Volume 5, Number 1 - Fall 1996
© Logo Foundation
You may copy and distribute this document for educational purposes provided that you do not charge for such copies and that this copyright notice is reproduced in full.
These are the major articles that appeared in Logo Update, Volume 5, Number 1 - Fall 1996. Minor changes have been made to update information such as addresses. Conference announcements and other items of transient interest that appeared in the original newsletter have not been included here.
Whatever Happened to the
Revolution? by Geraldine Kozberg
Understanding the Mathematics of Banking by Celia Hoyles and Richard Noss
In the halcyon days of the early 80s, the technology was new and euphoria high. Logo would transform our schools. From all over the world, the Logo community converged on MIT. Logo 84. Logo 85. Logo 86. We were ready for Star Wars.
At the final session of Logo 86, however, Brian Harvey and Bill Higginson stunned the audience with the question, "Whatever happened to the revolution?" Logo Exchange asked me to respond to Brian and Bill, an open letter of sorts. Here are parts of that letter:
Dear Brian and Bill,
Your presentation at Logo 86 was stunning. It was a piece of theater, a New Yorker postscript, a witty, sophisticated, calculated statement on education in general, and Logo in particular.
You speak of revolution. You ask, "Whatever happened to the revolution?" It depends. It depends on definitions and perspectives.
Nikolai Lenin: From a Lenin-like view of revolution, we know that we have the weapons. We know how to teach; we have the tools and the resources. We do not have what Lenin called an "intolerable level of discontent." People bitch a lot, but not enough to upset the status quo.
Martin Luther King: From the teaching and life of Martin Luther King, we know that revolutions are not created by the masses, rather by the small committed minority. We have a small committed minority within the Logo community. We do not have a critical mass.
Stephen Jay Gould: From the theoretical mind of Stephen Jay Gould, we understand cultural change in terms of "transformation." Change is preceded by a slow accumulation of stresses that the culture resists until a breaking point is reached, moving the culture to the next higher level of transformation.
Learning in a computer culture in 1986 is learning in a changing culture. I would say we are a culture in transition, not revolution. We experience resistance, but the more things change ... the more things change!
Logo teachers see change. What do they see? They "see" students thinking. The following learning behaviors there are eight were culled from teacher observations and anecdotal reports.
Students talk to themselves and to others. The act of languaging one's thoughts is a form of verbal mediation, a way of mediating and clarifying understanding.
Students move. Heads, hands, shoulders move with the turtle. Syntonic learning appears to be firmly related to how children process their worlds.
Students draw upon mental images. Memory is grounded in words and images. Visual imagery facilitates the storage and retrieval of information and instruction.
Students regulate their work. Casual explorations with the turtle give way to conscious control. Self-correction and evaluation are related to the sense of empowerment and control that we see in mature learners.
Students look for and create patterns. Pattern recognition is a basic strategy of organization and comprehension at the meta-cognitive state of learning awareness.
Students use Logo procedures as building blocks. The process of breaking problems into meaningful pieces of deconstruction has long been associated with formal problem solving.
Students compose and create. Artists describe the process of composition as impressionistic and generative. Ideas seemingly emerge, one idea triggering or guiding the next one, the subconscious driving the conscious.
Students present. Young people enter the adult world of presenters with confidence and poise. The work they present is their own.
These behaviors tell us something. Collectively, they give us an extraordinary new way of looking at learning. It may not be a revolution, but a new culture is emerging.
We call it a Logo culture. We are too close to it to fully understand it. The impact on learners and learning environments involves complex relationships. We have yet to tease out the nuances and subtleties. This is what I see:
Stronger Learners stronger in the sense of learners who use their intuition, who claim ownership of their learning, and are thus free to risk in order to learn more. I see students using learning modalities that are not tapped by the present curriculum.
Real Work a work intensity that is satisfying and worthwhile; a work ethic that encourages co-learning. Gender roles and age differences are blurred.
Expanded Vision in the shape of new visual forms that expand our vision and guide us toward a clearer understanding of our changing world. Throughout history, artists have operated at the edge of social and cultural change. The fluid rhythms and transformations of Logo images are visual expressions of a new aesthetic.
Thank you, Brian and Bill, for pushing us to new levels of consciousness and concern.
I leave you with the words of the poet, Robert Frost: "I bid you a one-man revolution."
Gerry Kozberg at a 1986 Logo Summer Institute in St. Paul, Minnesota
The above letter is dated August 10, 1986. Ten years have passed.
My enthusiasm and support for Logo has not diminished. I still see stronger learners, real work, and new visions, but the realities of our schools are harsh and compelling. There has been no revolution. The promise of systemic change is yet to come as the power of Logo waits its turn.
It is time for another letter. This one is addressed to you, the larger Logo community.
Whatever did happen to the revolution? The conditions for revolution were not present: There was no intolerable level of discontent, no critical mass, no breaking point in the larger culture; some cracks, perhaps, but no breaking point.
Revolution? In my school district we talk about test scores, violence, poverty, immigration. Technology? We talk about computers and the Internet. We buy computers and we inventory them.
