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,"

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.