Further to my post on the remarkable failure of Scandinavian education systems to develop their students to anywhere near the levels indicated by their IQ potentials, a professor of mathematics at a Wisconsin university sent me data on the percentage of respondents in the TIMSS who gave the correct answer to the following question:
Which shows a correct method for finding 1/3 – 1/4?
A (1 – 1)/ (4 – 3)
B 1/ (4 – 3)
C (3 – 4)/ (3*4)
D (4 – 3)/ (3*4)
Below are the results. Do bear in mind that these are 8th graders we are talking about.
A | B | C | D | |
Korea | 2.7 | 6.9 | 4.2 | 86 |
Singapore | 4.8 | 5.5 | 6.5 | 83.1 |
Taipei | 2.9 | 7.7 | 7 | 82 |
Hong Kong | 4 | 8.7 | 10 | 77 |
Japan | 15.4 | 11.1 | 8.2 | 65.3 |
Russia | 12.3 | 18.8 | 4.8 | 62.8 |
Average | 25.4 | 26 | 9.4 | 37.1 |
US | 32.5 | 26.1 | 10.7 | 29.1 |
Finland | 42.3 | 29.5 | 8.7 | 16.1 |
Sweden | 14.4 | |||
Chile | 11.7 |
Finally, an international ratings list on which those smarmy, goody-goody Scandinavians don’t come on top! They barely do better than Chile, a country that got 421 (equiv. IQ ~88) in the PISA 2009 survey. Here is what he has to say on the matter:
One interesting fact is that among the 42 countries which tested 8th grade students, Finland had the highest percent of students who picked answer A and the third lowest percent correct. Chile had 11.7 correct and Sweden had 14.4 percent correct. The Finnish result is likely a surprise to the people who have praised the Finnish school system for their results on another international test, PISA. However university and technical college mathematics faculty in Finland will not be surprised. See [this] article signed by over 200 of them.
Anybody who suggests the progressive/neoliberal education policies of the Scandinavian countries are worthy of emulation should be presented with these figures and laughed out of the room.
The results for individual American and Canadian states:
A | B | C | D | |
Mass. | 21.4 | 20.8 | 9.9 | 44.4 |
Calif. | 28.2 | 21.6 | 11 | 38 |
Minn. | 23.5 | 26.3 | 14 | 35.1 |
Quebec | 27.3 | 23 | 13 | 33 |
Ontario | 27.7 | 22.4 | 14 | 32.5 |
Conn. | 21.8 | 25.8 | 17.7 | 31.3 |
Alberta | 34.7 | 23.7 | 12.3 | 27.8 |
I agree that Sweden’s educational system should only be considered as a deterrent. I am very surprised and puzzled by Finland equally poor performance on this particular task. After all, Finland have been touted as a great educational success story. Their PISA math scores, is only surpassed by East Asian countries. Teachers also have high status and salary in Finland, in complete contrast to the situation in Sweden.
Despite awareness of the situation, I must say it is shocking only 1/7 of Swedish student could get this basic straightforward question right! Even a monkey would stand a 25% chance of picking the correct answer! US at 29% is also a shameful level in this regard.
This is a parsing problem but not a maths problem. How many 8 graders know that * means multiplication? My bet is that this number is very low in Sweden as they probably use the x for multiplications while it is commonly used in the far east
ps. i don't think i could have picked out the correct answer when i was in 8 grade but i certainly could have told you that a third minus a quarter was a twelfth.
Replies: @SH
Despite awareness of the situation, I must say it is shocking only 1/7 of Swedish student could get this basic straightforward question right! Even a monkey would stand a 25% chance of picking the correct answer! US at 29% is also a shameful level in this regard.
Replies: @charly
This is a parsing problem but not a maths problem. How many 8 graders know that * means multiplication? My bet is that this number is very low in Sweden as they probably use the x for multiplications while it is commonly used in the far east
ps. i don’t think i could have picked out the correct answer when i was in 8 grade but i certainly could have told you that a third minus a quarter was a twelfth.
Charly, unfortunately that is incorrect in the Swedish case. We very rarely use × as some other countries do (never seen it in any math textbook) so it cannot be blamed on misconception. (× is primarily used in everyday Swedish when input factors represent orthogonal dimensions of space that should be kept separate and not counted together)
However, I am also sure that the vast majority of the Swedish 8th graders would understand either * or × as multiplication in this context. (For example × appears on all calculators). This is not a parsing problem.
Replies: @charly
When I studied in Sweden I was surprised at how weak ethnic Swedes math skills were (As a Yank, I had a low bar for being impressed too). Foreign students from former Warsaw Pact countries were much stronger in math from the small sample I saw.
A waste of talent for sure, though it is simply not true that it’s due to the privatized schools. Public school students do even worse (when matched for population differences) from the studies I remember seeing.
Fixing math class should be at the top of Western countries’ lists, but I’ll stop before I put a blog post into your comments….
Many years ago I read that there were no grades in Swedish schools. They thought it was cruel to give kids grades or something like that. I later spoke with a Swede who not only confirmed this, but actually defended it.
What is the use of grades? They are obviously cruel and counterproductive
Replies: @Seeki
Replies: @charly
What is the use of grades? They are obviously cruel and counterproductive
When my son is a boy, I brought him to the clinic centre to check his health yearly. They measure many things. One of them is his body height. He's always below the 96% level. It shows that he is far shorter from the average. Is this kind of measurement cruel? This is one of the metric in measuring his health. Grades are the accurate measurement of how successful of the kids in learning. Youth have to face the truth. If they are bring up in the green house, they cannot be strong enough to stand for the reality. People need to get hurt for growth.
