You can ramble on as much as you like for mine, I have no personal objection to a low ‘information content to word count ratio’ :-) and I’m sure Harry will close the comments when he’s sick of us. But before I launch into my own dithyramble, some thoughts.

Firstly, thanks for the feedback on cuisenaire, though I’m not sure quite where that leaves things – however, it obviously wasn’t the all-conquering success it was originally spruiked as. The impression I get from your reply is that somehow they forgot to take the ‘average’ teacher along with them. Not the first time for that I guess, and won’t be the last time.

Secondly, mathematics and the teaching thereof. Apart from my continuing objection to what is, in fact, basic arithmetic being grandiously labelled as ‘mathematics’, there is a quote I remember from way too long ago to the effect that “For most students, the teaching of mathematics is too much too soon, for those who will go on to higher mathematics, it is much too little too late.” I think that is very true, and it certainly isn’t helped by teachers who “hate hate hate hated mathematics”. But I don’t see anything much that can be done about it. Unless, perhaps, something useful can be done by way of ‘self teaching’ – is there any research about things like that ?

Not that I’m expecting kids of age 10 or 11 to be able to spontaneously come up with, say, a simple formula for the sum of the integers from 1 to n as Gauss supposedly did [ n(n+1)/2 ] at that age. But mathematics is a ‘prodigy capable’ field (unlike, say, serious novel writing), and it’s never too early to start (I only wish I’d understood that).

Anyway, on to the main game. You’re right, it does sometimes help to take a little time to think. So, here is what I think:

Teaching, in all its manifestations, is a microcosm of humanity. So, there are a range of competencies, interests, ideologies, politics, economics etc etc. But teaching, at whatever level and of whatever students, is also a ‘microbusiness’ – in short, each educator is, in effect, a small business person, and their results and outcomes depend mostly on what they, personally, put in. The nearest profession to educating, IMHO, is Management – the same ‘individualness’ about it, and the same ‘Fad Surfing in the Boadroom’ (qv) approach to research and theory amongst the practitioners.

Trouble is, the students are very much a set of ‘captive customers’, and unlike just buying a pair of jeans, they are supposedly buying their very future. So we all would really very much like to see lots of very good results and outcomes at every level, and in every aspect:

• good research into human cognition and learning and teaching approaches and techniques that are relevant and can be applied in a range of circumstances

• good communication between researchers and educators at every level – and especially at the ‘educating the educators’ level, because then good stuff gets a cascade effect out into the nation’s, and even the world’s, classrooms

• well educated, competent, capable and emtionally sound teachers at every level (no more ‘hates mathematics’, for instance).

Then I remember that this is plain, old, bumble along, piecemeal approach, private agendas, politics and ideologies, emotional stumbling blocks homo sapiens, and the lovely daydream fades back into the quotidian reality of ‘situation normal’.

And then it annoys the hell outta me that nothing has really changed for the better in the 50 or so years since I received my ‘education’. There were Harrys and Salients and Lucy Mathiaks back then striving very hard and being basically blocked at nearly every key point and they still are.

There’s at least two ways I know to answer that, rephrasing the question as “how’s the ongoing implementation of cuisenaire rods going?” The good news is, it’s not that anything “happened” to cuisenaire rods, ETA hasn’t gone bankrupt, etc.

1) Somewhat like Unifix cubes, some districts have never owned any. It depends heavily on which state and which district—and (as I learned recently) on whether teachers in the district know about the manipulatives and have been exposed to how to use them, because in many cases, the answer to “whatever happened to cuisenaire” in a given district is they were never introduced—e.g. nobody teaching math in District X has ever worked extensively with cuisenaire rods, perhaps only 1-2 math teachers in the district even know what they look like firsthand or recognize the brand name, and so they wouldn’t know quite what to do with them if they were supplied, administration aren’t aware, etc, etc.

So there’s a question of exposure—not just to the manipulatives, but to effective lesson plans and unit plans that incorporate them consistently. That’s somewhat common—I mean, my first semester as a teacher, I had access to algebra blocks, but didn’t have a solid plan for how to use them, so they weren’t implemented nearly as often nor nearly as effectively as they could have been.

Last semester I had about half a dozen seniors majoring in elementary ed in my class, IIRC, & only one of them was aware of cuisenaire rods when I mentioned them, and it was because that student used the generic version as a kid (I’m not sure what the generic varieties are called, proportion rods, I guess?) And nobody in class knew what I meant when I started talking about algebra blocks. (They do now!) On the other hand, they were pretty uniformly excited to learn about these tools, and especially excited to learn there are good, battle-tested corresponding lesson plan and unit plan resources readily available. (And that there exist semi-transparent rods that can be placed on overhead projectors for ease of instruction.)

