Elementary education, or not.

Warning, long post...

I need advice; I'm hoping that this is the right group to ask:

I'm a member of a citizen's advisory group for my local school district (rural Orgeon); while the district has a very good academic record (compared to other schools in the state), top to bottom, I am really concerned about the state of math education, especially since the current 'testing' craze started in the US.

Specifically, I'm worried about elementary education; I've taken samples of all the standardized test (3rd-8th and 10th grades; Oregon's are federally certified for 'No Child Left Behind') and am very worried about the a number of observations:

1. Although NONE of the tests has a time limit, even the 3rd grade tests allow calculator use.

2. The problems are heavily slanted toward application (story problems, ukkk), and away from arithmatical proficiency.

3. Average scores and school performance start out well at the elementary level; e.g. the local elementary school starts out at roughly the 95% percentile at the 3rd grade level. But there is a steady trend downward toward high school and the 10th grade performace sits just BARELY above the federal target (60% this year, 70% next year). The local high school is one of the top performers in the state in this regard.

4. Anecodotal evidence with local employers and current/former students suggests that general ability and, more importantly, comfort level with math has probably declined rather than improved since my own time here (20+ years ago). More and more GOOD math students are taking remedial math classes at the university level.

5. My personal experience, both as the adult advisor for a science-based (rocketry) 4-H club and while tutoring my nephews/neices, suggest that even kids who LIKE math are getting short changed: they have the ability but they are not being exposed to concepts that keep up their interest.

6. Arthimetic, in particular, has moved from an abstract subject (i.e 100% proficiency on basic arithmetic) to a weird kind of 'applied' thing. This has resulted in situations where, for instance in my 4H club, I had to help 7th and 8th graders with long division problems: the kids were FAMILIAR with this but had very little experience with it.

7. The 'comfort' trend: Every 2nd-3rd grader I've talked to LOVES math, even if they have a hard time with it; by the time they are in 6th grade almost all of the hate it.

-----------

So. Have you noticed this? Does this matter? If so, what is going on here? Is this as bad a trend as I think it is?

I realize it's probably a systemic problem, and there is very little that can be done at a local level; but from the content of the tests I can see one thing right off: high school math becomes NECESSARILY abstract at some point, and this concentration on applications as opposed to arithmatical competence in the earlier grades leaves them ill-prepared for algebra, geometry, and the higher classes that come in high school.

This is always a tricky subject, and I don't think there's really been conclusive research on the relative long-term effectiveness of various pedagogical methods in mathematics. And people tend to have strong personal feelings about methods, which can further obscure the issue.

So I'll content myself with one tiny observation about your long and thoughtful post: I'm not at all convinced that teaching long division is useful, either for the skill itself or as an example of algorithmic thinking. I still remember with amusement the day I realized I had forgotten how to do long division. I was a doctoral student in mathematics at the time, standing in front of a classroom of undergraduates, trying to teach them calculus.

This is always a tricky subject, and I don't think there's really been conclusive research on the relative long-term effectiveness of various pedagogical methods in mathematics.

Yeah, and that's what worries me. It's not so much that math education was ever *good* before; it's just that it hasn't improved at ALL since I've been there.

As for long division: ultimately, yeah, it's pretty useless I guess, but I'm a promoter of the 'Learn to do it the hard way; that way you'll really appreciate the easy way.'

Really, my main question is: is abstraction the problem? If so, should we give up teaching it or should would make it MORE important earlier?

Caveats: It's really hard to say, especially since I suspect those of us who are reading this are probably above average. I went to the somewhat enriched grade school (top 5%) in my district from the late 1980's on.

Things I might note:

* Our school combined the "traditional" texts (times tables and long division) with pieces of CSMP (comprehensive school mathematics program archived at http://ceure.buffalostate.edu/~csmp/ ) and things like doing puzzle activities as treats and discussing the Math League contests. It helped me to make connections between the different areas.

* Most of our teachers had somewhat of a clue, not at all a certain thing in a country where people often seem proud of their innumeracy. In one case two teachers traded off so they could play to their strengths: one taught language arts and social studies and the other taught math and science.

