Sunday, December 25, 2011

Jerrid Kruse, Hainish, FedupMom, Barry Garelick, and LynnG on Between the basic elements and the fuzzy abstractions


Jerrid Kruse said...

My first reaction was that you haven't defined structure well enough so it ends up being this abstract concept - like the ones you lament. Then, I figured out what you mean by structure and your post makes much sense. (Examples are great, but are not a definition, sometimes the abstract is a very very good thing).
However, this structure you long for is most often reduced to a set of rules to be memorized rather than a set of understandings. Because the structure is interpreted this way, it rarely leads to the big ideas you are talking about. But hey, I was just glad to see you care about the big ideas.

Hainish said...

I don't think it's that teachers forget or want to forget structure, I think that it's below the radar for them. It doesn't even register. Instead of seeing the structure, they see the "lower-level" things--the names and dates and individual words--so they write it off as being too trivial to care about.

But, maybe I'm being uncharitable.

FedUpMom said...

At my younger daughter's school, they use Trailblazers math.

They've had a lot of complaints, of course, so in response they make the kids do a lot of very low level stuff -- computerized drill of adding and subtracting whole numbers, stuff like that.
The problem is that there's a whole layer of conceptual stuff that they still don't cover.
So they've got the airy-fairy "what's your favorite number?" level covered, and the bog-level drills covered, but none of the necessary concepts in between.

Barry Garelick said...

However, this structure you long for is most often reduced to a set of rules to be memorized rather than a set of understandings.
You don't teach a "set of understandings". The understandings are first about the procedures--presented in proper context, the understandings will follow. Sometimes the conceptual underpinnings will occur in a later grade/course. It wasn't until I took algebra, for example, that I understood why the invert and multiply rule for fractional division worked. In the meantime, however, I certainly knew when to apply fractional division to solve problems.

The hang-up over "big picture" is a continual confusion between epistemology and pedagogy. Novices don't learn to become experts by being given problems that only experts can solve.

Sweller talks about this in his article "Why Minimal Guidance During Instruction Doesn't Work":
According to Kyle (1980), scientific inquiry is a systematic and investigative performance ability incorporating unrestrained thinking capabilities after a person has acquired a broad, critical knowledge of the particular subject matter through formal teaching processes. It may not be equated with investigative methods of science teaching, self-instructional teaching techniques, or open-ended teaching techniques. Educators who confuse the two are guilty of the improper use of inquiry as a paradigm on which to base an instructional strategy.

LynnG said...

The cumulative building of knowledge in the various disciplines (and they are called "disciplines" for a reason) is under-appreciated in k-12 education.

In an effort to make everything relevant right now, the structure of knowledge has been lost.

I asked a couple 9th graders "which came first, the Vietnam War or the Civil War?" they had no idea. But they could all reflect at length about the evils of slavery and discrimination, without reference to any of the major events in the history of civil rights.
Our children have been taught that their feelings about big issues is far more important than knowledge of the these things.


Jerrid Kruse said...

Barry, as does Sweller, misunderstands inquiry-based teaching as discovery learning in which the teacher does very little. This has always been argued against by informed educators & because of the nuance required to both explain & understand the teacher's very involved, yet not lecturing role, the confusions have been seen time & again in those looking to reinforce the sit & get teaching model.

Also, Barry might want to consider that the reason he didn't understand the "why" of some procedures until much later is for developmental reasons or even that because the procedure was taught divorced from understanding, he had no hope until later. Barry, is clearly an intelligent individual who was able to figure out the understanding from the procedures. Unfortunately, not all are so intelligent so we ought to be explicitly addressing conceptual understanding as well as procedural knowledge. Of course, conceptual understanding lasts much longer than procedural knowledge so of we were to "lean" one way, I'd suggest toward conceptual.

Barry Garelick said...

Understanding and procedures are an iterative process. Sometimes the understanding can come right away, but other times, it takes time for students to have the tools to fully understand what's going on. (See "Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process" by Rittle-Johnson, et. al, Journal of Educational Psychology, 2001; Vol. 93, No. 2; 346-362) My being able to figure out the understanding from the procedures was not because I'm an "intelligent individual" as you kindly describe me. Even Singapore Math, noted for blending understanding withe procedures, does not explain why the invert and multiply rule for fraction division works when they initially present it in 6th grade. They present an explanation in 7th grade after students have had more experience with algebraic symbol manipulation. If the student does not have the tools for understanding, they will not understand. Getting the knowledge of the tools can be a function of development.

Even at the seventh grade level, such understanding of the invert and multiply rule may or may not happen. I just recently taught the rule with the reasons for why it works. I would say that some students may have understood it, but most did not. Most if not all understood how the procedure works. With increased use in solving algebraic equations such as 3/4(x) = 7/9, the understanding becomes more transparent. There were many things about arithmetic that I understood when taking algebra. Similarly, there were many things about calculus that I understood when taking differential equations and real analysis. I find it inaccurate, however, to claim that math as it was taught in earlier years was devoid of the conceptual context, or with no understanding at all. I believe many educators are misinformed about this.

As for the difference between inquiry-based teaching and discovery learning, I argue that it's a distinction without very much difference. Tomlinson and McTighe, in their book about differentiated instruction and "understanding by design" talk about students constructing their own meaning at their own pace, by being immersed in what they term “contextualized grappling with ideas and processes”. What does this mean? In general, it means giving students an assignment or problem which forces them to learn what they need to know in order to complete the task. It is a “just in time” approach to learning, in which the tools that students need to master are dictated by the problem itself. Educators who favor this technique feel that it does not burden the student’s mental inventory with the so-called “mind numbing” drills for mastery of a concept or skill until it is actually needed. The result is an approach that is like teaching someone to swim by throwing them in the deep end of a pool and telling them to swim to the other side. For the students who may already know a bit about swimming, they may choose to take that opportunity to learn the butterfly. The teacher might advise the weaker students to learn the breast stroke and provide the much needed direct instruction which they may now choose to learn. Or not.

Jerrid Kruse said...

You are right about the futility of Tomlinson's approach. This is NOT guided inquiry learning as it falls into all the traps you & others have noted. however, Consider a continuum between discovery & lecture - teachers should do a lot in the middle. You & this blog spend so much time arguing against one end of the continuum, that readers may
Think the other end of the continuum is "correct" when the lecture side of the continuum is just as bad as the discovery side. You & this blog may not mean to argue for passive teaching methods, but you do so implicitly so be careful.

Barry Garelick said...

Jerrid, I understand what you're saying. But I and others have said on this and in other places that traditional math teaching is/was not just lecture but involved scaffolding and asking questions (guiding, if you will). People like Tomlinson will defend their techniques by saying they use a "balanced approach". What Tomlinson and others do is not what I consider "balanced" but many do.