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Oh dear. When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of a

Oh dear. When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of algorithms. Whoever gives such an answer does not connect classroom learning with the real world, he does not understand what the answer means. You see this often, even in university students. The problem isn't solved with more accountability or more teacher autonomy. The only solution is better teacher training and a different syllabus. Also this: http://www.inference.phy.cam.ac.uk/sanjoy/benezet/ http://www.inference.phy.cam.ac.uk/sanjoy/benezet/

deleted 12y ago
[deleted]
21echoes 12y ago
or, as explained quite clearly in the article, a simple misunderstanding of symbols: the kid interpreted it as 12/7 instead of 7/12.
- HarryHirsch 12y ago
  Two misunderstandings: the kid also thought that 1.5 means 1 remainder 5. That's a profound lack of knowledge, he does not understand decimal notation. If you do not know what these number symbols mean, how can you communicate using those symbols? It all degenerates into futile pattern matching.
  - yuliyp 12y ago
    Yes, yes it is a profound lack of knowledge. But he's still a fifth grader, and once you diagnose the problem (pattern matching gone wrong), you can fix it. In this case, the teacher identified that there was a sloppiness error (note it and allow them to work on being less sloppy using grades as motivation), as well as the misunderstanding of "remainder" vs "decimal part". The goal isn't just to grade the student, but to actually fix the things that are blocking their learning.
  - aptwebapps 12y ago
    I think you're being too extreme. It quite possible to have a less than complete grasp of something and write down the wrong answer. Heck, you can give the wrong answer even when you know something cold, all thought that's obviously not the case here. We don't know how much or how little attention the kid is putting into it and this is a 5th grader, after all. I understand what you're getting at but some kids don't get all of this all at once in a blinding flash - they become more and more familiar with it over time.
  - judk 12y ago
    A typo is not a profound non understanding. When a C++ programmer misuses Java notation, it isn't a profound none understanding, it's a mistake.
  - 21echoes 12y ago
    > he does not understand decimal notation. which, again, is a symbol misunderstanding. it's extremely presumptuous of you to say that two chained symbol misunderstandings by a fifth grader new to a subject are "the problem [of] rote learning of algorithms [by someone who] does not connect classroom learning with the real world [or] understand what the answer means" when you're a hammer, everything looks like a nail
- EpicEng 12y ago
  Which, of course, is still wrong... again, not understanding what he is doing.
- thaumasiotes 12y ago
  to be fair, that isn't necessarily a misunderstanding of symbols -- if you retested that kid several times, you might find him interpreting 7/12 as 12/7 and 12/7 also as 12/7, on the grounds that obviously you divide the larger number by the smaller number. That kind of "reasoning" is common, but amounts more to ignoring the symbols than misunderstanding them.
mikeash 12y ago
Excellent point. It happens again in the article with the 55MPH example. The original answer of 18 miles in 15 minutes is clearly wrong because it's more than a mile a minute, and a mile a minute is 60MPH. I saw this sort of thing a lot when I tutored physics in college. My students would crank through the formulas and come up with an answer. I would point out that their answer says the roller coaster is seventeen light years tall, or the bowling ball weighs less than a grain of sand, and this couldn't possibly be right. I always tried to get them to look at the shape of the problem first, and then look at the details. Otherwise you're just pattern matching, and don't know when you've gone horribly wrong. It's tricky, though, because you need to strike a balance. Seeing the big picture really well doesn't get the job done if you don't have the mechanics to back it up. If the goal is a precise answer, then knowing that 7/12 is around 0.5 and is definitely not greater than 1 doesn't get you there. But it seems that schools have gone way too far the other way, focusing on the mechanics almost exclusively. The "word problem" seems to be the one concession to the big picture, but even those are taught as pure pattern-matching exercises.
bottled_poe 12y ago
The human brain is not good at arithmetic. It is an outdated topic to be including in a modern curriculum.
thaumasiotes 12y ago
> When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of algorithms. This doesn't seem right. What algorithm did you learn in school that would produce 1.5 from 7/12?
- nmrm 12y ago
  Immediately means "without executing an entire algorithm".
  - thaumasiotes 12y ago
    Perhaps, if the student had done some rote learning of algorithms, they wouldn't have come up with 1.5 at all.
    - nmrm 12y ago
      > if the student had done some rote learning of algorithms, they wouldn't have come up with 1.5 at all. When you see "7/12 is 1.5", is your first thought "one sec, let me calculate that and see if the answer is correct"? Or do you just know, without any computation, that the statement must be false? If you understand division, it's far easier to explain why this answer is wrong without appealing to an algorithm than by appealing to an algorithm. And besides, the very example we're discussing proves this isn't the case! The student was able to use the algorithm correctly; his error was at the boundaries of the standard algorithm -- correct input and interpreting output properly. Simply practicing arithmetic will not prevent you from ever making a silly (or not-so-silly) mistake. Rote practice of the algorithm without also acquiring an intuition for the meaning of division won't help. With correct guidance, practicing the algorithm can help students form this intuition, but that requires an a priori realization that we need to teach something other than just the rote algorithm, even if we teach it by discussing executions of the algorithm on specific inputs and outputs.
titter 12y ago
> Oh dear. When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of algorithms. Not really, no. 1. Students have to have a good concept of division, as you suggest. 2. Students have to know a method of division that accurately gets a precise answer. (whether this is via an algorithmic method or via a calculator) 3. Students have to know that their answer to (2) should correspond to their concept in (1) 4. Students have to recognise that they can check their answer using (3), and then remember to actually check this. A good teacher will teach all four parts of this process. However, it's not possible to teach part 4 without first teaching parts 1 and 2. Every single child makes this same mistake at some point in their learning. This is not really a "problem" and it doesn't indicate a failure of teaching - in fact, the opposite here: the important thing is that the teacher has identified it and can advise the child on how to improve their understanding. That's what teaching is. As you say, many adults have not consistently achieved part 4 - in fact, it's not immediate at all: it has to be learned.
javajosh 12y ago
It's funny, a physicist is going to say "it's a half". A math person is going to roll their eyes, bored, and not answer at all. Only a mechanical engineer or a carpenter or an accountant are going to need an exact answer. Most of us are well served by the physicist approach to estimation, because even in the rare case we do need the precise answer our estimate will prevent obvious errors. And having a "feel" for OOM estimates is handy almost everywhere. And, actually, the approach even gives you a short cut to the precise answer. It's about 1/2, because it's so close to 6/12. In fact, it's only 1/12 away from half. Which means the answer is 1/2 + 1/12 (which is actually a better answer for, say, a carpenter). Note also that this problem is really crappy to do in decimal, as 1/12 yields a repeating decimal (.08333...), which is natures way of saying "use fractions".