* Also, while it probably wasn't exactly "cool" to like math, I never heard of anyone getting beat up about it. The environment was such that even though I did not see myself as a math person at all I had plenty of internal incentive to enjoy what was handed to me and very little discouragement from outside.

* Some people seemed scared of "story problems." When I did a little tutoring later on I realized that it is because they required *more* abstract thought, not less. They required the student to recognize the mathematical structure and set up the calculation.

* Family support can matter. I know I have a few things in my background I chalk up to my mother. She is a SAHM and while I don't remember her ever reading aloud to me or anything, I do remember her teaching me about binary numbers and about working Dell logic problems and word arithmetic when I was in elementary school.

* This probably served me in good stead when I got to geometry. I'm told they sometimes don't do proofs in geometry, nowadays, either, but really I think there should be more logic and proofs, starting early wherever opportunities for naive treatment present themselves and not all-but-abandoning them after geometry. Touch on the foundational/philosophical ideas a little in high school. I know I felt cheated when I first read about Peano's axioms in 9th grade and thought, too bad this doesn't have anything to do with math (I was pleased to learn I was wrong!).

* Things I've read about engineering education, learning styles, etc. seem to suggest the importance of cycling between the abstract and concrete--so neither the back-to-basics nor the holistic learning movements may have the whole answer, despite the the oft-noted swing of pedagogical fashion. Both are needed, and on a micro level.

>4 chellerystick:

Thanks for the input!

- Cycling; now that sounds like a good idea. It is true that a lot of kids have trouble just shifting gears.

- Cross-training: another problem in rural areas is that math teachers are often the absolute dregs ('who'd want to live there?' they say) or don't have a specialization in math at all. We take what we can get and hope for the best.

- Logic. Yeah! Teach a pure logic course at elementary level. Could it be done? I had some wonderful instructors in college for Logic, luckily...one in the Maths and one in the Philosophy dept. Hmmm. Lewis Carroll!

- Family support: I suppose it varies depending on where you are, but... in lots of rural areas sports are so much more important than, well, anything that parents could care less WHAT their kids learn as long as they graduate. We're talking about generational innumeracy/illiteracy here.

- And there is the way the Professional Education Management has hi-jacked education. This is one of my frustrations on the board I'm on: the knee-jerk action of ANY of the district staff is simply to roll their eyes if this type of subject is mentioned. They want us to man the sports' booster clubs, plan Christmas parties, etc., but leave teaching to the professionals. Professional what, may I ask?

(I hope I am not offending any school district staff, here. A lot of districts are hamstrung by budget, state guidelines, testing req., blah, blah. And some of them just say so to shut us up.)

While I hesitate to send people to that site, I think this article on Kuro5hin is squarely on-topic:

If We Taught English the Way We Teach Mathematics...

Here's an excerpt:

By failing to address how mathematics works, how it speaks broadly about the world, and what it means, we hobble children's ability to appreciate mathematics -- how can they appreciate something when they never learn what it is? The formulas and manipulations children learn, while a necessary part of mathematics, are ultimately just the mechanics of the subject; equally important is why those mechanics are valuable, not just in terms of what they can do, but in terms of why they can do so much.

>6 Amtep: thanks for the link... and here is another great excerpt:


So why is it that this broader view is so rarely taught? There are, of course, many reasons, and it is not worth trying to discuss them all here. Instead I will point to one reason, for which clear remedies to exist, and immediate action could be taken. That reason is, simply, that far too many people who teach mathematics are unaware of the this broader view themselves. It is unfortunately the case that it is only at the upper levels of education, such as university, that any broader conception about mathematics becomes apparent. Since it is rare for people going into elementary school teaching to take any university level mathematics, the vast majority of elementary teachers -- the math teachers for all our children in their early years -- have little real appreciation of mathematics. They teach the specific trees outlined in textbooks...

Yeah. This is where I begin to think I'm beating my head against a wall...*grin*.

