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Recently, I read a paper about student's understanding of fraction. Most students mentioned in the paper had given $5/8$ as their answer to the following question:

  What fraction is shaded? 

enter image description here

For reasons that only an "educated eye" might see, the author of the paper was expecting $5/4$ as the answer. Surprised of the conviction of our educated eyes, I put a voting question (on social media) where "normal" people could vote for $5/8$, $5/4$, or $other$. The result, here and here, is telling. To make the voters more curious, I used different quantifications of the shaded area to "prove" here that $1=2$. For a similar argument, see also Benjamin's answer below.

I am now in the position of explaining the problem to more than 400 voters and the aim of this question is to get help to do so. Obviously, the question is about what fraction of what is shaded. More generally, the question is about unit conversion in disguise. So far so good. The problem started when I was going to write about the meaning of the equal sign in unit conversion in comparison with its meaning in arithmetic. Then I realized that I don't know (and this is my question) how these meanings are similar to and/or different from each other. Going into more details, I even don't know, I should think of $5/8$ and $5/4$, or of $5/8$ or $5/4$ as possible answers. Similarly, in writing $1m=100cm$ for a length, is the length quantified by $1$ and $100$, or, $1$ or $100$?

Just in case that you are wondering why I have asked this question here, I shall add that because we are the only community that rightly cares about different meanings of equal signs and the difficulties that they cause for our students, and of course, sometimes for us :)

Added. It's worth mentioning why I perceive this post related (at least partly) to unit conversion. I hope the following equalities tell it all:

$1 REC(tangle) = 2 SQU(are)$

Thus, $5/8 REC = 5/8 * 2 SQU = 5/4 SQU$

Of course, you might rightly question what is measured by these fictional units. For this, we might create some fictional story as well, say measuring pizza in Pizzaland where asking for 2 RECs pizza or equivalently 4 SQUs pizza is quite meaningful.

PS. I hope it can be seen now why $1=2$ in the original title, and in the body of the post, is not a contradiction if understood in the context of unit conversion.

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    $\begingroup$ The 'what fraction?' question is ill-posed. You haven't said what it is you want to take a fraction of. $\endgroup$ – Jessica B May 6 '16 at 12:33
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    $\begingroup$ @JessicaB You are right. But this is exactly the irony of the situation. First, notice that I just copied the question from the paper that I read. Second, I checked some resources available on the net, they all has taken "what it is you want to take a fraction of" for granted. Third, from more than 400 voters, only one explicitly mentioned what fraction of what. All it all, that is why I used "the conviction of an educated eye". $\endgroup$ – Amir Asghari May 6 '16 at 13:17
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    $\begingroup$ -1 For burning my eyes with the contradiction 1=2 in the title. Equating distinct options in a survey is not logical math or statistics or anything. $\endgroup$ – Daniel R. Collins May 6 '16 at 16:26
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    $\begingroup$ @DanielR.Collins I did my best to respect both your concern and the purpose of the question. $\endgroup$ – Amir Asghari May 6 '16 at 22:50
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    $\begingroup$ It would seem that you're using unit to refer to the whole and unit conversion to refer to mixing ups units or wholes. Half of cake is not as much as half of a a cupcake. However unit conversion usually refers to converting from inches to feet or kg to g. Perhaps it would be clearer to talk about confusing wholes and not unit conversion. $\endgroup$ – Amy B May 8 '16 at 8:36
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You write the following:

Obviously, the question is about what fraction of what is shaded. More generally, the question is about unit conversion in disguise.

I believe what you have recorded here is the importance of specifying the unit when broaching fractions. This is one of the principal sources of difficulties for students (and some teachers) in dealing with fractions. One of the places in which this is specified is in the NSF-funded Elementary Pre-service Teachers Mathematics Project (EMP) where I find the following sample item:

enter image description here

From my perspective, a description of your finding should be as simple as specifying what constitutes a unit (or whole or one -- this last term sharing an etymology with unit). Thematically, this arises repeatedly as students (and teachers) work their way through not only the equivalence and comparison of fractions, but also in carrying out fraction arithmetic. It occurs in the example you give and in the EMP example I have given with respect to parts of a whole when dealing with discrete items, but it can also rear its head, e.g., during the process of partitioning a number line and using that to add or subtract fractions, or partitioning an area diagram in the process of multiplying fractions.

