The Case for Computing
By Gary S. Stager

A chapter from the book, Snapshots! Educational Insights from the Thornburg Center (2004)

The personal computer is the most powerful, expressive and flexible instrument ever invented. At its best, the PC offers learners a rich intellectual laboratory and vehicle for self-expression. Although computing has transformed nearly every aspect of society, schools remain relatively untouched.

This chapter is not about predicting the future. It is about the learning opportunities that exist today and may be overlooked. Computers and creativity are in dangerously short supply. The dearth of compelling models of using computers in deeper ways has created a vacuum now filled by a Dickensian approach to schooling.

When I read the growing mountain of educational technology standards I can’t help but wonder if these objectives could be satisfied without the use of a computer. The unimaginative use of school computers is symptomatic of larger crises in schooling, including what Seymour Papert calls, “idea aversion.” Over the past few decades I have enjoyed working at key moments in the intersection of learning and computers. My daily work is guided by an optimism rooted in experiences learning with computers and observing children doing the same. As much as this is the story of great promise and great disappointment, the children we serve sustain our enthusiasm to work harder to realize the learning potential of the digital age.

Ancient History – My Early Years of Computing

In 1976 I got to touch a computer for the first time. My junior high school (grades 6-8) had a mandatory computer-programming course for seventh and eighth graders. More than a quarter century ago, the Wayne Township Public Schools in New Jersey thought it was important for all kids to have experience programming computers. There was never any discussion of preparation for computing careers, school-to-work, presentation graphics or computer literacy. Computer programming was viewed as a window onto a world of ideas given equal status as industrial arts, music appreciation, art and oral communications.

The scarcity of classroom computers made programming a highly social activity since we were often leaning over each other’s shoulders in order to get in on the action.

Mr. Jones, the computer programming teacher, was scary in a Dr. Frankenstein sort of way. However, I was attracted by the realization that this guy could make computers do things!

Mr. Jones knew how elaborate computer games worked and would show us the code afterschool if we were interested. Once I understood how to read a computer program, I could THINK LIKE THE COMPUTER! This made me feel powerful.

The feelings of intellectual elation I experienced programming are indescribable. The computer amplified my thinking. I could start with the germ of an idea and through incremental success and debugging challenges build something more sophisticated than I could have ever imagined.

The self-awareness that I was a competent thinker helped me survive the indignities of high school mathematics classes. Mr. Jones helped me learn to think like a computer. The ability to visualize divergent paths, anticipate bugs, and rapidly test mental scenarios is the direct result of computer programming. This gift serves me in everyday life when I need hack my way through a voicemail system to reach a knowledgeable human, or get my car out a locked parking structure.

Perhaps Mr. Jones was such a great teacher because he was learning to program too – maybe just slightly ahead of us. (This never occurred to me as a kid since Mr. Jones knew everything about computers.)

A strong community of practice emerged in the high school computer room. We learned from each other, challenged one another and played with each other’s programs. We altered timeshare games, added ways to cheat and programmed cheap tricks designed to shock classmates. I even ran after school classes in BASIC for kids interested in learning to program.

Computers were to be used to make things at my high school, not as a subject of study. There was never a mention of computer literacy and owning a computer was unthinkable. The school computers were a place to lose our selves in powerful ideas.

We never saw a manual for a piece of software although we treasured every issue of Creative Computing – working hard to meticulously enter hundreds of lines of computer code only to have every single program be buggy. Since we had little idea what was impossible, we thought anything was possible. We felt smart, powerful and creative. We took Fortran manuals out of the public library for no other reason than to hold a connection to a larger world of computing – a world we were inventing for ourselves.

Bill Gates and Steve Wozniak, were involved in similar little ventures at the time. Many of the computing visionaries who changed the world had similar early experiences with computers. I remember the explosion of thinking and creativity I experienced programming computers and try to recreate the spirit of that computer-rich learning culture in every school I visit. Kids deserve no less.

In the mid-80s I was welcomed into the global “Logo community” and asked to present papers at places like MIT. This was pretty heady stuff for a failed trumpet player and mediocre student. Logo programming offered a vehicle for sharing my talents, expressing my creativity and engaging in powerful ideas with some of the leading thinkers in education. Seymour Papert’s scholarship gave voice to my intuitions visa-a-vis the tension between schooling and learning.

