Apparently sometimes $1/2 < 1/4$?

Question

My son brought this home today from his 3rd-grade class. It is from an official Montgomery County, Maryland mathematics assessment test:

True or false? $1/2$ is always greater than $1/4$.

Official answer: false

Where has he gone wrong?

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Addendum, at the risk of making the post no longer appropriate for this forum:

Questions about context are fair. This seems to have been a one-page (front and back) assessment. Here is the front, notice the date and title:

Based on the title, it seems to me that this is an assessment about the number line in which case my son's picture and written proof are inappropriate and better would have been to locate $1/2$ and $1/4$ on the line and state something like "No matter how many times you check, 1/2 is always to the right of 1/4." However, based on the teacher's response it seems the class has entered into a quagmire and is mixing up numbers with portions.

Answer 1

Simply put this is a travesty. The question asks, what appears to be a simple question about two real numbers, $\frac{1}{2},\frac{1}{4} \in \mathbb{R}.$ In particular, it appears to ask if $\frac{1}{2}>\frac{1}{4}.$ The answer to this question is clearly, under normal construction and ordering of the reals, a resounding YES.

What the question meant to ask is a question about fractions of potentially different quantities. In particular, from the teachers drawing, the question meant to ask if $$\frac{1}{2}\cdot x>\frac{1}{4}\cdot y, \qquad \forall x,y \in \mathbb{R}.$$ The answer to this is again, very obviously no, but this is not what the question asked...

Answer 2

Awful, awful, awful question. The intended wording ought to have been along these lines:

Is $1/2$ of something always greater than $1/4$ of another thing?

Your son reasonably interpreted the question as referring to the ordering of real numbers $1/2$ and $1/4$, as David mentioned.

What to do now? Well, certainly bring it to the educator's attention. They may be unaware of the flaw, or they may have seen it and decided to fudge it (not cool in that case!). But more importantly, there is a lesson to salvage from the wreckage. This a learning moment. There was a major failure in the question. Ask your son why he thinks he is correct, if he sees the intended purpose, how would he would correct the wording, how he thinks the mistake could have been avoided, and what he feels about the whole debacle as it stands.

It's a good point to recognize that mathematics is not only a toolkit to handle computation and understand nature, but a form of communication. Language is essential.

Answer 3

What is missing in the question is the notion of units, or dimensions: greater in what? An inequality like $\frac{1}{4}<\frac{1}{2}$ should assume an equivalent unit on both sides. Speaking of the volume of a liquid, one "could" say:

$$\frac{1}{4} \textrm{(of a liter)}<\frac{1}{2} \textrm{in dm}^3$$

because one liter is one cubic decimeter. But saying

$$\frac{1}{4} >\frac{1}{2} $$

when the LHS is in kilometers and the RHS in micrometers, without mentioning it, is a twist to inequalities: they "should be" unit-independent. Or at least equipped with an order relation that axiomatizes the expectations. You could as well define "greater" as the largest on the denominator of reduced fractions. But this would be a very mundane "greater" definition.

What this can teach you is to be aware of logical fallacies in the real world, like a merchant offering you a price so good that he loses money on it. This could incite you in a duel in the manner of the barometer question:

A physics student at the University of Copenhagen was once faced with the following challenge: "Describe how to determine the height of a skyscraper using a barometer."

The student replied: "Tie a long piece of string to the barometer, lower it from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building."

an anecdote mocking the stereotypical answers sometimes required from students.

Answer 4

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.

Answer 5

This is a ludicrously badly worded question.

They do have a point. They are not making it intelligible.

Here is what the question could have been (and, as Thomas Andrews pointed out, it should have been in a chapter on units):

Joe's object weighs 1/2.

Pete's object weighs 1/4.

Therefore, Joe's object is heavier than Pete's, because $\frac 1 2 \gt \frac 1 4$.

What is wrong with this logic?

Or just use an intelligently designed book that teaches this basic level of logic in a way that is easily absorbed by kids.

Example question types from the linked book:

• Show an object (like a bus) and two units of the same type (like "inches" and "feet") and ask which unit would make the most sense for measuring that object.

• Give a math word problem that gives one unneeded measurement. Ask the student to cross out the information that's not needed to answer the question.

