2

In Anderson et. al 2010, "Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns", the experimenters taught people how to solve a novel system of arithmetic problems, which they termed 'pyramid problems'.

The basic idea is that a pyramid expression, such as $m\$n$, represents a pyramid with a base of $m$ blocks that is $n$ levels high. Each level of the pyramid is one block smaller than the one below it. The value of the expression is the number of blocks in the pyramid, and the goal of the task was to learn to evaluate these expression. For example, if the expression is $4\$2$, then it represents a pyramid with four blocks as a base, and with a second level composed of 3 blocks, for a total of 7 blocks.

Participants in this study successfully learned how to solve problems of this sort, and also generalized such expressions to handle negative arguments.

I'm interested in finding out if there are other such arithmetic systems. It seems like the kind of thing recreational mathematicians would be interested in. Has anyone heard of similar games?

http://link.springer.com/article/10.3758/s13415-010-0011-0

Nathan
  • 421
  • I mean that's cute, but in the end $m$n$ is still just short-hand for $\sum_{k=m-n+1}^m k=m \cdot n-\cfrac{n\cdot(n-1)}{2}$ – Zach466920 Oct 30 '15 at 15:45
  • Yeah, it's pretty simple, and it's really easy to come up with a different way to think about it. But the point is that it's an example of the kind of 'novel arithmetic' I'm curious about. – Nathan Oct 30 '15 at 15:49
  • Its a novel problem not a new arithmetic. You can make up any definition you like for a binary operation and give it any fancy symbol you like. E.g. let $x♥y$ be the number of bipartite graphs possible using $x$ black nodes and $y$ white nodes. – Ian Miller Oct 30 '15 at 15:53
  • @Nathan Is it really novel though? I mean, I evaluate $4$2$ by thinking $4$2=4+3=7$. IMHO, if it were truly novel, I wouldn't even be able to use the operators I know to evaluate the expression. – Zach466920 Oct 30 '15 at 15:53
  • I think you're getting stuck on aspects of the question I don't really care about. I'm not claiming that this is an unprecedented mathematical system totally new in mathematics, or that there's no way to reduce it to pre-existing systems. Just that, for the people who were learning the system, it was something they hadn't seen before. – Nathan Oct 30 '15 at 15:58
  • If there examples of truly novel arithmetic, I'd be really excited to hear about them. This game is really, really limited, and admittedly not exciting. Are there other things that are exciting? – Nathan Oct 30 '15 at 16:00

0 Answers0