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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?

enter image description here


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:

enter image description here

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.

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    $\begingroup$ The answer to the last question makes not any sense at all, mathematically seen. But the combination of those makes even less sense: if $1/2 < 1/4$ is possible, why woudn't $4/4 < 4/8$ be possible ? It's ridiculous. However, your son is not the only one who was punished for being correct: see e.g. math.stackexchange.com/questions/379927/… $\endgroup$ – wythagoras Mar 30 '17 at 16:16
  • $\begingroup$ I get that the teacher was trying to instill some kind of "out of the box thinking" or whatever but that was most likely not to be mixed with rigorous maths there in such a question... $\endgroup$ – xji Apr 7 '17 at 11:03
  • $\begingroup$ I don't have kids, but if a teacher would say something like this then someone is in trouble. Big trouble! Lol $\endgroup$ – Shashi Mar 14 '19 at 12:39
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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...

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    $\begingroup$ It's not what the student answered either, if you read his explanation. $\endgroup$ – Martin Argerami Mar 28 '17 at 9:49
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    $\begingroup$ @MartinArgerami: Note that the true/false answer has been marked separately from the explanation. There's objectively no way that marking is correct -- and the teacher's "what about this?" proves that they are misinterpreting their own question. This teacher needs remediation because they're corrupting a large number of students. $\endgroup$ – Daniel R. Collins Mar 28 '17 at 10:21
  • $\begingroup$ @DanielR.Collins: A lot of mathematics teachers need remediation in logic and the subject they are teaching, especially at the high-school level, but don't realize it (perhaps due to the Dunning-Kruger effect). The question then becomes; how does one as a teacher figure out whether or not one needs remediation? $\endgroup$ – user21820 Mar 28 '17 at 10:29
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    $\begingroup$ @DanielR.Collins: The problem is that a BA in math education is really insufficient in my experience, and hardly any them nor those who teach them realize it. This flawed grasp of logic can easily be tested and verified, but cannot be easily remedied. In my opinion teachers up to high-school level are generally incapable of correctly explaining what they are teaching. However, that is a topic for another time and thread. $\endgroup$ – user21820 Mar 28 '17 at 10:47
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    $\begingroup$ @DanielR.Collins: Oh didn't intend to exclude lower levels by saying "high-school" lol. Careless imprecision on my part. =P $\endgroup$ – user21820 Mar 28 '17 at 17:08
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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.

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    $\begingroup$ And precision can be essential, especially in math. A common amusing example is "Let's eat, grandma" versus "Let's eat grandma". $\endgroup$ – Nate 8 Mar 28 '17 at 2:52
  • $\begingroup$ If the son "reasonably interpreted the question as referring to the ordering of real numbers $1/2$ and $1/4$", how do you make sense of the son's explanation, and in what sense is it correct? $\endgroup$ – Martin Argerami Mar 28 '17 at 9:51
  • $\begingroup$ @MartinArgerami: I understood the explanation as meaning "if the thing-you-took-1/4/-of is smaller than or the same size as the thing-you-took-1/2-of" - i.e. $x \leq y \implies \frac{1}{4}x \lt \frac{1}{2}y$, which is true. It's not the most general statement that can be made, and it's not very clearly expressed, but it's not bad for an 8-year-old! $\endgroup$ – psmears Mar 28 '17 at 10:25
  • $\begingroup$ @MartinArgerami I obviously don't know for sure, but from your statement it doesn't appear you've ever worked with small children. There is often a chasm between what they understand and what they can articulate on paper. It is unclear what they were trying to express in the written portion, but the True/False, and picture, say a lot about what they meant. Moreover, the issue isn't with the student's understanding, it is with the teacher's understand/curriculum. $\endgroup$ – David Mar 28 '17 at 17:22
  • $\begingroup$ @Nate8: More such examples are here: public-domain-materials.com/… $\endgroup$ – Mike Jones Mar 29 '17 at 19:32
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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.

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    $\begingroup$ Student goes to super's office. Student, "Would you like to have this barometer?" Super, "Sure." Student, "How high is this building?" $\endgroup$ – ttw Mar 28 '17 at 14:36
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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.