I used to think that teachers just weren’t exposed at all to some of these tools, in part because it’s a gap in some education curricula—which class is supposed to teach about this?—but one thing I’ve noticed is that many (new) teachers are actually exposed, e.g. through advertising sent to schools, to an overwhelming amount of resources and manipulatives they could ask their district to purchase, and it’s not exactly clear how to recognize the good from the mediocre, etc. There’s just a ton of products out there.

2) Many schools are on a rather strictly/tightly set math curriculum, which implicitly (but perhaps not explicitly) discourages breaking from the set lesson plans in order to incorporate other activities. If new-ish teachers have a book to follow, it’s common for them to follow the book. So it’s proving important to place lesson and unit plans in the teachers’ hands as well as the manipulatives themselves, and provide some guidance as to common pitfalls, and ideas for remedies that seem to work.

This is especially true for teachers who didn’t minor in math, who never ever ever would touch a minor in math, who in fact hate hate hate hate hated math, who get literal stomach cramps at the thought of studying math, and who want to teach math by plowing through worksheets. The Chicago Consortium, as with basically everything else they do, says it quite well: “Teacher knowledge must be greater than merely knowing how to demonstrate routine procedures if students are to learn how to solve novel problems.”

That includes a deep comprehension of how concrete manipulatives represent abstract concepts, as well as a deep understanding of the mathematics…
—-

I know there are teachers in at least two schools in my city who use cuisenaire rods, to good effect and with satisfactory anecdotal results, and it’s my understanding that the school that is switching to the Singapore curriculum will be investing in them too.

What ropty was talking about is likely to be supplementary to, rather than a replacement of, proportion rods; there’s an emphasis on discussing mathematics with concrete objects, like apples, in the early grades. I don’t know if I’d say “increased emphasis” because concrete manipulatives like fake money have been used since before I was born. The “draw someone eating the three apples” strikes me as a tiny bit problematic—because, who eats three apples in one sitting?—but it sounds like a neat example of incorporating mathematics into drawing and storyboarding. (One theoretical purpose of such an approach is to help kids see, from an early age, that math is not this independent thing divorced from everything else that people do in life.)

Well you certainly have quite a lot to communicate about education and educating, that’s for sure.

Probably not a lot content-wise. I’m not very good at being concise. Or clear, for that matter.

I do hope the comment in brackets in your second response to me wasn’t too seriously intended

Wait, no, that parenthetical was a response to what ChrisB said in comment #1, not what you said. :-)

I’m sorry that I haven’t read your prior communications with Harry

Oh, whatever you do, don’t regret that. However much I try to be coherent and sensible, I end up saying quite a lot of stupid stuff that wouldn’t have been worth the time to read, unfortunately. (In part because the blog medium encourages speculative thinking and conjecture, and in part because I’m rather bad at speculative thinking.) I’m rather young and mostly still learning, which means a rather large proportion of what I say on blogs is unintentionally b.s. :-)

I’m pleased to have been able to extract the core takehome, since I have no connection whatsoever with the theory and practise of education (merely an interested bystander who once underwent some ‘education’ himself and found it very wanting) and I’m looking forward to your “more later” when you can get to post it.

Well you certainly have quite a lot to communicate about education and educating, that’s for sure. I may have to make the ‘minimum necessary time for proper understanding’ plea myself. However, I do hope the comment in brackets in your second response to me wasn’t too seriously intended: I do enjoy deep and meaningful discussions about such real-world matters as ‘personal preference for information content to word count ratio’, it’s just that I take it to be a light minded diversion, and I wasn’t in the mood for one (Grim by name, grim by nature).

But since you are an education professional, and given your diverting interlocution with ropty on the mechanics of arithmetic, perhaps you could answer a small query of mine: whatever happened to cuisenaire ? It was supposed to be the be-all and end-all of numeracy, though it came after my time so I was never introduced. Has it disappeared entirely such that ropty is talking about drawing whole and eaten apples instead ?

Before I do temporarily retire to reconsider what you’ve written in the hope of improving my understanding, here’s the issue as I see it: that it’s all about forming and shaping the growth of synapses in a human brain such that said brain is of some use to its owner, and his/her family, tribe, nation, and even, hopefully, species. Further, that the synapses so formed are adaptive over time so that the brain can and will continue to improve its usefulness to its owner, family, tribe, nation and species.

Now since that ‘forming and shaping’ happens in part during the very rapid synapse growth period (from early age to about puberty) , and in part after the rapid synapse growth ends, and that the owner of the synapses is a very complex creature, and that any effect that can be had on synapse formation is indirect, then the process is complex and indeterminate. And we need all the help we can get if we’re to have happy endings.

