Maths is a particular branch of philosophy, but you don’t have to teach kids that explicitly.
You do have to teach them the fundamental tools of maths, though, which include deductive reasoning. If the system isn’t doing this, that would explain why it’s not having great outcomes. “If 4/6ths of Marty’s pizza is a more generous serve than 5/6ths of of Luis’s, what does that suggest?” is a way of doing precisely that. When I saw the question, I assumed this was the point that it was intended to convey, and I was (and still am) genuinely baffled as to how the teacher did not see this.
Context is everything. I’m accustomed to a convention for discussing mathematics in which facts given in the problem are, well, given. The question explicitly stated that Marty ate more pizza than Luis, and an answer which says “that’s impossible” would look like smart-assery to me, when both common experience and simple deductive reasoning show that it is eminently possible. But maybe the question was included in a test where students were told that some of the information given in the questions was bogus, and they were invited to try and identify it. (Even then, the question was poorly written; the equivalence of the two pizzas should have been specified.)
Absent this, though, the student’s answer to the question was correct and reasonable and looks to me like the one the question was designed to elicit; the teacher’s answer was wrong.
There is one single word that was left out that would have made a world of difference, and that word is “their”.
The question, AS GIVEN, implies that Marty FACTUALLY (already) ate more pizza than Luis, so therefore any answer along the lines of “Sorry, 5/6th is bigger, so that means Luis ate more” is total bullshit. The question (IMO) automatically implies that the whole thing is over and done with and that Marty is the clear winner in “who ate the most pizza”. So, given that solid premise…there is only way way for Marty to have eaten more, despite only eating 4/6ths and that’s if Luis’ pizza is a little 10 inch one and Marty’s is, like FORTY INCHES across. Yes, Marty ate more pizza. BY FAR.
Now had the question been “Who ate more of THEIR pizza?” rather than “Who ate more pizza?”…then yeah, there is no way Marty’s 4/6th beats Luis’ 5/6. No matter how you cut it (heh), Luis ate more of HIS personal pizza, even if it was a really tiny one and Marty’s was the size of a tire.
But the question as you put it did not include the word “their”…it wasn’t “who ate more of their pizza?”…it was “who ate more pizza?” and giving the already finished results (I.E. Marty ate more pizza than Luis). At this end, we’re supposed to take the question at its word. There is no reason to doubt the question is lying, after all…or that it’s a trick question. If you ask someone something like “Okay, Ellen ran a mile in four minutes and Peter ran that same mile in 3 minutes and fifty nine seconds, yet Ellen was declared at having ran the mile FASTER than Peter…how is this possible?” you’re not going to be looking for or expecting an answer of “A haw haw…it’s not possible! Because three minutes and fifty nine seconds is FASTER than four minutes, ya numbskill! A haw haw! I got you good!”…what is going to happen is that they’re going to sit down and try to puzzle out how it’s possible, because they figure there’s an aspect that they’re not thinking of yet.
It’s like the joke “What did one elephant say to the other?” and the answer is “Nothing. Elephants can’t talk!” A hahaha…you got me. See, now that’s funny… BECAUSE IT’S A JOKE! You expect it to be a joke in that case, though, but here the question was put forth seriously.
So…final opinion: The kid is clearly right, the teacher is a moron and can’t seem to admit that he’s wrong, and everyone should always think before making questions like this, just to make sure there’s no loopholes if you’re clearly trying to go for just one answer.
Yes, and in this case, the horse is that the pizzas are two different sizes.
Pizzas coming in different sizes is an everyday thing.
Math questions where the answer is ‘the question is actually impossible’ are not.
I remember getting questions worded similarly to this one when I was in elementary school, and the kid’s answer was the expected one - we were expected to demonstrate knowledge that fractions weren’t just absolutes in a void; it’s 4/6 or 5/6 of SOMETHING, and that depending on what that something is, 4/6 can be bigger than 5/6 of something else.
I’ve been a teacher for 27 years and have designed successful exam questions.
