# Math Wars

Over at MathNotations, there is yet another raging debate about math education. I wasn’t going to get involved, but I finally broke down. π 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.

Very well written! You are right on track about elementary teachers, at least from my experience here in the south. I have worked with students who were halfway through college, 2-3 semesters away from their student teaching who couldn’t convert decimals to percents, identify basic shapes, perform long division, change improper fractions to mixed numbers, etc. And they’re going to be teaching our kids?!? Yes, they are! And they’ll probably be great, caring, organized, energetic teachers, except they can’t pass any love of math on to their students because they don’t possess any.

I also agree with your statement that number sense is supremely important. It is. The only thing I would say as a former middle school teacher is that the sense of number often is an outgrowth of proficiency at hand calculations. That is, if they are thinking about why the hand calculation works, their sense of number should increase.

That being said, they do need to know how to use calculators, and it’s not as natural a skill as one might think. They have to be taught how to do that, too. (things like order of operations & powers of ten must be understood & they must know how to communicate those things to the calculator) And calculators can help take some of the tedium out of doing things that aren’t easily done by hand (such as powers, roots, and factorials).

I was a sub in high school math, and had to fight with students to do simple math without a calculator. See the new book on amazon.com: “Teaching and Helping Students Think and Do Better”.

Thanks for your comment, Alane!

As for kids developing number sense by developing proficiency at hand calculations, I think that depends how it is taught. When I was taught the algorithms for carrying, borrowing, long multiplication, long division and even doing square roots by hand, they were just that, algorithms. There was never any thought to why they worked. They were just a series of steps to carry out to magically derive a correct answer. I loved math, was top in my class, but it never occurred to me to try to figure out why these algorithms worked. I think a teacher has to raise the question, and either provide the answer, or guide the kids to finding the answer.

Now I am hearing a lot of “back to basics” arguments that say that it’s more important for kids to be able to do the algorithm than it is for them to know why it works, and I don’t agree. Because, just as you said, thinking about why the algorithm works is part of developing number sense, which is so critical.

That’s also why teaching alternate methods of multiplication, like Russian Peasant Multiplication, is worthwhile, as long as time is taken to try to figure out why they work. (I’m not sure it’s reasonable to expect elementary kids to understand why the lattice method works, which is why I’d be reluctant to teach that one, except as a cool aside for kids who already understand the standard method and some of the more transparent alternatives.)

You are right that kids do need to learn how to use calculators effectively. But I would argue that that can wait until middle school or even high school. Until then, I prefer to reduce the tedium and work on those numeracy skills by using numbers that work out reasonably easily, so that calculators are not needed.

I agree. I didn’t get involved either, even though I have taught math in primary and secondary schools. I think that calculators shouldn’t be allowed and math needs to be stronger so that kids can actually like math.

Here in Quebec Canada, things have changed in the last 15 years since I have been in college. The kids see higher level mathematics and I’m happy to see this. It prepares them well for their university classes if they go into the sciences.

Then there are the kids who go into human sciences who haven’t done any mathematics since Grade 11.

University level math is hard, especially for math students. At least in our university, I find them challenging compared to other universities that I’ve been to. That doesn’t change the fact that I love math.

Anyway, these are just my two cents.

As always, Mathmom, you captured the essence of what is taking place over on my blog. I still haven’t come to the realization that it’s futile to try to change anyone’s mind. I keep beating my head against the wall and all I get from it is a huge headache!

You are correct that I have steadfastly refused to attribute the problems to lack of quality instruction. When I worked as a Staff Developer in K-5 classrooms I gained a deep appreciation for the overwhelming challenges faced by elementary teachers. This feeling was further reinforced when one of my older daughters became a 4th grade teacher, both in regular and special education. I still go back to her for her opinions on many of these issues and she always keeps me grounded. She is so perceptive and she has no difficulty in acknowledging that most elementary teachers feel more comfortable with language arts and, certainly, for K-2, feel that getting their youngsters to read will always be of paramount importance. She does feel that having math specialists would be helpful.

I appreciate your comments and, of course, I agree wholeheartedly. All of the existing curricula out there have their strengths, but full and effective implementation of htese program is easier said than done. In most instances, teachers receive minimal preparation and are thrown into it to sink or swim.

Jonathan’s phrase, “Enriched Basics,” comes fairly close to what I envision, but part of me believes math curriculum can be done in a new way that is still evolving. Writing a 4th grade curriculum, for example, would be a daunting task, one that I would be hard pressed to do all by myself. Several parents have emailed me or posted on my blog asking me to write it, so maybe I need to take a hiatus for awhile and do this. Another thought is to develop samples of what I envision and post these. If anyone is interested they can let me know and I would continue to develop more. And, yes, I would have to publish these myself – I am definitely not in the mainstream!

