[NKS 2004: Todd Rowland, NKS Curriculum] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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NKS 2004: Todd Rowland, NKS Curriculum
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Posted by: Catherine Boucher
Todd Rowland
Wolfram Research
NKS Curriculum
Session:
NKS in Education
Part 1
2:15pm, April 24, 2004, Eden Vale B
Abstract:
http://www.wolframscience.com/confe...s/index_58.html
Posted by: Todd Rowland
Teaching NKS should be simple. We at Wolfram Research have been working on classroom materials that introduce the basics behind NKS. The basic idea is to make something that teachers can easily use and understand.
Let me start by talking about why we would want to introduce NKS concepts to kids in the first place. It is not hard to imagine that because simple rules are simple they should be easy to explain. But at the same time, NKS tells us that simple rules are important, and fundamental. And it can also be appreciated on several levels, from graphics and other visualizations to logical formulas and other manipulations. It is possible to be rigorous, simple, and useful, all at the same time.
And there is a need to teach kids how to use technology in a fundamental way.
So it is basically an obvious thing for us to be doing.
I am going to show you some of the lessons we have made, but first here is a picture showing how I think of exploring NKS and how to decide what should go into introductory materials.
Here is a huge tree-like graph. At any point there are several directions one might take. For instance, one might start studying rule 45 or rule 73. While studying rule 73, one notices some patterns and further investigates those. While there are interconnections between different branches, they can take a fair number of edges to come together. For instance, one might discover a way for rule 73 to emulate rule 45, and that would connect the branch studying rule 73 to the branch studying rule 45.
At the core of NKS, there are some basic concepts which sort of form the trunk to the tree. They fall into what we have been calling Pure NKS. The simple rules without the modeling aspects. So we are going to start with the elementary cellular automaton. Along the way, basic concepts will be introduced: computational irreducibility, intrinsic randomness, and finally the Principle of Computational Equivalence.
So it is somewhat clear where we should start.
By sticking to concrete, simple examples, the NKS concepts will be understandable by teachers not already familiar with NKS.
Almost any educational approach asks students questions which can be answered. So we will concentrate on cases of computational reducibility. As a by-product, one develops an intution for irreducibility and randomness. In fact, to keep things simple we are going to start with the simplest cases, the neighbor independent rules, before moving onto the rest of the 256 elementary cellular automata.
These graphics here show the progression in the course, covering the most common basic behaviors found in the elementary cellular automata.
The core lessons form approximately two weeks of material. Our alpha version is for high school students, but we are also making less analytical materials for younger ages. We are also collecting material which can be put together, which extends the core lessons, to form a semester or year-long course.
One thing I am beginning to realize is that there is a really large body of material which is at the right level for students with only a beginners ability and understanding. Often it is already known, or possibly immediately obvious to an expert. Some of this material gets a passing mention in NKS, but most of it is not even there.
A sample core lesson
Here is the table of contents for the core 10 lessons. After introducing the idea of digital information and simple rules, we proceed by introducing rules which require more and more information, until we hit the elementary cellular automata and computational irreducibility.
Let's look at a sample lesson. Here we introduce the elementary cellular automata by looking at Rule 250...
We show how to evolve the elementary cellular automaton...
The basic pattern becomes obvious pretty quickly...
Discussion questions can center around the main pattern. Why does it happen? How can we specify it using precise language? Can we prove that the checkerboard pattern is the result of a single black cell? Can we be sure the checkerboard pattern continues indefinitely?
Other questions can be done individually. Given a picture of an ECA with cyclic behavior, fill in a region of its evolution.
What about various measurable quantities? The number of black cells makes a nice function. We can graph it. Or we can try to prove it is a linear function.
An easy observation to make is that the right thing to do, even in this case, is to describe the pattern by a rule instead of by a list of quantities.
A sample lesson beyond the core
So we have this core lesson on Rule 250. In order to give a brief survey of NKS which will fit into a two week course, we have to move on.
But there are lots of things one can continue to study with Rule 250, if one gets intrigued. These could be used by teachers who want to continue the core lessons in a longer course, or maybe they just want to stick with the simple cases. Or it can help a teacher who wants to give guidance to an advanced student.
Rule 250 is reducible. In fact, one can give a one liner in Mathematica which speeds up its evolution to predict the bit at {x,t} depending on the initial condition. But this is not immediately obvious, and makes a good problem for the students to work out.
There are several ways one might approach it, but as a hint, the simplest way is to first discover in what way Rule 250 has generalized additivity.
What NKS says about learning and what it says about learning NKS
It is probably a mistake to undergo such a project without a philosophy of learning. Most theories in education are based on ideas about why things are the way they are, in addition to conceptions of knowledge and understanding.
For the sake of this project, I take a naive view of why things are the way they are, and rely on an emphasis on simplicity, which is a common theme from the book.
But still it is an interesting question to look at what NKS has to say about learning.
Chapter 10 of the NKS book is about perception and learning, and a common theme in this chapter are hash codes. Basically, what they do is take a large amount of data and convert it into a simpler form. Then two objects are considered similar if they have the same hash code. So one learns that this is a chair and that that is also a chair. Even though the low-level details are not identical, the brain learns to associate the two objects as chairs.
And, not surprisingly, one of the ways this is most successful is when looking at simple programs like the elementary cellular automaton.
From this I conclude that learning patterns arising from simple rules is easier than learning those from more complicated rules. So NKS has a good chance to make sense to students, and seem like a sensible thing to be learning.
On the other hand, the act of learning about NKS is talked about on p.855. There are three basic ideas. Read the book. Perform your own experiments. Learn Mathematica. Obvioulsy, these are not quite independent. At least in the core lessons we have here, the only one of these which shows up is experimentation to investigate properties.
What the book is talking about is using experiments for exploration style learning. But these lessons we have made are for more structured learning environments, for students who aren't yet able to, or even interested in, performing their own experiments. Still I mention these three basic ideas as possible pillars for alternative NKS programs, for students who are more advanced and motivated.
Where this fits in and how to get it
While we are trying to stay away from certain educational debates, which I know little about, there are certain stands that we are taking.
The use of technology is obviously important to studying NKS. For instance, one could spend a bunch of time learning the skills of evolving Turing machines by hand on a piece of paper. While that makes sense in the introductory lesson, it is not a good way to explore Turing machines.
One way to approach teaching is to think about what are the tools students need to learn. What is needed to learn NKS is not the skill of being able to compute an evolution quickly, but ultimately, one needs to know some basic programming.
At the early levels, before high school, there is lots of room in the traditional curriculum for learning more about simple rules. The typical high school curriculum is fairly structured, so we need to think about how it can fit in.
So these lessons can be taught in a math class. One feature of teaching NKS material in a math class is that it is possible to ask questions requiring justification, which require thought instead of algebraic manipulation. Indeed, depending on the level of the students, they can try to make rigorous proofs.
But they can also be taught in a science class, and that would change the emphasis. Instead, one can emphasize the experimental aspects, such as what is involved in doing a rigorous survey. Or discuss levels of certainty. For instance, how certain can you be that Rule 30 doesn't degenerate into a regular pattern?
One novel way these lessons can be used is as a course called pre-computer science. There are many ideas at the core of computer science which can be learned through NKS, such as emulation and universality.
It seems that there is a great need for a course on NKS, at the very least to teach the principles for significant use of technology.
Let me finish by saying how one can get a hold of the alpha version. We will post a notebook version online soon...
Posted by: Todd Rowland
We have a very alpha version of the educational materials we have been working on.
There are probably various types of bugs. Please send your comments and suggestions, or post a reply. Or even better, post some educational materials.
We hope that you will find them to be useful.
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