>7 wyrdchao: My impression that a lot of the reason this wasn't taught was because it was, in the 60s, as New Math, and that it was a failure. (Wikipedia backs me up here: http://en.wikipedia.org/wiki/New_Math )

"2. The problems are heavily slanted toward application (story problems, ukkk), and away from arithmatical proficiency."

Having worked in education for many years, I can speak to this one. The trend right now is towards pushing students to use their higher level thinking skills. Story problems force learners to apply math rather than simply do math (at least that's the tghinking of those on high). Is it working? Not from what I've seen.

8> no queston, there. I talk to my 90+ year old grandmother about this, and what amazed me is that Oregon had proficiency testing in the 30's! Mostly REALLY tough arithmetic problems but she also had to know some algebra. I'm not saying things were better then, but....

9> I think the key word there is 'force'! 2nd/3rd/4th graders are so universally brilliant, they already KNOW how to think, but then this 'application' fever gets into them and teaches them that its hard. Aarghh! Kids are smart enough to figure out HOW to apply this stuff, if math is taught correctly.

I often speculate about whether math should simply be taught as a very odd, but very useful, foreign language. Everybody knows that the best way to teach languages is:

1. at as young an age as possible

2. intensively at first - rote memorization to get familiar with it

3. immersion after, as soon as they are comfortable

4. repeat 2 and 3 as often as necessary

Computer languages are easiest to learn this way, too, so...

Well, Keith Devlin proposes that math is like gossip (a network of stories and relationships of soap-operatic complexity), so it might make sense to teach it as a language. I'm not convinced by Devlin's arguments there but I am intrigued. (Been a few years since I read The Math Gene.)

Note that in natural language learning, children learn more by immersion and "language instinct" (approximately) and not at all by abstraction (grammar etc.). I'm not sure how to translate that, especially how to create a sufficiently mathematically rich environment for learning that would also get children's attention. (Communication is intrinsically compelling; how do we access that motivation?)

Stumbled across another thought about story problems/problem solving at http://elementaryteacher.wordpress.com/2007/09/30/why-the-united-states-lags-oth...

#2's mention of the day he realized he'd forgotten how to do long division reminded me of something I read somewhere about something, that went something along these lines: the whole point of progress {or civilization, or education, or whatever} is to be able to do things automatically, without everyone having to think them through from first principles every time. So, to the extent that I've thought about this, I agree with #10, that the basic mechanics of calculation should be drummed in as early as possible, so that kids can concentrate on applications later.

As to word problems, I think they involve as much interpretation as math -- you have to be able to identify what modifiers go with what entities (which speed with which train, for example). So my first response would be that kids shouldn't be confronted with word problems until they're comfortable with reading. This does not mean that they couldn't be given more complicated problems until they can read well, just that they shouldn't be presented with problems in paragraph form.

Another thing, of course, is the eternal complaint of 'but what am I ever going to need this for?' What with GPS, there's very little call for most people to use trig while navigating. But lots of kids are interested in music. And geometry: what is with all those damned proofs? Proofs are an exercise in logic, not in geometrical principles. If we want kids to learn logic (and I like the idea), it should be taught explicitly, not on the sly in an ostensible math class (it's not like geometry is the only math-like subject that uses proofs, so why is that they only time they're taught to most people?). The result is that most kids leave knowing little about either.

I say all of this as someone who never grasped a branch of math in fewer than four semesters (but usually needed six), who sort-of understands what calculus is for but couldn't do it to save her life, who has wondered for years what it is that mathematicians actually do, and has, in the past couple of years, found a great deal of pleasure in rediscovering math on her own. I am currently working my way through Precalculus Mathematics in a Nutshell, a bare-bones introduction to the basic elements of algebra, trig, and geometry, without all of the 'irregular verb'-type special cases that make textbooks balloon up to 400 pages. Once I work through that, I will undertake Mathematics for the Nonmathematician, which contains much of the same but works through it in the chronological order of math history, and is as much a history of mathematical applications as of mathematical tools.

