Because most philosophies that frown on reproduction don't survive.

Thursday, April 08, 2010

Waiting On Math

This article has been on my brain ever since seeing it on a friend's Facebook page a while page. The claim: teaching math can be delayed entirely until junior high and students in the end learn it better.
In 1929, the superintendent of schools in Ithaca, New York, sent out a challenge to his colleagues in other cities. "What," he asked, "can we drop from the elementary school curriculum?" He complained that over the years new subjects were continuously being added and nothing was being subtracted, with the result that the school day was packed with too many subjects and there was little time to reflect seriously on anything....

One of the recipients of this challenge was L. P. Benezet, superintendent of schools in Manchester, New Hampshire, who responded with this outrageous proposal: We should drop arithmetic! Benezet went on to argue that the time spent on arithmetic in the early grades was wasted effort, or worse. In fact, he wrote: "For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child's reasoning facilities." All that drill, he claimed, had divorced the whole realm of numbers and arithmetic, in the children's minds, from common sense, with the result that they could do the calculations as taught to them, but didn't understand what they were doing and couldn't apply the calculations to real life problems. He believed that if arithmetic were not taught until later on--preferably not until seventh grade--the kids would learn it with far less effort and greater understanding.

Benezet followed his outrageous suggestion with an outrageous experiment. He asked the principals and teachers in some of the schools located in the poorest parts of Manchester to drop the third R from the early grades. They would not teach arithmetic--no adding, subtracting, multiplying or dividing. He chose schools in the poorest neighborhoods because he knew that if he tried this in the wealthier neighborhoods, where parents were high school or college graduates, the parents would rebel. As a compromise, to appease the principals who were not willing to go as far as he wished, Benezet decided on a plan in which arithmetic would be introduced in sixth grade.

As part of the plan, he asked the teachers of the earlier grades to devote some of the time that they would normally spend on arithmetic to the new third R--recitation. By "recitation" he meant, "speaking the English language." He did "not mean giving back, verbatim, the words of the teacher or the textbook." The children would be asked to talk about topics that interested them--experiences they had had, movies they had seen, or anything that would lead to genuine, lively communication and discussion. This, he thought, would improve their abilities to reason and communicate logically. He also asked the teachers to give their pupils some practice in measuring and counting things, to assure that they would have some practical experience with numbers.

In order to evaluate the experiment, Benezet arranged for a graduate student from Boston University to come up and test the Manchester children at various times in the sixth grade. The results were remarkable. At the beginning of their sixth grade year, the children in the experimental classes, who had not been taught any arithmetic, performed much better than those in the traditional classes on story problems that could be solved by common sense and a general understanding of numbers and measurement. Of course, at the beginning of sixth grade, those in the experimental classes performed worse on the standard school arithmetic tests, where the problems were set up in the usual school manner and could be solved simply by applying the rote-learned algorithms. But by the end of sixth grade those in the experimental classes had completely caught up on this and were still way ahead of the others on story problems.

The article goes on to speculate on why this might be, and why it hasn't caught on. In addition to children of younger ages often (I've certainly known extreme exceptions) possessing limited abilities to grasp mathematical concepts, the article seems to lay much blame at the feet of elementary teachers, who are often required to be generalists and often personally do not like math and know little more than what they are trained to teach. (In one example, the author of a recent study goes through an entire elementary teaching staff and fails to find anyone able to explain to him correctly how to calculate the area of a rectangle, despite being responsible for teaching multiplication.)

Also important to note, it seems to me, is that despite the audacity of the overall suggestion, the proposal tested by Benezet did not actually involve not teaching math in the younger grades (at least, not as we would think of it) but rather not doing drill in arithmetical operations (addition, subtraction, multiplication, division, fractions.) Instead, it says he asked his teachers to, as well as focusing on recitation, "give their pupils some practice in measuring and counting things, to assure that they would have some practical experience with numbers." Obviously this kind of more practically focused work has since 1929 become much more common in better classrooms and certainly in the books used by most homeschoolers.

Still, this does tie well with my own reaction to "math class" when I was in parochial school, which was that underneath all the drill we learned very, very little between second grade and fifth. And come to that, very little more was done up until seventh or eight grade when the curriculum started taking baby-steps into algebraic concepts. I wasn't a math wiz, but I was at least able to latch on to what we were doing within the first couple days after a new concept was introduced (up till now you've done "short division" up through twelve, we will now do "long division" of larger numbers) and proceed to be bored for the next two to three months while we repeated it. (We now move to from long division of five digit number to long division of six digit number. And now, we will do word problems involving long division. And now we will do long division of decimal numbers. Buckle your seatbelts!)

