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Mindstorms

Mindstorms

Thoughts on Seymour Paper's Mindstorms: Children, Computers, And Powerful Ideas.

Chapter 1 Computers and Computer Cultures

Forunately, there is a weak link in the vicious circle. Increasingly, the computers of the very near future will be the private property of individuals, and this will gradually return to the individual the power to determine patterns of education. Education will become more of a private act, and people with good ideas, different ideas, exciting ideas will no longer be faced with a dilemma where they either have to "sell" their ideas to a conservative bureaucracy or shelve them. They will be able to offer them in an open marketplace directly to consumers. There will be new opportunities for imagination and originality. There might be a renaissance of thinking about education.

Personal computers have certainly become privately owned. But so far I don't know whether education has become more of a private act. Perhaps the recent uptick in homeschooling post Covid has been a trend in this direction.

Of course the Turtle can help in the teaching of traditional curriculum, but I have thought of it as a vehicle for Piagetian learning, which to me is learning without curriculum.

I'm not familiar with "Piagetian learning". I've heard vague (usually critical) comments about schools without grades where the students' interests determine what they'll study. But I've never exposed myself to real instantiations of these ideas.

My image of myself as a write includes the expectation of an "unacceptable" first draft that will develop with successive editing into a presentable form. But I would not be able to afford this image if I were a third grader. The physical act of writing would be slow and laborious.

"Writing" refers to materially different activities here. The first is, "Discovering, refining and communicating ideas." The second is "Stringing letters together consistently and coherently into sentences." The first depends on the writer's ability to the second with "automaticity". Math Academy refers to automaticity in their FAQ:

What is the difference between fluency and automaticity?

Fluency, when referring to math skills, generally means that a skill has been memorized, such as a student knowing their multiplication facts. Automaticity is one step beyond that, meaning, the skill has been stored into long term memory and can be recalled instantly- without thinking about it.

The reason this is important is once a student moves on to more complex skills that require problem solving, the original skill won't take up brain processing power that will be needed in order to solve the problem.

Math Academy's learning philosophy first builds automaticity and then moves on to complex, problem solving skills. This is one reason why we have timed diagnostic assessments and quizzes. We are measuring not just a right or wrong answer, but the student's level of automaticity.

I think "automaticity" is underappreciated, especially in conversations about test-taking anxiety. It's OK to say, "We don't just need the student to be able to get the right answer eventually. We need for them to know the right answer effortlessly." Otherwise, you risk building on rocky foundations.

An example of BASIC idealogoy is the argument that BASIC is easy to learn because it has a very small vocabulary... Perhaps the vocabulary [of a special language to help children learn to speak] might be easy to learn, but the use of the vocabulary to express what one wanted to say would be so contorted that only the most motivated and brilliant children would learn to say more than "hi".

APL-inspired languages left a big impression on me, even though I've only ever learned to write trivial programs with them. One of the key ideas is to start with a larger vocabulary. When you have a large vocabulary, programs can shrink. I've sometimes fantasized about a game that gradually introduces you to the meaningful symbols of APL, such that by the end of the game you can express complex ideas into very short "sentences", which are runnable programs.

Geometric Algebra is an expressive mathematical system that aims to simplify / unify several number systems used in physics, graphics, and modeling.

I would love to experiment with a high-vocabulary language that combines

Chapter 2 Mathophobia: The Fear of Learning

This great divide [between "humanities" and "science"] is thoroughly built into our language, our worldview, our social organization, our educational system, and, most recently, even our theories of neurophysiology.

Before wordcels and shape rotators, we had humanities and science.

Many more people have not completely given up on learning but are still severely hampered by entrenched negative beliefs about their capacities. Deficiency becomes identity.

A family friend and her daughter in 9th grade recently explained to me how they "weren't math people." I've heard this plenty of times, but seeing the belief passed from parent to child is especially frustrating. The daughter seemed to have adopted her anti-math identity in 6th grade or perhaps earlier. The parent lauded her daughter's attitude, as if it vindicated her own shortcomings.

I agree with the book's general assertion that our beliefs about education are ultimately to blame, but it's hard not to feel frustration with the parent, who I perceive as limiting her daughter's achievement and immediate satisfaction with a subject that ultimately produces powerful tools of thought.

Math Land

I've frequently imagined "Math Museums" where concepts come to life in interactive exhibits. Explanations for maths is built on top of such small, steady increments, that if someone "inhabited" a place where these steps were laid out in front of them, I believe achievement and understanding would be significantly higher.

