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 teachers in my market area cannot produce. Mathematician Patricia Kenschaft's article from the Notices of the American Mathematical Society "Racial Equity Requires Teaching Elementary School Teachers More Mathematics,"
reports on her work in teacher training programs for in-service teachers in New Jersey. "The understanding of the area of a rectangle and its relationship to multiplication underlies an understanding not only of the multiplication algorithm but also of the commutative law of multiplication, the distributive law, and the many more complicated area formulas. Yet in my first visit in 1986 to a K-6 elementary school, I discovered that not a single teacher knew how to find the area of a rectangle.
"In those innocent days, I thought that the teachers might be interested in the geometric interpretation of (x + y)^2. I drew a square with (x + y) on a side and showed the squares of size x^2 and y^2. Then I pointed to one of the remaining rectangles. 'What is the area of a rectangle that is x high and y wide?' I asked.
. . . .
"The teachers were very friendly people, and they know how frustrating it can be when no student answers a question. 'x plus y?' said two in the front simultaneously.
"'What?!!!' I said, horrified."
Professor Kenschaft's article includes other examples of the mathematical understanding of elementary schoolteachers in New Jersey. In this regard, New Jersey may actually set a higher standard than most states of the United States, so all over the United States, there is risk of learners being misled into incorrect mathematical conceptions by their schoolteachers.
The problem is not ideally written, to be sure. In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"
in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics.
But if you put a poorly written textbook into the hand of a poorly prepared teacher, you get bad results like that shown in the submission here. Those bad results go on for years. Poor teaching of fraction arithmetic in elementary schools has been a pet issue of mathematics education reformers in the United States for a long time. Professor Hung-hsi Wu of the University of California Berkeley has been writing about this issue for more than a decade.
he points out a problem of fraction addition from the federal National Assessment of Educational Progress (NAEP) survey project. On page 39 of his presentation handout (numbered in the .PDF of his lecture notes as page 38), he shows the fraction addition problem
12/13 + 7/8
for which eighth grade students were not even required to give a numerically exact answer, but only an estimate of the correct answer to the nearest natural number from five answer choices, which were
(a) 1
(b) 19
(c) 21
(d) I don't know
(e) 2
The statistics from the federal test revealed that for their best estimate of the sum of 12/13 + 7/8,
7 percent of eighth-graders chose answer choice a, that is 1;
28 percent of eighth-graders chose answer choice b, that is 19;
27 percent of eighth-graders chose answer choice c, that is 21;
14 percent of eighth-graders chose answer choice d, that is "I don't know";
while
24 percent of eighth-graders chose answer choice e, that is 2 (the best estimate of the sum).
I told Richard Rusczyk of the Art of Problem Solving about Professor Wu's document by email, and he later commented to me that Professor Wu "buried the lead" (underemphasized the most interesting point) in his lecture by not starting out the lecture with that shocking fact. Rusczyk commented that that basically means roughly three-fourths of American young people have no chance of success in a science or technology career with that weak an understanding of fraction arithmetic.
The way this is dealt with in other countries is to have specialist teachers of mathematics in elementary schools. Even with less formal higher education than United States teachers,
teachers in some countries can teach better because they develop "profound understanding of fundamental mathematics" and discuss with one another how to aid development of correct student understanding. The textbooks are also much better in some countries,
and the United States ought to do more to bring the best available textbooks (which in many cases are LESS expensive than current best-selling textbooks) into many more classrooms.
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.
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 :)
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.
I've been tutoring 1 on 1 middle school kids (at the 8th grade level) in underprivileged areas and what I find is that they have little understanding for what fractions are/represent at all. For example, I asked a student what the decimal value of 1/2 was (I had explained what decimal values were beforehand) and she didnt know. I was shocked (maybe its because we're cs people we have a special affinity for 2s). As a further test I gave her a piece of paper and asked fold the paper in half and she knew it instantly. I then asked where on the paper the 1/4 mark was. Again, she didnt know. This came from a further problem of not understanding that (1/2)/2 is 1/4. After playing with the paper folding it in so many ways she started to internalize what these fractions meant.
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.
This is something I find tremendously useful in programming, but at the same time find a lot of other developers amazed when it's used.
I don't care if the dataset in memory is 553MB or 632MB - what I really need to know is whether it's "a few tens of MB", "a few hundreds of MB", or a "a few thousand MB".
I don't care if the API server can service 7321 simultaneous requests or 6578 - I just need to know if its "a few hundred", "a few thousand", or "a few tens of thousands".
You can solve an enormous number of engineering and architecture problems with a reliable order-of-magnitude estimate - at the very least you can quickly exclude solutions that are vastly under (or over) provisioned for the problem you're trying to solve.
A good order-of-magnitude estimate is also a great error check for a more detailed calculation, if my quick estimate said "5000-ish plus or minus 50%", and your calculation says "24,152", one of us has got something wrong.
I remember this from my first university physics class. We would derive a movement equation for a cannonball, to find the optimal angle to shoot a cannon for maximum travel. Everybody knew the answer of course, but we'd always just used the formula. This time we'd start with the obvious integration equation, movement + attraction between 2 point masses, integrate over flight time, and find the point where it crosses the ground plane.
And then the teacher just took the range from the integration, and the formula, multiplied the two and put a ~= sign between them. I believe I actually stood up and said you can't do that and we had the first of many discussions about exactness.
That was scary.