Also, at a minimum, this kind of finding strikes me as reassuring as a homeschooling parent, since a couple times a year MrsDarwin and I take time out to sit down and freak out that we aren't getting through enough stuff with the girls. Being assured that there's still plenty of time to get people up to full mastery later is thus a plus.

In the end, though, while the viewpoint it interesting, I'm hesitant to commit to actually holding off on introducing arithmetical concepts (and drill) until that late. Homeschooling parents often see their children as being like those of Lake Wobegon, with all of them above average. And MrsDarwin and I do both know very well how to calculate the area of a rectangle. So surely, we can get these concepts in sooner and avoid the mental chloroform trap, right?

I do, however, find the article very interesting. And I remain convinced there must be something applicable we can get out of it, though I am unsure as to what.

UPDATE: From one of the links The Other Sherry provided, some more info on Benezet's approach:
. . . Benezet in Manchester, N. H., carried out a study from which he concluded "If I had my way, I would omit arithmetic from the first six grades . . . . The whole subject could be postponed until the seventh year . . . and mastered in two years’ study." This led many people to conclude erroneously that all arithmetic could be deferred until the seventh grade. However, closer observation showed that there was much arithmetic taught in grades I to VI. Thiele visited the Manchester, N. H., schools and said: "Firsthand observation leads me to conclude that Benezet did not prove that arithmetic can be taught incidentally . . . . Instead, he provided conclusive evidence that children profit greatly from an organized arithmetic program which stresses number concepts, relations, and meaning. Buswell found that Benezet had only deferred "formal" arithmetic, and that all other aspects of a desirable arithmetic curriculum were present. Of the formal arithmetic, Buswell said, "I should like to eliminate it altogether." On the same topic, "deferred arithmetic," Brueckner says, "From these studies the conclusion should be drawn not that arithmetic should be postponed, [page 18] but that the introduction of social arithmetic in the first few grades does not result in any loss in efficiency when the formal computational aspect of the work is introduced later on, say in grade three." — What does Research say about Arithmetic? By Vincent J. Glennon and C. W. Hunnicutt, National Education Association, Washington D. C., 1952, page 17. [source]

16 comments:

Rebekka said...

I hated math. Ironically, I use it daily at work at a practical level since I need to calculate percentages, dosages (for example by surface area) and administration rates of medications - and it's really, really important that I don't screw it up. And I actually LIKE doing those things and seeing things turn out sensibly. Life is weird.

mandamum said...

I think St Ignatius of Loyola suggested leaving math ed to older grades as well.

There was an article in one of the homeschool mags a while back about how a group of unschoolers learned most of the highschool curriculum in 6 weeks or so, with a good teacher and lots of motivation (an exam they wanted to pass--and they went out and recruited the teacher). I'm guessing they got the measuring and the parts-of-a-pizza stuff from life earlier, as a bedrock on which to build....

So coming from having read that article, I find it very interesting to see this being tried in a school setting.... THANKS for highlighting this!

--Amanda

Kate said...

Thanks for posting the link to this article! As a homeschooling mom to a young child, I appreciate the 'permission' to take it easy for a few more years. I started 'schooling' with my oldest when he was 3, because he was anxious to learn to read, and now that he is 5 I felt like I should be doing more with him.

We were working through a grade 1 math workbook, but a couple of weeks ago I put it away because it was such a grind working through even 5 simple addition problems a day. Instead we're working at recognizing and estimating groups of objects, working addition and subtraction into stories and puzzles, sorting larger groups into smaller groups, and things like that. We have more fun, and I really enjoy watching him grasp a development appropriate concept (like learning to recognize a group of 8 by noticing that it consists of 2 groups of 4).

I doubt we'll wait until he's 12 before picking up the workbooks again, but I am certainly more comfortable putting them off for another year.

the other Sherry said...

Some additional takes on this issue:

http://www.triviumpursuit.com/articles/building_a_firm_foundation.php

http://www.triviumpursuit.com/articles/research_on_teaching_math.php

A Philosopher said...