I recently visited an exhibit in New York by MoMath, the National Museum of Mathematics. It was great. It's just 10,000 times too small (in terms of content), and it's not in a format that can be distributed around the country or the wider world.

Very few people ever suspect that the reason for what is included and what is not included in school math might be as crudely technological as the ease of production of parabolas with pencils! This is what could change most profoundly in a computer-rich world: The range of easily produced mathematical constructs will be vastly expanded.

What a fascinating observation. I have not heard of any schools where students are producing mathematical systems and simulations instead of hand-sketching graphs of various functions. Why not? Shouldn't they?

There are tons of mathematical phenomena that are easily produced and understood via systems and algorithms that are difficult to express in "closed form" equations.

As with the comments in Chapter 1 about the act of writing, students who find it tedious to produce graphs will never get onto the more interesting task of "thinking and experimenting with functions as a way to explain and understand observations, measurements, phenomena".

Here is an example of the kind of thing I wish more people would learn: People commonly call graphs that go up and to the right, "exponential". But claims about which function underlies these phenomena should be met with strong skepticism. Vaclav Smil illustrates this idea well in Growth. If I recall correctly, he dedicates a long chapter to showing how a large number of curves can match with arbitrary precision the beginnings of an "S curve," but that choosing one curve or another to interpolate forward leads to vastly different conclusions.

Chapter 3 Turtle Geometry: A Mathematics Made for Learning

Figure 3. An Actual Transcript of a Child's Early Attempt at a Square

Wow, I love the idea that we don't tell a child "a right angle is 90 degrees", and that instead they naturally want to use right angles (even if they don't have a name for them), so they must discover the number that gets them that result.

The Turtle circle incident illustrates syntonic learning. This term is borrowed from clinical psychology and can be contrasted to the dissociated learning already discussed. Sometimes the term is used with qualifiers that refer to kinds of syntonicity. For example, the Turtle circle is body syntonic in that the circle is firmly related to children's sense of knowledge about their own bodies. Or it is ego syntoinc in that it is coherent with children's sense of themselves as people with intentions, goals, desires, likes, and dislikes. A child who draws a Turtle circle wants to draw the circle; doing it produces pride and excitement.

Turtle geometry is learnable because it is syntonic.

Interesting term and ideas. It seems pretty important to find out how accurate this mental model of learning is.

Turtle Geometry lesson illustrated

This was cool to see and read through. I might not have the patience to be a teacher every day, but it looks and sounds like fun to be either the teacher or the student in that scenario.

Chapter 4 Languages for Computers and for People

Children often develop a "resistance" to debuggin analogous to the resistance we have seen to subprocedurizing.

I have had to overcome "resistance to debugging" even as a professional programmer. It's rarely an issue for me anymore, but in my first few years I would get myself in trouble by making too many changes to the program before running / testing its behavior. One of the worst feelings comes from doing too much and then finding that you cannot back-track or understand what you've done. It feels like a big waste of time. It puts you in the position to risk spending even MORE time debugging than you would if you started over from scratch.

I attribute the resistance earlier in my career to a few feelings:

The instructor and a child were on the floor watching a Turtle drawing what was meant to be a letter R, but the sloping stroke was misplaced. Where was the bug? As they puzzled together the child had a revelation: "Do you mean," he said, "that you really don't know how to fix it?" THe child did not yet know how to say it, but what had been revealed to him was that he and the teacher had been engaged together in a research project. The incident is poignant. It speaks of all the times this child entered into teachers' games of "let's do that together" all the while knowing that the collaboration was a fiction. Discovery cannot be setup; invention cannot be scheduled.

...Can an adult and a child genuinely collaborate on elementary school arithmetic?

I love this story. And I love the idea that children are exposed to the reality that in life, we don't have answers to many important questions. We have more knowledge than any single human can realistically know, so we (adults) do not try to teach everything. Instead, we teach things we think are so valuable and core that it empowers them to navigate the unknown.

Sharing the problem and the experience of solving it allows a child to learn from an adult not "by doing what teacher says" but "by doing what teacher does."

I love this. And it makes me of apprenticeship, mentorship, and studying under a master for prolonged periods.

Chapter 5 Microworlds: Incubators for Knowledge

Just as students who prefer to do their programming using Newtonian Turtles with third law interaction are making Newton their own, children making a spectacular spiral in a non-Newtonian microworld are no less firmly on the path toward understanding Newton. Both are learning what it is like to work with variables, to think in terms of raios of dissimilar qualities, to make appropriate approximations, and so on. They are learning mathematics and science in an environment where true or false and right or wrong are not the decisive criteria.