What is the use of grades? They are obviously cruel and counterproductive
Replies: @Seeki
When my son is a boy, I brought him to the clinic centre to check his health yearly. They measure many things. One of them is his body height. He’s always below the 96% level. It shows that he is far shorter from the average. Is this kind of measurement cruel? This is one of the metric in measuring his health. Grades are the accurate measurement of how successful of the kids in learning. Youth have to face the truth. If they are bring up in the green house, they cannot be strong enough to stand for the reality. People need to get hurt for growth.
“Which shows a correct method for finding 1/3 – 1/4?
A (1 – 1)/ (4 – 3)
B 1/ (4 – 3)
C (3 – 4)/ (3*4)
D (4 – 3)/ (3*4)”
Seems like a weird question to ask. Why not just make the kids compute the answer?
And did they really use that exact notation in the TIMSS question? We Finns are not used to such notation, nor to multiple-choice questions.
Plus the method seems unfamiliar. We were taught in school to first convert both of the fractions to have the same denominator, then to add together the fractions.
I’d be willing to bet that if the TIMMS test had just asked the pupils to calculate the answer, Finnish kids would’ve scored much better.
But I also think the Finnish educational system is way overhyped.
Canadian/American here and +1. This is a really bizarre way to ask the question. I could easily calculate the answer but would never visualize the process like D. Perhaps this is how fractions are taught in the East?
In fact D is exactly the way as you mentioned above. 1/3 - 1/4 = 4/3*4 - 3/3*4 (convert the fractions to have the same denominator) = (4 - 3)/3*4. I feel strange that you don't notice it.
Replies: @charly, @Teemu from Finland
Here's how we're taught to do fractions in school (both the notation & the method):
http://imageshack.us/a/img545/940/qh5.gif
More explicitly: http://imageshack.us/a/img203/338/o2r9.gif
Replies: @SH
This is a parsing problem but not a maths problem. How many 8 graders know that * means multiplication? My bet is that this number is very low in Sweden as they probably use the x for multiplications while it is commonly used in the far east
ps. i don't think i could have picked out the correct answer when i was in 8 grade but i certainly could have told you that a third minus a quarter was a twelfth.
Replies: @SH
Charly, unfortunately that is incorrect in the Swedish case. We very rarely use × as some other countries do (never seen it in any math textbook) so it cannot be blamed on misconception. (× is primarily used in everyday Swedish when input factors represent orthogonal dimensions of space that should be kept separate and not counted together)
However, I am also sure that the vast majority of the Swedish 8th graders would understand either * or × as multiplication in this context. (For example × appears on all calculators). This is not a parsing problem.
x looks different from *. I would expect them to know x but not *. But even knowing that * means multiplication doesn't mean that the answer is not more difficult to parse
Replies: @SH
A (1 – 1)/ (4 – 3)
B 1/ (4 – 3)
C (3 – 4)/ (3*4)
D (4 – 3)/ (3*4)"
Seems like a weird question to ask. Why not just make the kids compute the answer?
And did they really use that exact notation in the TIMSS question? We Finns are not used to such notation, nor to multiple-choice questions.
Plus the method seems unfamiliar. We were taught in school to first convert both of the fractions to have the same denominator, then to add together the fractions.
I'd be willing to bet that if the TIMMS test had just asked the pupils to calculate the answer, Finnish kids would've scored much better.
But I also think the Finnish educational system is way overhyped.
Replies: @XVO, @Seeki, @Teemu from Finland
Canadian/American here and +1. This is a really bizarre way to ask the question. I could easily calculate the answer but would never visualize the process like D. Perhaps this is how fractions are taught in the East?
A (1 – 1)/ (4 – 3)
B 1/ (4 – 3)
C (3 – 4)/ (3*4)
D (4 – 3)/ (3*4)"
Seems like a weird question to ask. Why not just make the kids compute the answer?
And did they really use that exact notation in the TIMSS question? We Finns are not used to such notation, nor to multiple-choice questions.
Plus the method seems unfamiliar. We were taught in school to first convert both of the fractions to have the same denominator, then to add together the fractions.
I'd be willing to bet that if the TIMMS test had just asked the pupils to calculate the answer, Finnish kids would've scored much better.
But I also think the Finnish educational system is way overhyped.
Replies: @XVO, @Seeki, @Teemu from Finland
In fact D is exactly the way as you mentioned above. 1/3 – 1/4 = 4/3*4 – 3/3*4 (convert the fractions to have the same denominator) = (4 – 3)/3*4. I feel strange that you don’t notice it.
It is the same method but it is not the same notation. If you don't understand the notation how can you understand the question?
"In fact D is exactly the way as you mentioned above. "
No, it's not.
***
Perhaps the expression in D is a shorcut taught in some countries for summing fractions.
BTW, are the students given some kind of incentive to do their best in these TIMSS/PISA tests?
Replies: @SH
A (1 – 1)/ (4 – 3)
B 1/ (4 – 3)
C (3 – 4)/ (3*4)
D (4 – 3)/ (3*4)"
Seems like a weird question to ask. Why not just make the kids compute the answer?
And did they really use that exact notation in the TIMSS question? We Finns are not used to such notation, nor to multiple-choice questions.
Plus the method seems unfamiliar. We were taught in school to first convert both of the fractions to have the same denominator, then to add together the fractions.
I'd be willing to bet that if the TIMMS test had just asked the pupils to calculate the answer, Finnish kids would've scored much better.
But I also think the Finnish educational system is way overhyped.
Replies: @XVO, @Seeki, @Teemu from Finland
Here’s how we’re taught to do fractions in school (both the notation & the method):
More explicitly:
Charly, why do you assume *-symbol was actually used in the question. They most certainly used notation familiar to the student. The asterisk* is conveniently used in computer and programming context. Unless you have evidence to support they used asterisks, it is safe to assume they did not.
In fact D is exactly the way as you mentioned above. 1/3 - 1/4 = 4/3*4 - 3/3*4 (convert the fractions to have the same denominator) = (4 - 3)/3*4. I feel strange that you don't notice it.