To this day, my work with adults and kids is centered around using computers as intellectual laboratories and vehicles for self-expression. To experience the full power of computing, the tools need to be flexible extensible and transparent. The user needs to be fluent in the grammar of the system whether it is text based, symbolic or gestural.

Laptops

In 1989, Methodist Ladies’ College, an Australian PK-12 school already recognized for its world-class music education, committed to every student having a personal laptop computer. By the time I began working with MLC a year later, 5th and 7th graders were required to own a laptop. The “P” in PC was taken very seriously. Personal computing would not only solve the obvious problems of student access, low levels of faculty fluency and the costs associated with the construction of computer labs – the PC would embody the wisdom of Dewey, Vygotsky and Piaget. Logo, because of its open-endedness and cross-curricular potential, was the software platform chosen for student learning. The potential of Logo as a learning environment that would grow with students across disciplines and grade levels could only be realized with access to ubiquitous hardware. This justified the investment in laptops.

MLC principal, David Loader, understood that the personal was at the core of any efforts to make his school more learner-centered. He was not shy in his desire to radically reinvent his school. Bold new thinking, epistemological breakthroughs, sensitivity to a plurality of learning styles, increased collaboration (among teachers and children) and student self-reliance were expected outcomes of the high-tech investment. Teachers learning to not only use, but program, computers would acquaint themselves with the type of “hard fun” envisioned for student learning.

If the computer were to play a catalytic role in this educational shift, it was obvious that the computers needed to be personal. Truly creative and intellectual work requires freedom and a respect for privacy. Quality work is contingent on sufficient time to think, to experiment, to play. The laptop can only become an extension of the child when it is available at all times. Therefore, there was never any debate about laptops going home with students. Time and time again, the most interesting work was accomplished during the student’s personal time.

Laptops were a way to enable student programming “around the clock” and make constructionism concrete.

MLC was a magical place during the early nineties. Every aspect of schooling was open for discussion and reconsideration.

When I expressed concern over the gap between classroom reality and the rhetoric proclaiming the school’s commitment to constructionism, the principal supported my desire to take dozens of teachers away for intensive residential professional development sessions. After all, constructionism is something you DO as well as believe. You cannot be a constructionist who subcontracts the construction. “Do as I say, not as I do,” would no longer cut it.

A renaissance of learning and teaching catapulted MLC and the subsequent Australian “laptop schools” to the attention of school reformers around the world.

We were ecstatic when “laptop” students began to adorn their computers with their names written in glitter paint. This signaled appropriation. The computers mattered. Success.

The early success of MLC and the many other “laptop schools” to follow were a realization of the dream Seymour Papert and Alan Kay held for decades. In 1968, computer scientist Alan Kay visited Seymour Papert at MIT. Papert, a protégé of Jean Piaget, a mathematician and artificial intelligence pioneer was combining his interests by designing computing environments in which children could learn. Kay was so impressed by how children in Papert’s Logo Lab were learning meaningful mathematics that he sketched the Dynabook, a dream of portable computers yet to be fully realized, on the flight home to Xerox PARC, a leading high-tech thinktank.

Kay set out to design a portable personal computer for children on which complex ideas could come alive through the construction of simulations. Dr. Kay recently remembered this time by saying,  “More and more, I was thinking of the computer not just as hardware and software but as a medium through which you could communicate important things. Before I got involved with computers I had made a living teaching guitar. I was thinking about the aesthetic relationship people have with their musical instruments and the phrase popped into my mind: an instrument whose music is ideas.”

Kay’s poetic vision resonated with my memories of Mr. Jones, summer camp and my own experiences programming in Logo.

“One of the problems with the way computers are used in education is that they are most often just an extension of this idea that learning means just learning accepted facts. But what really interests me is using computers to transmit ideas, points of view, ways of thinking. You don’t need a computer for this, but just as with a musical instrument, once you get onto this way of using them, then the computer is a great amplifier for learning.”

At-risk and high tech

For three years, beginning in 1999, I worked with Seymour Papert to develop a high-tech alternative learning environment, the Constructionist Learning Laboratory (CCL), inside the Maine Youth Center, the state facility for adjudicated teens. This multiage environment provided each student with a personal computer and access to a variety of constructive material. The experience of trying to reacquaint or acquaint these previously unsuccessful students with the learning process teaches us many lessons about just how at-risk our entire educational system has become.