The Logo community has been unable or unwilling to confront the larger social issues that are tearing at public education. In 1981, I wrote: "Logo is one part of a larger change effort designed to serve as an intervention in learning and learning environments."
For the most part, this has not happened. The problem is not the technology, certainly not Logo. The problem is one of equity. Logo is for all kids, but the kids who need Logo the most have no access to it. They are relegated to educational games and instruction in the basic skills.
Logo remains the province of a small, somewhat special group of schools, teachers, and students:
Logo is for all kids. That does not mean all kids are the same, or that the Logo experience is the same for all kids. Rather it talks to the power of Logo as a language of learning, an instructional process appropriate for a wide-range of learning behaviors.
Which brings me to my second point, learning and learning environments. We have not done a good job of helping teachers understand the learning process, not Logo, but learning itself. We have not paid serious attention to the social determinants of achievement, focusing instead on cognition and the technology. For some students, social and behavioral skills may be a pre-requisite to classroom participation and learning.
Papert taught that the context for human development is always a culture, never an isolated technology. School is a social institution and Logo understands the principles of learning within the culture of the school. Learning takes place in a social context. Learning is grounded in our social and cultural histories. Why, then, doesn't Logo learning transfer? Why doesn't the integrated learning of Logo pervade the culture of the school?
In Czarist Russia, Jewish children were not allowed to attend the public schools. When Marc Chagall was thirteen, his mother bribed a local school official and Chagall entered the secular, academic world. In his autobiography, Chagall remembers a wondrous time in his geometry class of "lines, angles, triangles, squares." They carried me far away to enchanting horizons. And during those hours of drawing, I lacked only a throne."
Piet Mondrian, the Dutch artist. When we examine the structural evolution of Mondrian's work, we begin to understand the deconstruction of the process into its most elemental forms. Reconstruction takes on different forms.
What could be more relevant for today's young people than the visual learning of Chagall, Mondrian, Logo?
The findings of Logo studies in Saint Paul suggest that improved achievement is more likely to occur at the lower end of the scale. Kids at the top already have their set of "gears" to work with. Kids at the bottom are still looking. The odds are that when they find their "gears," the learning process will be strengthened. It may be that Logo facilitates the development of natural learning systems, and especially for those youngsters who have not had the freedom to make learning connections in their own unique ways.
On Monday, we begin our 15th year of continuous Logo teacher education in Saint Paul. I hope the next fifteen years will concentrate on issues of equity and the dynamics of learning. Some will argue that I am asking too much of Logo. Not so. I would ask why have we denied access to Logo for significant populations? I would ask why have we lost the educational context to technocentric commercial interests?
Logo takes on social and political perspectives as schools search for solutions that will act upon the forces that create and maintain inequity in public education.
At the end of the year, we plan to open five Logo centers in Cambodia, a small country half way around the world, ravaged by war and social upheaval. Phnom Penh ... Battambang ... Siem Reap ... Takeo ... Ratanakeri. Strange and exotic sounds that translate into people and places.
Phnom Penh, a city of one million. The Logo center will be housed in a high school serving 6,500 students in three shifts.
Siem Reap, a rice village, site of the ancient ruins of Angkor Wat, on the edge of the jungle hiding Pol Pot and the Khmer Rouge guerrillas.
Ratanakiri, a remote mountainous province bordering Laos and Vietnam, accessible only by air. Electricity was installed last year.
Our challenge will be to apply the lessons of the first fifteen years: helping teachers understand the learning process, learning environments, and why Logo is for all kids.
I end this letter with a story: In old Cambodia, the people tell of three monks who were looking at a banner flying from the top of their temple. The first monk said, "The banner is moving." The second monk said, "No. The wind is moving." The third monk said, "It is the mind that is moving."
I heard this story in 1988 in a makeshift temple in a refugee camp on the Thai-Cambodian border. The image of "the moving mind" helped me transcend the suffering and deprivation of the camp, and moved me to focus on the wonder of the human mind and spirit. The mental image of the mind in motion has stayed with me. It is a fitting metaphor that helps me understand Logo and learning in a computer culture.
Gerry Kozberg may be contacted at:
740 River Drive
St. Paul, MN 55116
The Swiss Bank Corporation came to us in May 1994 with a problem. Many employees did not have a robust grasp of the mathematics unerpinning much of their work they had little "feel" for the mathematics which would enable them to appreciate the models underlying the financial instruments they were using and to recognize their limitations. More generally, there was a lack of a mathematical culture in the bank. Many felt uncomfortable with mathematical ideas, and would certainly be unlikely to talk about them, or to display their ignorance of them.
We had no previous contact with banking and finance. It seems that one of the senior managers had come across a review of our book*, which tells how programming computers can empower users (our "users" are mainly children) to learn mathematics. Our own work, and that of others in the research community, had made it painfully evident that many of those working in a range of commercial and industrial settings are often insufficiently qualified mathematically, and are sometimes unaware of the mathematics they encounter. Almost all this previous work has involved rather elementary mathematics (e.g. basic number operations). The situation described by the Swiss Bank was similar in essence but different in degree. It was similar in that it highlighted the problematic relationship between using a mathematical tool and understanding it; it differed in the content and the level of sophistication of the mathematical ideas involved.
