Last week I called out this problem (whose existence was implied by a web search that brought a reader to my blog):
In how many ways can 7 peanuts be shared among 3 people?
In particular, I argued,
Any problem that uses everyday language [such as “share”] or imagery that will mislead if taken seriously is a bad one in my view.
Readers took me to task for too narrow a view of the verb share.
I concede.
Not all sharing is equal sharing. I probably use the word sharing to mean equal sharing far too often. This makes my point while simultaneously implicating me. Sweet.
But no way am I going to let that crummy problem off the hook.
Chris Hunter argues in the comments that the problem has gotten the implicit stamp of approval from the National Council of Teachers of Mathematics (about which more in the next couple of weeks), by way of being in an article published in Teaching Children Mathematics:
Danny, Connie, and Jane have eight cookies to share among themselves. They decide that they each do not need to get the same number of cookies, but each person should get at least one cookie. If the children do not break any of the cookies, in how many different ways can they share the cookies?
But that’s not the peanut problem.
Danny, Connie and Jane are likely to be satisfied with their share of eight cookies. Indeed (equal sharing aside), it is likely possible to find some way to share these cookies so that everyone’s appetite is sated.
Were they sharing peanuts, it would be tougher going. When was the last time you stopped at the seventh peanut?
And by the way, what’s the unit here? Is this one peanut or two?
The sharing will proceed differently in each case, I would imagine.
Here’s what I’m saying. Context matters. Dan Meyer will argue that context matters for motivation and for intellectual honesty. Karim Ani will argue that context matters for motivation and so that kids understand that math is power.
All true.
But I want to argue that context matters because people bring intuitive mathematical ideas to class. More often than not in K-12 schooling (and beyond), those intuitive ideas are based on their experiences in the real world. If we don’t build on those ideas, then we alienate students from mathematics.
If we do build on those ideas, then we’re helping students to make their ideas better. There is efficiency in this, but also an opportunity to avoid the well-documented effects of instruction that doesn’t connect to everyday experience. Namely, that said instruction has absolutely no effect on people’s views of the world, nor on their ways of operating in it.
It’s not so much that students end up choosing not to use their mathematics education in their lives, it’s that it never occurs to them to do so.
Because no one shares seven peanuts among three people.
Overthinking my teaching 
but what if it was the last seven peanuts and the three survivors of a plane crash in the outback someplace?
Good point, Eva. I am humbled.
But back at ya’…
What if one of those survivors had called ahead about a peanut allergy? Now there are no peanuts to share.
In the real world one person eats all seven peanuts, then pretends to be surprised that there aren’t any left for anyone else.
I don’t think that it is necessary for all problems to be based on “real-world” scenarios. Certainly some should be, but some can be simple puzzles, just for fun.
gasstation:
Agreed. Totally and completely agreed.
But then why make it about sharing peanuts? Why not stick to balls in boxes?
And btw, how much fun was that person Googling peanut sharing having solving that problem?
At least with balls and boxes, he/she might have gotten directly to a helpful site.
I think that defeating the “google the answer” cheat is a good feature of the peanut-sharing problem.
I remember reading about a sociology experiment where person A is given $1 and asked to share the money with person B. A gets to decide the split, while B gets to decide whether to accept or not. If he doesn’t, neither person gets anything.
What they found was that person B would often decline when they felt the split wasn’t fair. For instance, if A splits the $1 50/50, B will likely accept and they’ll both get $0.50. But if A decides to give B $0.10 and keep the rest, then B is likely to decline the offer, even though he’d be worse off in the end. Which is to say, [our notion of] fairness has value.
Not sure how directly this applies to the peanuts, but surely it does. Putting aside Chris’ absolutely spot-on observation that nobody shares seven peanuts, anyway, it could be that a 5-1-1 split is mathematically possible but practically (literally) not. Interesting.