My 8 year old has been given a worksheet of numeric sequences, e.g. "what are the next numbers in the sequence 11, 12, 14, 15, 17, ..." and "make your own number pattern" and "make your own colour repeating pattern". I've had no problem helping her with these.

The final exercise is "Make your own turning pattern". No other explanation is given.

Apart from sending her back to her teacher to ask "what's a "turning pattern"?", what would you say this is?


My guess would be they want something like this (as examples): the "turning pattern" $$\underbrace{\fbox{$\,\uparrow\,\strut$}\;\fbox{$\rightarrow\strut$}\;\fbox{$\,\downarrow\,\strut$}\;\fbox{$\rightarrow\strut$}}_{\large\mathtt{part\, that\, repeats}}\;\fbox{$\,\uparrow\,\strut$}\;\fbox{$\rightarrow\strut$}\;\fbox{$\,\downarrow\,\strut$}\;\fbox{$\rightarrow\strut$}\;\cdots$$ corresponds to walking in a snaking line, and $$\underbrace{\fbox{$\,\uparrow\,\strut$}\;\fbox{$\,\uparrow\,\strut$}\;\fbox{$\rightarrow\strut$}}_{\substack{\large\mathtt{part\, that}\\ \large\mathtt{repeats}}}\;\fbox{$\,\uparrow\,\strut$}\;\fbox{$\,\uparrow\,\strut$}\;\fbox{$\rightarrow\strut$}\;\cdots$$ corresponds to repeatedly moving like a knight can in chess.

It's worth pointing out that "guess what comes next" exercises like

Find the next numbers in the sequence: 11, 12, 14, 15, 17, ...

are really not very good problems (in my opinion). All they judge is whether you can guess what the teacher is thinking of; they do not ask a real mathematical problem. This is because any finite sequence can be extended in literally any way you want; there is always a justification that can be made for it. There is no answer for what the sequence "should" be.

There is a rigorous sense in which what I've said above is true, but sometimes it can be a little hard to get one's head around for someone not mathematically inclined; it might be easier to note that there is a similar problem with analogies. If you were given an analogy to complete,

$\mathsf{grass} : \mathsf{green}\, ::\, \mathsf{ocean} : \mathord{?}$

you can guess that the answer should be "blue", because the teacher is likely thinking of the relation "grass is green", but there is just as much reason to say any other color. For example, "orange" works, because for all we know, the intended relationship between green and grass is "they start with the same letter" (of course, other colors may require more convoluted explanations).

  • $\begingroup$ sounds plausible $\endgroup$ – Jeffrey Kemp May 20 '13 at 11:18
  • $\begingroup$ With regards to your comments about the "guess what comes next" questions: true, but at this level it's assuming a certain limited range of operations (simple addition or subtraction), plus the application of Occam's Razor. In this case the problems were very simple - e.g. "3,6,9,12,...", "...,33,44,55,66,..." and "10,11,13,16,20,25,31,..." Mind you, I personally love these sorts of problems because they provide a challenge to find alternative interpretations that are both consistent and plausible :) $\endgroup$ – Jeffrey Kemp May 20 '13 at 11:36
  • $\begingroup$ If one has been reading Homer, it’s clear that the answer to the verbal analogy question is wine-dark. :-) $\endgroup$ – Brian M. Scott May 20 '13 at 17:30
  • $\begingroup$ Her teacher explained it to the class again and you were right. They just need to pick a symbol (Chloe drew a "B") and then turn it in a repeating sequence of -90 or 90 degree turns. $\endgroup$ – Jeffrey Kemp May 22 '13 at 7:58

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