Reviewed by Alex Feinman

- Constructivism, Interactionism
- Involving the student in the learning process.

- Sensorimotor - learning how to affect the environment
- Preoperational - development of symbolic thinking
- Concrete operational - mastery of classes, relations, etc.
- Formal operational - "mastery of thought"

The two cultures - humanities vs. sciences

Balkanization of knowledge; the plane of knowledge thought to be is split into distinct areas, which interferes in transference of knowledge from area to area, and causes people to think that they're "no good" at some topic that they might otherwise be able to find another path to learning.

Papert suggests that mathematics, in particular, needs to be presented in a context where it has connection to other topics -

"a Mathland -- where mathematics would become a natural vocabulary".

Furthermore, Papert argues, improving teaching methods by extending current methods would be as fruitless as perfecting the horse-drawn carriage as a mode of transportation; it would completely ignore technological advancements that would allow quantum leaps in teaching efficacy.

So Papert is trying to jump ahead to the "educational automobile", the next step in the evolution of teaching.

Papert also gives a few examples of how competence in one area can depress skills in another area, either by shifting the student's concentration, or through more complex interaction such as the failure of English spelling to follow rules of logic, hence alienating a student who excels in logic. I'm not sure if I agree with his logic.

Papert gives an example of the leap in interest and therefore competence of a student when an academic subject is grounded in a real-world problem. So-called average students were instructed to write a program to generate random, grammatical sentences. Because solving this problem required them to internalize rules of grammar, and then feed that information into the computer and see the results of their learning, the students suddenly saw uses for the previously abstract rules they had been taught.

2 * 2 = 4 2 * 3 = 6 2 * 4 = 8 ...Why? Because I told you so.

Papert argues that the existing systems for teaching math, and the curriculum selected, are based on historical context, and are out-dated in the modern day. Both subject matter and teaching methods were constrained by the needs of the day, and the tools available.

With the advent of cheap, available computers and simple robotics for use as teaching tools, a variety of new teaching methods becomes available; old methods (such as rote drilling of information) are enhanced, but that's not what Papert is interested in. He has instead created Turtle geometry, discussed in length later.

Finally, there is a discussion of feedback and credit assignment, a problem which all of you from the AI community should be intimately familiar with. Papert gives the two examples: the first is a living language, where other practitioners of the language provide feedback as to the progress of the student, either implicitly or explicitly. The second is math sums, where a teacher grades the student's paper once he or she is done, marking the student's work as correct or incorrect, with no feedback as to the context of the work. He then proceeds to put down New Math as a failed attempt to break out of the old 'dead math' curriculum.

The Turtle is a point with a direction. However, it also has an
**identity**; almost everyone who uses Turtle geometry at
some point or another identifies
with the turtle as a way to understand what is going on, whereas I've
never heard of anyone identifying with a Euclidian point.

The turtle is commanded with a handful of simple commands, called
TURTLE TALK; the usual set includes:
`
`

- forward X
- left turn X
- right turn X
- repeat X

...

end - to function :arg1 :arg2 ...

...

end - pen up
- pen down

The turtle makes a line as it moves, drawing with a pen. By using these commands in sequence, the student can experiment with the turtle, drawing various shapes. While the basic controls are learned, the student is also introduced to the concept of state (the state of the turtle), modularity (using the 'to' command allows students to encapsulate functions and reuse them), and the idea of debugging a problem.

Because there is no single correct answer to solving most problems, and the students are free to fiddle with the turtle to solve problems until they solve them, the censure of failure is less, and students tend to experiment with the turtle. In addition, getting the turtle to obey the student's commands turns into a challenge, which most students find interesting enough to rise to.

Turtle geometry also allows incremental movement along the way to a solution; the student does not simply get the problem 'wrong' or 'right'. Papert gives an example of a student attempting to combine a square and triangle into a house, and some steps along the way. This allows gradient learning to take place, another concept you AI-types should be very familiar with.

Papert proposes that turtle geometry teaches only a small corner of
mathematics, but sheds light on the larger topic of *mathetics*,
the study of teaching. He claims that the turtle allows syntonic learning,
as opposed to the dissociated or disconnected learning that is the usual
method of teaching math. He then spends quite some time investigating the
areas of mathematics that turtle teaching covers, with some claims that
it allows the intuiting of differential calculus, teaches variable
algebra, and so on.

In summary, he claims that the turtle is:

- powerful - the student can use it to do something
- general - the student can use it do lots of things
- intelligible - the student can understand what it's trying to do
- personal - the student can identify with the program
to these I would add:

- incremental - small changes in the program generally have small effects
- responsive - changes in the program are apparent immediately

Should learning be done only in school?

First Papert gives the example of Brazilian samba schools, where what looks like spontaneous learning occurs. I hate this example. It's nearly as far-fetched as the algolgol example from last week.

A more concrete example I came up with involves what companies call 'on-the-job training' - note the modern avoidance of the word "teaching", which is loaded with connotations of the unpleasantness of primary school, in favor of the more mild "training". In this training, for example on a construction crew, a more learned individual will instruct a less knowledgable one on the method to do something - let us say hammering in a nail. First the teacher demonstrates the skill, annotating his actions as he feels necessary. The student is free to interrupt and ask questions at any point. Then, the student tries out the skill, and the teacher critiques. This cycle is repeated until the student and teacher feel that the skill has been transfered sufficiently. The learning episode is then over, and work resumes. The student is left to experiment with the skill, and discover more about it.

In this example, it is obvious that learning has occured, but the roles of teacher and student are fleeting, and no syllabus is set out ahead of time. Furthermore, almost all people are willing to learn this way, provided the information is not presented in such a form as to make the student feel embarassed not to have known it. This is a very different form of social structure than that of the classroom, where learning is usually a one-way, non-stop, non-correcting flow of information whose course is set out by people who are not even present when the material is taught.

Papert then tries to defend his thoughts against all comers. He also looks forward to "computational samba schools", which to a certain extent have come about in the form of discussion groups on UseNet and other such facilities. But in all cases he emphasizes the importance of the culture surrounding the learning process.