And no, I’m sorry that I haven’t read your prior communications with Harry, I’m a very recent partaker of this blog, and as I’m sure you’re aware “what we learn from history is that we don’t.”

And utilizing addition is an important skill, but until you know how to carry the one, you are in for a world of trouble.

Really? Why? I never learned to “carry the one” and was capable of adding essentially arbitrarily large numbers in the first grade using counting-on, and just a little later in life, partial sums (so long as you let me write down each number if they were truly huge). I mean, adding three digit numbers in my head was really boring by this time of year 1st grade, and the only real limit to the number of digits was memory. (And no, I wasn’t some kind of atypical freak.)

The methods I was taught were fast, reliable, made concrete sense, could be diagrammed if necessary, carried over smoothly to rational-number arithmetic (not true of the RTL CPA) and to subtraction (only marginally true of the RTL CPA), generated reasonable, increasingly accurate estimates at each step of the procedure (not true of the RTL CPA), and allowed me to pwn that Number Crunchers videogame a full decade and a half before the word “pwn” meant anything. I mean, we’re not nineteenth-century bookkeepers, so why focus in-class attention on outdated methods?

I’m curious, how do you perform addition of, say, a handful of four-digit or five-digit numbers in your head using the RTL CP algorithm? I can’t do it reliably (I have to jot the number on paper as I go, or I forget what should’ve gone in the ones column), etc. It’s just not an optimal algorithm, is it?

Certainly, there’s empirical evidence to support the appropriateness and effectiveness of flexibility in teaching algorithms, with emphasis on student input and student generation —for one of many examples, see the excellent “Longitudinal Study of Invention and Understanding in Children’s Multidigit Addition and Subtraction” by Thomas Carpenter and Elizabeth Fennema at U.W. (along with other authors whose names escape me but are surely google-able).

[For those who are wondering about the acronyms I’m using because I’m too lazy to type whole words, I think they’re mostly covered here; CP means carry-propagate, i.e. whenever you have overflow in a digits column you carry the overflow.]

…Maybe I am? I’ll be perfectly honest: I don’t know if I’m playing devil’s advocate. I do know that I haven’t said anything insincere, or anything meant to be sarcastic or flippant, and I know that I don’t plan to do any such thing here.

That having been established, I’m frankly willing to dispute the notion that students at your kid’s school are not being taught to add. They may not be being taught to add in a manner that you acknowledge to be teaching addition, but they’re being taught to add in a manner that I’d acknowledge to be teaching addition. (And if they’re not and won’t be, sure, that is a problem, but it has nothing to do with whatever particular math program is being used—the NCTM standards are rather clear about this. Kids learn to group, associate, and add, and to apply addition to solve problems, etc, etc. The point I’m attempting to make is, the “teach my kid to add” is a straw man; nobody suggests it should be otherwise, and your kid is indeed being taught to add.)

But unless you have encountered primary education issues (grades k-5) and the silly stuff that goes on there, you are argument is askew.

Heh, I should stop assuming that people on this blog have read my previous comments on Harry’s previous posts, especially since so many were long and boring and horrible. :-)

I’ve done done my time as a public school teacher, and I currently teach math and math education courses at university, and in my spare time engage directly with schools and with legislators in my current U.S. state (and previously in the state of Wisconsin when I lived there), developing educational resources, etc, etc. More accurately, I help my bosses/advisers/whatdoyoucallits knowledgeable experts do these things, and learn from ‘em. It would be silly for me to be pseudonymous and yet attempt to rely on any kind of claim of prestige or claim to expertise, but I do take full responsibility for being reasonably aware of the “silly stuff” of which you speak, and I can probably assert that I’m directly responsible for some little piece of it, at least in a handful of classrooms. So feel free to hold me accountable, etc.

“What sound does ‘O’ make?”

Depends, doesn’t it? And that might be part of the problem with the “sound it out” model. Or it might not be. Where’s the research that backs it up? How comprehensive or universal is it? How widely studied? Does it consistently provide greater benefits than alternative approaches? What benefits, precisely? Or is it just that because Generation A was taught to sound it out, and it served (some) Generation A persons’ development of literacy, all future generations therefore ought to learn to sound it out?

This is the kind of thing I’m talking about, above, when I say there’s a widespread (and IMO false) impression that we’ve basically had everything figured out, with respect to what’s appropriate in education and what’s best practices and so on, for a couple generations now. It’s a very small-c conservative mindset: the system I grew up through worked for me and people like me, and change is risky, ergo, let’s continue with that system.

