So I will play the devil's advocate even if I personally abhor this "new math" business as much as anyone. The first professor I TA'd for in the US had a very strange style of teaching and one of his tenets was "Answer the question I wanted to ask not the question I asked." What he meant was use all the information you have at your disposal when answering questions. If you are asked to differentiate $f(x)=x^4+ax^3+b$ and not told that $a$ and $b$ are constants point out that you assume it (since the usual convention is that we use letters early on in the alphabet for constants) and differentiate with regards to $x$ even if it's not specifically spelled out.

The reason he gave for this was that in life people rarely asked fully defined and reasonable questions and the trick was to answer correctly even when the questions were ill-defined. I wasn't really convinced then, but having since TA'd for many other professors I found that the students he taught were head and shoulders above the rest in their mathematical understanding both as measured by the exams we gave and as measured by success in further mathematics courses they took.

The point here being that while the questions is certainly ill-posed and strange, if the students so far never really talked about what a real number is, but rather spent the time talking about 1/2 and 1/4 of bigger and smaller things and how 1/2 of a orange can weigh less then 1/4 of a melon, the answer seems much less nonsensical. The fact they are asked to "prove" the answer gives further credence to this possibility.

**EDIT:**
To further clarify: The **context** of the question is important. What lecture time had been spent on lately and what the other questions on the exam are might have a big impact on just how crazy this question really is. A perfect example was given in the comments below the OP question.

If you ask "Does an elephant weigh more then a cat?" in the course of normal conversation the appropriate response is surely "Yes." If on the other hand the same question is posed as part of a physics lecture on the difference between weight and mass the correct answer is almost certainly "Depends on where each of them is. If the elephant is on the ISS and the cat on earth the cat weighs more."

**EDIT 2**:

Furthermore rereading the "proof/explanation" it seems a little as if the student might be trying to give the "correct" reasoning for exactly this strange interpretation:

"1/2 is always greater than 1/4 if 1/4 is smaller than 1/2 or same size"

Could be parsed as "a half is bigger then a fourth if [what we are taking a] quarter [of] is smaller than [what we are taking a] half [of] or [the] same size."

Certainly that is a lot of extra words but having graded many attempts at proofs of students at a much higher level (college) I can attest that even then this kind of butchering of language is common and the students in those cases had argued that it was obvious that's exactly what they meant.

**Edit 3**

With more context posted in the OP I must withdraw most of my objections. The context is apparently comparing fractions as rationals. In the light of the extra context, the picture seems to imply not that $1/2<1/4$ but rather that the picture proof was not deemed appropriate. This still does not explain marking the circled "True" as wrong though.