What’s Wrong with the World

You know, I remember having cubes and sticks that represented 1's and 10's. In kindergarten. For like a few months. Until we had the concepts down. And yet in second grade, my daughter is still hitting the sticks and blocks. We've gone to doing 'stacking' (er, adding?) at home and it's so much easier for her. And less messy, really.

The basic mistake is to think that educators should even teach "understanding" in this subject. The goal should be to teach children to calculate at the grocer's level. The "understanding" in this case doesn't help them do that. The "understanding" is really pure math. I'm all for teaching pure math (in reasonable amounts), but there lots of other kinds of pure math that are more interesting, more beautiful, and less confusing than this. Like geometry, for instance, or topology. Let them play with a Moebius strip.

I also learned the New Math, I think in its first wave (late 1960s). We did arithmetic in all kinds of different bases other than base ten, in order to learn the concepts. I think I did learn the concepts, correctly even, but so what? What point is there to knowing them?

To a surprising extent, I agree with you, Aaron, though I do believe in teaching the concept of place value. Some understanding of place value is a good thing, but this approach is implying that a kid has to be thinking deeply about place value at every moment when doing every operation. That, of course, simply isn't true. Nobody thinks about place value constantly (like, "oooh, and that's the thousands' place!") when balancing the checkbook.

The traditional math curriculum I use _does_ teach about place value and goes over it again every year. Borrowing and carrying are discussed in terms of place value. The child may indeed forget some of that in between reviews, and most of the time they just add in the usual way (by "stacking" and carrying), during which time, yes, perhaps they are to some degree, depending on the child, forgetting the concepts of place value. So you review it again the next year. Some kids will remain "recipe workers" all their lives, because they just don't care about such things and promptly and deliberately forget any conceptual work you do with them. Some will love it. So part of what we have here is the idea that no child should work by recipe, that every child should be a conceptual worker and a child-level mathematician, and they have an idea that if they develop a sufficiently esoteric method, they can force that outcome. It ain't gonna happen, no way, nohow.

What point is there to knowing them?

Some conceptual understanding is helpful, hence why kids often count on their fingers.

The people against teaching history as a bunch of facts have a point, since history is necessarily a story that requires a telling (and biases, as all stories have, but leaving that aside). But instead of replacing bad history education with good history education, they replaced mediocre history education with bad history education that gets the story wrong anyway.

I don't know about calculators though. Realistically, they are ubiquitous, and for low-performing students the proficient use of a calculator is probably about as much as they are ever going to learn. I know that I, an engineer working for an aerospace company, haven't done an on-paper calculation for years. Just as books rendered the memorizing of long stories unnecessary, computers are doing the same to mental math. Knowing some times tables so you can perform math on the fly is useful sometimes, but computer proficiency is much more useful now.

I don't know if I would call it science envy so much as a consequence of scientific knowledge being the only legitimate type in today's world. Maybe that's another way of saying the same thing. A good part of it now is entrenched bureaucracies looking for a reason to exist/seeking rents.

What ever happened to the abacus?
Nothing illustrates place values more clearly and quickly.

Matt, I think it's disastrous to _replace_ mental math or scratch-paper math with the use of calculators. It's not that I want some kid to go through a math major in college or become an engineer while never using a calculator. It's that I think they need to learn the basics, and the basics do include pencil and paper arithmetical operations. These are greatly aided, though, by memorizing math facts to begin with--the times tables, basic addition facts, and so forth.

The use of a basic adding machine or calculator for simple operations is easily learned. We're not talking there about learning to program computers. It would be foolish to use a graphing calculator to do simple addition. So obviously, if they need to do basic arithmetic quickly later on, they can easily use a calculator. It doesn't take years to teach that to them. It does take years to make them fluent with basic arithmetical operations, and the years when they are young and find memorization relatively easy are _the_ years to use for that purpose. So once you miss that opportunity, you've pretty much lost it. Also, if you have never learned real basic math, you won't have any idea if your answer is wildly wrong because you happened to punch the wrong numbers into your calculator.