I assume that you are playing devil’s advocate a bit, and that’s OK. But unless you have encountered primary education issues (grades k-5) and the silly stuff that goes on there, you are argument is askew.

It is useful to recognize when adding should occur, it is a useless skill if you don’t know how to add. And I do not mean that metaphorically, I really mean 7 + 5.

And I really mean sounding it out. I do not mean encountering a new vocabulary word, I really mean a word like “dog.” As in, the teacher saying things like “what is in the picture? Is it a cat? No, it is a …” as apposed to “what sound does a ‘D’ make? What sound does ‘O’ make” etc.

The problem as I see it, is that most of the reasonable arguments are higher level ones—for example looking at context, maybe pictures, similar words, etc to learn a new word—are taught to kids that really need to know what sound letters make, or what each letter looks like. And utilizing addition is an important skill, but until you know how to carry the one, you are in for a world of trouble. Or, making kids think that is important to draw a picture to do the math homework. And I don’t mean that metaphorically, I really mean that you have to draw 5 apples, and then somehow illustrate someone eating three of them. And then draw two of them. And write a sentence like “When you eat 3 apples, you are left with 2” instead of practicing a wide verity of single digit equations. (1+2, 3+5, 4+7, 6+4, etc etc )

As for the skill vs knowledge question, let’s say I’d lean on the knowledge side. I’d rather go to a doctor that knows stuff about medicine, then someone who has great skill but does not know about the germ theory of disease (or believes in the flow of Chi causes arthritis).

I’m neither a reading specialist nor a cognitive scientist, but my gut feeling is that, while accessing general background knowledge helps with reading comprehension, accessing personal background knowledge does indeed lead you astray.

(Which is not to say I disagree with everything said on that blog, several posts were quite good.)

I think the AP exams are a better example, since we are at least talking about primary (and secondary) education.

No, they’re not a good example, because AP exams don’t cover what we want students to be able to do. They cover, in part, what we want students to be able to know, but they don’t even cover that comprehensively, and furthermore, they can’t: hopefully a hundred hours’ worth of learning can’t be boiled down to a single three-hour test, right?

It is impossible to write a single test which satisfactorily measures what we want students to be able to do as a result of having taken the AP coursework. One, the test would be several days long with time only for sleep, and two, even that test would probably only cover knowledge—it wouldn’t be able to test for the broader skill set that we intend for the students to develop.

I think the central dispute I’d take to your post is that you are attempting to assert that the only thing kids get from school, or ought to be getting from school, is knowledge. Yet the one specific complaint you raise, addition, is with regard to a skill, not knowledge.

There might be value to arguing over whether it is important for a student of grade X knows the capital of France, or knows how to find the capital of France.

Well the answer to that is obviously no, right? Try to name one fact (not a skill, a fact) that is sufficiently important to know that we couldn’t possibly drop it from the elementary school curriculum without devastating our students. Valuating knowledge one microcosmic piece at a time just belies the value of accumulated knowledge.

I just want the freaking school to teach my kid to add.

Won’t help much if the kid doesn’t recognize the situations where adding will give them the answer they want—and that difficulty is surprisingly common when it comes to subtraction. Sure, five apples minus two apples is three apples, but it’s not like a problem that simple comes up often—usually some interpretive thinking (what gets called “algebraic thinking” nowadays) is required in order to reformulate an encountered problem into a math problem.

And frankly, many of my college students have difficulty knowing when to add: for example, recognizing that they need to know the total length of some complicated thing like the perimeter of an object, and that they can add up the length of each piece to get that total perimeter. That’s the kind of thing most first graders “should” comprehend how to do, but it requires teaching more than board work addition with numbers. So there’s that.

And why do my kids now think that if you don’t know a word, you should try looking at the picture and guess (instead of sounding it out)?

Not sure what you’re saying here, and I’m not sure that any of this is worth a long discussion, but sounding out a word would help you pronounce it, not understand what it means. As well, your strategy doesn’t exactly work for words kids often struggle with, like “through” versus “rough” versus more obscure words like “bough”.

Looking at how the word is used and guessing a meaning based on context clues is a valuable strategy.

Anyhow, I agree that automaticity is important, but if by the word “just” in “I just want the freaking school to teach my kid to add” you mean “and I don’t want them to teach them all this other weird stuff that I never learned” then I’d have to disagree.

