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A “simple” 3rd grade problem

math.stackexchange.com

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Re: A “simple” 3rd grade problem

#111

Post by jamessb »

derleth wrote: post #80
jrabone wrote: post #27

Earlier quoted context omitted.

It is ABSOLUTELY open for debate, and part of the clue is in the question "if she works just as fast" ie. the cutting rate is constant. Then, it is ambiguous since the SIZE of the pieces is not mentioned. It's not the teacher's fault, per se; the question is unanswerable. The student picked one interpretation but the (likely) correct one is shown in the answer http://math.stackexchange.com/a/380007

If it is ambiguous, there is no answer. There must be an answer. Therefore, it cannot be ambiguous. The answer given is the only one it is possible to give. Therefore, it must be the correct one. The context isn't so much "third grade" as it is "math test", and very, very few math tests allow "Question ill-formed as posed" as a valid answer. Maybe more should.

The first round of the UK Maths Challenge * is multiple choice, and does often include questions with "not enough information provided" as one of the available answers. However, this isn't a mechanism for identifying badly phrased questions.

* (the feeder competition for the British Mathematics Olympiad, and then the International one)

Re: A “simple” 3rd grade problem

#112

Post by eridius »

Total_Meltdown wrote: post #98
downandout wrote: post #15

The student is absolutely correct. I don't think it's even open for debate. Cutting anything in half requires exactly one cut; cutting in thirds requires two. It's as simple as that. The teacher that crafted the question, or worse yet, the publisher of a textbook that may have provided the test question, needs to take a hard look at whether or not they are in the correct profession. The fact that the teacher not only…

It is open for debate. The question does not say cut "into thirds," it says "into three pieces." This - http://i.stack.imgur.com/kEjP0.png - is a perfectly reasonable answer which, assuming the rate of cutting is constant, would result in 15 minutes. It's a bad question. Edit: That said, I would have given the same answer as the student, because I think that's the most reasonable interpretation, especially considerin…

But even with your picture the answer can be 20 seconds. You're assuming the person is starting at the top of the line and cutting all the way through the board to the bottom. But they could just as easily rotate the board 90° and cut across a different axis. Assuming a 1" thick board, this means they're cutting through 1" of wood on each cut, meaning both cuts take the same amount of time.

Re: A “simple” 3rd grade problem

#113

Post by SiVal »

DanBC wrote: post #100
tokenadult wrote: post #97

I read all the comments on the math.stackexchange.com submission and all the comments here before starting to type this reply. There are a lot of issues here, and I will try to add the perspective of a mathematics teacher. The reason I can gain paying clients for my mathematics lessons even though I have no degree in mathematics and no degree in teaching is that I can produce results that many elementary school teach…

Educational books is something that really could work fantastically well with open source models. Some group of people prepare best current practice chapters for a single topic. This group includes educators (to know where children get confused and make mistakes) and experts (to spot subtle errors, and to 'foreshadow' knowledge needed later). These are released. People can make corrections. For something like math th…

The joke would be that if you don't know the answer, the "I don't know" answer is completely correct.

More seriously, yes, I think the open source books (actually teaching materials that include books) will eventually replace commercial materials in almost all cases except those tertiary (college/uni) level classes where the book is written by the teacher. Financial pressure, if nothing else, will have this effect. Many of the open source books could be primarily the work of a single Benevolent Dictator For Life, of course.

Re: A “simple” 3rd grade problem

#114

Post by rmrfrmrf »

dfc wrote: post #42
utopkara wrote: post #24

This is a classical question I ask to children (and I was asked as a child too). It was/is fun, because it is easier to answer if you haven't yet started arithmetic, or if you can manage to step outside the pressure of this new thing that you are being taught at school. How many cuts do you need to make in order to split a board into 2? How about 3? How about 4? In this case, the teacher has failed. But, everybody mu…

Answer: 1, 2 and 2 cuts.

Unless you have some kind of vice holding that piece of wood together, you're making 3 cuts.

Re: A “simple” 3rd grade problem

#116

Post by dfc »

rmrfrmrf wrote: post #114
dfc wrote: post #42

Earlier quoted context omitted.

Answer: 1, 2 and 2 cuts.

Unless you have some kind of vice holding that piece of wood together, you're making 3 cuts.