True anecdote from some friends of ours, overheard: Two kids are talking. One says to the other, "What's zero times five?" (Or zero times something else. I don't remember exactly what the other number was.) Second kid rolls his eyes and says, "I don't know! I don't have my calculator with me."

Not the outcome we're looking for.

"Computer proficiency" is a vague phrase, and it's been a real blow to education to press computers on kids when they are quite young and don't have more basic cognitive skills in place in the name of such a goal.

I don't know how remedial math is taught now, but it will certainly *be* taught the way we teach remedial reading: go back the traditional way that always worked for pretty much everybody. So why not just DO IT THAT WAY IN THE FIRST PLACE!!!!

Not that I'm upset by this continual trend or anything.

That would be "go back *to* the traditional way."

Also, we already have a large percentage of college students who take remedial math. How many more will need it in the future?

Here's the comment that really took my breath away:

Teaching kids to memorize meaningless (to them) algorithms for the last 150 years has not exactly catapulted our country to the top of the global math class.

Right. Because everybody knows how terrible America's track record has been in math, and the sciences that depend on them, over the last 150 years. And everybody knows how we've been catapulting past everybody ever since we abandoned what we were doing when we put a man on the moon.

This is a depressing level of idiocy we're dealing with. But I think you're on to something, Lydia, and that this is as much about the guild mentality and "brand" maintenance of professional educators.

The most obvious question here is why is it that every country that is kicking our national ass on math scores uses the traditional algorithms and still succeed. It probably has something to do with the fact that their kids are forced to take school seriously and their teachers are not composed primarily of the dreck of college graduates.

Obtaining an education degree ought to be a capital offense. It's as simple as that.

The best part of this is the "non-educators" line. This is what it's about: parents can't be trusted with children, but the people who received grades so poor that the only department that would graduate them was education and the only employers who would hire them were grade schools, those people should be trusted implicitly with the intellectual and moral development of children.

This is a lie. Neither the state nor its lackeys are responsible for the care and education of children. That is a sacred duty of divine origin resulting from God's design of the human race and imposed by the begetting of children.

And on the nitty-gritty level, yes, the blocks are a handy teaching tool. But they're appropriate for roughly ages 2--5. (As an aside, how do these people give tests? You would need whole bluebooks just to give a ten-question math quiz.) As for memorization, the object of derision by the inferior, it is the grist in the mill of the mind. It is the key that unlocks, in mathematics, the capacity to work complex arithmetic, and in writing and other work that allows proficiency of technique to be displayed. A mere talent for using words means nothing in the hands of a person who does not know any (or who is ignorant of history, literature, or art about which to write).

This is what a society in decline looks like. Proceed at your own risk.

Many of those countries have educational environments that resemble the "bad old days" of American education, and some of them are even big on single-sex ed, which to my way of thinking is a very big deal and would make a huge immediate impact in improving performance among American high school kids. But they'll never do it, for purely ideological reasons, no matter how many times it's demonstrated to work.

It's that I think they need to learn the basics, and the basics do include pencil and paper arithmetical operations.

I agree if "the basics" means conceptual understanding. A person must know what addition is in order to use a calculator correctly to add something. But this doesn't include memorization of tables, since the ubiquity of calculators has made it unnecessary. It may be a moot point, since kids spend enough time in school that they can learn both, and because learning concepts necessarily entails a bunch of repetitive practice that involves memorization anyway.

All I'm saying is that once you are out of school, you will never do e.g. long division. The skill of long division is only useful inasmuch as it supports the learning process.

Also, we already have a large percentage of college students who take remedial math.

Yeah, that might be me...I got ten points below what's needed on the SAT to make normal college algebra. That's the minimum - basically, one question.

My point, though- Basic Math in College is literally addition and subtraction. It was ridiculously easy. I'm currently in Intermediate Algebra, also ridiculously easy.