The intent of the project was to create a rich constructionist learning environment in which severely at-risk students could be engaged in long-term projects based on personal interest, expertise and experience. Students used computational technologies, programmable LEGO and more traditional materials to construct knowledge through the act of creating a personally meaningful project. The hypothesis was that the constructionist philosophy offers students better opportunities to learn and engage in personally meaningful intellectual development. The computer was the magic carpet that would allow these children to escape their history of school failure.

Students in this alternative learning environment routinely suffered from what Seymour Papert called,“the curious epidemic of learning disabilities.” Kids with low or non-existent literacy skills were able to invent and program robots capable of making decisions and interacting with their environment. Robo Sumo wrestlers, interactive gingerbread houses, card dealing robots, luggage sorting systems and temperature-sensitive vending machines capable of charging a customer more money on hot humid days were but a few of the ingenious inventions constructed with programmable LEGO materials. Students also designed their own videogames, made movies and explored the universe via computer-controlled microscopes and telescopes. They wrote sequels to Othello and published articles in programming journals. These kids proved that computing offered productive learning opportunities for all kinds of minds.

One child, said to be completely illiterate, wrote a page of program code the night before class because an idea was burning inside of him. Another “illiterate” youngster, incarcerated for more than half of his life, was capable of building dozens of mechanisms in the blink of an eye and installing complex software. His ability to program complicated robots presented clues about his true abilities. A week before he left the facility, this child, so accustomed to school failure, sat down and typed a 12,000-word autobiography.

Tony’s adventure is also a tale worth telling. He had not been in school since the seventh grade and indicated that none of his peer group attended school past the age of twelve or thirteen. In the CLL he fell in love with robotics and photography at the age of seventeen.

During the spring of 2001, the MYC campus was populated with groundhog holes. To most kids these familiar signs of spring went unnoticed, but not for the “new” Tony.

Tony and his new assistant, “Craig,” spent the next few weeks building a series of what came to be known as “Gopher-cams.” This work captured the imagination of the entire Maine Youth Center. Tony and Craig learned a great deal about how simple unanticipated obstacles like a twig could derail days of planning and require new programming or engineering. These students engaged in a process of exploration not unlike the men who sailed the high seas or landed on the moon. While they never really found out what was down the hole, they learned many much more important lessons.

Robotics gives life to engineering, mathematics and computer science in a tactile form. It is a concrete manifestation of problem solving that rewards debugging, ingenuity and persistence. The LEGO robotic materials promote improvisational thinking, allowing even young children to build a machine, test a hypothesis, tinker, debug, and exceed their own expectations.  As often experienced in programming, every incremental success leads to a larger question or the construction of a bigger theory.  This dialogue with the machine amplifies and mediates a conversation with self.

Digital technology is a critical variable in the transformation of reluctant learners. Self-esteem, or even academic grades, might have been enhanced through traditional activities. However, the availability of computationally-rich construction materials afforded the learners the opportunity to experience the empowerment associated with the feeling of wonderful ideas. For the first time in their lives, these children experienced what it felt like to be engaged in intellectual work. This feeling required a personal sustained relationship with the computer and computationally-rich objects to think with such as LEGO and MicroWorlds. All students deserve the chance to make important contributions to the world of ideas, and must be given the means to do so.

State of the art?

Much needs to be done to ensure that all students enjoy the quality of experience offered by the best laptop schools, online environments and the CLL.

Somewhere along the line, the dreams of Kay, Papert and Loader were diluted by the inertia of school. Detours along the road to the Dynabook were paved by the emergence of the Internet and corporate interest in the laptop miracle.

Until the explosion of interest in the Internet and Web, individual laptops offered a relatively low-cost decentralized way to increase access to computers and rich learning opportunities. The Net, however, required these machines to be tethered to centralized servers and an educational bureaucracy pleased with its newfound control. Computing costs soared, data and children were either menaced or menaces. Jobs needed to be protected. The desires of the many often trumped the needs of the learner.

