The meaning of equal sign in unit conversion

Question

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? 

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.

Answer 1

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:

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:

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:

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.

Answer 2

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.

Answer 3

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.