So if the lesson is on ratios and a series of samples are shown of ratios the moment a bottle and a cup are used instead of a plate and cake, wagons and firewood, feathers and buzzards, etc. the children will be like “hold up!” and proceed to play 20 questions?
In all my years of experience of filling cups from bottles, I’ve always used one bottle to fill multiple cups. Maybe the first bottle ran out and I had to open a second, but never, in decades of experience, have I poured the contents of m bottles into n cups where m > n.
Many of us mathematically literate Dopers (including myself) were confused by the question. It presented us with a setup (“John’s mom has 4 water bottles. She needs to fill three cups with water”) that led me to picture something that didn’t fit with what the question’s author apparently had in mind.
I suppose it’s possible that the question was originally accompanied by an illustration that cut down on the confusion, but I’m thinking probably not.
Maybe, maybe not, but that’s not how the initial question was presented, or worded.
The wording muddles the distinction between the abstract and the concrete (which in real life would focus on the volume/content).
I teach third grade math, and I think you know from other threads that if I have a bias, it’s to defend teachers from uninformed criticism.
Consider that backdrop when I say what an awful question this is.
If kids are left to make assumptions, they’re going to assume the items in a problem work the way they work in the kids’ experience. They’re gonna assume bottles of water are big and cups are small. Whenever you need to break those assumptions, you need to break them explicitly. Asking kids to get into the heads of a lead math coach and decide what problem she was trying to model and determine what assumptions about the items in the problem she had? That’s neither appropriate nor a request they can fulfill.
A good math curriculum goes out of its way to present such problems. A problem about how many legs there are on five cats should begin with “Cats have four legs.” Setting aside the rare cases where that’s not true in the real world, it makes it clear what the assumptions for the problem should be.
In addition, part of math education is learning to question assumptions that might throw off your answer. If I ask, “Who ate more, the kid who ate 1/2 of a candy bar or the kid who ate 1/3 of a candy bar?” I want my students to respond, “Were the candy bars the same size?” If that’s not their response, they’re opening themselves up to error.
Granted I read this problem immediately on waking up, pre-coffee, but I couldn’t make heads or tails of it–and remember, I teach third grade math.
My guess? Some teacher tired of the classroom applied for the position of district math curriculum leader or something; talking a good game during the interview, she got the job. This problem is one of many she makes in packets for all grade levels; she made it quickly, and nobody in authority (her or anyone else) has reviewed it prior to its distribution to teachers. Teachers are not empowered to edit these problems, since “Common Formative Assessments” is such a gigantic buzzword in education.
My favorite story along these lines, for certain values of favorite: years ago I was at a training for a new math textbook. The trainer showed us a problem that asked students, “What is the difference between 60 and 37?” The trainer told us how there could be multiple answers: one difference is that 60 is a multiple of 10, another is that 60 is even and 37 is odd, etc. I objected that “difference” is a mathematical term with a precise definition, and that the difference is 23. That goddamn trainer would not admit she was wrong.
Common formative assessments? Man, I’m glad I haven’t encountered that particular bit of nonsense. Common summative assessments, for all that they’re often implemented poorly, make sense. But of all sorts of assessments, formative assessments are the one that most demands individual customization.
(For the non-teachers in the audience: A summative assessment is for determining how much a student has learned, especially for purposes of determining whether they meet some standard (do they pass the class, do they get an A, do they get into the school they want, etc.). A formative assessment is an assessment a teacher uses to gauge the success of teaching as the teaching is being done, so as to customize the teaching for the needs of the students, and can be as informal as just looking to see whether the kids have confused looks on their faces. If kids are struggling more than usual on one particular point, you can spend extra time on that, and if they’re getting some other point quicker than usual, you can speed through that section.)
Yeah, I think that was the main stumbling point for me, too. There are, though, those little mini-bottles that are fairly popular that could easily work out to a 4 bottles to 3 cups of a reasonable size. At any rate, all that needs to be clearly stated to make this question work is that 4 bottles exactly fill the 3 cups. And perhaps change the word “cups” to “glasses,” because there seems to be some confusion that the actual volume matters.
I’m not making any assumptions. I’m trying to explain how the wording of the problem can mislead the reader into forming a mental image which is completely incompatible with the intent of the problem.
It’s a typically poorly worded elementary school math question. It forces the person answering the question to make assumptions about the problem that are not explicitly stated (and often not particularly implicit, either). These questions are the bane of elementary school students, their parents, and high school math teachers who are forced to try and unscramble the mess that such questions make of the mathematical thinking processes of the students they eventually get. :mad:
IF one MUST pick a result, one should pick d), because it’s the answer that fits the simplest unstated assumption (that no water be left over, and the cups be filled full).
As for the assertion that 1.33, or 1.333, or 1.33333333333, or any other written expression of a specific number of threes is equivalent to the fraction 4/3, well, that, too, is a bane of high school teachers. :mad: :mad:
Actually, the simplest unstated assumption would lead to the answer 1/3
The question was:
“John’s mom has 4 water bottles. She needs to fill three cups with water. What fraction of water bottles will she put into each cup?”