We decided to accept the challenge to try to "do something" about the problem, and adopted an approach to teaching which uses Logo. We designed tasks around banking which exploited the folowing ideas:
Modelling: computer models of financial situations which encapsulate the relevant mathematical ideas, the "big" issues. Building models like this sowed seeds of familiarity with mathematical ideas by trying them out, and watching the feedback and in the process made them explicit, "friendly" and subject to change; and
Debugging: an idea familiar to programmers, but relatively uncommon in the teaching of mathematics. Trainees would construct their own programming models, observe how, for example, cash flows are calculated and be confronted with the consequences of any mathematical misunderstandings they might have built into their models.
We have summed up our approach as:
If you want to understand, teach someone else.
If you really want to understand, teach a computer.
We have used the modelling/debugging approach successfully with learners of all ages and experiences. We decided (maybe rather courageously) to treat bankers and infants alike although clearly the mathematical content would be rather different! Thus our attempt to help SBC was built on four foundations:
In a nutshell, we were not prepared to simply offer sets of "how to" rules and mathematical formulae in lecturing situations. The nature of the problem necessitated courses which were intensive, spread over time, and based on a substantial element of learner activity with computers.
We have developed two modules entitled Time Is Money and Rates and Change . The first deals with the mathematics of simple, discount, and compound interest a surprisingly rich and interesting area of mathematics, with plenty of scope for surprises and extensions. The second focusses on the growth of money, particularly bonds, in different situations. Each course comrpises six half days spread over five to six weeks.
We piloted Time is Money in November and Deccember 1994 and, having made modifications, repeated it several times during 1995. To date we have noticed a considerable range of background and expertise amongst the participants, but all felt the need to have a better grasp of the mathematical theory underlying the financial instruments. This is interesting in itself. Some simply "knew" what was the right way to approach tasks in the banking context but not why. Others had forgotten or even felt themselves to have been "deskilled" by their reliance on machines at work. Judging by the comments of participants, everybody learned from the course and felt more empowered and confident after the experience. We have come up with three Big Ideas which we intend to use as guiding principles in future developments in the Bank:
We view all financial instrumnents in terms of their common mathematical structure, that of the notion of mathematical function (an idea which is under-rated in schools, but turns out to be surprisingly accessible). This viewpoint provides the glue by which different financial instruments can be compared, contrasted, and modelled.
We provide the tools and windows the trainees extend them, exploit them for their own work and in their own projects, and reflect on the results of their application.
It is often said that to do mathematics, there is no room for intuition think of the expression "cold calculating." We disagree. We start from situations with which people are familiar (mostly personal finance) and use these as motivation for more formal, rigorous, and most importantly generalized mathematical relations through computer modelling.
We'll end with an example of the approach. We asked trainees a "simple" question: How is it that if 10% is added to a bill say £100 the answer is £110, yet reducing £110 by 10% does not leave £100? First we had better admit that not everyone saw we were right! But one trainee answered, somewhat disarmingly: "Yes. I've always wondered that too. I just think it's because numbers are funny!"
We will cut a long story short: this trainee concentrated hard on our activities, learned to write short but succinct programs, and above all, had fun with mathematical ideas. She reports that this was the first time she had ever connected fun with mathematics! On the way, she sorted out for herself how the simple percentage question worked, and most importantly felt proud of her discovery.
It's too early yet to know if this case is merely an isolated example, or a symptom of something more general, the start of building mathematical understandings in the bank which begins to change the mathematical culture of the bank. Only time will tell but the quotes below from some of the participants last year describing a good experience in the course give us grounds for optimism:
"The ability to predict the way a graph might look from a particular function. At last I nearly have a feel for numbers. I don't feel quite so "mathematically challenged"!
"Graphing the equations plugging in various values and visually interpreting the effects, seeing differences between simple and discount graphs. Mapping the math onto banking truisms."
"It was great remembering algebra. Plotting graphs to show the risk associated with a 10% movement in the underlying rate and discovering that over time there was a maximum movement, regardless of the size of the underlying rate it would be great to understand why!"
*Hoyles, Celia and Noss, Richard, editors, Learning Mathematics and Logo, 1992, MIT Press, Cambridge, MA
Exploring the 10% Problem with Logo
to 10%more :bill.amount
output :bill.amount + .1 * :bill.amount
print 10%more 100
print 10%more 50
to 10%less :bill.amount
output :bill.amount - .1 * :bill.amount
print 10%less 100
print 10%less 110
print 10%less 10%more 100
print 10%more 10%less 100
Celia Hoyles and Richard Noss may be contacted at:
Institute of Education
University of London
20 Bedford Way
London WC1H 0AL UK
The authors have written a more extensive piece on the same subject: "The Visibility of Meaning: Modeling the Mathematics of Banking," which appears in the International Journal of Computers for Math Education, Vol. 1 No. 1, 1996, Kluwer Academic Publishers, The Netherlands