Date: Jun 8, 1995 8:44 AM
Author: Dax Mitchell
Subject: Re: Spiraling at the Elementary Level



I've really enjoyed the postings on spiraling. I'm wondering if
anyone who has experience using this technique has opinions about
the following issue; namely, when a topic is revisited after a period
of a week (or perhaps several weeks?) at a more sophisticated level,
do the students have to _relearn_ what they studied on the previous
visit? Since on the previous visit, they studied the topic briefly
and then moved on, perhaps the basic ideas within the topic have not
soaked in. I don't mean to ask this as a skeptic, because the spiraling
idea has a lot of appeal for me personally. Nevertheless, in my own
mathematical learning, I know that if I see a new idea sketched in, say,
a seminar and then don't see that idea for a few weeks (or sometimes,
unfortunately, days), my feeble mind usually takes a while to retrace
the steps--I usually have to do some "reviewing" in my own mind to get
brought up to speed. And it seems that many times when I'm "reviewing"
the speaker has moved on to bigger and better things, leaving me in the
dust.

Thanks for your kind attention.

Dax Mitchell
Duke Math Dept.

{the rest is a clipping}

> From: Ronald A Ward <ronaward@henson.cc.wwu.edu>

> Recently there have been several posts regarding spiraling, one of which
> asked for curricula that effectively and properly use this approach. In
> the interest of time, I am forwarding a post that I wrote last November
> on the subject of the UNIFIED SPIRAL. I believe it is responsive to the
> questions and issues currently being discussed.
>
> Ron Ward
>
> ---------- Forwarded message ----------
> Date: Wed, 2 Nov 1994 13:30:13 -0800 (PST)
> From: Ronald A Ward <ronaward@henson.cc.wwu.edu>
> To: nctm-l@sci-ed.fit.edu
> Cc: Multiple recipients of list <nctm-l@sci-ed.fit.edu>
> Subject: Re: Unified Spiral
>
> On Tue, 1 Nov 1994, Marie Revak wrote:
>

> > What is "unified spiral" ?
>
> In reply, I will try to (1) describe this curriculum-organization approach.
> Then, I will (2) cite one of the elementary curricula that appear to use
> this effectively, and (3) give you a contact person for that program.
>
> (1) When I worked at the CEMREL lab in St. Louis, we found that children
> learn thru many interrelated experiences. But no experience,
> particularly one math lesson, is an end in itself. It is neither
> intended nor expected that each child will meet the full challenge of
> each situation in every lesson. Furthermore, we found that it is most
> effective to vary the situations and topics from day to day rather than
> to continue with one type of situation or piece of content until
> so-called "mastery" has occurred. Students at the elementary level do
> not seem to enjoy staying on the same topic for a very long time, but
> prefer to be involved in a variety of situations. Thus, the lessons were
> scheduled so that several topics would be studied during each week and
> similar situations would appear again and again at different times and
> levels in a spiral development.
> If a spiral seems strange at first, consider an illustration
> (which I can't show you on here): imagine an academic subject that could
> be divided into five subtopics. Then imagine a child's progress being
> represented by a spiral, starting at the center and traveling outwards in
> the direction of increasing degrees of maturity and sophistication. In
> such a diagram, an intersection point between the spiral and one of the
> five topic lines indicates a lesson on that topic. The student is first
> introduced to Topic A, then to B, then to D, followed by a slightly more
> sophisticated lesson on A, and so on. Topics C and E might not be
> introduced until a little later in the spiral, while Topic B may be of a
> more terminal nature and is considered complete after a few lessons.
> This spiral development, in which a child experiences each of several
> ideas a little at a time and then proceeds thru increasing levels of
> sophistication as the situations become more challenging, is wholly
> consistent with Caleb Gategno's "Pedagogy of Situations," with which many
> of you are no doubt familiar.
> Children learn at different times and at different rates. Only
> part of the learning of a given topic actually takes place during a
> lesson on the topic. In between the times when the topic occurs on the
> learning "spiral," the child is mentally digesting the idea, sometimes
> consciously, usually unconsciously. In this way, the spiral development
> gives each student a new chance to "catch on" at each stage.
> We found it highly effective to follow this kind of development
> rather than continuing on and on with a topic until so-called "mastery"
> has occurred.
> Sticking with the spiral idea requires an act of faith on the
> part of teachers, particularly if they have been used to teaching a given
> topic until they were satisfied that all (or at least most) of the class
> knows it "cold." Belief in the spiral approach implies that some lessons
> may be stopped before some (or even a majority of) students appear to
> have caught on, or in a lesson involving worksheets or workbooks, before
> everyone has successfully completed all of the pages. It involves
> knowing that the topic will reappear again and again, and that varying
> degrees of understanding will come at varying rates and times. It
> entails a belief that an idea planted now may not sprout until much
> later, and that it is best not to force its development. All our
> experience at CEMREL, which included five years of extended pilot testing
> at each grade K-6, indicated that this approach really works.
> The reason I used the name UNIFIED spiral is that the curriculum
> must be very carefully structured to make sure that when you are working
> out of the probability strand on Thursday, and want to use a "geometric
> area" model or fractions to calculate a probability, then the children
> must have previously seen those related ideas in the geometry and numerical
> strands, which you may be doing on, say, Tuesdays and Wednesdays. Constructing
> such a curriculum is not for the fainthearted! :)
>
> (2&3) The unified spiral approach is utilized in the "CSMP/21"
> [Comprehensive School Mathematics Program for the Twenty-First Century]
> curriculum which is centered at the McREL Educational Laboratory in
> Colorado. The Project Director is Clare Heidema (cheidema@mcrel.org).
> This curriculum is described in the Educational Programs That Work
> catalog of the National Diffusion Network and is a nationally validated
> program. Among other things, this means that, in every state, there is a
> State Facilitator who can help schools learn about that curriculum. [I'm
> sure that Clare could provide individual contacts for every state to
> interested readers.]
>
> Hope this helps.
>
> Ron Ward/Western Washington U/Bellingham, WA 98225
> ronaward@henson.cc.wwu.edu
>
>
>