As I stated downthread: If you ask me to make these cuts I am plugging in my chop saw (or mitre saw if you prefer) I am making 1, 2 and 2 cuts. Where one cut is defined as pulling the trigger on the saw arm and pressing the handle down.

Your "vice" will be my left hand pushing the wood against the fence and towards the stop block. Do you do a lot of woodworking?

Re: A “simple” 3rd grade problem

#117

Post by SiVal »

dos1 wrote: post #106
sillysaurus wrote: post #101

Earlier quoted context omitted.

It seems likely that no one taught those students how to think about math. I.e. teaching students the steps to solve a math problem is not teaching them how to think about the problem. I instantly knew 12/13 + 7/8 was ~2 because I visualize two pie charts in my head, both of which are mostly full. This is in contrast to the other way to solve the problem, converting the fractions to a common denominator and then divi…

It's always interesting to hear how people go about solving math problems. You mention a pie chart visualization and then the much more labor intensive (but maybe "correct"?) method. I used a third way, which was thinking that 13/13 would be one, so 12/13 is pretty close, so that's ~1. And 8/8 would be 1, so 7/8 is pretty close and also ~1. 1 + 1 = 2 :) I imagine there are myriad other ways people approach estimation…

I use the heuristic that many of us here probably use, consciously or not, after our years of experience with math problems: if it's a math problem, as opposed to problem in some other domain that ends up requiring math (science, accounting, carpentry, etc.), there will be some degree of artifice in the problem. Somehow, the numbers will just happen to end up being integers or perfect squares or exact multiples or whatever, so that there is an easy way to solve this specific problem (not a general problem of this sort but this specific instance).

In this case, you examine the numbers and spot that they are both just "one off from one" fractions, so the sum is roughly 1+1. The test givers will then see to it that there is only one answer that matches the result of the "trick" they were testing to see if you could find.

Kids who get a lot of math internalize this heuristic, which actually trips them up briefly when they start having real science classes, because they think they've done something wrong if the answer turns out to be 5.6293 or 0.07291 instead of 4 or 9 or 5/8 or sqrt(10). They assume they missed the trick.

Re: A “simple” 3rd grade problem

#119

Post by king_jester »

alexvr wrote: post #19

I'm impressed that the student thought it through, but people are giving the grader too much of a hard time. If the question was instead, "If a machine can produce 2 cars in 10 minutes, how long does it take to produce 3 cars?" the teacher would be correct. If you've ever taken a standardized math test, it's easy to assume that the question is just a variation of that classic question. If I were a third-grader, I wou…

> "If a machine can produce 2 cars in 10 minutes, how long does it take to produce 3 cars?"

This question is also ambiguous, because there is no info about how long the operation takes, e.g. the machine may be parallelized and produce a 3rd car in 10 minutes along with the 2 others or that the machine may obey a non-linear increase in production time per unit.

Re: A “simple” 3rd grade problem

#120

Post by Periodic »

SiVal wrote: post #117
dos1 wrote: post #106

Earlier quoted context omitted.

It's always interesting to hear how people go about solving math problems. You mention a pie chart visualization and then the much more labor intensive (but maybe "correct"?) method. I used a third way, which was thinking that 13/13 would be one, so 12/13 is pretty close, so that's ~1. And 8/8 would be 1, so 7/8 is pretty close and also ~1. 1 + 1 = 2 :) I imagine there are myriad other ways people approach estimation…

I use the heuristic that many of us here probably use, consciously or not, after our years of experience with math problems: if it's a math problem, as opposed to problem in some other domain that ends up requiring math (science, accounting, carpentry, etc.), there will be some degree of artifice in the problem. Somehow, the numbers will just happen to end up being integers or perfect squares or exact multiples or wh…

When I did my undergraduate degree in physics I think one of the best things I learned early on was estimation skills. I was used to doing things precisely and finding the tricks. Our professors made jokes about things just needing to be right to "within an order of magnitude", and it wasn't for two years that I internalized that.

When you deal with the real world there are always a lot of errors and uncertainty in measurement. Simply being within 10% of the right answer is generally sufficient and quickly getting that answer over getting the 99.99% accurate answer is better if it takes you one-tenth the time.

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