Recently, while browsing through Barnes & Noble, I ran across a translation of Copernicus' On the Revolutions of the Heavenly Spheres. Now, I knew, abstractly, that calculus wasn't developed until the 1600s. But that fact didn't make much contextual penetration; so I was floored to realize, flipping through Copernicus' book, that he did it all with geometry. Even given the point about it being useful to calculate without thinking about it, I think math is too often presented as something with no history. Of course mathematical laws govern the universe regardless of whether humans are aware of them, but humans have had to work at identifying those laws through the centuries. I think that should be explained to kids at some point.

And as long as we're wishing about things to teach primary &/or secondary schoolkids, I want to plug basic probability. And the relativity of percentages – that a 5% decrease in each of four elements of 25% does not add up to a total decrease of 20%!!!!

I'm not sure about just "teaching a pure logic course." I don't think the "teach one topic and then leave it behind" way of structuring courses is optimal--revisiting a topic often from different points of view seems preferable--but it's a concession to the pragmatic difficulty of planning a flow of content with students moving in and out, students mastering content in different orders, teachers having different styles and preferences, administrators looking to order a few standard textbooks, and so on.

#13: Good on you! I'd recommend Mathematics in Western Culture rather than Mathematics for the Nonmathematician, though, as far as Morris Kline's books go. Similar material, but the former has a more cohesive narrative, I think.

Thanks all, for some more great links and comments...

11> True, kids are much better at immersion learning (which is why those 3rd graders love math so much); the problem is that math has very little redundancy - it's a lot less forgiving of small errors, and that might be why the abstraction part is so important. (But they all seem to manage those extremely complicated video games without a problem, huh?)

13> There are a lot of professional fields that the US is sorely lacking bodies for, and very many of these require more than just a familiarization with algebra. These are very high-paying jobs that we have almost given up recruiting for domestically; something like 70-80% of graduate students in science/engineering are foreign students. I imagine those students are the only reason those departments manage to stay open at all, considering budget constraints on education now days.
My mother bought the book Algebra for Dummies recently and I was pretty impressed with it; my 10-year old nephew has it now, but he's already at the math denial stage. He's good at it but hates it.

14> Logic: Just and idea off the top of my head. My philosophy course in college concentrated on translating a log of English sentences with logical content into symbols; we sometimes did proofs both ways. Don't know if this could work for kids, but some of Lewis Carroll's sorites are silly enough that the 'fun' factor would be there.

>13 drbubbles: drbubbles
One thing I noticed about Kline's books is that they're really slanted towards Eurocentricity. They really do underexagerrate (is that a word?) the contributions of Arabics, Hindus in ancient times - no, he thought it was all about the Greeks.
So take it with a grain of salt then go read The Crest of the peacock.

>15 scottja:, 17:

Thanks for the suggestions. The main advantage to Math for the Nonmath..., though, is that I already have it.

>13 drbubbles::

You also should check Eli Maor's books. He emphasizes the historical development of mathematics to provide explanation as well as perspective. You may also find that trigonometry still has many, many uses. Electronics and signal processing would be rather difficult without it. He also explains the use of geometry in the development of proofs; remember that Euclid's Elements are often listed as the first "official" proofs...

The use of long division? Seems to me I just HAD long division in a Calculus II course. We were dividing one polynomial into another and it was ----long division! This complaint reminds me that I've had several professors complain that nobody understands fractions - they stumble over them in Calculus. I have a clunky way of adding fractions that I learned back in the 50s in Catholic school. Dull as dish water, it was a vertical method that we used to work problems, many worksheets full of problems. Borrrring... But it stuck with me and I can take a relatively sticky looking bunch of rational polynomials and add them, for example, using this method - and when I show it to classmates, it's all new to them. By the way, the 'horizontal' way the professor adds fractions (polynomial or otherwise) is just fine and it is nifty and better but I just can't kick that old habit.

I'm not an educator, or at least, not a trained educator.

I'm not a natural mathematician. In High School the
consensus was that I was that I should probably NOT
plan to be a mathematician. Good advice.

On the other hand, I'm pretty good at math in the sense of being able to learn and to apply math that's already been invented for me by people who DO have math talent.