I agree with all the above posters - the student answered the question correctly (and cleverly) whilst the teacher was wrong on several counts:
shouldn’t have designed the question that way if if the idea was to simply to test if students knew that 4/6 was less than 5/6
shouldn’t have included the phrase ‘Marty ate more pizza than Luis’ as a statement of fact if it was a lie
should have congratulated the student for an excellent answer
Yes. This is a certain genre of question, and it probably appeared as part of a whole section of questions, all of which are phrased in this way:***REASONABLENESS: [circumstance] + [proposition that is prima facie not possible]
***The idea, as you say, is to train students to avoid making obvious mistakes out of sloppiness, without having to go through lengthy proofs.
HOWEVER, the language pragmatics of this particular test item themselves are sloppy; it probably was written by a mathematician, and not someone well-trained in test writing.
The test item should have been phrased something like this:***Martiy ate 4/6 of a pizza and Luis ate 5/6 of a pizza. Marty says, “I ate more than you.” What’s wrong with his statement?
***That doesn’t avoid the presupposition that both pizzas are the same, but it more clearly portrays the proposition as an assertion to be questioned, rather than a given circumstance.
I’ve helped my 4th grader with a number of these lessons that are designed to get kids to be able to check their work quickly by applying a “reasonableness” test. In all cases that I remember, the questions were worded with a 3rd party. For instance,
“Marty ate 4/6 of his pizza and Luis ate 5/6 of his pizza. Sally says that Marty ate more pizza than Luis. Is Sally correct? Why or why not?”
Or a more abstract version,
“Marty ate 4/6 of his pizza. Luis ate 1/6 more of his pizza than Marty. Sally has calculated that Luis at 5/6 of his pizza. Is Sally’s answer reasonable? Why or why not?”
Although in that 2nd example, the arithmetic is usually much more challenging to discourage the kids from simply checking it in their heads. Still, it’s easily possible to word these types of questions in a way that makes it clear what the student is supposed to be doing. I agree with glee that including a blatant lie as a statement of fact is… bizarre at best. Also, on preview, ninja’d by guizot.
The premise there is that zebras as (presumably) rare in your neighborhood whereas pizza coming in different sizes is incredibly common. I would assume that the pizzas were different sizes because that’s the most logical answer (more so than “trick question!” in my book).
In fact, if given the question, I would have given the same answer (different sizes) and assumed that the point of the question was demonstrating that percentages are useless without establishing “percentage of what?”
The student was right. Nothing about the question indicates that the pizzas were the same size. It’s the same as saying Marty went on a diet and lost 15 lbs, Luis went on a diet and lost 20 lbs, but Luis still weighs more than Marty. A pizza is not an absolute amount of food.
Perfectly reasonable answer because he explained the logic of it. It’s not a question that has a single correct and best answer.
What if the question was “What do George Washington, Thomas Jefferson, James Madison, James Monroe, William Henry Harrison, John Tyler, Zachary Taylor and Woodrow Wilson have in common?” and the student answered “They’re all dead”.
Context is also important here. My Lincoln question was following up on a lesson about his politics, so most students understood what I was driving at. If your "eight presidents"question came at the end of a unit entitled “Presidential Facial Hair,” then the expected and reasonable answer would be “none of them had beards.”
If the question was on a cumulative exam for a “Survey of US History” course, then it’s too vague to be useful except, perhaps, as a very open writing prompt.
I agree that the question was poorly written. That’s why I feel seeing the overall test is important before judging this one question.
That said, I’m sticking with the zebras metaphor. When two people sit down to eat pizza in a math problem, your base assumption should be that the pizzas are equivalent. Somebody who ate 5/6 of a pizza generally ate more than somebody who ate 4/6 of a pizza just as somebody who ate three cheeseburgers generally ate more than somebody who ate two cheeseburgers.
I’ll also note I didn’t say the student’s answer was wrong. I said both the student’s answer and the teacher’s answers were correct. To use my hoofbeats metaphor, I said hoofbeats could be coming from horses or zebras. Saying horses are more common than zebras and therefore the hoofbeats must be coming from horses is wrong. But saying zebras are possible so the hoofbeats must be coming from zebras is even more wrong.