Replies: @charly, @Teemu from Finland
It is the same method but it is not the same notation. If you don’t understand the notation how can you understand the question?
Charly, unfortunately that is incorrect in the Swedish case. We very rarely use × as some other countries do (never seen it in any math textbook) so it cannot be blamed on misconception. (× is primarily used in everyday Swedish when input factors represent orthogonal dimensions of space that should be kept separate and not counted together)
However, I am also sure that the vast majority of the Swedish 8th graders would understand either * or × as multiplication in this context. (For example × appears on all calculators). This is not a parsing problem.
Replies: @charly
x looks different from *. I would expect them to know x but not *. But even knowing that * means multiplication doesn’t mean that the answer is not more difficult to parse
Why do you assume *-symbol was actually used in the question. They most certainly used notation familiar to the student. The asterisk* is conveniently used in computer and programming context. Unless you have evidence to support they used asterisks, it is safe to assume they did not.
Here's how we're taught to do fractions in school (both the notation & the method):
http://imageshack.us/a/img545/940/qh5.gif
More explicitly: http://imageshack.us/a/img203/338/o2r9.gif
Replies: @SH
Charly, why do you assume *-symbol was actually used in the question. They most certainly used notation familiar to the student. The asterisk* is conveniently used in computer and programming context. Unless you have evidence to support they used asterisks, it is safe to assume they did not.
x looks different from *. I would expect them to know x but not *. But even knowing that * means multiplication doesn't mean that the answer is not more difficult to parse
Replies: @SH
Why do you assume *-symbol was actually used in the question. They most certainly used notation familiar to the student. The asterisk* is conveniently used in computer and programming context. Unless you have evidence to support they used asterisks, it is safe to assume they did not.
In fact D is exactly the way as you mentioned above. 1/3 - 1/4 = 4/3*4 - 3/3*4 (convert the fractions to have the same denominator) = (4 - 3)/3*4. I feel strange that you don't notice it.
Replies: @charly, @Teemu from Finland
“In fact D is exactly the way as you mentioned above. ”
No, it’s not.
***
Perhaps the expression in D is a shorcut taught in some countries for summing fractions.
BTW, are the students given some kind of incentive to do their best in these TIMSS/PISA tests?
D (4 – 3)/3*4 is simply the most direct method of calculation in this task. These objections makes no sense.
Replies: @Anonymous
I was taught the same way as Teemu above (Swedish school, Finnish math teacher). The question is phrased in a way that’s non-typical for Swedish and probably Finnish schools. If you assume that the Finnish students began by first calculating the correct answer (1/12) and then tried to match that answer to one of the given choices, the results make sense.
The Finnish students would calculate the correct answer — 1/12 — and then “clearly” see that the correct choice must be either A or B, as both of them contain a “1” in the numerator.
“Well, B is 1-1 or possibly 1/4 – 3, a negative number, so the correct choice must be A. What an odd question! These international tests sure are silly.”
So I messed up my fictional quote but I assume that you understood my point.
Replies: @SH
The Finnish students would calculate the correct answer -- 1/12 -- and then "clearly" see that the correct choice must be either A or B, as both of them contain a "1" in the numerator.
"Well, B is 1-1 or possibly 1/4 - 3, a negative number, so the correct choice must be A. What an odd question! These international tests sure are silly."
Replies: @Matti
So I messed up my fictional quote but I assume that you understood my point.
Obviously, 0/1 must equal 1/12 , makes sense(?)
Instead of looking for 1's, students should learn to perform simple calculations.
This is a straightforward question and every 8th grader should be able to figure out that (4 – 3)/ (3*4) = 1/12 . I am from Sweden and studied here, and these objections makes no sense to me.
Replies: @Matti
So I messed up my fictional quote but I assume that you understood my point.
Replies: @SH
Obviously, 0/1 must equal 1/12 , makes sense(?)
Instead of looking for 1’s, students should learn to perform simple calculations.
This is a straightforward question and every 8th grader should be able to figure out that (4 – 3)/ (3*4) = 1/12 . I am from Sweden and studied here, and these objections makes no sense to me.
Teemu said that it was a weird question to ask. XVO called it a bizarre way of asking the question. I wrote that the question was phrased in a non-typical way.
That, along with the heavily skewed choices by the Finnish students, should tell you that the question isn't as straightforward as you seem to think it is.
Replies: @SH
Obviously, 0/1 must equal 1/12 , makes sense(?)
Instead of looking for 1's, students should learn to perform simple calculations.
This is a straightforward question and every 8th grader should be able to figure out that (4 – 3)/ (3*4) = 1/12 . I am from Sweden and studied here, and these objections makes no sense to me.
Replies: @Matti
Teemu said that it was a weird question to ask. XVO called it a bizarre way of asking the question. I wrote that the question was phrased in a non-typical way.
That, along with the heavily skewed choices by the Finnish students, should tell you that the question isn’t as straightforward as you seem to think it is.
They could have phrased the question as follows: Which alternative equals 1/4 - 1/3. But it's not reading comprehension that is the issue here. If you know your math you can apply it to this situation even if you are not used to this kind of question. Despite what some believe, there are no Asian or Scandinavian fractions or mathematics. It is universal. What differs here are the students' ability to think in a few simple steps.
What's bizarre here is that so many Finnish students choose A where nominator is (1-1) or B for that matter (1/1). That if anything, reveals an incredible lack of understanding for basic mathematics.
Replies: @Teemu
Teemu said that it was a weird question to ask. XVO called it a bizarre way of asking the question. I wrote that the question was phrased in a non-typical way.
That, along with the heavily skewed choices by the Finnish students, should tell you that the question isn't as straightforward as you seem to think it is.
Replies: @SH
They could have phrased the question as follows: Which alternative equals 1/4 – 1/3. But it’s not reading comprehension that is the issue here. If you know your math you can apply it to this situation even if you are not used to this kind of question. Despite what some believe, there are no Asian or Scandinavian fractions or mathematics. It is universal. What differs here are the students’ ability to think in a few simple steps.