Microsoft generously offered to bring the laptop message to American schools, but their promotional videos pushed desks back into rows and teachers stood at the front of classrooms directing their students to use Excel to calculate the perimeter of a rectangle. Over emphasis on clerical “business” applications – were manifest in elaborate projects designed to justify (shoehorn) the use of Excel or Powerpoint in an unchanged curriculum. Many of these projects have the dubious distinction of being mechanically impressive while educationally pointless. Our gullible embrace of false complexity increases as the work is projected in a darkened classroom.

I’ve developed Murray’s Law to describe the way in which many schools assimilate powerful technology. “Every 18 months schools will purchase computers with twice the processing power of today, and do things twice as trivial with those computers.”

There is a fundamental difference between technology and computing, which can be seen in the words themselves. One is a noun, the other a verb, What we saw students do with technology at the CCL was active, engaged, compelling, sophisticated learning.  They were computing, and similar experiences for all students can transform the experience of school.

What are you really saying?

I know that many of you must be thinking, “Does Gary really believe that everyone should be a programmer?” My answer is, “No, but every child should experience the opportunity to program a computer during her K-12 education.” Critics of my position will say things like, “Not every person needs to program or will even like it.” To these people I suggest that not every kid needs to learn to write haiku or sand a tie rack in woodshop. However, we require millions of children to do so because we believe it is either rewarding, of cultural value or offers a window onto potential forms of human expression.

Despite our high-tech society’s infinite dependence on programming and the impressive rewards for computing innovation, many people find the notion of programming repulsive. Everyone wants their child to earn Bill Gates’ money, but only if they never have to cut a line of code. Educators especially need to get past this hysteria rooted in fear and ignorance for the sake of the children in our care. (this sentence is optional if you feel it is inflammatory)

I do not understand why anyone would question the value of offering programming experiences to children.

It is unseemly for schools to determine that a tiny fraction of the student population is capable of using computers in an intellectually rich way. The “drill for the test” curriculum of the A.P. Computer Science course serves only a few of the most technically sophisticated students. That is elitism.

Children enjoy programming when engaged in a supportive environment. The study of other disciplines may be enhanced through the ability to concretize the formal. For example, complex mathematical concepts become understandable through playful manipulation, graphical expression of abstractions or the application of those concepts in service of a personal goal. It would be difficult to argue that mathematics education, at the very least, would not be enriched through programming.

Schools need to make a sufficient number of computers with powerful software available for the transparent use of every child across all disciplines. Schools also have an obligation to offer a more inclusive selection of courses designed for a more diverse student body interested in learning with and about computers. Courses in software design, digital communication, robotics, or computer science are but a few options. The Generation Y program, in which students lend their technological expertise to teachers who want to integrate technology into their lessons provides another outlet for authentic practice.

Whither computing?

I wonder when the educational computing community decided to replace the word. computing, with technologyThe Computing Teacher became Learning and Leading with TechnologyClassroom Computer Learning begot Technology and Learning Magazine. Conference speakers began diminishing the power of the computer by lumping all sorts of objects into the catch-all of technology. Computers are in fact a technology, but they are now spoken of in the same breath as the blackboard, chalk, filmstrip projector or Waterpik. Computing was never to be mentioned again in polite company.

I recently read the conference program for a 1985 educational computing conference. The topics of discussion and sessions offered are virtually the same as at similar events today. The only difference is that all mentions of programming have disappeared from the marketplace of ideas.

It seems ironic that educators fond of reciting how kids know so much about computers act as if the computer was just invented. We should be unimpressed by breathless tales of children web surfing or using a word processor to write a school report. My standards are much higher. We will cheat a second generation of microcomputer-age students if we do not raise our game and acknowledge that so much more is possible.

If we concur that kids are at least comfortable with computers, if not fluent, then teachers have a responsibility to build on the fluency of computer-savvy kids. This is a classroom gift, like an early reader, a natural soprano or a six year-old dinosaur expert. It is incumbent on schools and their personnel to steer such students in more challenging and productive directions. Teachers have an obligation to respect the talents, experience and knowledge of students by creating authentic opportunities for growth.

If the youngest children can “play” doctor, lawyer, teacher or fireman, why can’t they imagine themselves as software designers? Open-ended software construction environments designed for children, like MicroWorlds, make it possible for children of all ages to view themselves as competent and creative producers of knowledge. Too few students know that such accomplishments are within reach.  This failure results from a leadership, vision, and professional knowledge deficit.