If I wanted to go into Calc and beyond, I probably could. I just don't want to. Computer programmers need to take Calc. I know many computer programmers. None of them have EVER had to use it. It's pointless busy work, and probably does a lot to drive people away from programming, which is one field that actually needs people.

But this doesn't include memorization of tables, since the ubiquity of calculators has made it unnecessary.

I strongly disagree. If you don't already know a simple math fact such as 6 x 7 = 42, there are many calculator manipulations you could accidentally do wrong, just bumping a wrong number, that would end up with a wildly wrong answer, and you would have no little bell going off in your head saying, "That doesn't look right." See also my example above of the kid who couldn't tell you what 5 x 0 was without his calculator. This is extreme dependence. This is not knowledge or basic competence. If you received an even halfway decent math education, you don't even realize how valuable it has been to you. The fact that you don't sit down with a paper and pencil and do long division problems in your daily job is definitely the wrong way to measure the value of traditional K-8 math learning.

I also have to say I strongly disagree that basic knowledge of multiplication tables is non-essential because of the ubiquity of calculators. In my own case, I was completely crippled in mathematics by my own refusal to learn them when they were first presented to me, forever setting me back a year in math, and the idea that a person can be considered competent in mathematics without knowing the answer to 5 x 7 = x, without the assistance of a machine, is just strange. A person who leans on a calculator to know such elementary operations is going to be incompetent to carry out basic, daily uses of math that should not call for a calculator.

I'm sorry to go on about it, but this claim is shocking to me. I use basic multiplication almost every day as a matter of course. Think of all the extra things that I would be forced to memorize if I had not memorized the multiplication tables. It is how I understand that four quarters equal a dollar without actually memorizing that fact, or that three twenty-minute increments come to an hour. I cannot imagine my mental universe without such basic knowledge as 3 x 2 = 6, and the idea that I could master the same information simply by entering it into a calculator thousands of times is absurd.

Moreover, when I learned my tables, I learned them in one day. This is not something that crowds out other learning in the least, and it is only the most lame and self-indulgent excuse-making that leads teachers to try to avoid it. It takes more effort by a mile to come up with a "better" way of learning to carry out basic math.

When the student is taught the addition/ subtraction facts, 1-12, by rote, and then taught multiplication tables, 1-12, by rote, the relationships among these operations and division, along with insight that helps in the use of number theory, become clear in an almost intuitive way, that requires no teacher-led "critical thinking" at all.
Institutionalized education is sowing distrust within the individual of his own mental processes, and undermining individual confidence. It's disgusting.
It seems to be about making more work for teachers, teaching to the test and from the theory.
Mental processes have a concrete existence; they don't need any theorizing to be activated. They need only one thing-- a problem to solve. Period.

I actually did relatively worse in higher mathematics classes in high school due to my lack of mathematical aptitude (relative to the boys who mostly scored 750+) and my anxiety and self-doubt while being in the same class as them adversely affecting my performance along with a below-average work ethic. I was particularly strong in memorizing math facts and performing other simple arithmetic operations such as 2x2 digit multiplication and addition and subtraction in elementary school, mostly due to my exceptional memory, but also in part to my "understanding" of those math facts beyond concretia. In the second and third grade, I already knew that 7X4 = 7 + 7 + 7 + 7, and it has a visual representation as a 7 unit by 4 unit rectangle, 9 x 8 is (9 x 4) x 2, and 9 x 9 = (9 x 8) + 9 (in that case just add an additional ninth row of "9" to eight rows of nine by simple adding 9 to 72). The math facts were not simply material to be memorized by rather truths that I could tangibly access and visualize, and soon enough, I just had enough faith in them that I no longer need to represent them in my mind or question them to verify their truth. I took some pride that I could breeze through the multiplication quizzes with such celerity and accuracy while other children struggled with the threes. In this case, memory complemented understanding, and it would indeed be unlikely that I would regard these math facts highly if I did not know why they are true.