Note bolded. The fraction of the 4 water bottles she would put into each cup would be 1/3.
The effort to create “real world” problems has created a lot of situations like this one. Perhaps it is to prepare third graders for the people they will encounter who have no clue how to effectively communicate. Don’t even get me started on the irrelevant details some of these questions contain; many almost make my librarian head explode, and undermine my efforts to help students become effective consumers and producers of information.
My concern is that is scares students away from math and physics, because it makes applying math hard and worse says someone is wrong when a test forces the student to provide the wrong answer.
The real mathematically correct answer would be:
But is the source this problem book publishers or just overworked teachers that are generalists and probably didn’t like math or word problems either.
The one feeble defense I can offer is this: coming up with elegant word problems is not easy.
By “elegant,” I mean:
-Novel (kids won’t read it and mutter, “Jesus Christ, how many times are you gonna ask me to share pizzas with friends?”)
-Unambiguous (they don’t rely on unstated assumptions, as this problem did)
-Concise (they don’t require kids to read a paragraph of explanatory text, where kids will get lost)
-Focused (they deal only with mathematical principles the kids have accessed)
-Relevant (they don’t require background knowledge of something like parking decks or mortgage rates, which kids at the age being taught don’t all know)
Coming up with a few such problems isn’t too bad, but doing it day in and day out is tricky.
Now, that’s not an excuse: it’s tricky, not impossible, and people who do it like this need to find something else to do. But if you think it’s easy, think again.
Q: John’s mom has a pitcher that contains four pints of water. She needs to pour equal amounts of it into three containers. How much water must she pour into each container?
A: 1 1/3 pints.
An answer that should clarify the only possible presumed meaning of the very badly phrased question in the OP, an answer that should be clear to any adult and third-grader alike, and an answer that demonstrates how easy it is to set out this particular math problem without having to wonder about the capacity of the bottles, the capacity of the cups, or boggling the mind with confusing bullshit like what the hell “fraction of water bottles” is supposed to mean.
I’ll defer to your expertise. It’s just when I’m helping my children with math I give them some pointers I’ve learned over the years in my own education such as don’t bring in unnecessary assumptions yet use simplifying ones if need be. This is probably my own adult educated bias showing when I respond to questions like this.
For example when we do ratios at the house I make them keep the units explicit. And we deal with the unit ratio as a quantity so they can get used to the concept of working with things like banana/plate. I think it will be helpful for physics and other sciences in the future.
I only have my children so what works for us might be unwieldy or inappropriate for other children of that age.
I seem to recall way back in my elementary education nonsense terms being used. Such as 1 squazit = 5 mublut. I’m wondering if abstraction is better if it eliminates assumptions.
Keeping units explicit is key, and I applaud you for doing that. I’m forever having kids tell me the answer to a perimeter question is 50, and I say, “Fifty zebras, right?” until they learn that they better say 50 inches. Knowing your units becomes especially important in multi-step problems, where confusing the unit is a major cause of missed problems.
Definitely true–but I think in the problem given, the units are not clear.
My other bias is that I teach at a school with an experiential education magnet theme, so in general I don’t like working with nonsense. Some kids, especially kids who love math and take to it like a fish to water, can work well with nonsense terms. Other kids use the concrete elements of the problem to visualize what’s going on and to navigate their way to an answer.
As a physicist, let me just say thank you. All too often, I see students dropping the units as soon as they start working on a problem, and then getting to the end and saying “What are the units of electric field?”, or the like, so they can plug them back in (or, worse, leaving them off even at the end). I don’t know if stressing it a young age like this works or not, but I certainly applaud the effort.
Update: I wrote to the teacher and I have pasted her explanation below.
In 3rd grade, they are learning how to partition items to share equally. There were 4 water bottles and 3 cups that would be used to share the water bottles. They always need to first see if each cup could get a full water bottle - YES. This leaves 1 bottle left that needs to be partitioned/divided into thirds, since there are 3 cups. Each cup would receive a 1/3 of the 4th bottle. So, each cup would get 1 1/3 of the bottles. …end
As most of you have so elegantly explained , nowhere does it say to distribute ALL the water amongst the cups.
What kind of kid were you in 3rd grade, Xema? 
When I first read it, I partly and not coherently but not incoherently (:)) was thinking along these lines, and then rejected as not getting anywhere within three seconds, and deciding this couldn’t be where the question was going; I then grabbed onto this scenario and conclusion:
Of course, this whole thread is about the tolerable ambiguity in arithmetic equations.
Real question: Given that, at what grade should the first answer cited here by octopus (“4/3 bottle/cup”) be acceptable? What is the name (out of many, of course) of that aspect of mathematics that that unassailably correct answer? A simple move from arithmetic to algebra?
While my children were at a Montessori school I saw children using their materials to do this sort of thing mechanically at ages 4 and 5. They’d count out 24 glass beads and divide them into or onto something and say 6 beads for each box or plate or bowl. So they can see what they are doing at a relatively young age. I don’t know if they abstract it as x/y or a/b as a typical pre-k or kindergartener.