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.
---------- Forwarded message ---------- Date: Wed, 2 Nov 1994 13:30:13 -0800 (PST) From: Ronald A Ward <email@example.com> To: firstname.lastname@example.org Cc: Multiple recipients of list <email@example.com> 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 (firstname.lastname@example.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 email@example.com