Newtonian mechanics isn't our current best theory of how the world works. Its principles don't hold at high speeds or small scales. We shouldn't teach Newton just so students can solve Newtonian physics problems. Newtonian physics is a useful tool, but being able to inspect, adopt, and invent tools, systems, and simulations like Newtonian physics is even more useful.

A monkey and a rock are attached to opposite ends of a rope that is hung over a pulley. The monkey and the rock are of equal weight and balance on another. The monkey begins to climb the rope. What happens to the rock?

I'll admit -- I somehow never had a Newtonian physics class, and don't know how to answer this question. I wish I knew Newtonian physics. Maybe it's good that I don't, because it gives me an opportunity to learn extremely mainstream material with "fresh eyes".

Chapter 6 Powerful Ideas in Mind-Size Bites

If students are given such equations a f = ma, E = IR, or PV = RT sa the primary models of the knowledge that constitutes physics, they are placed in a position where nothing in their own heads is likely to be recognized as "physics... A different sense of what kind of knowledge constitutes physics is obtained by working with Turtles: Here a child, even a child who possesses only one piece of fragmentary, incompletely specified, qualitative knowledge (such as "these Turtles only understand changing velocities") can already do something with it.

"Being able to do something with it" is repeated several times in the book so far. I'm reminded again of "automaticity", though here the emphasis seems different. It's more about "getting familiar" with the general idea of these systems.

I'm also reminded of the phrase attributed to Oscar Wilde, "play gracefully with ideas". Apparently, the full quote is:

“The unfortunate accident—for I like to think it was no more—that you had not yet been able to acquire the “Oxford temper” in intellectual matters, never, I mean, been one who could play gracefully with ideas but had arrived at violence of opinion merely.”

― Oscar Wilde, De Profundis

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The cultural assimilation of the computer presence will give rise to a computer literacy. This phrase is often taken as meaning knowing how to program, or knowing about the varied uses made of computers. But true computer literacy is not just knowing how to make use of computers and computational ideas. It is knowing when it is appropriate to do so.

This and the preceding section had a lot to say about procedures. Arithmetic via place-value digits is a procedure we teach kids. Difficulties in executing the procedure, Papert argues, are difficult for the kid to debug when they are unfamiliar with procedures themselves. They need to see "addition-with-carrying" as a procedure, with steps, that allows them to sum numbers.

I think there's another kind of literacy which goes beyond this description of computer literacy, which has to do with dynamic, stateful systems. Systems that are too difficult and stateful to carry out in the head. Systems that are easier to understand through computer simulation.

I think the distinction between these two types of literacy is analagous to the "writing of a third grader" vs "writing of the adult Papert" described earlier. Without basic "procedural literacy", it's hard to imagine someone gaining "systemic literacy". And while both are useful, "systemic literacy" seems like the more powerful kind of thought enabled by computers.

People often fear that using computer models for people will lead to mechanical or linear thinking: They worry about people losing respect for their intuitions, sense of values, powers of judgement. They worry about instrumental reason becoming a model for good thinking. I take these fears seriously but do not see them as fears about computers themselves but rather as fears about how culture will assimilate the computer presence.

This chapter's hypothetical dialog between imaginary characters GAL and ARI, as well as its discussion of the "string around the circumference of the earth" problem, were particularly interesting. I feel like these two examples are probably among the ones that people "misinterpret" by focusing too much on the mathematical details, rather than their epistemological aspects.

I want to reread them and meditate on what he's trying to communicate beyond these specific examples.

Imagine a string around the circumference of the earth, which for this purpose we shall consider to be a perfectly smooth sphere, four thousand miles in radius. Someone makes a proposal to place the string on six-foot-high poles. Obviously this implies that the string will have to be longer. A discussion arises about how much longer it would have to be. Most people who have been through high school know how to calculate the answer. But before doing so or reading on, try to guess: Is it about one thousand miles longer, about a hundred, or about ten?

Most people who have the discipline to think before calculating - a discipline that forms part of the know-how of debugging one's intuitions - experience a compelling intuitive sense that "a lot" of extra string is needed. For some the source of this conviction seems to lie in the idea that something is being added all around the twenty-four thousand miles or so of the earth's circumference.

The book continues by imagining similar, simpler problems that a student might think about before attempting to directly calculate the answer.

How about an earth that is a square? Well, along the sides of the square, the length of the string is the same. It's only the corners that are "extra", and those 4 "extra" corners are a circle, with a six-foot radius.