Later on,

I choose to study molecular biology precisely to avoid a quantitatively-heavy curriculum because I was intimidated by my relative inferiority in my high school mathematics classes where there were three 3-sigma boys who dominated every exam and got high (780+) SAT-M scores. Although I underperformed on the SAT in high school relative to my “true” ability, I realized then and now that the SAT-M, in most cases, measures “real” innate quantitative ability, and I simply do not have elite mathematical ability (which is not needed in most STEM programs), even if I was tested under optimal conditions, without anxiety or sleep deprivation. Nevertheless, I still had “enough” mathematical ability (low 700) to appreciate and comprehend the concepts, not merely plug in numbers in equations or apply procedural techniques to get a correct answer to a problem, of calculus and statistics if I applied reasonable effort; I just needed more time to learn them as opposed to my intellectually superior classmates and a solitary environment so I would not be frustrated when I see them learn at a faster pace. Then, in high school, I realized that I lack raw abstract ability to flourish in mathematics and decided to focus my intellectual efforts in other disciplines emphasizing more memorization yet demanding mastery and comprehension of concepts and models. Molecular biology fits this quite nicely since it is still intellectually cognitively inaccessible for most people since it requires a bit of abstraction to conceptualize the interactions, mediated by the known laws of physics and chemistry, of numerous molecules in living systems into more cognitively manageable mental representations (models) of the underlying physical phenomenon such as a set of differential equations that describes the rate of flux through a given metabolic pathway.

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As I discussed here before, math fluency isn't really that important because it isn't that "g-loaded".

I understand the thesis that math instruction cannot be entirely extricated entirely from memorization, especially during the abecadarian stages of addition and multiplication, but the reason that emphasizing "understanding" usually fails is simply that most students lack the requisite "g" to understand mathematics facts and on its own memory will serve them better.

the reason that emphasizing "understanding" usually fails is simply that most students lack the requisite "g" to understand mathematics facts and on its own memory will serve them better.

Again, as above, I think this may be somewhat over-stated, but I'm inclined to agree with the thrust of it. The educators aren't willing to accept anything remotely close to this. They want everybody to be in deep communion with the underlying mathematical meaning at all times. Which is stupid anyway, because even very g-loaded people don't do that. They don't have time when they're trying to figure out how much to tip the waitress. Life can't be carried on that way. Math education should offer a certain amount of understanding but not try to force-feed it by elaborate methodology, since as often as not the latter will actually work against accurate understanding, and memory can provide a basis for understanding.

It doesn't take a Ph.D. in education to realize that there is something wrong with a program of math education that insists on teaching children, every single year for 10 years, the basics of set theory, and then dropping the ideas after 3 weeks and never once using the concepts again - until next year. And that teaching the underlying concepts behind place value (a way of grouping) should shift immediately into showing how such groupings are carried out in place value notation so that you NEVER perform operations in other formats. Only someone really trained beyond sense into non-sense could imagine this new approach is worthwhile. As with the silly-clevers, giving a teacher training in "education" at higher levels is likely to ruin her.

Of course, one of the modern failures is the failure to realize that most "conceptual" learning doesn't occur by teaching the concept directly, it generally occurs by getting hip-deep embroiled in stuff in which the concept is embedded, and teasing the concept out by either induction or by deduction. Induction is what we do all the time in earlier years, by examples, by illustrations, etc. A child grasps the difference (conceptually) between cats and dogs not by being taught the theoretical principles that separate the canine from the feline, but by experiencing many cats and many dogs. A child learns the names of animals and plants with his mom repeating the names until he catches on - rote memorization that can be made fun with songs or other sounds ("What does an elephant say?") He doesn't need to know why the words are that way, with prefixes suffixes and root words shared, he just needs to memorize them first. Later on he will discover the concepts behind them - using what he has already learned by rote as material upon which to erect the conscious grasp of principles that he already sees imperfectly. That's why you teach early math by DOING simple problems over and over. And by rote memorization of the specific values of the simple arithmetic problems of which others are made up. The concepts come out of the base material being touched, handled, pawed over, tried this way and that, until total familiarity with them leads to comprehending them.