What’s bizarre here is that so many Finnish students choose A where nominator is (1-1) or B for that matter (1/1). That if anything, reveals an incredible lack of understanding for basic mathematics.
"They could have phrased the question as follows: Which alternative equals 1/4 – 1/3."
Yes, that would've been a much better wording.
"Despite what some believe, there are no Asian or Scandinavian fractions or mathematics. It is universal."
Well, there are different _METHODS_ and notations that are taught in different countries, for example for long multiplication, long division, for calculating with fractions, etc. How fricking hard is this to understand? All of them work equally well in the end (i.e. give the correct result in the end), but they are not the same methods.
"What differs here are the students’ ability to think in a few simple steps."
So you say.
Replies: @SH
They could have phrased the question as follows: Which alternative equals 1/4 - 1/3. But it's not reading comprehension that is the issue here. If you know your math you can apply it to this situation even if you are not used to this kind of question. Despite what some believe, there are no Asian or Scandinavian fractions or mathematics. It is universal. What differs here are the students' ability to think in a few simple steps.
What's bizarre here is that so many Finnish students choose A where nominator is (1-1) or B for that matter (1/1). That if anything, reveals an incredible lack of understanding for basic mathematics.
Replies: @Teemu
“They could have phrased the question as follows: Which alternative equals 1/4 – 1/3.”
Yes, that would’ve been a much better wording.
“Despite what some believe, there are no Asian or Scandinavian fractions or mathematics. It is universal.”
Well, there are different _METHODS_ and notations that are taught in different countries, for example for long multiplication, long division, for calculating with fractions, etc. How fricking hard is this to understand? All of them work equally well in the end (i.e. give the correct result in the end), but they are not the same methods.
“What differs here are the students’ ability to think in a few simple steps.”
So you say.
Then enlighten me about these different methods for calculating " 4 - 3 " and " 1 - 1 " and " 4 × 3 ".
Whatever method the Nordic students used for calculating the fractions on top do not matter as long as they arrive at the correct number. In this case, it is really just about finding the least common denominator then subtract. It is elementary.
About notation, as I wrote earlier we do not know which (·/×) notation that was used. Either · or × would be comprehensible to all or the vast majority of students, so why use it as a poor excuse. Also, TIMSS likely adjusts notation for the specific country.
"They could have phrased the question as follows: Which alternative equals 1/4 – 1/3."
Yes, that would've been a much better wording.
"Despite what some believe, there are no Asian or Scandinavian fractions or mathematics. It is universal."
Well, there are different _METHODS_ and notations that are taught in different countries, for example for long multiplication, long division, for calculating with fractions, etc. How fricking hard is this to understand? All of them work equally well in the end (i.e. give the correct result in the end), but they are not the same methods.
"What differs here are the students’ ability to think in a few simple steps."
So you say.
Replies: @SH
Then enlighten me about these different methods for calculating ” 4 – 3 ” and ” 1 – 1 ” and ” 4 × 3 “.
Whatever method the Nordic students used for calculating the fractions on top do not matter as long as they arrive at the correct number. In this case, it is really just about finding the least common denominator then subtract. It is elementary.
About notation, as I wrote earlier we do not know which (·/×) notation that was used. Either · or × would be comprehensible to all or the vast majority of students, so why use it as a poor excuse. Also, TIMSS likely adjusts notation for the specific country.
“Then enlighten me about these different methods for calculating ” 4 – 3 ” and ” 1 – 1 ” and ” 4 × 3 “.”
Okay, I’m done with you.
Teemu, the only methodological reason I can see that may explain the poor results, is that the students simply do not know how to calculate with parentheses. I simple assumed it was universally taught by 8th grade. It is strange that Finland that scored well in PISA fail so miserable at this question.
sorry for the bluntness, but there isn't much methodology to this, except for basic counting rules
Okay, I'm done with you.
Replies: @SH, @SH
Teemu, the only methodological reason I can see that may explain the poor results, is that the students simply do not know how to calculate with parentheses. I simple assumed it was universally taught by 8th grade. It is strange that Finland that scored well in PISA fail so miserable at this question.
Okay, I'm done with you.
Replies: @SH, @SH
sorry for the bluntness, but there isn’t much methodology to this, except for basic counting rules
To expand and clarify on my posts above. When learning fractions, it’s likely that Finnish students are told to always try to give the answer expressed *as the simplest possible fraction*. The failure to do so when solving problems results in mark-ups from the teachers, or when doing tests, the loss of points.
Given the above problem, many students would begin by finding the simplest possible answer to “1/3 – 1/4” using the “method” outlined by Teemu above.
At this point they point they have
1) Correctly calculated the fraction to 1/12
2) using the method that they’ve been taught.
3) And, importantly, they have the fraction expressed in the simplest way possible, as 1/12. The fraction can’t be simplied further.
If this was a typical test in a Swedish or Finnish school, they would be done with the problem at this point.
THEY HAVE CORRECTLY CALCULATED THE FRACTION AND HAVE THE ANSWER IN ITS SIMPLEST FORM — just as they’ve been taught.
Now you are asking them to do something highly un-intuitive:
— Multiple-choice questions aren’t typically used in the same way as they seem to be used in other countries.
— “A correct method for finding..” They have already used the method they’ve been taught. Why would you call any of the possible choices a method? None of those are “methods”.
— Again, method??? The simplest answer has already been found. A method has already been used.
— In a slightly confused state, they are trying to find the best fit of “1/12” to one of the choices available. This “matching-up” step is done in “pattern matching mode” rather than in “mathematical mode”. AT THIS POINT MOST OF THEM ARE NO LONGER CALCULATING. Look at the “form” of “1/12” and the “form” of the choices given — and the results make sense.