Steve has a lousy program in his district, and I understand his anger. But that in no way gives him the right to be abusive, which is what happened. And at Dave? Maybe you guys know something I don’t, but I’ve never heard Dave push Everyday Math, TERC/Investigations, or any of those sorts of programs.

There is a math war (tail end, but all too real in some districts), but this is not it. This is just one angry man.

I think there is a tremendous problem with elementary teachers who can’t handle math. I am not sure how I gave anyone the impression otherwise; it certainly was not my intent.

There are two crucially important questions:

1. Do we admit defeat. Do we recognize that math-weak teachers will be in elementary schools forever, and build curricula that they can’t screw up too badly?

I’ve asked it provocatively, but I think the answer has to be “sort of.” One of Everday Math’s and especially Investigation’s greatest faults was that they can only be taught with a math-knowledgeable, math-competent teacher (moreso for Investigations, but really true for both)

The closer our base curricula looks like what the teachers themselves learned, the easier it will be for them to handle it.

Do I have a text in mind? No. But roll back the clocks a couple of decades, pre-Principles and Standards, and that might be a good starting point.

2. What do we actually want school math to look like?

Here I’d like to start with that base curriculum. Traditional. Traditionally ordered. And I would, to the extent possible, like to see it regularly enriched. Deepened from time to time. Room for extended explorations. One model could be Dave’s stuff.

Not instead of the curriculum. But as well as the curriculum. And not the day before vacation. I could imagine extra questions showing up during discussion, or in an exercise, from time to time. And I could see a rich activity, once every other week, or maybe once a month.

This is (albeit under extraordinary circumstances) exactly what my school does. We use algebra, geometry, and algebra II books that predate the standards. We add those extra questions regularly. And we bring the rich activities into our classroom (the ones that eat half a period or an entire period) every 2 – 4 weeks.

This could happen, tomorrow, in most high schools in this country, if we decided that this is the way we want math organized.

In middle schools? Not tomorrow, or the next day. But there is a pool of math-knowledgeable middle math teachers. Emphasis, training, maybe a different focus in hiring, and I think we could make this happen in middle schools as well. It would just take longer.

Could this happen in the elementary school around the corner from my house? Not with most of the classroom teachers with their current level of mathematics. Could it eventually? There are much bigger questions about “who teaches” that go beyond just the math question. But is there anything else we could do?

Specialists, for push-in? Schoolwide or districtwide training to support some of the more common activities? I don’t know about that, when training for basic skills may take priority.

— — —

I’d rather focus on high school, because it is easy to modify. But maybe the most fruitful discussions are to be had around middle school.

But “discussion” is the key word.

Jonathan,

I agree that Steve’s abusiveness was unwarranted. Dave has not advocated Everyday Math, or Investigations, but he has defended aspects of those programs, as have I.

I think there is a tremendous problem with elementary teachers who canβt handle math. I am not sure how I gave anyone the impression otherwise; it certainly was not my intent.I’ve found both Dave and especially you to be very defensive when teachers are “blamed” for problems in education. But here, I think that part of the problem really is many of the teachers, and some of the curricula being developed or “solutions” being recommended are a backlash response to that.

(From what I’ve heard about “Reading First” the backlash problem is not restricted to math education. I’ve had teachers of that program tell me that they were required to “cover” certain pages each day, regardless of whether students were way ahead or way behind. A gifted teacher was not even allowed to choose books for her kids to read for enrichment that weren’t on the grade-level approved list for her grade. She could not go up a grade under any circumstances!)

I don’t know how effective it is to focus on high school, when the kids are arriving with such a shaky foundation from elementary and middle schools. I’m not saying we shouldn’t do anything in high school — we definitely should. I just think that without widespread changes earlier on, it’s not sufficient.

I doubt teacher training at the elementary level is the right approach. If the teachers hate math, that is going to be passed on no matter whether you slightly improve their teaching skill or not. It would take a remarkable training program to break through old prejudices and have teachers suddenly realize how great math could be.

My state started requiring math problem solving portfolios as part of their standardized testing package some time ago. At that time, they started having a specialist come in and teach problem solving to the 3rd and 4th grade classes. It seemed to work very well. They still had their classroom teacher for most of the time, since there was only the 1 specialist for several 3rd and 4th grade classes, but the specialist appeared to be specially trained to teach problem solving, and seemed to enjoy doing so. I thought most kids really gained a lot from that instruction. Imagining a world where all of an elementary student’s math education would come from a specialist math teacher sounds exciting and wonderful to me.

