you can do it in your head, quickly, without effort. There is no miracle involved. To sum up the numbers between 1 and 99, just pair up the numbers. You pair 1 with 99, 2 with 98… and so forth, up to 49 and 51. So you have 49 such pairs, and each pair, sums up to 100 (99+1, 98+2,…). So you have 49 times 100 which is 4,900. Then you have to add the remaining number (50), so that the sum is 4,950.
I did not know that, but now I can check off my “learn something new today” box.
Regarding the article as a whole though, I guess a big part of his premise is in the definition of “tools”. I guess I would agree that mental models are tools. But when you equate mental performance with tools, then it seems like the final bit goes from:
My answer is that acquiring new tools is the surest way to get smarter.
to:
acquiring new tools is the surest way to get more tools.
The triangle numbers (sum of 1 to n) are quite an interesting thing. When I was in 8th class I kinda discovered this little factoid on my own and simplified it into (n^2 + n) / 2 or n * (n+1) / 2. The interesting thing is that this is also the number of combinations of n+1 elements including with themselves. If you take n * (n-1) / 2 you get without themselves. They are also useful in table-manipulation.
It’s quite amazing how such a simple little operation can turn into something mindblowing and useful.
I did not know that, but now I can check off my “learn something new today” box.
Regarding the article as a whole though, I guess a big part of his premise is in the definition of “tools”. I guess I would agree that mental models are tools. But when you equate mental performance with tools, then it seems like the final bit goes from:
to:
I don’t think it’s that tautological. I read it as ‘acquiring new tools is the surest way to be able to solve more problems.’
The triangle numbers (sum of 1 to n) are quite an interesting thing. When I was in 8th class I kinda discovered this little factoid on my own and simplified it into
(n^2 + n) / 2orn * (n+1) / 2. The interesting thing is that this is also the number of combinations ofn+1elements including with themselves. If you taken * (n-1) / 2you get without themselves. They are also useful in table-manipulation.It’s quite amazing how such a simple little operation can turn into something mindblowing and useful.
Be careful not to redefine it. If you want to solve the original problem, you can not alter its definition.
In computer science we actually often have the luxury that we CAN change the problem and still get great results developers rarely do however.