(This the point where SH steps in and goes: BUT! BUT! BUT! It’s so wrong!!! BUT! BUT! How bizarre!!! BUT! BUT! BUT! They should.. why don’t they.. BUT! BUT! That’s incorrect…! Explicable! BUT! BUT! I’m so shocked!!! BUT! BUT! They must!!!)
— The smarter students will realize that there’s something weird going on and check their results by calculation. The other students will have moved on at this point, given that this is a test with other problems to be solved, presumably under some time pressure.
It is of course perfectly understandable that those of the Aspergic nature — some engineers, programmers and accountants — will have trouble understanding the above, as it requires a certain amount of empathy, and the ability to put yourself into the students’ shoes, the ability to figure out what kind of background knowledge and training the students are working from.
How I long for a edit button for Wordpress comments.
Inexplicable.
Replies: @Matti
Given the above problem, many students would begin by finding the simplest possible answer to "1/3 – 1/4" using the "method" outlined by Teemu above.
At this point they point they have
1) Correctly calculated the fraction to 1/12
2) using the method that they've been taught.
3) And, importantly, they have the fraction expressed in the simplest way possible, as 1/12. The fraction can't be simplied further.
If this was a typical test in a Swedish or Finnish school, they would be done with the problem at this point.
THEY HAVE CORRECTLY CALCULATED THE FRACTION AND HAVE THE ANSWER IN ITS SIMPLEST FORM -- just as they've been taught.
Now you are asking them to do something highly un-intuitive:
-- Multiple-choice questions aren't typically used in the same way as they seem to be used in other countries.
-- "A correct method for finding.." They have already used the method they've been taught. Why would you call any of the possible choices a method? None of those are "methods".
-- Again, method??? The simplest answer has already been found. A method has already been used.
-- In a slightly confused state, they are trying to find the best fit of "1/12" to one of the choices available. This "matching-up" step is done in "pattern matching mode" rather than in "mathematical mode". AT THIS POINT MOST OF THEM ARE NO LONGER CALCULATING. Look at the "form" of "1/12" and the "form" of the choices given -- and the results make sense.
(This the point where SH steps in and goes: BUT! BUT! BUT! It's so wrong!!! BUT! BUT! How bizarre!!! BUT! BUT! BUT! They should.. why don't they.. BUT! BUT! That's incorrect...! Explicable! BUT! BUT! I'm so shocked!!! BUT! BUT! They must!!!)
-- The smarter students will realize that there's something weird going on and check their results by calculation. The other students will have moved on at this point, given that this is a test with other problems to be solved, presumably under some time pressure.
It is of course perfectly understandable that those of the Aspergic nature -- some engineers, programmers and accountants -- will have trouble understanding the above, as it requires a certain amount of empathy, and the ability to put yourself into the students' shoes, the ability to figure out what kind of background knowledge and training the students are working from.
Replies: @Matti
How I long for a edit button for WordPress comments.
Inexplicable.
See?
An.
How I long for a edit button for Wordpress comments.
Inexplicable.
Replies: @Matti
See?
An.
I’ll try to *sum up* my thoughts later. But look at this:
Average overall TIMSS score
Russia 539
Finland 514
Chile 416
Correct answer to our question (%)
Russia 62.8
Finland 16.1
Chile 11.7
Doesn’t that seem a bit fishy? Are Finnish kids just extra bad at doing fractions?
Hahah, I understand what’s meant by method now.
The Finns are beginning by calculating the fraction to its simplest and most elegant form.
In Finland students are always assumed to give the most elegant version of an fraction as their answer.
The students have done this by the time they start looking at the multiple choices.
Then they become extremely confused with this method talk. So apparently, in some countries, you call the “middle-steps” taken towards the final answer a method?
You seriously ask for a “middle-step” — calling it a “method” — when it’s a question of doing simple fractions?
Hahaha, that’s actually pretty funny. 🙂
It’s the utter lack of elegance in the correct solution that trips the Finns up.
The students are assuming that this “method” thing is some advanced concept that they haven’t been taught, so they take a guess between A and B.
This actually explains the apparent lack of calculation.
“Method”!
Bwahaha!
When they look for the least common denominator they probably multiplies them 3×4. They also subtract (4-3) /12, So why A or B should make most sense is beyond me. Could you explain how or why?
They aren’t doing calculation at that point. They are not looking for mathematical equality. They have already calculated the most elegant version of the fraction.
The word “method” puts them in a state of confusion. They are trying to make sense of it, thinking that it’s some highly advanced concept that they haven’t learned yet. Perhaps it is some elaborate algorithmic transmutation that is only taught to advanced math students?
This changes the state of the students from “calculation mode” to something that we could call “look for patterns or similarity and make the best guess”.
The possibility that the students, being in a test situation, are being asked for a “middle-step”, the so-called “method”, never occurs to them because Finnish teachers insist that answers to fraction problems should always be given in their simplest and most elegant form — an answer that they’ve already calculated.
If you were to transmute the elegant and simple…
1/12
…to something even more elegant, something more higher-order, something more abstract — which option would you choose?
When the problem is posed that way 0/1 comes close to making sense. 🙂
Keep in mind that this occurs during time pressure. As you familiarize yourself with the problem, and spend more and more time reasoning about it, it becomes harder to see how it might look like for the average and slightly nervous 8th grader.
" The possibility that the students, being in a test situation, are being asked for a “middle-step”, the so-called “method”, never occurs to them "
Matti, Are you saying that Finnish students should not describe how they arrived at the answer when performing exams / tests, i.e which method they used? I find this very strange. I was always required to show my methodology, when I studied math in Sweden, especially during exams.
Replies: @SH, @Matti
That’s what I call contortions. I highlighted that there are obvious similarities between D and the calculation method (practically identical, in compressed format).
I am sure we can all agree on this 🙂 So are Scandinavian students oblivious to how they calculated the fractions on top?