deleted 12y ago

[deleted]

21echoes 12y ago

or, as explained quite clearly in the article, a simple misunderstanding of symbols: the kid interpreted it as 12/7 instead of 7/12.

HarryHirsch 12y ago
Two misunderstandings: the kid also thought that 1.5 means 1 remainder 5. That's a profound lack of knowledge, he does not understand decimal notation. If you do not know what these number symbols mean, how can you communicate using those symbols? It all degenerates into futile pattern matching.
- yuliyp 12y ago
  Yes, yes it is a profound lack of knowledge. But he's still a fifth grader, and once you diagnose the problem (pattern matching gone wrong), you can fix it. In this case, the teacher identified that there was a sloppiness error (note it and allow them to work on being less sloppy using grades as motivation), as well as the misunderstanding of "remainder" vs "decimal part". The goal isn't just to grade the student, but to actually fix the things that are blocking their learning.
- aptwebapps 12y ago
  I think you're being too extreme. It quite possible to have a less than complete grasp of something and write down the wrong answer. Heck, you can give the wrong answer even when you know something cold, all thought that's obviously not the case here. We don't know how much or how little attention the kid is putting into it and this is a 5th grader, after all. I understand what you're getting at but some kids don't get all of this all at once in a blinding flash - they become more and more familiar with it over time.
- judk 12y ago
  A typo is not a profound non understanding. When a C++ programmer misuses Java notation, it isn't a profound none understanding, it's a mistake.
- 21echoes 12y ago
  > he does not understand decimal notation. which, again, is a symbol misunderstanding. it's extremely presumptuous of you to say that two chained symbol misunderstandings by a fifth grader new to a subject are "the problem [of] rote learning of algorithms [by someone who] does not connect classroom learning with the real world [or] understand what the answer means" when you're a hammer, everything looks like a nail
EpicEng 12y ago
Which, of course, is still wrong... again, not understanding what he is doing.
thaumasiotes 12y ago
to be fair, that isn't necessarily a misunderstanding of symbols -- if you retested that kid several times, you might find him interpreting 7/12 as 12/7 and 12/7 also as 12/7, on the grounds that obviously you divide the larger number by the smaller number. That kind of "reasoning" is common, but amounts more to ignoring the symbols than misunderstanding them.