I wonder, do they do something like this in other countries whose math education we often try to replicate here?

Me defensive? Hmm. We had a discussion about grading on-line. It’s part of my (real life) job to defend against that (helps that I believe I am right) and I probably came off super strong.

But it’s really issue by issue with me.

I don’t know how having universal specialists would work. I think for the lower grades the specialist would have content, but the regular classroom teacher has all that bonding stuff…

— — —

How we got in this mess is easier to explain than how we get out. Teaching (in the lower grades) was women’s work, and paid accordingly. It was the only professional option open to most women.

But with changes that accelerated in the 1970’s, many more women were able to pursue a wider range of careers. Many of those with math skills? Somewhere else.

Further, elementary teachers today often choose lower grades in part, to avoid math. The ed schools emphasize, in general, everything but math (lots of reading). And then the methods classes emphasize activity over content.

Why does lattice multiplication always take such a beating?

I teach the area model, like 23 * 47 =

20 + 3

40

+

7

(I hope things line up!)

and then filling in the 800, 120, 140, and 21 in the relevant spots,

and then adding those up.

The lattice multiplication is just a way of leaving off the extra zeros and making the place values line up — a “shorthand” for the area model.

“Iβm not sure itβs reasonable to expect elementary kids to understand why the lattice method works” — Sure it is, it’s just the area model, it’s just the distributive property.

I think Russian Peasant is much subtler, because (at least as I know how to explain it) you first have to think of one number as a sum of powers of 2, and then use the distributive property.

But no matter what, the distributive property is where it’s at, and I think the area model is a good way of teaching that.

But back to what the topic is supposed to be here:

My goal for the next year is to continue working with middle school teachers and to find a way to get more involved in elementary school math (probably, as I did with middle school, first by doing enrichment programs for students, and then by reaching their teachers).

You’re right Joshua, the area model is a good one to show kids. I find the layout of the lattice multiplication to be obfuscating rather than clarifying, but maybe that’s just me.

Check out my explanation of Russian Peasant via the link in my comment above. I think it’s more accessible than thinking about powers of 2.

thanks, an excellent piece. I posted it on mine:

http://homeschoolmath.blogspot.com/2008/04/points-on-math-education.html

Maria Miller

Thanks for some great thoughts, MathMom.

You are so right about elementary teachers. It is rather unfair to expect them to be competent in areas as diverse as art, English, music, geography as well as math. But what is more important than competence is the ability to engender a love for the subject. That’s pretty hard to do when many elementary teachers frankly do not know how to do a lot of the math at elementary level. I agree with you that there should be more specialist math teachers in elementary schools.

You said that

…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..Sadly, in many countries there is a critical shortage of people wanting to go into mathematics teaching. As a result, many middle and high school math classes are taught by teachers who are not trained. (It’s often the art teacher who has a shortfall in their timetabled hours.)

This sets up an inevitable cycle where even more people don’t want to study math.

To me, the key issue is Jonathan’s point:

2. What do we actually want school math to look like?I expect everyone on the planet has a strong and different view on that one…

Hi Zac,

Do you think people would even disagree on such “basic” desiderata as:

We want math to be taught by people who enjoy and have more than a minimal understanding of the topic.You are probably right about even middle and high school math teachers, too. But my experience has been that most (?) kids do seem to get at least some quality math teachers during their high school experience, even if not every one is great.

“Do you think people would even disagree on such βbasicβ desiderata as:

We want math to be taught by people who enjoy and have more than a minimal understanding of the topic.”

Yes. I think there is a real sense among many that is is ok to have math-weak or even math-phobic teachers in the lower grades (K-2?) because the math there is so basic. I think there is a real belief in this country that it is ok not to be good at math. And I do not get a sense that this is being addressed by anyone, anywhere.

Jonathan

You’re right Jonathan. I was more thinking that people would still think that better qualified math teacher would be “better” even for the younger grades, but that would probably only last until they started having to pay for that for many people. I can appreciate that most people would think that “anyone” could teach elementary math competently, and not realize the importance of having good math teachers right from the start.

Teaching maths is no simple task. It involves alot of inspiration and patience. And I agree that maths learning does not restrict to classroom setting. Anywhere cans. But the ultimate goal is to enable students to like and have fun with maths. They must see maths as part of the logical thinking process.

I have just finished my first year of teaching in “Where The Wild Things Are”.

For me, that was “Where The Rich Folk Live”.

Understand, very much like the neighborhood I grew up in, but the polar (and to some degree i do mean northern european) opposite of the schools I have taught in over the years.