Sure time is limited but conditions are equal for all nations. How much time is really needed to calculate (1-1) or (4-3)?
About elegant. This is mathematics were you are expected to perform a couple of basic calculations and finding the correct answer. You are not supposed to sit and pick the numbers that looks pretty, elegant or ones you like most.
Minäkin taidan antaa periksi…
The word "method" puts them in a state of confusion. They are trying to make sense of it, thinking that it's some highly advanced concept that they haven't learned yet. Perhaps it is some elaborate algorithmic transmutation that is only taught to advanced math students?
This changes the state of the students from "calculation mode" to something that we could call "look for patterns or similarity and make the best guess".
The possibility that the students, being in a test situation, are being asked for a "middle-step", the so-called "method", never occurs to them because Finnish teachers insist that answers to fraction problems should always be given in their simplest and most elegant form -- an answer that they've already calculated.
If you were to transmute the elegant and simple...
1/12
...to something even more elegant, something more higher-order, something more abstract -- which option would you choose?
When the problem is posed that way 0/1 comes close to making sense. :-)
Keep in mind that this occurs during time pressure. As you familiarize yourself with the problem, and spend more and more time reasoning about it, it becomes harder to see how it might look like for the average and slightly nervous 8th grader.
Replies: @SH
” The possibility that the students, being in a test situation, are being asked for a “middle-step”, the so-called “method”, never occurs to them ”
Matti, Are you saying that Finnish students should not describe how they arrived at the answer when performing exams / tests, i.e which method they used? I find this very strange. I was always required to show my methodology, when I studied math in Sweden, especially during exams.
Method is this context should not be something mysterious. They are specifically told that method here should be understood as one of the four alternatives. They are also asked to match it with "1/3 - 1/4"
I cannot imagine that "method" would be such an unfamiliar and strange word in Finnish. It is unambiguous to me as a Swede.
Replies: @Matti
Did your examiners take into account your methodology when you answered multiple-choice questions?
Right.
When you studied math you were typically given free-response questions during exams, as is done in Finland.
Hence why Teemu above noted that the test item was of the, for the students uncommon, multiple-choice format.
Replies: @SH
" The possibility that the students, being in a test situation, are being asked for a “middle-step”, the so-called “method”, never occurs to them "
Matti, Are you saying that Finnish students should not describe how they arrived at the answer when performing exams / tests, i.e which method they used? I find this very strange. I was always required to show my methodology, when I studied math in Sweden, especially during exams.
Replies: @SH, @Matti
Method is this context should not be something mysterious. They are specifically told that method here should be understood as one of the four alternatives. They are also asked to match it with “1/3 – 1/4”
I cannot imagine that “method” would be such an unfamiliar and strange word in Finnish. It is unambiguous to me as a Swede.
"metod subst. planmässigt tillvägagångssätt för att uppnå ett visst resultat"
-- Nordstedts Svenska ordbok
This is identical to how "method" is typically used in English. It refers to an orderly process or procedure -- not an incomplete or a partially reached result. The students follow the procedure/process/method that they've been taught and arrive to the final and simplified fraction 1/12. At this point they stop calculating because there isn't, according to the method that they are using, anything more to calculate.
The *lack* of further calculation explains the weird response pattern. The questions isn't where they went wrong in their calculations, the question is why they *stopped* calculating during a math examination.
(Att beskriva svarsalternativen som utgörs av oförenklade bråk som "metoder" för att uppnå slutresultatet, som under typiska omständigheter utgörs av bråket i dess enklaste form, tillhör inte normalt språkbruk varken inom det svenska eller finska skolväsendet.)
Replies: @SH
" The possibility that the students, being in a test situation, are being asked for a “middle-step”, the so-called “method”, never occurs to them "
Matti, Are you saying that Finnish students should not describe how they arrived at the answer when performing exams / tests, i.e which method they used? I find this very strange. I was always required to show my methodology, when I studied math in Sweden, especially during exams.
Replies: @SH, @Matti
Did your examiners take into account your methodology when you answered multiple-choice questions?
Right.
When you studied math you were typically given free-response questions during exams, as is done in Finland.
Hence why Teemu above noted that the test item was of the, for the students uncommon, multiple-choice format.
Yes, It's not typical. However, a student could be expected to apply his skills and teachings to solve it regardless. I think we should aim to educate students that are capable of applying their skills to somewhat unfamiliar but simple situations.
As I mentioned, there are also very strong similarities between D (essentially identical) and how you calculate the top fractions. If you calculated it correctly matching should be straight-forward task.
Replies: @charly
Did your examiners take into account your methodology when you answered multiple-choice questions?
Right.
When you studied math you were typically given free-response questions during exams, as is done in Finland.
Hence why Teemu above noted that the test item was of the, for the students uncommon, multiple-choice format.
Replies: @SH
Yes, It’s not typical. However, a student could be expected to apply his skills and teachings to solve it regardless. I think we should aim to educate students that are capable of applying their skills to somewhat unfamiliar but simple situations.
As I mentioned, there are also very strong similarities between D (essentially identical) and how you calculate the top fractions. If you calculated it correctly matching should be straight-forward task.
but than it is a different question. It is no longer how many students can calculate 1/3 - 1/4 but how many can parse the answers in a particular country. And if they have use *, something which i expect, that that question is just a lot harder than calculating the answer.
ps. my guess is that more people know by heart that 1/12 is the answer to 1/3 - 1/4 that can name * as a sign for multiplications
Replies: @SH
Yes, It's not typical. However, a student could be expected to apply his skills and teachings to solve it regardless. I think we should aim to educate students that are capable of applying their skills to somewhat unfamiliar but simple situations.
As I mentioned, there are also very strong similarities between D (essentially identical) and how you calculate the top fractions. If you calculated it correctly matching should be straight-forward task.