Yes, it's a poor use of time trying to inculcate full mastery of arithmetic facts at a relatively young age. But the right response to that isn't to put math off until later - it's to ease off on the arithmetic, and spend much more time teaching algebra, geometry, probability, and combinatorics at an early age. Get a copy of a Mathcounts handbook, for example, and start picking out problems that are reasonbly light on the arithmetic.

Foxfier, formerly Sailorette said...

I enjoy most math-- I hate word problems with a passion, but that may just be that most of the ones I know of are irrational. (why is the rail totally straight between two cities? Why do I know how fast the trains are going, but not when they will arrive? Why are they going different speeds?)

Seems to me that practical math would work great for younger kids, but the math theory stuff needs older minds, maybe even taught in combination with classical logic. Of course, I'd probably also teach basic accounting pretty early... hm, I really need to get on to writing this stuff, Kit is nearly six months old and Washington schools aren't getting any better.

Anonymous said...

Fascinating.

Joel (former math teacher)

fernandobe96 said...

Wow, this blog is fantastic... I really was to find something like this. The understanding the authors constantly express that eternal immutable Truth is not to be held by constantly changing observational knowledge -- science -- is some missing maturity in catholic blogosphere. Congratulations -- and dan't swell, but rather keep merely doing your jobs

Mrs. Cranky said...

As a child I was a slow reader (my sister and my mother also). If we'd skipped math until jr. high (when we all became voracious readers) we would have been miserable. Perhaps we've got weird brains. I was a bit frustrated with high school math until I got to Calculus which I loved. Finally all those disparate math facts come together for a purpose. I thought if you could integrate some Calculus into lower level math, it might make it all make a bit more sense. But I'm no educator.

I do have a question for some home schoolers out there. I know two home school families with children approaching college. Both kids hated math and put it off, now they are way behind on their requirements for college. Both of them haven't gotten through Algebra I, which I did in 8th grade. Is this a common problem for homeschoolers? Are there solutions for motivating teens once being on a schedule really matters.

Darwin said...

Mrs Cranky,

I guess the question I'd have is: Where were the parents in all this? I know mine would certainly not have allowed me to get away with skipping a subject just because I didn't like it.

I don't recall knowing anyone in that kind of situation, though a lot of homeschoolers I knew who were planning on going on in the humanities didn't go beyond Algebra 2 since that tended to be the minimum requirement at many colleges if you were on a humanities track.

The Opinionated Homeschooler said...

I knew a girl in that situation. My oldest daughter's first tutoring job was for an 18-year-old girl who had scraped through Algebra I early in the high school years without really understanding it, and was now preparing for the ACT exam. She hadn't enjoyed or understood math, and had therefore just taken the minimum.

But she wasn't homeschooled; she was in our local public school system. My homeschooled 11-year-old got her up to speed sufficiently to do a respectable job on the ACT math portion.

Christina said...

I also found this article very intriguing-- thanks for posting! Might I add to the list of comments how this discussion lends itself very well to re-examining the place of music in a child's curriculum. I teach piano to kids (some of them pre-K) who understand the basics of counting and in some cases, arithmetic, but who stall completely when you try to explain 'measures', relative lengths of notes, etc. Rather, simply teaching them by example and repetition how to clap out rhythms and look for patterns in their music has enabled them to quickly and easily grasp the concept of dividing notes into groups of 4 counts, fill in the missing values when a measure is left incomplete (concepts we didn't get until pre-algebra!) and even figure out some of the arithmetic on their own! I join the many voices who would be hesitant to postpone formal teaching until such a late age, but it seems that all agree there are better ways to begin teaching math than rote memorization. I throw in 'music' as perhaps an exemplary way to do so.

Foxfier, formerly Sailorette said...

I seem to remember a theory of different kinds of learning-- by example, memorization, theory, music and doing. It makes sense to me....

Anonymous said...

I don't know any homeschoolers who totally skipped high school level math, but my husband (who went to public high school) never had to go beyond Algebra I (I believe) and never had a science lab.

--Elizabeth B.

Mrs. Cranky said...

I didn't mean to flog homeschooling. I'm generally pro-homeschooling. I only know two homeschooling families besides the Darwins and both of them have these way behind teenagers. I'm glad to hear that this is unusual.

Foxfier, formerly Sailorette said...

Found this in the comments over at Lilek's; it's online math drills:
AAAMath.com