My comment notes follow.
[Education Next]: Will the Common Core put an end to what has sometimes been termed the “math wars”? In your view, do the math standards resemble those recommended by the National Council of Teachers of Mathematics (NCTM), and what do you make of that similarity (or lack thereof)?
[W. Stephen Wilson]: The end of the math wars! You must be joking.
There will always be people who think that calculators work just fine and there is no need to teach much arithmetic, thus making career decisions for 4th graders [1] that the students should make for themselves in college. Downplaying the development of pencil and paper number sense might work for future shoppers, but doesn’t work for students headed for Science, Technology, Engineering, and Mathematics (STEM) fields.
There will always be the anti-memorization crowd who think that learning the multiplication facts to the point of instant recall is bad for a student, perhaps believing that it means students can no longer understand them. Of course this permanently slows students down, plus it requires students to think about 3rd-grade mathematics when they are trying to solve a college-level problem.
There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared.[2] Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older.
There will always be people who believe that you do not understand mathematics if you cannot write a coherent essay [3] about how you solved a problem, thus driving future STEM students away from mathematics at an early age. A fairness doctrine would require English language arts (ELA) students to write essays about the standard algorithms, thus also driving students away from ELA at an early age. The ability to communicate is NOT essential to understanding mathematics.
There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.
There will always be people who think that statistics and probability are more important than arithmetic and algebra, despite the fact that you can’t do statistics and probability without arithmetic and algebra and that you will never see a question about statistics or probability on a college placement exam [4], thus making statistics and probability irrelevant for college preparation.
There will always be people who think that teaching kids to “think like a mathematician,” whether they have met a mathematician or not, can be done independently of content. At present, it seems that the majority of people in power think the three pages of Mathematical Practices in Common Core, which they sometimes think is the “real” mathematics, are more important than the 75 pages of content standards, which they sometimes refer to as the “rote” mathematics. They are wrong. You learn Mathematical Practices just like the name implies; you practice mathematics[5] with content.
There will always be people who think that teaching kids about geometric slides, flips, and turns is just as important as teaching them arithmetic. It isn’t. Ask any college math teacher.
The end of the math wars! You must be joking.
[1] " making career decisions for 4th graders "
This is one of the big problems I have with the kids' school. Their curriculum makes it almost impossible to follow a path towards a STEM career. Our school uses the weak "Everyday Mathematics" curriculum, but even does that poorly--every year, they only get through about 2/3rds of the material. Year after year, those 1/3rds left behind add up. The entire 6th grade math curriculum is a review of 1st-5th grades, because the kids are so poorly prepared for pre-algebra. And even there, they are already 60% of the way through their school year (even counting the week-long trip that is coming up in March) and are only through chapter 5 of a 13 chapter text. Chapter 6 is the always-exciting "Using Multiplication"!! This is supposed to be 6th grade, third trimester work, and these kids have no idea how to work with fractions, calculate the area of a circle, or the volume of anything! This is simply not a track that will allow kids to take calculus senior year of high school, which is what many STEM majors in college need.
[2] "The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared."
Try the lattice method with large numbers or decimals! At open house for 4th grade this year, the parents were doing a good job of eating the teacher alive. Sure enough the lattice method came up, and someone asked about doing it with bigger numbers. My sister piped up and asked about doing it with decimals--The teacher didn't know how to do it! Another parent, with an older kid, started to explain, and my sister said--"Yes, we know how to do it; the point is she (the teacher) doesn't know how to do it!" Yet, they still spent several weeks doing it this year, and if you used the standard algorithm on your homework, you were graded down.
I remember when the 3rd grade teacher wanted to show his students how well and quickly the lattice method worked. He had someone come up and do the standard algorithm on the board while he did the lattice method. Lo and behold, the teacher won! Our kid came home defending the lattice as "quick!" I had to point out to him that the teacher had cheated; he had drawn the lattice in advance—the most time-consuming part was thus ignored. I showed our kid how long it actually takes to do the two methods. But it's a strange thing, kids can get very defensive about their teachers, even when the teachers are full of … The kid got mad at me for showing him the teacher was wrong.
[3] " you do not understand mathematics if you cannot write a coherent essay "
This is actually a bigger problem than it seems. There has been a push for many years to drive reading and writing into every class, including math. No similar push has been made to get math into reading and writing classes—leaving less math time for actual math. But that's not even the big problem.
Some kids excel at language, some kids excel at math. Not all the kids who are good at one are also good at the other—though many are. In particular, many boys hate the language parts of school (especially when they are forced to write or read self-reflective pieces about emotions, hopes and dreams—sometimes it's better when they are allowed to blow things up with words and read snot jokes.) For many of the boys who struggle with language (and it can be something as simple as have slower-to-develop fine motor skills which makes writing a (literally) painful chore) math is their salvation. They can be good at math without having to put any of it into words.
At least, they used to be able to. Now half the problems are word problems, and half their answers have to be too. This is one of the reasons boys are having so much trouble in school. For the more math-oriented boy, they no longer can escape their weak language skills.
Also, ask yourself this: which do you think is easier for kids to master, a set of 26 symbols with 42 different sounds, joined together in increasingly complex and confusing ways. Or a set of 10 well defined symbols that always mean the same thing and follow simple and easy to experience rules. "Wind" can be something that blows, or "wind" can be something that rolls up. 10 is always 10, 15 is always 15. There is some evidence that pushing arithmetic at an earlier age and backing off on the language skills could be beneficial for a non-negligible segment of kids. In essence, kids can develop a natural and strong understanding of numbers before they develop a similar understanding of phonics, vocabulary, language and the motor skills needed for reading and writing.
[4] " you will never see a question about statistics or probability on a college placement exam "
I'm not so quick to dismiss statistics. I think it is the most-important worst-taught subject in school. If it is not part of the regular math sequence, I think it should be a separate semester-long class at the high school level. Every single day we are bombarded with statistics, and without learning how they can be manipulated, exaggerated, and misused, we are all apt to fly off the handle when we hear our (completely insignificant) RISK HAS DOUBLED! Or 50% of people polled think you're an IDIOT (when they only asked your best friend and your twerp of a brother.) People need a far better understanding of statistics than they currently have. My high school required a one-semester economics course; they should also have required one semester of statistics.
[5] " You learn Mathematical Practices just like the name implies; you practice mathematics"
Amen. I keep telling the kids this one. You can only get good at math and science by doing as many problems as you can get your hands on. If the teacher assigns the even problems, and the book has the answers for the odd in the back, do the odd too! The more, the better.
This is also the biggest problem with the ALEKS online math program. They give a kid three problems, and if they get them right, that's it. Eventually, the kid will be given a test on whether they've really learned it or not, and often will have to repeat the section, again and again—with a feeling of failure instead of mastery. It would have been better to be given many more examples before letting the student cross that topic of their list. It takes a lot of practice to really learn long division or how to find a least common denominator. Practice makes perfect.
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