Replies: @charly
but than it is a different question. It is no longer how many students can calculate 1/3 – 1/4 but how many can parse the answers in a particular country. And if they have use *, something which i expect, that that question is just a lot harder than calculating the answer.
ps. my guess is that more people know by heart that 1/12 is the answer to 1/3 – 1/4 that can name * as a sign for multiplications
1) there is highly unlikely that they used *asterisk, since it's only used in computer language. Why do you expect * was used?
2) They should know how to calculate fractions. They should know and understand some method for this, not rely by heart. You can argue that this is not an important issue. But I consider it a basic test in mathematical knowledge and understanding. 14-16% is an incredibly low result. It reveals some gaps in the math-education in Finland/Sweden for sure.
3)It is not about parsing, it's about straightforward calculation in both instances. Why is this difficult to understand?
Replies: @charly
Method is this context should not be something mysterious. They are specifically told that method here should be understood as one of the four alternatives. They are also asked to match it with "1/3 - 1/4"
I cannot imagine that "method" would be such an unfamiliar and strange word in Finnish. It is unambiguous to me as a Swede.
Replies: @Matti
“metod subst. planmässigt tillvägagångssätt för att uppnå ett visst resultat”
— Nordstedts Svenska ordbok
This is identical to how “method” is typically used in English. It refers to an orderly process or procedure — not an incomplete or a partially reached result. The students follow the procedure/process/method that they’ve been taught and arrive to the final and simplified fraction 1/12. At this point they stop calculating because there isn’t, according to the method that they are using, anything more to calculate.
The *lack* of further calculation explains the weird response pattern. The questions isn’t where they went wrong in their calculations, the question is why they *stopped* calculating during a math examination.
(Att beskriva svarsalternativen som utgörs av oförenklade bråk som “metoder” för att uppnå slutresultatet, som under typiska omständigheter utgörs av bråket i dess enklaste form, tillhör inte normalt språkbruk varken inom det svenska eller finska skolväsendet.)
I prefer not to get into a dispute about semantics. Personally I wouldn't use the word "method" in this question. Instead asking which alternative is equal to = "1/3 - 1/4". But given the context method should be comprehensible. At least it seems obvious to me. Could almost everyone really misapprehend so severely?
Replies: @SH
Seeing how that question differs from all the others, it might be a problem with semantics.
It is not clear that D is “a correct method”, especially correct generic method.
D is a possible correct intermediate “step” after using a correct method and
performing 2 multiplications, but it is not a generic correct method for solving:
A/x-B/y
A correct method for solving that clearly wouldn’t be:
(y-x)/(x*y)
as the D might be interpreted to suggests, but
(y*A)/(x*y)-(x*B)/(x*y)
or
(y*A-x*B)/(x*y)
So perhaps using the word “method” leads them thinking more abstractly and generically than the question is intended to be interpreted.
Perhaps after failing to find any correct *method*, the students choose the simplest “least wrong” alternative, hoping that it is mistyped in the test, or is some kind of a trick question.
Here are all the top results from previous years
http://en.wikipedia.org/wiki/Trends_in_International_Mathematics_and_Science_Study
"A correct method for solving that clearly wouldn’t be":
(y-x)/(x*y)
This is simply incorrect. In fact (y-x)/(x*y) is a more straightforward and perfectly valid method, and I would say more elegant way to solve this task than (y * A) / (x * y) - (x * B) / (x * y) which is unnecessarily lengthy. If both numerators are 1 why not just use (y-x)/(x*y). However, I assume it is not the procedure that most Scandinavian students use.
but than it is a different question. It is no longer how many students can calculate 1/3 - 1/4 but how many can parse the answers in a particular country. And if they have use *, something which i expect, that that question is just a lot harder than calculating the answer.
ps. my guess is that more people know by heart that 1/12 is the answer to 1/3 - 1/4 that can name * as a sign for multiplications
Replies: @SH
1) there is highly unlikely that they used *asterisk, since it’s only used in computer language. Why do you expect * was used?
2) They should know how to calculate fractions. They should know and understand some method for this, not rely by heart. You can argue that this is not an important issue. But I consider it a basic test in mathematical knowledge and understanding. 14-16% is an incredibly low result. It reveals some gaps in the math-education in Finland/Sweden for sure.
3)It is not about parsing, it’s about straightforward calculation in both instances. Why is this difficult to understand?
1) Because it is the question/answers they give as belonging to the score. It also would explain the results as i don't think the average Scandinavian knows programing.
2) 14-16% is the number i would expect to get it the first time the teacher explains it. That would be second or third grade
3) It is about parsing. I have to think about what * means and i seriously doubt that would have known it when i was in 8 grade. Which would have made the whole question a guessing game
"metod subst. planmässigt tillvägagångssätt för att uppnå ett visst resultat"
-- Nordstedts Svenska ordbok
This is identical to how "method" is typically used in English. It refers to an orderly process or procedure -- not an incomplete or a partially reached result. The students follow the procedure/process/method that they've been taught and arrive to the final and simplified fraction 1/12. At this point they stop calculating because there isn't, according to the method that they are using, anything more to calculate.
The *lack* of further calculation explains the weird response pattern. The questions isn't where they went wrong in their calculations, the question is why they *stopped* calculating during a math examination.
(Att beskriva svarsalternativen som utgörs av oförenklade bråk som "metoder" för att uppnå slutresultatet, som under typiska omständigheter utgörs av bråket i dess enklaste form, tillhör inte normalt språkbruk varken inom det svenska eller finska skolväsendet.)
Replies: @SH
I prefer not to get into a dispute about semantics. Personally I wouldn’t use the word “method” in this question. Instead asking which alternative is equal to = “1/3 – 1/4”. But given the context method should be comprehensible. At least it seems obvious to me. Could almost everyone really misapprehend so severely?
