I've lost count of the number of times that Colin has posted an interesting article on mathematics education on Hacker News over a weekend. This submission here is particularly good. The problem indicated is quite stark, and very commonplace. Most pupils in a mathematics lesson in elementary school swiftly learn that "getting the right answer" is the point, and some check out and soon begin to doubt their own ability to REASON to the right answer.

The author of the submitted article writes, about a seventh grade class, "The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability." As the author makes clear, the particular lesson arises from the new Common Core State Standards in mathematics,[1] which are only recently being implemented in most (not all) states of the United States, following a period of more than a decade of "reform math" curricula that ended up not working very well. I am favorably impressed that the lesson asked students to put their numerical estimates of probability on a number line--the real number line is a fundamental model of the real number system and its ordering that historically has been much too unfamiliar for American pupils.

The author continues by elaborating on his main point: "When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one 'answer' and then call it a day." I like the author's discussion of that issue, but I think she misses one contributing causal factor--TEACHER education in the United States in elementary mathematics is so poor[2] that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included[3] and don't do anything to develop teacher readiness to respond to a different approach in a student's reasoning.

What I LOVE about the Singapore Primary Mathematics series,[4] which I have used for homeschooling all four of my children, is that the textbooks encourage children to come up with alternative ways to solve problems and to be able to explain their reasoning to other children. The teacher support materials for those textbooks are much richer in alternative representations of problems and discussions of possible student misconceptions than typical United States mathematical instruction materials before the Common Core. Similarly, the Miquon Math materials[5], which I have always used to start out my children in their mathematics instruction before starting the Singapore materials, take care to encourage children to play around with different approaches to a problem and to THINK why an answer might or might not be correct. (Those materials, both of them, are very powerful for introducing the number line model of the real number system to young learners, as well as introducing rationales as well as rote procedures for common computational algorithms. I highly recommend them to all my parent friends.)

I try to counteract the "what's the correct answer" habit in my own local mathematics classes (self-selected courses in prealgebra mathematics for elementary-age learners, using the Art of Problem Solving prealgebra textbook[6]). I happily encourage class discussion along the lines of "Here is a problem. [point to problem written on whiteboard] Does anyone have a solution? Can you show us on the whiteboard how you would solve this?" Sometimes I have two or three volunteer pupils working different solutions--which sometimes come out to different answers [smile]--at the same time. We DISCUSS what steps make mathematical sense according to the field properties of the real numbers and other rules we learn as axioms or theorems in the course, and we discuss ways to reality-check our answers for plausibility. We don't do any arithmetic with calculators in my math classes.

[1] http://www.corestandards.org/Math/

[2] http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma...

[3] When I last lived overseas, I had access to the textbook storage room of an expatriate school that used English-language textbooks from the United States, and I could borrow for long-term use surplus teacher editions of United States mathematics textbooks. They were mostly terrible, including no thoughtful discussion at all of possible student misconceptions about the lesson topics or of alternative lesson approaches--but they were all careful to show the teachers all the answers for the day's lesson in the margins next to the exercise questions.

[4] http://www.singaporemath.com/category_s/252.htm

[5] http://miquonmath.com/

[6] http://www.artofproblemsolving.com/Store/viewitem.php?item=p...