I have several degrees in electrical and electronic engineering, and thus have gotten some exposure to (and a middling mastery of) the applied side of a fair number of mathematical topics, possibly more than the ordinary math major.

I have to admit, however reluctantly, that I've come to agree with a certain family friend, once a high school teacher, and now a lawyer, who says "I have seen the past, and it works".

Though it bored me silly when I was a kid, I can see now the utility of learning all those !@#$!@#$ algorithms for basic arithmetic. How to add, subtract, multiply and divide whole numbers including zero and negative numbers, decimal numbers and fractions.

How to do yukky long division and how to find the Least Common Denominator among a handful of fractions so as to be able to add or subtract them.

They are mechanical methods, they dont take much thought, any dummy who isnt actually learning impaired can master them, and everybody needs to be fluent in the basics, even if later we spare ourselves the pain by using calculators.

Youngsters should be brought to an understanding of how our positional number system works.

The basic boredom of the Add/Subtract/mutlipy/divide curriculum might be jazzed up by introducing the notion of number systems with other bases besides the familiar 10, or perhaps some side trips into how the basic algorithms work on an abacus (chinese, japanese and russian).

Assuming we dont have a language problem (i.e. assuming all or the vast majority of students being native english speakers) I dont see what the objection to word problems would be.

What's the point of knowing how to do the four functions (add/subtract/multipy/divide) if you dont know WHAT to add or subtract or to multiply by to get a meaningful answer?

Kids might be introduced to elementary probability right there in grade school (why wait to graduate school or beyond for such an important part of real life?) A bit of combinatorics wouldnt hurt and it would
all support the "Learn the number system and the algorithms for the four function" basic program while also providing some usefull math skills that will come in handy in defending themselves against the worlds BS-ers, of which there are so many.

I LOVED geometry. We did Euclids program pretty much the way it had been done since the days of the ancient greeks. We learned to cross the Pons Assinorum. We learned what a logical argument was and we incidentally learned a bit of useful geometrical lore (whats the area of a rectangle, a triangle, a rhombus, a circle .. ) As a vehicle for a using a bit of formal logic and for constructing logical arguments, Geometry is a good choice because though it is abstract, it has concrete expression in a realm that most humans are equipped to understand intuitively;
so the burden of learning an abstraction at a relatively young age is relieved by the natural affinity of the human mind for geometrical relations.

Anybody want to take a whack at knocking all that down?

Working to pass the 1990 California Math framework, at the state board of education level, the final version that removed personalized views of great mathematics, even Archimedes, Newton and Gauss, and contributions of non-European mathematicians. George Joseph's "Crest of the Peacock" came out about this time, a clear view of criticaLmath threads (beginning with strong arithmetic foundations of LCMs, GCDs and prime numbers) that successful students need to consider at some time during their math educations.

When students are ready, offer them to read and discuss "Crest of the Peacock" as college Economic IA students are often asked to read the "Worldly Philosophers' view of the great economic theories (that have been censored and/or accepted into our modern economy). The same is true concerning the foundations of mathematics. Providing young math students access to the great mathematicians, and its censored and accepted threads personalizes student educations to the economic need level.

Two random thoughts.

First, students/kids should be allowed to enjoy mathematics even when they seem not very good at it. The school system often leads to 'prescribing' less mathematics for such students. This in my opinion is a very dubious practice. Obviously, considering the importance of mathematics, the students are not favored by this. It does make things more convenient and manageable for the school, teaching only the ones that don't actually need much teaching in the first place. The hidden message is that you better be good at mathematics, otherwise you can not possibly enjoy it and you better stay away from it. As if one would need to talent of a Picasso in order to enjoy drawing.

A second, somewhat related point, is that I have heard many people state that 'mathematics is not really supposed to be fun'. Similarly, these kind of people would often make a distinction for their kids between 'work time' and 'play time', the former time being strictly allotted for doing tedious worksheets, thereby usually ensuring a lifelong hatred toward the subject. Again, the school system contributes a lot to this apparent state of affairs.

Given the above, I do think that mathematics is of quintessential importance to elementary education, but that the school system has very poorly implemented this.