mikeash 12y ago

Excellent point. It happens again in the article with the 55MPH example. The original answer of 18 miles in 15 minutes is clearly wrong because it's more than a mile a minute, and a mile a minute is 60MPH. I saw this sort of thing a lot when I tutored physics in college. My students would crank through the formulas and come up with an answer. I would point out that their answer says the roller coaster is seventeen light years tall, or the bowling ball weighs less than a grain of sand, and this couldn't possibly be right. I always tried to get them to look at the shape of the problem first, and then look at the details. Otherwise you're just pattern matching, and don't know when you've gone horribly wrong. It's tricky, though, because you need to strike a balance. Seeing the big picture really well doesn't get the job done if you don't have the mechanics to back it up. If the goal is a precise answer, then knowing that 7/12 is around 0.5 and is definitely not greater than 1 doesn't get you there. But it seems that schools have gone way too far the other way, focusing on the mechanics almost exclusively. The "word problem" seems to be the one concession to the big picture, but even those are taught as pure pattern-matching exercises.

bottled_poe 12y ago

The human brain is not good at arithmetic. It is an outdated topic to be including in a modern curriculum.

thaumasiotes 12y ago

> When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of algorithms. This doesn't seem right. What algorithm did you learn in school that would produce 1.5 from 7/12?

nmrm 12y ago
Immediately means "without executing an entire algorithm".
- thaumasiotes 12y ago
  Perhaps, if the student had done some rote learning of algorithms, they wouldn't have come up with 1.5 at all.
  - nmrm 12y ago
    > if the student had done some rote learning of algorithms, they wouldn't have come up with 1.5 at all. When you see "7/12 is 1.5", is your first thought "one sec, let me calculate that and see if the answer is correct"? Or do you just know, without any computation, that the statement must be false? If you understand division, it's far easier to explain why this answer is wrong without appealing to an algorithm than by appealing to an algorithm. And besides, the very example we're discussing proves this isn't the case! The student was able to use the algorithm correctly; his error was at the boundaries of the standard algorithm -- correct input and interpreting output properly. Simply practicing arithmetic will not prevent you from ever making a silly (or not-so-silly) mistake. Rote practice of the algorithm without also acquiring an intuition for the meaning of division won't help. With correct guidance, practicing the algorithm can help students form this intuition, but that requires an a priori realization that we need to teach something other than just the rote algorithm, even if we teach it by discussing executions of the algorithm on specific inputs and outputs.

titter 12y ago

> Oh dear. When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of algorithms. Not really, no. 1. Students have to have a good concept of division, as you suggest. 2. Students have to know a method of division that accurately gets a precise answer. (whether this is via an algorithmic method or via a calculator) 3. Students have to know that their answer to (2) should correspond to their concept in (1) 4. Students have to recognise that they can check their answer using (3), and then remember to actually check this. A good teacher will teach all four parts of this process. However, it's not possible to teach part 4 without first teaching parts 1 and 2. Every single child makes this same mistake at some point in their learning. This is not really a "problem" and it doesn't indicate a failure of teaching - in fact, the opposite here: the important thing is that the teacher has identified it and can advise the child on how to improve their understanding. That's what teaching is. As you say, many adults have not consistently achieved part 4 - in fact, it's not immediate at all: it has to be learned.

javajosh 12y ago

It's funny, a physicist is going to say "it's a half". A math person is going to roll their eyes, bored, and not answer at all. Only a mechanical engineer or a carpenter or an accountant are going to need an exact answer. Most of us are well served by the physicist approach to estimation, because even in the rare case we do need the precise answer our estimate will prevent obvious errors. And having a "feel" for OOM estimates is handy almost everywhere. And, actually, the approach even gives you a short cut to the precise answer. It's about 1/2, because it's so close to 6/12. In fact, it's only 1/12 away from half. Which means the answer is 1/2 + 1/12 (which is actually a better answer for, say, a carpenter). Note also that this problem is really crappy to do in decimal, as 1/12 yields a repeating decimal (.08333...), which is natures way of saying "use fractions".