It wasn't until I was doing graduate math (abstract algebra, group theory) that I really started to grasp with confidence enough of the fundamental truths behind arithmetic to string together the bits and pieces I had seen before. The child of 6 or 8 doesn't need that, he needs to memorize the multiplication tables.

Calculators cannot begin to replace a capacity to do basic arithmetic. The muscles of the mind, imagination, memory, and interior senses are all used for developed thinking in any subject, and they require being stretched by mathematical exercises (as well as by language). The ability to engage higher forms of thinking require mastery of some simpler material, and basic arithmetic is part of that. Doing without mastery of basic arithmetic is effectively dooming the child to never achieving higher thinking on MOST subjects, not just math.

He doesn't need to know why the words are that way, with prefixes suffixes and root words shared, he just needs to memorize them first. Later on he will discover the concepts behind them - using what he has already learned by rote as material upon which to erect the conscious grasp of principles that he already sees imperfectly. That's why you teach early math by DOING simple problems over and over. And by rote memorization of the specific values of the simple arithmetic problems of which others are made up. The concepts come out of the base material being touched, handled, pawed over, tried this way and that, until total familiarity with them leads to comprehending them.

Let me say here that I had to learn the truth of what Tony is saying by experience teaching my own children. At the beginning, being young, idealistic, highly abstract in my own approach, and somewhat anxious to prove my mettle as a new home schooler, I was much taken with the craze for manipulables that was going around. I started with Unifix cubes (you can google them) and little counting bears and such and tried as hard as I could to teach my five-year-old child (and then, when she was six, my six-year-old child) really to understand addition and subtraction. Now, true confessions: I still think Unifix cubes are cool. I love the way they break apart and the way that you can show that the number seven can be thought of as being made of a five group and a two group or (alternatively) a four group and a three group. I really tried very hard to do addition and subtraction using the fullest comprehension with Unifix cubes.

She hated it.

It didn't gel. It didn't work. I don't think it added much of anything to her understanding of basic arithmetic. The cute little plastic bears were slightly more of a hit, but to this day I have little real confidence that they helped in her understanding of math.

We started doing an extremely traditional math curriculum, beginning with kindergarten level. Now, don't get me wrong. They had pictures of balls and blocks and showed take away and addition using them. But pretty darned quickly they moved to writing out "3 + 2 = 5." And it didn't take long before she got it. And I do mean "got it." Actually understood that if you have three things and you get two more you have five things altogether.

With the other two, the little bears have made a brief appearance at the beginning, but the Unifix cubes stayed pretty much in the cupboard until I finally gave them away. Kids are different. Maybe the mother I gave them to will find that they work better for her.

I'm not going to be dogmatic about this to the point of saying that manipulables are worthless or anything. I'd certainly play with blocks (or bears) with a very little child just learning addition and subtraction, but I wouldn't expect a *whole lot* from it. As an empirical matter, Tony is definitely onto something in the above comment, and it's something that the people who want to force understanding without memorization would do well to heed.

"Doing without mastery of basic arithmetic is effectively dooming the child to never achieving higher thinking on MOST subjects, not just math. "

I do not believe this.

See this for a counterexample. http://educationrealist.wordpress.com/2012/10/05/math-fluency/

I used to accept this as a given until seven years ago, when I ran into my first kid who knew his math facts cold but couldn’t solve 2x + 7 = 11, unless I asked him what number I could multiply by two and add seven in order to get 11 and got the correct response almost before I finished the sentence. By that time, I’d already met a few 600+ SAT students who growled in frustration and reached for the calculator when it came to knowing 6 x 9. I’ve also tutored a dozen or more ISEE/SSAT (private school test) fifth and sixth grade students who went to precious little snowflake schools and knew none of their math facts with any fluency yet easily mastered fractions, ratios, and solving for unknowns and scored in the top 90% of a highly skilled population.

