Over at MathNotations, there is yet another raging debate about math education. I wasn’t going to get involved, but I finally broke down. 
I started this as a comment on that post, but it is so long that I decided to post it as a blog post here.
So… here are some of my thoughts on the issue of math education, particularly for elementary and middle school kids. Let me say right up front that I am not arguing for or against any particular math curriculum here. My kids and I do not have extensive experience with any particular math curriculum. My own kids are not taught math from a single curriculum, but are rather taught using a wide variety of materials, exercises, explorations, etc. I have friends who are very pleased with Singapore Math. (I have used some of the workbooks for my younger sons as well, just for fun.) I have other friends whose school does an awesome job with Everyday Math, though I realize that that appears to be the exception and not the rule with that curriculum. What is key is great teachers, in either case. But that is all I am saying about particular curricula.
There is a problem that great teachers like Dave and Jonathan seem reluctant to admit, and that is that many otherwise great elementary teachers are poor math teachers. Elementary teachers are generalists, and for many of them, math is something to “get through” and to get kids through. It is not something they ever enjoyed, not something they are comfortable with, not something they are good at teaching. These teachers destroy curricula like Everyday Math that really require a great teacher. They would probably also be poor at teaching Singapore Math. So… as much as great teachers should not be restricted, or heaven forbid scripted, in the way they teach math, we have to do something about the fact that many elementary kids are getting their fundamentals from instructors who hate math and who are no good at teaching it. Some people think the answer is a highly scripted curriculum. I think that sounds horrible! I think the solution probabaly requires thinking outside of the box a bit more. Perhaps even elementary students need to have specialist math teachers, just as they often have specialist art and music teachers. (I’m focusing on elementary here because this is less of a problem in middle and high school where math teachers are generally specialists. There are better teachers and there are worse teachers, but few middle school, high school or college math teachers hate math.)
Steve H is concerned about kids who don’t have mastery of basic number facts (addition, multiplication, etc.) at an age where he thinks they should. He blames Everyday Math for the fact that kids in his child’s school don’t know their facts, but I’ve never seen any curriculum that builds this drill and practice in. In my experience, this has always been done in addition to following whatever “curriculum” the school is following, and I would think it could be done just as easily in conjunction with Everyday Math as it could with Singapore Math or any other curriculum. It also generally happens mostly at home. Teachers must expect students to study their facts at home, and must assess their progress, but parents must get out the flashcards, or get their kids onto a practice game or website, and get their kids to practice at home.
Class time should not be taken for memorizing number facts, no matter what curriculum you’re using. A few “mad minutes” a week are enough to assess how that is going and keep the kids motivated, and only takes a few minutes out of the math instructional time.
There is an argument that states that mastery of number facts and procedures (such as long division), performed accurately but without necessarily any understanding of why they work, is the most critical job of any math curriculum, and should be addressed before taking on anything else. These are certainly basic skills that all kids should learn. But to be honest, as much as I hate calculator use in school, in this age of calculators and computers, efficiency at hand computation is not, IMO, the most critical math skill for kids to learn. I am NOT saying that it should be ignored, or that kids should be allowed to skip it, and just use calculators in class (see rant linked above). But it is not, IMO, the be all and end all of math education, nor is it a prerequisite, IMO, for studying anything else.
What I consider even more important is a strong sense of number. I want kids who know immediately when the answer they got (either by hand computation or with a calculator) is way off. I want kids who have an instinctive understanding of the distributive law before it is ever formally taught or named (12 sevens is obviously the same as 10 sevens and 2 more sevens). I want kids who know when the amount of change handed to them makes no sense. I would rather have a kid who can multiply 64 x 25 mentally (by halving 64 twice and doubling 25 twice, to see that it’s equal to 16 x 100 = 1600) than a kid who can sit down and carry out the long multiplication with pencil and paper, by rote.