• Give a math word problem. Ask the student whether he was given enough information to answer the question.

Answer 6

'What about this?' is ABSURD. Fractions are real numbers and $1/2$ is NOT smaller than $1/4$. Period.

If the dumb teacher wants to compare a half of biscuit to a quarter of pizza, then they are no longer numbers, but physical quantities (masses or volumes), which have their units, and the stupid needs to consider bringing them to appropriate common unit of measure to compare. One should also consider the correspondence of the quantity type (say, not to compare a half of hour to a quarter of mile!)

Answer 7

Possibly another way to refute the counterexample given, is to reduce it to a statement that's more clearly untrue. Suppose $\frac{1}{2} < \frac{1}{4}$. Then, multiply both sides by 4 to get $2 < 1$, which is more clearly false.

So at this point point, the teacher either has to say that the multiplication doesn't preserve the inequality, or (less likely) that the resulting statement is false.

(Though I wouldn't be surprised if they try to refute modus ponens, or the law of the excluded middle, or so on)

Answer 8

Your student seems to understand the question as the teacher asked it, they just answered incorrectly, seemingly because they misunderstood the logical question and how to prove their answer.

The explanation provided by the student (emphasis added):

1/2 is always greater than 1/4 if 1/4 is smaller than 1/2, or the same size.

It's clear the student in question has been exposed to the concept that fractions from disparate objects/groups can have relations beyond the pure numerical answer. "... or the same size" is a nonsensical answer in a pure fractional context. Note the student's answer mirrors the language from the 3rd grade math standards:

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket...

The student's explanation also conflicts with their true/false answer: "A is always true if..." implies A can be false whenever the conditional is false. Which is correct, and would suggest the answer should be False.

As an aside, their explanation is a tautology, plus a superfluous contradiction: A if A or B, where B necessitates ~A*, this distills to A if A... which explains nothing.

*In this case A and B are not independent propositions; they are mutually exclusive:

A) 1/2 is always greater than 1/4

B) they are the same size

These are contradictory, ergo B necessitates not A (B iff ~A) and vice versa: A iff ~B

So in this case, despite not being able to solve the logical problem, it's clear the student understands this question in the proper context. What other context clues do we have?

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Always stands out in this question, as both the part the student stumbled on, and as something the majority of answerers just straight ignore. In a pure numbers sense, it would be superfluous. Let's consider the possibility that it was included intentionally, rather than accidentally:

Consider the two following questions:

Do dogs have four legs?

Do dogs always have four legs?

The inclusion of always completely changes the question. I think we can agree the answers to each of these would be "(Generally) Yes" and "No." Interesting we see an answer inversion which mirrors the question being asked here.

I'd wager this always framing is likely the exact exposure to the concept the students had in class. This is a context clue for the students that, without that classroom context, people are struggling to catch.

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We then see the student is asked to "Prove your answer with drawings and an explanation." If we accept the premise that this is a pure numbers question, the expected solution to this problem would likely then be "copy the number line from the first page and show that 1/2 is further right than 1/4." That's a nonsensical question from a testing perspective, which again provides evidence that this is not the correct approach.

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The totality then of the context shows that the context does not support the interpretation that this is a pure number line question:

1. The student themselves recognizes the proper interpretation of the question, even if they didn't get it right
2. The inclusion of always in the question changes the meaning of the question
3. The inclusion of a follow-on question to prove with a diagram and explanation is nonsensical within the proposed different context

The student's own answer proves that for said student at least, there was sufficient context to correctly interpret this question, and the other context clues simply reinforce this conclusion. It also shows that, with the additional context this student has from class, the confusion does not lie where the majority of the answerers see it.

And the answer, where the student went wrong:

The mistake is likely a misunderstanding of the word always and the logical effect it has on the question: if you have to use a conditional like 'if', then it is not always.

In the diagram/explanation portion, you should show an example of when it doesn't hold, rather than an example of when it does: giving a positive example does not disprove negative examples. You can either show you negative examples cannot exist, or show a single negative example (which would prove the statement true or false, respectively). This is exactly what the teacher added during grading: an example which disproves the statement asserted by the student.