That was my first run-in with what I considered the central article of my then faith : that you can derive the structure of the physical world from first principles. Throwing away terms in an equation in order to arrive at correct physics laws, I don't know, I considered it sacrilege or something. Of course I've since learned that deriving all of physics from it's own basic laws doesn't work, and the way we fix that is that we delete "inconvenient" terms in the equations when required. Deriving physics from a few mathematical laws is completely impossible. You can't even correctly derive the (mathematical) fields used in physics, so the very numbers that one uses to do physics aren't actually valid mathematical numbers.
So the relation between physics and mathematics is not that one is based on the other, because that was tried and didn't work out, and people have almost completely given up. So it was replaced by a marriage of convenience (this works ! Sure it won't validate mathematically but the numbers look really similar), ignoring at least a dozen elephants that stood in the way, and we just act like they don't exist.
As I read the question, I was prepared to work it out the hard way, and when I got to the part about not choosing the exact answer but only the closest, I was really dismayed because I thought I'd also have to work out how close each choice was to the actual value. Then I saw the possible answers and I thought, "Oh, 2."
I did it that way too. And while I agree that the best teacher for a person is that person themselves, this is such a basic understanding of math that I think that even the less motivated need to know this.
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 this could have significant impact not just in the US and EU but in the developing world too.
PS: About the fraction multiple choice: There's probably a bad joke about 24% being what we'd expect if we let the students chose at random. I'm not funny enough to think what it is. (The punchline being that there are 5 options, not 4.)
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.
It in no way detracts from your point, but I believe that asking a 'number sense' problem like the estimation or fraction problem you gave is a different issue than teaching the more algorithmic procedure of solving fraction problems.
One doesn't seem to preclude the other, nor does it seem to mean you won't have success in a science or technology career. I think you'll find a lot of people who know how to solve, say, 'circular motion problems,' but don't really understand what they are doing.
Just curious: which aspects of this elementary math education would you say could be taught by automated high-quality means, such as the combination of interactive games and questions that many people are working on right now? I assume that video lectures would not be effective at that age.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
reports on her work in teacher training programs for in-service teachers in New Jersey. "The understanding of the area of a rectangle and its relationship to multiplication underlies an understanding not only of the multiplication algorithm but also of the commutative law of multiplication, the distributive law, and the many more complicated area formulas. Yet in my first visit in 1986 to a K-6 elementary school, I discovered that not a single teacher knew how to find the area of a rectangle.
"In those innocent days, I thought that the teachers might be interested in the geometric interpretation of (x + y)^2. I drew a square with (x + y) on a side and showed the squares of size x^2 and y^2. Then I pointed to one of the remaining rectangles. 'What is the area of a rectangle that is x high and y wide?' I asked.
. . . .
"The teachers were very friendly people, and they know how frustrating it can be when no student answers a question. 'x plus y?' said two in the front simultaneously.
"'What?!!!' I said, horrified."
Professor Kenschaft's article includes other examples of the mathematical understanding of elementary schoolteachers in New Jersey. In this regard, New Jersey may actually set a higher standard than most states of the United States, so all over the United States, there is risk of learners being misled into incorrect mathematical conceptions by their schoolteachers.
The problem is not ideally written, to be sure. In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"
http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o...
in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics.
But if you put a poorly written textbook into the hand of a poorly prepared teacher, you get bad results like that shown in the submission here. Those bad results go on for years. Poor teaching of fraction arithmetic in elementary schools has been a pet issue of mathematics education reformers in the United States for a long time. Professor Hung-hsi Wu of the University of California Berkeley has been writing about this issue for more than a decade.
http://math.berkeley.edu/~wu/
In one of Professor Wu's recent lectures,
http://math.berkeley.edu/~wu/Lisbon2010_4.pdf
he points out a problem of fraction addition from the federal National Assessment of Educational Progress (NAEP) survey project. On page 39 of his presentation handout (numbered in the .PDF of his lecture notes as page 38), he shows the fraction addition problem
12/13 + 7/8
for which eighth grade students were not even required to give a numerically exact answer, but only an estimate of the correct answer to the nearest natural number from five answer choices, which were
(a) 1
(b) 19
(c) 21
(d) I don't know
(e) 2
The statistics from the federal test revealed that for their best estimate of the sum of 12/13 + 7/8,
7 percent of eighth-graders chose answer choice a, that is 1;
28 percent of eighth-graders chose answer choice b, that is 19;
27 percent of eighth-graders chose answer choice c, that is 21;
14 percent of eighth-graders chose answer choice d, that is "I don't know";
while
24 percent of eighth-graders chose answer choice e, that is 2 (the best estimate of the sum).
I told Richard Rusczyk of the Art of Problem Solving about Professor Wu's document by email, and he later commented to me that Professor Wu "buried the lead" (underemphasized the most interesting point) in his lecture by not starting out the lecture with that shocking fact. Rusczyk commented that that basically means roughly three-fourths of American young people have no chance of success in a science or technology career with that weak an understanding of fraction arithmetic.
The way this is dealt with in other countries is to have specialist teachers of mathematics in elementary schools. Even with less formal higher education than United States teachers,
http://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles...
http://www.ams.org/notices/199908/rev-howe.pdf
teachers in some countries can teach better because they develop "profound understanding of fundamental mathematics" and discuss with one another how to aid development of correct student understanding. The textbooks are also much better in some countries,
http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...
and the United States ought to do more to bring the best available textbooks (which in many cases are LESS expensive than current best-selling textbooks) into many more classrooms.