Also, 1+1+1 = 3 but 1 is one third of 3…and using building blocks…

Children will begin from the earliest of ages to understand fractional concepts more completely, and many will learn enough to appreciate the magic and will be eager to show their classmates/friends…

ok so now you’ve gotten the second of 2 tenths…is it too early to show that 2/10 = 1/5…can somebody make suggestions here…

If this can be shown, a whole new world of mathematicians might be born who may end up teaching their parents!!!

Can I get an AMEN?

Hope this helps!

JM

PS: this is only about 1/10 of what I have to say on the subject

I’m not a regular reader of the blog, but I do check in from time to time. In particular, I think the dissent postings are some of the best things on the blog.

Along those lines, I’d like to point out how well Dan furthers the conversation by engaging with Sean rather than piling on the reflexive criticism. Implicitly, Dan assumes that he and Sean have the same goal, viz., improving math instruction. Many comments above seem to feel that any discussion of alternatives to WCYDWT or discussion of the spaces in between present theories are bricks thrown through the windows. My experience (which is therefore not universal or even provably extrapolatable) is that the best teachers are always teaching *and* learning.

“If you need to know whether drug X reduces blood-pressure, you may fairly safely draw a negative conclusion from a “treatment model” experiment in which hospitalized patients were given X and no change in blood-pressure was observed. On the other hand, you would not deduce that drug Y does not increase fertility from the simple fact that hospitalized patients who received it had no babies. You would want to know more about other conditions that are known to be necessary. Nor would you deduce that ice is a bad material for building dwellings if you heard that I tried to build an igloo in Boston in mid-summer and failed. The right environment and, I presume, a high degree of special skill are necessary. Such a failed experiment would say much more about me than about whether “igloos deliver what they promise.”

Papert, S. (1987) Computer Criticism vs. Technocentric Thinking.

A version of this piece was published as “M.I.T. Media Lab Epistemology and Learning Memo No. 1″ (November 1990). Another version appeared in Educational Researcher (vol. 16, no. I) January/February 1987.

Of course, when we’re told (by people from University of Oregon’s Direct Instruction cabal) that Project Follow-Through definitively settled this question a long time ago, it’s obvious that we’re hearing from folks who have a dog in the fight. The methods used to similarly “prove” the superiority of instruction in the studies that have appeared lately seem so contrived and absurd that it matters little whether the folks who did them also have a dog in the fight or merely are sloppy researchers.

However, the fact is that Sweller, Kirschner and Clark don’t do much, if anything, to hide their disdain for what is loosely (and, to my mind, inaccurately) termed “constructivist” methods of teaching, and it is obvious that their 2006 piece and the rest of their research in this arena is focused on discrediting something they dislike. Unfortunately, the manner in which they attempt to do so is yet another case of loading the dice to “prove” that one’s biases are “objectively true.”

Reading some of the comments found at http://halfanhour.blogspot.com/2007/11/kirschner-sweller-clark-2006-readings_12.html , I was struck by comments from Barak Rosenshine (University of Illinois) regarding what he alleges is “romanticism” on the part of those curriculum folks at universities who favor constructivism, vs. the allegedly “rational” views of the researchers in question. Of course, Rosenshine’s description of constructivism is completely off the mark, but even so, he loads the dice with his dichotomy so perfectly that any other nonsense he offers up is completely predictable.

There are some good critiques of the 2006 article included at the above-cited URL, so rather than go through the arguments for and against Kirschner, et al., I’ll let interested readers check for themselves.

I’m still highly skeptical that their work has anything important to say about Dan’s teaching. Indeed, I consider their work an enormous red herring, the pursuit of which will not prove enlightening. Your mileage may, of course, vary.

Sean: My impressions are that WCYDWT falls almost entirely under guided discovery (‘give me a guess you know is too low…’, etc.) with some faint shades of pure (having students create the question/solution strategy).

I agree. Here’s the thing, though: if a teacher asks the students to ask a question that interests them about a video, to predict the answer to the question, to set bounds on the answer, and to come up with a list of information that will be essential for solving the problem, I can’t bring myself to care, really, if the teacher just works out the problem explicitly (a la Sweller, Kirschner, and Clark) for the students to take down in their notes.

Personally, I want to give the students a moment to start in on a solution themselves, if only to identify the students who are so advanced they’ll experience a negative effect (according to Sweller et al) from worked examples. My working theory is that, even if I just start lecturing there, students will see gains over the status quo. Certainly in motivation (they care about the answer) if not ability also.

Sean: Lastly, a lot of commenters have rightly defended WCYDWT based on the high enjoyment levels their students have while engaged in a lesson. My small concern about this is ‘enjoyment’ as a paramount criterion for the value of an activity.

Agreed again. Enjoyment shouldn’t be paramount but it should figure somewhere into the cost-benefit analysis. I expand on this in the post I was writing when Sean’s comment came in, all of which has been one colossal diversion from the paper I’m supposed to be writing right now. (Which happens to be about PBL, though, so perhaps I’m less off task than I thought.)

Sean: In Dr. Ferguson’s research in urban schools, students rated enjoyment as considerably less important than ‘does this teacher have control of the class?’ and ‘am I being challenged?’ Seems noteworthy.

Interesting. Can I get the reference on that? I would have expected a pretty strong interaction effect between “am I enjoying myself?” and “am I throwing a chair at the teacher?”

Shari: ‘ I think if more math skills were taught in context that is meaningful and relevant to students, we’d find it unnecessary to repeatedly re-teach skills.’

On a purely procedural level, though, are they more likely to remember how to solve a quadratic equation six months later if they had a really memorable problem that required a quadratic equation?

I really don’t know. I think exposure to- and a rigorous discussion about the efficiency of- a range of different solution strategies may be effective. Where WCYDWT fits in here I’ll leave to Dan.

Michael: ‘ First, who gives a rat’s patootie what is the “most efficient” way to teach mathematics? I’m reasonably certain that “God’s pedagogy” isn’t going to be discovered by any human, and certainly not by anyone in my lifetime.’

Asking questions about the effectiveness of one part of a curriculum and seeking a Deified Pedagogy I think are different. I’m also not convinced efficiency is a bad word. Sweller and Kirschner’s critiques may not have been as nuanced as we’d like, but there’s something to be said that their findings indicate direct instruction and worked-out examples are clearly the most effective way of learning. Dan’s right in that the literature is thorny, and Marty seems to have a much better grasp of the competing viewpoints, but I think a dispassionate review is better than a dismissal of an idea based on how it makes us feel.

Lastly, a lot of commenters have rightly defended WCYDWT based on the high enjoyment levels their students have while engaged in a lesson. I have had similar experiences with students, particularly in a summer school class. My small concern about this is ‘enjoyment’ as a paramount criterion for the value of an activity. In Dr. Ferguson’s research in urban schools, students rated enjoyment as considerably less important than ‘does this teacher have control of the class?’ and ‘am I being challenged?’ Seems noteworthy.