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A “simple” 3rd grade problem

math.stackexchange.com

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Re: A “simple” 3rd grade problem

#101

Post by sillysaurus »

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…

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 dividing by the denominator. It would take me some time to do the latter, whereas I can instantly do the former.

I don't think the students who got that wrong (nor some who got it right) do any kind of visualization in their heads.

Teachers need to realize that it's the operations in the head that count the most, not rote memorization of steps to solve a problem.

Re: A “simple” 3rd grade problem

#102

Post by downandout »

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…

As someone said below, it's only open for debate if you want to be pedantic. The Dr. Sheldon Cooper's among us may debate it, but it's pretty obvious what the question was looking for. There is even an illustration showing the cut, which would take an identical amount of time.

Re: A “simple” 3rd grade problem

#103

Post by Androsynth »

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…

You are correct, but you can trisect that piece of wood an infinite number of ways. The logical equilibrium point is 3 even pieces.

The student chose a ratio of 1,1,1; which is the logical equilibrium point. your image shows 1.5,0.75,0.75; which is the second most logical ratio because it is in the form x + 2y = 3 (which can be trisected an infinite number of ways while maintaining that ratio). The third form would be x + y + z = 3; which can also be trisected an infinite number of ways and would be the least intuitive.

i am agreeing with you, i am just trying to show that it is illogical for it to be 'open for debate'.

There is a game theory term for this type of equilibrium, but i forgot its name. Its the same type of equilibrium as "there are three colors and a number, which one is different?" type sesame street problems.

Re: A “simple” 3rd grade problem

#104

Post by Fargren »

jneen wrote: post #87
derleth wrote: post #80

Earlier quoted context omitted.

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.

That's actually a great idea. If I were a math teacher, I would teach my class that IFQ is a reasonable answer to a question, and I'd throw in a few plainly ill-formed questions just to keep them on their toes. Actual thinking > correct answers.

I think for most questions that are not very straight forward, IFQ would be a valid answer with only very little argumentation. That's why formal languages are needed.

Re: A “simple” 3rd grade problem

#105

Post by mratzloff »

DanBC wrote: post #47
downandout wrote: post #36

Earlier quoted context omitted.

True, but I think that maybe 1/3 or more of the students would have answered this correctly. At some point during the grading process, you would think the teacher would begin to wonder why all of these students would have answered such a seemingly simple question "incorrectly" and take a second look.

...because 'trick' questions are (at least, were) part of math. The justification is to make sure people read, and understand, the question.

It's not a trick question, just a bad one. It doesn't include enough information to give an informed answer. Telepathy is required to suss out the author's intention.

In the real world, you can ask more questions and get a more complete picture. On an exam, generally you must accept what you are given.

Re: A “simple” 3rd grade problem

#106

Post by dos1 »

sillysaurus wrote: post #101
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…

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 problems like this. In response to the rest of your post, I was never taught how to "think" about math. I was educated in a decent school system, but it was all rote memorization of multiplication tables. I think most people who are interested in learning will come up with their own tricks regardless of curriculum. Of course, imagine how much better I'd be at this stuff if I had math teacher's who were competent :)

Re: A “simple” 3rd grade problem

#107

Post by kilburn »

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…

I didn't see the picture at first, and reasoned just as you exposed.

However, the teacher corrects it by writing "4 = 20". This is plainly wrong and with no possible explanation, since following the above reasoning, cutting in 4 pieces would require: 10 + 10 / 2 + (10 / 2) / 2 = 17.5 minutes.

Re: A “simple” 3rd grade problem

#108

Post by Avshalom »

s_kilk wrote: post #26
auctiontheory wrote: post #20

A friend of mine teaches school in rural North Carolina - here's what she tells me. Her school has to meet certain percentage-based "standards" - I forget the exact numbers, but let's say 75% is the cutoff. So now when Joey gets 5 answers right out of 10, the resulting 5/10 is defined as "75%." We're doomed.

Wait... How on earth do they justify redefining 50% as 75% ? (or whatever the actual numbers are)

Because if they don't they get defunded.

Re: A “simple” 3rd grade problem

#109

Post by lifeformed »

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…

I can cut a piece of wood into thousands of pieces in 5 seconds. I just slice it across the top a few times with my saw, and all the sawdust that comes off counts as separate pieces.

Re: A “simple” 3rd grade problem

#110

Post by auctiontheory »

GuiA wrote: post #70
s_kilk wrote: post #26

Earlier quoted context omitted.

Wait... How on earth do they justify redefining 50% as 75% ? (or whatever the actual numbers are)

It's called grading on a curve, and it's (imo, unfortunately) very common in undergraduate courses in the US :)

I understand grading on a curve. This is not that.

What we're talking about here is remapping a fraction (what the student scored) to a higher-than-equivalent percentage (what the "standard" requires).

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