I am a defeatist: there is simply no pedagogical method to make children of low g understand abstractions and rote memorization will serve them well, but society does not reward one for proficiency with arithmetic. Moreover, there does not seem to be much evidence that knowledge of mathematical facts are essential for doing more cognitively demanding and abstract mathematics. Now show me evidence that knowledge of mathematical facts affects one's SAT-M performance independently from any correlation of mathematical facts knowledge and g. SAT-M performance is all that matters since it is an important credential signal; if one can do well SAT-M without a mastery of arithmetic, then [f-bomb] arithmetic, and I saying this as a person who as a young girl did exceptionally well on math facts.

Black_Rose, I have taught college-age students remedial math (7th and 8th grade math), algebra, algebra with trig, pre-calc, and calculus. I probably dealt with some 250 students over several semesters - not a huge number and not a statistical zilch either. In my experience, I never had a student who clearly demonstrated the capacity for grasping either geometric or algebraic material without at least moderately good grasp of basic arithmetic facts. Or, if I did, (trying to be open minded here) they at most hinted at the potential to grasp geometric or algebraic material. Just for example, factoring a trinomial into two binomials generally requires recognizing in the coefficients of the trinomial the factors that can be factored out. If you don't recognize that the 42 in 42x^2 can be resolved into 6 times 7, you won't try to factor
42x^2 + 3x - 6 into two binomials with terms 6x and 7x. And a student that has to use a calculator to discover this resolution of 42 into two factors 7 and 6 is soon going to stop trying because there are too many other options to run down and not succeed with. Maybe they understand the underlying concept of "resolve into binomial factors using factors of the coefficients" but if they can't show me that by actually doing it, I can't discover that they understand the concept.

That's my experience, anyway. I suppose that I might have discovered a few students demonstrating better grasp of the algebraic concepts if I had kept the numbers below 5. That's possible. I never tried.

By that time, I’d already met a few 600+ SAT students who growled in frustration and reached for the calculator when it came to knowing 6 x 9.

I have never had that experience. I have had experience of 700+ math people, including professors, get 6 x 9 wrong when they were concentrating on something else, like solving an integral. I don't think that demonstrates that they were "shaky" on basic math facts, I really don't, because when they took the effort to focus on the actual numbers they didn't have any trouble with them. Alternatively, I have seen 600+ math people struggle with math facts over 10 x 10, such as 11 x 12, but I ascribe that to the fact that many schools only taught math facts up to 10 x 10 so they never learned higher numbers by rote. Oddly enough, in my (very anecdotal) experience of people good at math, the more adept they are at higher math, the more likely they are to have picked up things like the squares up to 25 x 25 in spite of never being rote-trained on these facts in school. And I wouldn't have any reason to expect this more likely to run true up to some IQ limit like 150, and then break down so that real genius level math people demonstrate all sorts of quirky inability to name 11 x 12 or even 8 x 9 readily.

but society does not reward one for proficiency with arithmetic.

Yes and no. Demonstrate that you can correctly provide change for a dollar, can correctly figure out the right pitcher size for mixing drinks, and so on, and you can get by normally in society with no social handicaps. If, however, you can do none of these, you might still get by but will suffer for it. If you have to get out a second pitcher because the first is too small to hold all the components, it's an inconvenience. And if you get out one that is 3 times larger than needed, it will both be inconvenient AND be s social gaffe. If you have to stop at a gas station when you are at 1/2 of a tank because the next station is 100 miles away and you don't know how to figure out your current range in your head, you inconvenience nobody but yourself (and your passengers). The complexity of modern life is significant. I am convinced that a significant share of the people who "check out" from that complexity by simply doing what they are told all day (by TV, by the union, by anyone and everyone), as well as street people who live off of handouts, are people who find it a little too strenuous to manage the low level math facts and manipulations that the rest of us do without even noticing, and possibly other comparable thinking in non-math areas of thought.