**In STEM education it is common to talk about "spiraling" in the context of topics being taught progressively over a series of grades. Our Grade Level Stratification work supports a precision approach to the search, modeling, analysis and visualization of this important practice. **

**What we are finding is that individual topics vary relatively little from state to state. But what is taught in a given grade varies a great deal. These big differences are due to different ways of spiraling, or jumping from topic to topic, not to different topics being taught.**

**This "jumping problem" is explained below, in terms of a simple 3D spiral model.**

**First divide the science to be taught into, say, 30 topics, each of which is taught progressively in several grades over the K-12 period. The science of light, sound and electricity are three such topics. Let a vertical tube 60 inches high represent each topic. Place the tubes in parallel on the floor, making a cluster of tubes. Next divide each tube into vertical segments, say 20 per tube, each of which will represent a group of concepts that are normally taught together.**

**Label the segments in sequence so that the concepts taught earliest for each topic are closest to the floor, and progress upward to the last concepts at the top end of each tube. The sequence of segments represents the fact that many concepts have to be taught sequentially. Assume for now that there is only one such sequence for each topic. We now have 30 tubes with 20 segments each, or 600 segments in all.**

**Now let a string represent the actual sequence of teaching of each segment in a given curriculum. Attach the string to each of the 600 segments in the order in which these are taught. The basic model for that curriculum is now complete. Spiraling refers to the fact that the string will leave a given tube for a period of time then return, then leave again, return again, and so on.**

**The amount of spiraling can vary enormously from curriculum to curriculum, or school to school, depending on the overall sequence of teaching. Different colored strings could be used to compare them. (Of course these differences can also be analyzed without visualization.) This is because, while the sequence of concepts in each tube may be fixed, there is in general no required sequence between tubes.**

**At one extreme, suppose each tube is completely taught before another is begun. In this case the string would only jump 29 times, from the top of each tube (except the last) to the bottom of another. At the other extreme the string would jump to another tube after every segment, or 599 times.**

**In any given curriculum the amount of jumping is normally somewhere in between these two extremes, perhaps several hundred jumps. All of this jumping probably contributes to the difficult of learning. **

**Moreover, the number of possible paths for the string, or sequences of concepts taught, is enormous. This means that two different curricula can have very different string sequences. Students moving from one to another will be taught some concepts twice and others not at all. Missing concepts can be a very big problem, given that later concepts depend on earlier ones.**

**This simple spiral model shows just how complex science education is. Note too that this model applies to math as well as science, in fact some science concepts require a certain amount of math. Combining the two gives a 60 tube model. The overall model is very complex, but that is just how the reality is. Science and math are not simple and spiraling is a big part of challenge.**

**Comments or questions?**

**David Wojick, Ph.D**** <dwojick@hughes.net>**