You can’t just sit down and teach kids “number sense”. Certain mental math tricks can be taught and practiced, but the way to achieve real numeracy involves lots of experience playing with numbers, with manipulatives, with measurements, etc. (Ok, I said I wasn’t going to talk about curricula, but I will say that from what I’ve seen of Singapore Math, it seems to do very well at guiding kids toward the development of good number sense.)
“Mastery” is also a slippery concept. I’ve seen kids “master” skills and then promptly forget them. I’ve seen kids who can easily do a page of long division quickly and without errors, who 6 months later will have no clue how to do long division… This is part of the reason why schools often utilize spiral curricula. Because if you have gifted kids, it’s easy to think “master one skill and then move on to the next” is the only sensible way to approach mathematics education. But most kids need more repetition than that, and many kids don’t fully “get” something the first time, even if they appear to have “mastered” it. Steve is right that a spiral curriculum can lead to a lax attitude of “it’s ok if they don’t master this now, because they’ll see it again later” that goes on ad infinitum, and the kid never masters anything. This is clearly no good. But the solution isn’t necessarily to take away the spiraling for those who need it, IMO. The solution is to have limits — for example, it’s ok if they don’t completely “get” long multiplication when it’s previewed in 3rd grade, or even when it’s introduced more formally in 4th, but they have to get it when it’s reviewed in 5th, or they shouldn’t move on. It needs to be clear where in the spiral one is, and whether this is a preview, or core instruction, or a last chance review, and make sure that kids really “get it” before moving past that “last chance”.
Dave is advocating and providing samples of “non-routine” problems. These often considered the domain of math contests, and to be reserved for only the most gifted math students. In my opinion (and I know Dave agrees), this type of problem solving is important for math students of all levels of ability for many reasons.
First, it provides a fabulous way of helping students to appreciate the uses of the procedures and skills they have learned or are learning (or in some cases, motivates a procedure yet to be learned). It provides a great way for teachers to re-assess students’ continued mastery of multiple skills, to see which need further review or clarification. This addresses that slippery slope of mastery. Invariably, some previously “mastered” skills are shown to be weak, and must be re-visited. In this way, we avoid just “checking off” topics, and make sure that students can recognize when a particular method or procedure is called for, and
remember how to use it long after the first time that they supposedly “mastered” it.
Second, this is the kind of thing that “real mathematicians” do! If one of the goals of K-12 math education is to prime future mathematicians, this is a valuable opportunity to do so. A “mathematician” does not sit down and solve 25 ratio and percent word problems, knowing exactly which skills are required to perform the computations. Instead, she investigates “puzzles”, looks for interesting patterns makes new discoveries, generalizes results. For students who might have the inclination to pursue mathematics further at some point, this kind of experience early on may spark their interest, and excite and motivate them.
Third, it develops self esteem and confidence. This may seem surprising, since the problems are very hard for most students. But the students that I work with know that the problems I bring them are meant to be hard. That they aren’t meant to be able to solve them all on their first try. That they may need help. But, when they do solve one correctly on their own, they are so very proud of themselves, and rightly so. A student gains so much more self-esteem and confidence from struggling and succeeding at something hard than they do from breezing through something easy. (Not to say that the rest of the math curriculum is easy, of course, but there is a persistent fallacy that I’ve seen many times, that the way to develop self-esteem in kids is to make sure to give them lots of easy work that they can effortlessly succeed at.)
Fourth, it builds transferrable problem solving skills. Math class is not the only place where people will have to solve difficult problems, problems where the best approach isn’t obvious at first glance. Practice in solving problems like this transfers to many different areas of the curriculum, and to “real life” as well.
Personally, I’d love to see what Dave would come up with as a Math Curriculum. However, I don’t think it’s so easy to just write a great curriculum and have the world beat a path to your door. There are huge corporations with a lot invested in selling math curricula, that he’d be competing with.
![5^{\frac{1}{3}} = \sqrt[3]{5}](http://l.wordpress.com/latex.php?latex=5%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D+%3D+%5Csqrt%5B3%5D%7B5%7D&bg=ffffff&fg=000000&s=0 "5^{\frac{1}{3}} = \sqrt[3]{5}")