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    $\begingroup$ Rather depends how good one is at mind-reading... $\endgroup$ – Tim Mar 28 '17 at 6:48
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    $\begingroup$ @tim Yeap but that's exactly what you spend the rest of your life doing in the real world. Actually even in academia and even in mathematics you spend a surprising amount of time filling in the blanks. $\endgroup$ – DRF Mar 28 '17 at 7:05
  • $\begingroup$ Is it any wonder I've been so misunderstood?! Or not quite understood so many instructions. $\endgroup$ – Tim Mar 28 '17 at 7:23
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    $\begingroup$ I think it is normal to differentiate a function of x along x. It is considerably less common to interpret a fraction like ¼ to be "any multiple of one quarter". Unconventional use of notation should be explicitly defined. $\endgroup$ – timuzhti Mar 28 '17 at 8:40
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    $\begingroup$ This is dumb. A better response is "if the question is ambiguous, then it's your responsibility to verbally ask and it's my responsibility to clarify". Going through life with a cycle-on-requirements mindset is fine; being trained to make assumptions left and right is not (old saying, etc.). End result of not being careful like that: Look, here we are and you're frustrated thinking that you're not being understood (you're understood, just not correct). $\endgroup$ – Daniel R. Collins Mar 28 '17 at 10:37
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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.

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    $\begingroup$ Still, I would say nothing is wrong with that logic. If you weigh two things I assume you'll be using the same units. I don't understand this fascination with questions whose answer is obvious, so to make it non-obvious you have to intentionally misguide the reader. If you want to test if the student knows that $\frac x2 > \frac y4$ could be true or false depending on $x,y$, just ask so $\endgroup$ – Ant Mar 28 '17 at 9:28
  • $\begingroup$ Changing the book does nothing if the teacher doesn't understand basic math (or writing). Unfortunately, in the U.S. we neither select, admit, train nor hire K-6 teachers on that basis. $\endgroup$ – Daniel R. Collins Mar 28 '17 at 10:28
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    $\begingroup$ How interesting. If you weigh Joe on Earth but Pete on the moon, and both readings are 70N, do they weigh the same? Seriously, the question in the OP is clearly about (unit-less and real/rational) numbers and numbers only. Why mess with units or dimension analysis? $\endgroup$ – user1551 Mar 28 '17 at 15:10
  • $\begingroup$ @user1551, if you have another response to the teacher's blue marker question, I'd love to hear it. That question seems to be pretty obviously about units and dimensions. $\endgroup$ – Wildcard Feb 17 '18 at 2:45
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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)

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    $\begingroup$ It doesn't refute the point they are trying to make at all. 2 inches is indeed less than 1 mile. They're just not intelligibly making that point. $\endgroup$ – Wildcard Mar 28 '17 at 2:16
  • $\begingroup$ @Wildcard: They have actually written a perfectly clear question, it's just that they have no damned idea as to the meaning of what they wrote. $\endgroup$ – Daniel R. Collins Mar 28 '17 at 2:21
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    $\begingroup$ The issue is that the question is different when you leave off units. As Daniel said above, it's not ambiguous, it's incorrect. $\endgroup$ – Nate 8 Mar 28 '17 at 2:49
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    $\begingroup$ I would be surprised if they recognized the terms modus ponens or law of the excluded middle. $\endgroup$ – chrylis -cautiouslyoptimistic- Mar 28 '17 at 3:26
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    $\begingroup$ "Then, multiply both sides by $4$ to get $2<1$, which is more clearly false." — What about this? ${\small 2}\quad{\Huge 1}$ ;-) $\endgroup$ – celtschk Mar 28 '17 at 9:21
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'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!)

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    $\begingroup$ One point to add is that some years ago, the LSAT had a question, "Which is larger the volume or the surface area of a cube with side 2?" There were 4 answers: 1.Volume, 2.Surface Area, 3.Equal, 4.Incomparable. My students all answered (correctly!) 4. The book answer was 2. $\endgroup$ – ttw Mar 28 '17 at 9:41
  • $\begingroup$ @ttw Then the book was a trash. It's hard to dicuss with a book. If it was on a lecture, they might have asked a teacher 'But in what units do we measure the volume and the surface?' as measuring the same cube in square and cubic inches may make different answer than in square and cubic miles. Or they might recall the measuring units are arbitrary and simplify by 'Assuming the edge length as a new length unit...' $\endgroup$ – CiaPan Mar 28 '17 at 10:22
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    $\begingroup$ The instructions in the question clearly say, "prove your answer." The student's comments prove nothing. The "What about this?" annotation could be the teacher's attempt to goad the student into thinking more deeply about a proof. As in, "Ok, you drew a picture. Well here's my picture that contradicts yours. What about this? What's wrong with my picture?" $\endgroup$ – Solomon Slow Mar 28 '17 at 12:41
  • $\begingroup$ @ttw - at least there was mention of base units - although how to compare square whatevers with cubed whatevers is a question begged. That's a great example of what I call a 'non question'. In the OP's question, since no base unit was mentioned, it's reasonable to assume the same will apply to both fractions, surely? $\endgroup$ – Tim Mar 28 '17 at 14:41
  • $\begingroup$ @JimLarge, perhaps—but what kind of proof do you expect in third grade? Sheesh. $\endgroup$ – Wildcard Feb 17 '18 at 2:47

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