For an example with the number line, one sometimes sees the assertion that fractions placed on two number lines are equivalent if (and only if) they are directly above (or below) one another. The basic idea of this is pretty accurate, but a bit more needs to be said about alignment; consider the following:

enter image description here

As you can see from this example, literal adherence to the earlier description of equivalence would yield $1 = 0$. Students are generally able to argue back that the number lines need to be "lined up" somehow; the key in ensuring you have accomplished this, though, involves attending to the notion of the unit or whole that is partitioned:

enter image description here

In this picture, the latter ? can be seen to correspond to $1/2$, but there is not enough information to identify the former ?: Even though it "lines up" with the bottom number line, the top number line has not established what a unit is; so we might (as teachers) assign it the value of $1$ and ask whether the two number lines now assert that $1 = 1/2$. Clearly this is not the case, but the true question is what is still missing from our notion of equivalence. As I perceive matters, the issue is once again analogous to the one that you have pointed to: The top number line and bottom number line are using a different representation of one unit.

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    $\begingroup$ Thanks a lot Benjamin for saving the question :) Indeed, I was about to delete it, since I felt it is misunderstood. Most people missed that part that I mentioned "the author of the paper was expecting 5/4 as the answer". My small experiment (on my Facebook with a lot of educated friends) was a reaction to the claim made by the author. My assumption was, if we don't specify the unit, most people go with 5/8 rather than 5/4, unless they have a trained eye. It was a simple experiment out of curiosity without any controls on the variables involves. $\endgroup$ – Amir Asghari May 6 '16 at 21:36
  • $\begingroup$ I've not come across fractions treated this way, and I can't see why you would want to. I don't see why you would want to talk about fractions and units together, when they are both easier to understand alone. $\endgroup$ – Jessica B May 7 '16 at 15:33
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    $\begingroup$ @JessicaB Units are another word for the whole. If you don't know what the whole is you can't figure out the fraction. If you teach fractions and there is no mention of the whole (or the unit), then you won't know what is a fraction of what. All halves would be equal and half a cherry would be as much as half a watermelon - which is clearly wrong. $\endgroup$ – Amy B May 8 '16 at 8:33
  • $\begingroup$ @AmyB I'm used to the term 'units' being about whether you are measuring in kilometres or miles, for example, or alternatively in the context of 'tens and units'. I would find using 'whole' much clearer than using 'unit' here. $\endgroup$ – Jessica B May 9 '16 at 15:12
  • $\begingroup$ @JessicaB I agree and suggested that to the OP that he edit the question - see my comments above. I am glad my clarification helped $\endgroup$ – Amy B May 10 '16 at 4:58
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Your findings, as far as I'm concerned, have nothing particular to say about fractions and nothing at all to say about unit conversion.

You started with a question that was not well-posed. Since we are generally expected to answer questions even if they are not well-posed (mathematicians not doing so annoys people), people voting chose what they interpreted as the most likely intended question, and answered accordingly. What you have actually done is a psychology experiment into the metric people have on a particular type of maths problem.

My guess would be that the key determining factor is what people have been taught at school. It is not uncommon for practice questions to also be somewhat ambiguous, and typically we have drummed into us how to interpret certain wording to remove the ambiguity. I would also expect the details of how they were presented with the two questions to have some bearing on the results - the two different answers become more obvious when the two diagrams are compared. Also, presenting only two possible answers (it is implicit that 'other' is almost certainly not the 'right' answer) is probably not the best way of testing what people see.

There is no need to explain to anyone why they are wrong, because no-one is wrong. The question was ambiguous, so there is no unique answer.

PS. There is a standard proof that $0=1$, from which $1=2$ is easily deduced. In the book Zero: the biography of a dangerous idea this proof is used to show that Winston Churchill is a carrot.

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  • $\begingroup$ I assure you that I am not going to explain to anyone that they are wrong. Yes, the question was ambiguous. But, please consider that I haven't designed the question, I just use it as it is customary. A simple search for "what fraction is shaded" gives you plenty of similar tasks. I used 1=2, just to make people curious and cautious, as "the standard proof" of 0=1 traditionally is used to make students aware of the danger of dividing by zero. By the way, I haven't yet wrote the full story for the voters. The aim of my post here was just to have some help to do so. $\endgroup$ – Amir Asghari May 6 '16 at 22:03
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You might get better answers by:

  • drawing some crust around the edges and asking "how many Detroit pizzas?"
  • drawing some crust around new circular borders and asking "how many pies?"
  • drawing 100m for comparison and asking "how many hectares?"

Turning this into a word problem develops a more useful skill with an easier question.

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