What if the earth were a bigger square? Well, we get the same result. Along the sides, the same length of string is the same. And the quarter-circles at the corners of the square are just as small even if the square is bigger.

What about an octagon? It's a bit more work, but we can see the same phenomena. Along the sides of the polygon, no more string is needed. But at each corner, we have an eighth of a circle, with a six-foot radius.

We can extrapolate for an 100-gon, 1000-gon, or any n-gon. A circle is like an n-gon where n is infinitely large. So it seems that we'll get the same small amount of string no matter what: (2 * pi * 6 feet).

Chapter 7 LOGO's Roots: Piaget and AI

Ontogeny recapitulates phylogeny

I had never heard of this. https://en.wikipedia.org/wiki/Recapitulation_theory

We recall that children up until the age of six or seven believe that a quantity of liquid can increase or decrease when it is poured from one container to another. Specifically, when the second container is taller and narrower than the first, the children unanimously assert that the quantity of liquid has increased. And then, as if by magic, at about the same age, all children change their mind: They now just as unequivocally insist that the amount of liquid remains the same.

This is so strange to me. It is definitely in a category that is "so obvious" to me, that I can't believe this until I ask my brother to test this on my nephews (who are just around that age).

Children at all the levels of development anthropologists have been able to distinguish, and in over a hundred different societies from all the continents, have been asked to pour liquids and sort beads. In all cases, if conservation and combinatorial skills came at all, conservation of numbers was evidenced by children five or more years younger than those evidencing combinatorial skills. Yet this observation casts no doubt on my hypothesis. It may well be universally true of precomputer societies that numerical knowledge would be more richly represented than programming knowledge. But things may be different in the computer-rich cultures of the future. If computers and programming become a part of the daily life of children, the conservation-combinatorial gap will surely close and could conceivably be reversed: Children may learn to be systematic before they learn to be quantitative!

This book was written decades ago. I wonder if the prediction came true.

First, it provides a specific psychological theory...

Second, it shows us the power of a specific computational principle, in this case the theory of... procedures that can be... used in a modular way.

Third... different languages can influence the cultures that can grow up around them.

And from earlier:

The "theory" might appear to be nothing but anthropomorphic talk. But we have already seen that anthropomorphic descriptions are often a step toward computational theories. And the thrust of the society-of-mind theory is that agents can be translated into precise computational models.

This refers to a mental model involving how various agents ( A_width, A_height, A_history, and A_geom ) in the child's mind might develop to eventually come to understand conservation of fluids.

More so than any other chapter so far, I find myself reading with skepticism. My "default" bias when reading it is to discard most of it. Perhaps another read through or time while it rolls around the back of my mind will change this. But I am definitely noticing myself disengaging and becoming less interested in this chapter compared to others.

This chapter seems like it's trying to "make sense" of the work they're doing with kids. And I'm not convinced that they've nailed.

I definitely like the "turtle" device and programs. But whether they're effective for the reasons they think is up for debate. School is usually tedious and boring. Playing with the turtles is much closer to playing games. Were the students more engaged because of the Turtle programs matched Piaget's ideas, or just because the alternative classroom activities were dreadfully dull?

Chapter 8 Images of the Learning Society

The samba school has a purpose, and learning is integrated in the school for this purpose. Novice is not separated from expert, and the experts are also learning.

These descriptions immediately bring to mind dynamicland's hyper-local approach to community spaces.

The samba school has rich connections with a popular culture. The knowledge being learned there is continuous with that culture. The LOGO environments are artificially maintained oases where people encounter knowledge (mathematical and mathetic) that has been separated from the mainstream of the surrounding culture, indeed which is even in some opposition to values expressed in that surrounding culture.

I played soccer in clubs growing up. Most of the clubs made no attempt to bring the children or parents of different age groups together. But that was the vision of Coach Craig, who was the best coach / manager I've known. They were small things, but they were meaningful: He organized the clubs' summer camps and Friday afternoon "shooting clinics", which were optional, co-ed practice sessions focused on scoring goals. They were tremendous fun. I wish we had done even more of them.

Funding agencies as well as universities do not offer a place for any research too deeply involved with the ideas of science for it to fall under the heading of education and too deeply in an educational perspective for it to fall under the heading of science. It seems to be nobody's business to think in a fundamental way about science in relation to the way people think and learn it.

I didn't go to grad school, but I've read that for people pursuing Math PhD's, it's a jarring experience to transition from classes to research. In the first two years, they are rapidly acquiring vast amount of knowledge. Then, as they pursue novel research, the pace at which the make tangible progress slows to a crawl. This can feel very demotivating to some students.