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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

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/ http://www.corestandards.org/Math/ [2] http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html 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 http://www.singaporemath.com/category_s/252.htm [5] http://miquonmath.com/ http://miquonmath.com/ [6] http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra http://www.artofproblemsolving.com/Store/viewitem.php?item=p...

analog31 12y ago
My kids are now in middle school. The curriculum that they used in K-5 encouraged kids to come up with their own approaches. I'll admit that I was rather shocked, as it seemed quite messy and inefficient, but I discussed it with a neighbor who's a high school calculus teacher. He told me: "Be patient, just you wait, they'll surpass you in math." And he was right! Also, our school kept several sets of the Singapore Math books on hand, and loaned them out to families who wanted to give it a try at home. So I think the "correct answer" is not the problem, because that's just math. The "correct algorithm" is the problem. Or maybe, the algorithm is the answer. Oddly enough, the one course where the algorithm is indeed the answer -- high school geometry -- is the one that most people remember fondly.
- deleted 12y ago
  [deleted]
slowmotiony 12y ago
Homeschooling as in not sending your kids to school and teaching them yourself at home? This is such an exotic concept to me as a european, would you mind telling my why you'd do that?
- mercer 12y ago
  I can't answer for the OP, but I was homeschooled and I was around a lot of people from other cultures that were also homeschooled. In the case of the Americans, they were homeschooled primarily because the parents felt they could do a better job than the US school system, and because they wanted to use Christian-friendly learning materials. For social contact and 'real-life' social education, they'd put their children into a variety of (sports) clubs. In the case of my parents, it was a practical necessity, as we lived in a developing country. The nearest international school was a long drive away, it was expensive, and reintegration into the Dutch school system would have been much more difficult. Instead, their solution was the get a just-out-of-school teacher to have one or more exciting years volunteering and practicing their craft on me and my siblings, which my parents would augment with personal attention and long-distance help from retired teachers back in Holland. The teacher would use a Dutch 'long-distance learning' approach that was the defacto system for Dutch expats. For high-school material, this meant a book that explained, day by day, what should be read and what exercises should be done. Now, I personally feel the results were so good, that I would do everything to make it possible to either home-school my children (if I have 'em), or to get them into some alternative school (montessori, etc.). Aside from some difficulties adjusting to Dutch society, my siblings and I did quite well, and I believe on the reasons we are all active learners and autodidacts is a direct result of not having been part of the normal system. The primary argument I hear against home-schooling when normal schooling is possible is that it hampers a child's social development. I never bought that. Kids get a lot of that from playing out in the street, or being part of clubs. Furthermore, for a significant portion of kids I believe their 'normal' schooling can have a negative social impact, through various degrees of bullying, and the group pressure to believe that learning is stupid. I did not have a clue that voluntary study was stupid and nerdy until I returned to high school at age seventeen... (of course, I also understand that for many people home-schooling isn't a practical option. I think that's a shame. And I also understand that some parents might not be any good at teaching, and that a mandatory educational system can be a benefit in that case.)

analog31 12y ago

My kids are now in middle school. The curriculum that they used in K-5 encouraged kids to come up with their own approaches. I'll admit that I was rather shocked, as it seemed quite messy and inefficient, but I discussed it with a neighbor who's a high school calculus teacher. He told me: "Be patient, just you wait, they'll surpass you in math." And he was right! Also, our school kept several sets of the Singapore Math books on hand, and loaned them out to families who wanted to give it a try at home. So I think the "correct answer" is not the problem, because that's just math. The "correct algorithm" is the problem. Or maybe, the algorithm is the answer. Oddly enough, the one course where the algorithm is indeed the answer -- high school geometry -- is the one that most people remember fondly.

deleted 12y ago
[deleted]

slowmotiony 12y ago

Homeschooling as in not sending your kids to school and teaching them yourself at home? This is such an exotic concept to me as a european, would you mind telling my why you'd do that?

mercer 12y ago
I can't answer for the OP, but I was homeschooled and I was around a lot of people from other cultures that were also homeschooled. In the case of the Americans, they were homeschooled primarily because the parents felt they could do a better job than the US school system, and because they wanted to use Christian-friendly learning materials. For social contact and 'real-life' social education, they'd put their children into a variety of (sports) clubs. In the case of my parents, it was a practical necessity, as we lived in a developing country. The nearest international school was a long drive away, it was expensive, and reintegration into the Dutch school system would have been much more difficult. Instead, their solution was the get a just-out-of-school teacher to have one or more exciting years volunteering and practicing their craft on me and my siblings, which my parents would augment with personal attention and long-distance help from retired teachers back in Holland. The teacher would use a Dutch 'long-distance learning' approach that was the defacto system for Dutch expats. For high-school material, this meant a book that explained, day by day, what should be read and what exercises should be done. Now, I personally feel the results were so good, that I would do everything to make it possible to either home-school my children (if I have 'em), or to get them into some alternative school (montessori, etc.). Aside from some difficulties adjusting to Dutch society, my siblings and I did quite well, and I believe on the reasons we are all active learners and autodidacts is a direct result of not having been part of the normal system. The primary argument I hear against home-schooling when normal schooling is possible is that it hampers a child's social development. I never bought that. Kids get a lot of that from playing out in the street, or being part of clubs. Furthermore, for a significant portion of kids I believe their 'normal' schooling can have a negative social impact, through various degrees of bullying, and the group pressure to believe that learning is stupid. I did not have a clue that voluntary study was stupid and nerdy until I returned to high school at age seventeen... (of course, I also understand that for many people home-schooling isn't a practical option. I think that's a shame. And I also understand that some parents might not be any good at teaching, and that a mandatory educational system can be a benefit in that case.)