As I’ve said here before—and today in an earlier comment!—it’s not like we had it all figured out X years ago and had this ideal educational system that best met students’ needs.

not being a lawyer I did not know that, so perhaps the bar exam is a bad example (after I posted I wondered about if the bar was a good example, with all those stories about prisoner who study for the bar and become their own lawyer). But I think my general point stands—being a lawyer can require many things, and their are many types of lawyers, some of whom are trial lawyers, corporate lawyers, etc. A good law school might prepare students to do any number of types of law, but surly is a baseline for a basic understanding of law. I think the AP exams are a better example, since we are at least talking about primary (and secondary) education. There might be value to arguing over whether it is important for a student of grade X knows the capital of France, or knows how to find the capital of France. On the other hand, I think it is not a useful argument that the students of grade X should learn things that cannot be tested, like having spirit, or a strong moral sense, and so it is not useful to ask if they know the capital of France, or what a square root is or what not.

Oh I don’t know. There is a lot of angst about this sort of thing (see some edu-blogs, http://oilf.blogspot.com/ and the like). I just want the freaking school to teach my kid to add. Why the hell should I worry about if they are going to teach borrowing for subtracting, or how to multiple without getting drawing a bunch of boxes. And why do my kids now think that if you don’t know a word, you should try looking at the picture and guess (instead of sounding it out)? Arg.

OK, Grim, this one’s for you! :-) Here’s a kind of provocative comment that’s unfortunately at best partway relevant to the discussion (but which I’m thinking about as a direct result of having read the above), I don’t have anything deeper to share quite yet. It’s probably repetitive, sorry, and it’s definitely overgeneralized and fudgy, but here goes:

A lot of what gets taught in education courses is philosophy and theory, frameworks for thinking about knowledge and learning, which is (this is the problematic or dangerous part) presented as empirical fact. The net effect is that my students aren’t very good at identifying what aspects of their theory are grounded in experiment; they see the things they are taught as a kind of gospel truth about human knowledge (and they protest rather vocally when this notion conflicts with other course material—in particular, they get mad that they’re not being taught “the truth” about people—e.g. why did we cover Piaget if it’s not the “right” model?).

My working hypothesis is that many education students have a hard time coming to understand that we haven’t figured it all out yet.

Many of my students generally want to go do their jobs as teachers, and do those jobs well, and they want a model to follow that is known to work, and that’s that. Neat and tidy, and possible to memorize and follow conscientiously, please. (I don’t blame them for wanting this! I’d love to have such a theory myself!) As a consequence, often, the earliest theoretical models they learn about and understand are interpreted in that light: as Truth inviolate. It becomes part of their political orientation.

One consequent problem is that distinctions, in my students’ minds, get rather blurred, between what is known to be generally true about particular subpopulations of human beings from empirical research, and what is thought to be true about human beings as part of an organizational framework.

Both are crucial tools, but they ought to be contemplated, engaged with, and revised in different ways.

In particular, though, there’s less direct engagement with empirical research than I’d like. (I’m using the “than I’d like” phrase to weaken/soften the assertions here.) There’s also less understanding of the general scientific process of discovery, experimentation, and revision than I’d like: how do these studies get formulated? (This strikes me as one of Harry’s points: future educators should receive more training in how to interpret research studies and judge their value.)

But also: future educators need more training in how to interpret philosophy and theory and engage with non-empirical ideas. In particular, we probably need more emphasis on the fact that we don’t know everything yet—my education students very much tend to learn about a given theorist, like what s/he has to say, and attach themselves emotionally to that theory. So it becomes political.

E.g. a student may take Piaget at face value and see anyone who offers a competing theory as challenging or attacking Piaget. The discussion of new theorists, or challenges to the model both from incompatible theories and from countervailing empirical evidence, can become very emotive, as students feel they need to devote themselves to the promotion of the “right” side.

There’s so much more to say, but hopefully my comment makes Grim a little less grim and encourages her/him to return! The answer to your question “Am I alone in thinking this is a significant criticism of the process of educating educators?” is: no, you’re not alone, even among educators of educators! :-)

Nah, that’s somewhat a good sign (# of comments is a rather poor indicator of # of total readers, or total # of hours people have spent thinking over a post). It’s in part because quick/flippant/fun comments are easy to make on the books threads, whereas this post requires some serious digestion. Not to be dismissive of the other threads on book length and productivity software, but the points and conjectures contained in this post (and the stuff I’m looking into as a result) have been occupying much more of my thought space throughout the days. There’s just more to contemplate.

I am of the mind that the problem with schooling is the false belief in scalability. What Harry lists as some reasoning behind why research has not taken root in schools gets at this root. No two schools or even classrooms are alike, not even the same teacher teaching the same prep one period after another. Research helps perspective; without collaboration among school leaders, teachers, students and parents about the purpose and goals of the whole education enterprise, ideologies compensate for thought.