To the point, I met more parents this year than I’ve met in all the others combined. So there’s my sample.

Many professed to being “good in math” when they were in school. None of them were ‘professional’ educators. Math is a stepping stone for financially rewarding careers, not teaching.

We’ve just adopted a new, and very scripted math program. Different from the other scripted program, which has (hypothetical) scripted class discussions complete w/(hypothetical) student names. Because I am crass, I would joke (with my lower-SES-campus colleagues) about how idealistic these were. (hypothetical) Nguyen always had some brilliant things to add to the (hypothetical) discussion, for example.

I fell in love with math (in college) through learning how the Maya were trying to work it all out mathematically, but hated it in public school. Too defiant to learn the mult. facts, factoring equations was nearly impossible, I fell further & further behind. First days of (minimun req.) college math classes, I wondered why the guys are so huge. Oh, that’s the football team. Of course they were on to more financially rewarding careers, but I digress.

Lim Ee Hai is right, and both inspiration and patience are best tailored to individual student needs. Homogeneous grouping have to allow for for rapid individual growth, and are really too clunky to do that. Also difficult politically in a parent-rich (no pun intended) environment.

Class size is the crucial element. Especially as curricula become more scripted. The difference between 20 and 30 kids is (what %) increase in workload with (what %) increase in compensation?

If I were just babysitting, let’s see:

$5.00 per kid, per hour, and….

I wouldn’t even be expected to teach them math.

Having returned to the world of Math teaching after a stint in the corporate world, I was shocked to see that in the last 15 years we are still arguing the same points. Class size does matter; the teacher teaching the subject of maths should have the passion and the joy of maths (yes, it is fun to teach) and technology plays a key part in the exploration of maths.

I am a firm believer in there is a need to the rote learning of the basics (yep, those time tables) and then a need to explore in order to understand most maths concepts. But the exploration must then be back up with the practice (rote?) of the concepts. Most students hate the textbooks we use, but stick it on a powerpoint slide and all of a sudden it is not so bad, just just sums to work through and look we’ve done it! Is it putting it into the digital world that makes it work for them. My school is still bnehind when it comes to ipods, podcasts and the digital world, but hopefully only for a brief period of time π

My classes enjoy the use of technology – I have a smart board for a few classes and the students love the different options I can give them for learning to one concept. They are all individuals and have different ways of learning – they should not be hindered by the limited teaching styles of the teacher.

what is the good suggestion of teaching probability-permutation? Any teaching tyles on how to attract of the student interest ?

I prefer listing things for quite a while, even after they have “the formula.”

I feel one of our greatest challenges is creating environments where students are naturally asking questions about math. I don’t know that traditional curriculum does that as well as we wish it did. I’m building an online math game for middle school

students called Ko’s Journey- a story based approach and the more beta testing we do the more we realize how much story context we need to orient students thinking mathematically.

Hey, sorry to reply to a post that’s a year old (especially when you haven’t updated in nine months and as such have probably abandoned this), but I saw your blog, came across this article…and instead of seeing an article ranting about the math curricula in a purely political way I got somebody who had real things to say. As somebody who did well in math throughout school and is now a grad student in the subject, having people with some degree of expertise at lower levels would have been great: in fifth grade, I knew more math than any of the teachers at my elementary school. In tenth, I knew more than any of the teachers at my high school. When did learning really begin for me? College, of course, the place where there were people to learn from.

Hi Mike,

I haven’t totally abandoned the blog. I just haven’t had the time/inspiration to post. I appreciate your comments. In my perfect world, gifted kids shouldn’t have to wait until college to be challenged!

Thanks for commenting!

This is an excellent article, I will definitely be sure to add this blog to my morning routine π

Calculation is the language of mathematics. No one dares make a case that someone could read Shakespeare without learning the alphabet, people are continually trying to make the case that children can tackle higher mathematics without becoming experts at calcualtion.

When we were in elementary school we learned addition, subtraction, multiplication, and division – and memorized our multiplication tables. We were attacked by waves of worksheets until we could calculate backwards, forwards, upside down, and asleep. With this background, algebra was easy. With that background, everything else became easy.

Any curriculum that dares ask the question (as Everyday Math does) “What color would this number be?” amounts to educational malpractice. The “new new math” is worse than the “new math” and represents an even greater departure from what works than the new math did.

revisiting after years. great stuff here!

i’ll’ve been impressed already (in particular)

by JD’s remarks… but i’m even more

impressed now. this’ll mark a place

for *me* to come back to.

thanks, mathmom!

“math wars”, alas.

take a sad song and make it better. v.

I enjoy reading an article that can make people think. Also,

many thanks for permitting me to comment!