> That math homework you're trying to help your child muddle through might include problems with no possible solution. It could be that key information or steps are missing, that the problem involves a concept your child hasn’t yet been introduced to, or that the math problem is structurally unsound for a host of other reasons.
That sounds an awful lot like reality. Not that I think it's a great way to teach maths (though maybe it is a great way to teach problem solving) but an educated adult should have little trouble spotting the missing information or contradictions and either guessing, pointing out, or assuming them away.
I realize that it wasn't intentional on the part of the authors (neither are real problems), and that it's not the point of the article but merely a hook.
Open-ended questions and problems are awesome. However, typically book work is intended to be rote practice of an algorithm. Already it is not a simulation of the real world - typically the real world open ended problems are cross-disciplinary and much larger in scope than a context-free simple arithmetic algorithm. That is not the point of book work, and when students go in expecting problems to be internally consistent, sound, solvable, and to behave the way the teacher told them, it can hamper learning as they have had the proverbial rug pulled.
> However, typically book work is intended to be rote practice of an algorithm.
I see your point, I see the article's point and you're both right -- a maths textbook probably shouldn't look like that. It's a mistake, an oversight, a bug, and, as someone noted in the comments, it would be great if books were being improved instead of written anew every few years. It's why I can run the most current Linux kernel and actually have a better experience than with a stable one -- it was improved, not rewritten.
What I didn't like was this idea that a parent could be scared because of some imperfection in the way a homework problem is framed, pandering to this light version of maths anxiety many people seem to have that actually prevents them from learning maths. If the problem is poorly stated, reframe it, point out the problem, guess or find the missing information, assume whatever needs to be assumed to get some sort of workable solution (which you can then improve upon :) There is no reason to panic, be afraid, or even really bothered by that.
Again, I know, it was just a hook but it bothered me enough to post these two comments.
Oh, yeah, I understand that. If it's something you consider yourself to be educated in already then you should be able to skip around some of these issues and make a note to send the publisher. I go off half-cocked in these education discussions sometimes.
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.
Problems in textbooks that are open-ended by accident or broken in one way or the other are surely a great example for real world problems but usually get the pupil that is spotting the problem into trouble. A lot of teachers face high pressure and often just graduate by comparing the student's results to the supplementary material's results.
A cynical person might say that the student now has learned another lesson: That actual thinking isn't necessarily rewarded in the real world.
Math literacy is generally poor, so we want schools to do as good as they can.
Not all children have "educated" adults around to help them with their homework, and we especially want these children to have good education to help lift them from poverty.
Someone is paying money for those text books, so that person at least has a reasonable expectation of suitability for purpose.
And why do we accept[1] educational methods without insisting on good quality research to back them up? Why is it acceptable to churn out a book that not only has small errors but which causes confusion and delay in children?
[1] If we do accept them. I'm not a teacher, so maybe there's lots of research and it's all rigorous and great.
That sounds an awful lot like reality. Not that I think it's a great way to teach maths (though maybe it is a great way to teach problem solving) but an educated adult should have little trouble spotting the missing information or contradictions and either guessing, pointing out, or assuming them away.
I realize that it wasn't intentional on the part of the authors (neither are real problems), and that it's not the point of the article but merely a hook.