Open-ended questions and problems are indeed awesome. Moreover, they are an essential part of a sound education in mathematics, even at the K-12 (primary and secondary schooling) level of learning. But open-ended questions used for teaching purposes should be carefully written for sound teaching points, and teachers using them should have sufficient background in mathematics to guide student approaches to grappling with them. One of my favorite authors on mathematics education reform (Professor Hung-hsi Wu of UC Berkeley) began writing on that issue in 1994 with his article, "The Role of Open-ended Problems in Mathematics Education,"
and he followed up on that article with a wonderful article in the fall 1999 issue of American Educator, "Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education."
Since then, Professor Wu has written many more useful articles on mathematics education, including guides for parents, teachers, school administrators, and teacher educators on how to apply the new Common Core State Standards in mathematics better to improve mathematics education in the United States.
A good example of a beguiling textbook by a world-famous mathematician with lots of open-ended problems is Algebra, by the late Israel M. Gelfand and Alexander Shen.
Some of the problems in this book are HARD, but they are generally well posed problems of actual research interest to mathematicians, that just happen to be accessible to pupils just beginning to learn algebra.
AFTER EDIT: answering the question kindly posted below, one example I had in mind is that Gelfand asks students to figure out how many different ways there are to group terms in an expression with parentheses as the number of terms increases. This essentially asks the students to discover the Catalan number sequence.
> Some of the problems in this book are HARD, but they are generally well posed problems of actual research interest to mathematicians
Could you share a few examples? I've looked through the books from Gelfand's correspondence course (which are indeed excellent) but don't remember any problems that fit your description. Some of them would certainly be challenging for young children--I'm more interested in the second half of your statement.
I'll volunteer one potential example. There was a sequence of problems that dealt with the solvability by radicals of palindromic polynomials. That certainly motivates some ideas of Galois theory in no small way, but it's very basic and of no interest to research mathematicians.
Addendum: Now that I have looked it up in the book, I see it was a single problem, Problem 270, not a sequence of problems. That sequence of problems was from a mathematics competition for young children.
Parent meant problems that were in the past of interest to research mathematicians. Someone(s) published the first papers exhibiting Catalan numbers underlying various counting problems. Obviously we don't expect elementary students to be at the forefront of modern cutting edge research.
If so, that's a much weaker and not terribly interesting claim. The concrete example of counting with Catalan numbers is more compelling; it would be very challenging for children who lack experience with recursive definitions and inductive proofs. Systematic enumeration, albeit elementary, is distinctly modern.
Open-ended questions and problems are indeed awesome. Moreover, they are an essential part of a sound education in mathematics, even at the K-12 (primary and secondary schooling) level of learning. But open-ended questions used for teaching purposes should be carefully written for sound teaching points, and teachers using them should have sufficient background in mathematics to guide student approaches to grappling with them. One of my favorite authors on mathematics education reform (Professor Hung-hsi Wu of UC Berkeley) began writing on that issue in 1994 with his article, "The Role of Open-ended Problems in Mathematics Education,"
http://math.berkeley.edu/~wu/open-role.pdf
and he followed up on that article with a wonderful article in the fall 1999 issue of American Educator, "Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education."
http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf
Since then, Professor Wu has written many more useful articles on mathematics education, including guides for parents, teachers, school administrators, and teacher educators on how to apply the new Common Core State Standards in mathematics better to improve mathematics education in the United States.
http://math.berkeley.edu/~wu/
http://www.corestandards.org/the-standards/mathematics
A good example of a beguiling textbook by a world-famous mathematician with lots of open-ended problems is Algebra, by the late Israel M. Gelfand and Alexander Shen.
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
Some of the problems in this book are HARD, but they are generally well posed problems of actual research interest to mathematicians, that just happen to be accessible to pupils just beginning to learn algebra.
AFTER EDIT: answering the question kindly posted below, one example I had in mind is that Gelfand asks students to figure out how many different ways there are to group terms in an expression with parentheses as the number of terms increases. This essentially asks the students to discover the Catalan number sequence.
http://www.geometer.org/mathcircles/catalan.pdf