I should also clarify that D;(4 – 3)/ (3*4) is indeed a method/procedure (in fact, the most direct) to calculate 1/3 - 1/4 as long as the numerators are identical. If the numerators are 2 or more you simply multiply with the actual numerator; 2(y-x) / (xy). I hope this can provide some clarification.
It is not clear that D is "a correct method", especially correct generic method.
D is a possible correct intermediate "step" after using a correct method and
performing 2 multiplications, but it is not a generic correct method for solving:
A/x-B/y
A correct method for solving that clearly wouldn't be:
(y-x)/(x*y)
as the D might be interpreted to suggests, but
(y*A)/(x*y)-(x*B)/(x*y)
or
(y*A-x*B)/(x*y)
So perhaps using the word "method" leads them thinking more abstractly and generically than the question is intended to be interpreted.
Perhaps after failing to find any correct *method*, the students choose the simplest "least wrong" alternative, hoping that it is mistyped in the test, or is some kind of a trick question.
Here are all the top results from previous years
http://en.wikipedia.org/wiki/Trends_in_International_Mathematics_and_Science_Study
Replies: @SH
“A correct method for solving that clearly wouldn’t be”:
(y-x)/(x*y)
This is simply incorrect. In fact (y-x)/(x*y) is a more straightforward and perfectly valid method, and I would say more elegant way to solve this task than (y * A) / (x * y) – (x * B) / (x * y) which is unnecessarily lengthy. If both numerators are 1 why not just use (y-x)/(x*y). However, I assume it is not the procedure that most Scandinavian students use.
I prefer not to get into a dispute about semantics. Personally I wouldn't use the word "method" in this question. Instead asking which alternative is equal to = "1/3 - 1/4". But given the context method should be comprehensible. At least it seems obvious to me. Could almost everyone really misapprehend so severely?
Replies: @SH
I should also clarify that D;(4 – 3)/ (3*4) is indeed a method/procedure (in fact, the most direct) to calculate 1/3 – 1/4 as long as the numerators are identical. If the numerators are 2 or more you simply multiply with the actual numerator; 2(y-x) / (xy). I hope this can provide some clarification.
1) there is highly unlikely that they used *asterisk, since it's only used in computer language. Why do you expect * was used?
2) They should know how to calculate fractions. They should know and understand some method for this, not rely by heart. You can argue that this is not an important issue. But I consider it a basic test in mathematical knowledge and understanding. 14-16% is an incredibly low result. It reveals some gaps in the math-education in Finland/Sweden for sure.
3)It is not about parsing, it's about straightforward calculation in both instances. Why is this difficult to understand?
Replies: @charly
1) Because it is the question/answers they give as belonging to the score. It also would explain the results as i don’t think the average Scandinavian knows programing.
2) 14-16% is the number i would expect to get it the first time the teacher explains it. That would be second or third grade
3) It is about parsing. I have to think about what * means and i seriously doubt that would have known it when i was in 8 grade. Which would have made the whole question a guessing game
This kind of fraction calculation should be taught in grade 4 or latest grade 5. But even in the top performer countries, a lot of 8th grader don’t know the answer (from the top table given above: Korea 14%, Singapore 14.9% and Taipei 18%). This means that no matter how good your teaching method is, if the student don’t learn, they will not know. The global fashion trend in education to dummyfied the exam questions in order to have no child left behind is of no use at all.
"In fact D is exactly the way as you mentioned above. "
No, it's not.
***
Perhaps the expression in D is a shorcut taught in some countries for summing fractions.
BTW, are the students given some kind of incentive to do their best in these TIMSS/PISA tests?
Replies: @SH
D (4 – 3)/3*4 is simply the most direct method of calculation in this task. These objections makes no sense.
American here. This is how I was taught to solve this kind of problem:
1/3 - 1/4 = 1(4)/3(4) - 1(3)/4(3)= 4/12 - 3/12= 1/12
In most of Europe (in Latin Europe for certain) fractions are quite overlooked at school while a lot of time is devoted to the understanding of decimals. In US/Canada fractions are a drill that pupils have to undergo for many many years and the use of decimals is not as well taught as in Europe.
I bet that most US/Canadian pupils would have significant difficulty in picking up the correct answer to the question: 0.333-0.250 = A) 0.583 B) 0.313 C) 0.023 D) 0.083
Just as they can very easily visualize a length like 6′ 1″ 3/8 which is utterly meaningless to most Europeans, while 1.864 m means little to most US/Canadians.
This is not IQ it is culture.
Fractions are useful for artisans. School learning is not aimed at being helpful to artisans. It is infected by science. Scientists use decimals.
FWIW, all possible confusion over asterisks aside (TI calculators use those, btw), the kids should’ve been able to use basic rules to eliminate the first three, even if the method of D was a bit unorthodox for them.
A’s numerator would equal 0, B would equal 1, and C would yield a negative number, which is obviously not possible since .33 – .25 is not negative.
As a Finn and someone who graduated from Helsinki University of Technology, I can tell that this 16% represents those who actually _understand_ some math and are capable of analytical thinking.
I grew up with a tight group of 5 nerdy friends who always were best in their class at math and physics from the first grade to the last grade of high school. Three of us graduated from Helsinki University of Technology, one from University of Helsinki, one from a polytechnic college.
All five of us tried to actually _understand_ math and physics. I can tell that nearly everyone else in our class were quite bad at math and struggled if it dealt with something abstract like fractions. Therefore I’m not surprised with the results. Having studied in Finland and dealing with people in every day’s life I can tell that the majority of Finns (80-90%) are incapable of analytical thinking.
I suppose whites are simply intellectually inferior. I propose that we cull the bottom 50% since whites already consume more resources than everyone else.
D (4 – 3)/3*4 is simply the most direct method of calculation in this task. These objections makes no sense.
Replies: @Anonymous
American here. This is how I was taught to solve this kind of problem:
1/3 – 1/4 = 1(4)/3(4) – 1(3)/4(3)= 4/12 – 3/12= 1/12