The comments already posted here are quite interesting. It takes well prepared teachers to serve up engaging problems that will excite young learners about mathematics. I just learned about the 2010 Teacher Education Study in Mathematics (TEDS-M)

http://www.educ.msu.edu/content/sites/usteds/documents/USTED...

a few days ago, as I discovered the book Teacher Education Matters: A Study of Middle School Mathematics Teacher Preparation in Six Countries

http://www.amazon.com/Teacher-Education-Matters-Mathematics-...

at my alma mater university library as I searched for books about mathematics education, my occupation. (The book, in turn, appears to be based on a publication from the study

http://www.educ.msu.edu/content/sites/usteds/documents/MT21R...

that I was able to view in one Web browser but not another. Perhaps most of you HN participants can read the study publication directly online.)

The study found and the book reports that "Putting more resources into U.S. middle school mathematics teachers' education could significantly raise future teachers' mathematics skills but may not be sufficient to equal those in countries where mathematics skills are substantially higher or produce sufficient numbers of more highly skilled middle school mathematics teachers, for two reasons. Average mathematics knowledge among U.S. college students is much lower than in Taiwan, South Korea, or Germany, and because of the relatively low salaries and prestige of teaching in the United States, the college students enrolled in teacher education are likely to average much lower mathematics skills than the large number of students in science, engineering, and economics/business." (Pages 278-279) The book also reports, especially relevant as a commment on the submitted article here, "South Korean and Taiwanese future teachers included both simple and complex examples in their lessons, usually including these in the beginning and middle of the lesson. By contrast, sampled U.S. future teachers tended mostly to use simple examples and to include them at the very end of the lesson." (Page 289)

Teachers in the early grades having adequate mathematics preparation to help young learners advance in their understanding is a very severe problem in the United States, where it has been reported that most elementary school teachers in a sample of teachers in New Jersey did not know a general rule for finding the area of a rectangle if the side lengths of the rectangle are known.

http://www.ams.org/notices/200502/fea-kenschaft.pdf

The dramatic differences in teacher preparation result in dramatic differences in mathematics achivement between countries.

http://pirls.bc.edu/timss2007/PDF/T07_M_IR_Chapter1.pdf

(See Exhibit 1.1 on pages 34 and 35 of the .PDF document for an example of an excellent use of parallel boxplots to compare the centers of various groups.) In general, United States "average" students are at the bottom level of top-performing countries, while even "average" students in those countries are at a "gifted" level for the United States.

The FAQ page for Epsilon Camp collects some other writings about producing challenging (and thus engaging) lessons for mathematics learners, preparing them to go far in mathematics with a love for the subject.

http://www.epsiloncamp.org/FAQ.php