While school computing fades from memory, keyboarding instruction inexplicably remains a K-12 staple from coast to coast. Computer assisted instruction, schemes designed to reduce reading to a high-stakes race and low-level technical skills dominate the use of computers in schools. In the hands of a clever curriculum committee, “uses scroll bars” can be part of a nine-year scope and sequence.

Examples of kids composing music, constructing robots, or designing their own simulations are too hard to find. More than a quarter century has passed since Mr. Jones taught me to program. Yet, children in that school are now compelled to complete a keyboarding class. There can be no rational justification for so blatant a dumbing-down of the curriculum.

Computing Changes Everything

There are so many ways in which children may use computers in authentic ways. Low-cost MIDI software and hardware offers even young children a vehicle for musical composition. The 1990 NCTM Standards indicated that fifty percent of mathematics has been invented since World War II. This mathematics is visual, experimental and rooted in computing. It may even engage kids in the beauty, function and magic of mathematics.

In Seeing in the Dark: How Backyard Stargazers Are Probing Deep Space and Guarding Earth from Interplanetary Peril, author Timothy Ferris describes how amateur astronomers armed with telescopes, computers and Net connections are making substantive contributions to the field of astronomy. For the first time in history, children possess the necessary tools to be scientists and to engage in scientific communities.

MacArthur Genius Stephen Wolfram has written a revolutionary new 1,280 page book, A New Kind of Science. The book illustrates his theory that the universe and countless other disciplines may be reduced to a simple algorithm. Scientists agree that if just a few percent of Wolfram’s theories are true, much of what we thought we knew could be wrong and many other cosmic mysteries may be solved. Wolfram believes that a human being is no more intelligent than a cloud and both may be created with a simple computer program.

A New Kind of Science starts with very a big bang.

“Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world. My purpose in this book is to initiate another such transformation, and to introduce a new kind of science that is based on the much more general types of rules that can be embodied in simple computer programs.”

You do not have to take Wolfram’s word for it. With the $65 A New Kind of Science Explorer software, you and your students can explore more than 450 of Wolfram’s experiments. The visual nature of cellular automata – the marriage of science, computer graphics and mathematics – allows children to play on the frontiers of scientific thought while trying to prove, disprove or extend the theories of one of the world’s greatest scientists. The intellectual habits required to “think with” this tool are rooted in computer programming.

I recently told Alan Kay that while I was hardly a mathematician, I knew what it felt like to have a mathematical idea. He generously replied, “Then you are a mathematician, you’re just not a professional.” The work of Seymour Papert shows us that through the explicit act of computing children can too be mathematicians and scientists.

“If you can use technology to make things you can make a lot more interesting things. And you can learn a lot more by making them. …We are entering a digital world where knowing about digital technology is as important as reading and writing.  So learning about computers is essential for our students’ futures BUT the most important purpose is using them NOW to learn about everything else. “ (Papert 1999)

We can neutralize our critics and improve the lives of kids if we shift our focus towards using school computers for the purpose of constructing knowledge through the explicit act of making things – including: robots, music compositions, digital movies, streaming radio and simulations. Children engaged in thoughtful projects might impress citizens desperate for academic rigor. Examples of competent children computing bring many current educational practices into question. Emphasizing the use of computers to make things will make life easier for teachers, more exciting for learners and lead schools into what should be education’s golden age.

SIDEBAR

Why Should Schools Compute?

Computing offers an authentic context for doing & making mathematics
Traditional arithmetic and mathematical processes are provided with a genuine context for use. New forms of mathematics become accessible to learners.

Computing concretizes the abstract
Formal concepts like feedback, variables and causality become concrete through use.

Computing offers new avenues for creative expression
Computing makes forms of visual art and music composition possible for even young children while providing a canvas for the exploration of new art forms like animation. A limitless audience is now possible.

Computer science is a legitimate science
Computer science plays a revolutionary role in society and in every other science. It should be studied alongside biology, physics and chemistry.

Computing supports a plurality of learning styles
There are many ways to approach a problem and express a solution.

Computing offers preparation for a plethora of careers
There is a shortage of competent high-tech professionals in our economy

Computing grants agency to the user, not the computer
Rather than the computer programming the child, the child can control the computer.

Debugging offers ongoing opportunities to enhance problem-solving skills
Nothing works correctly the first time. The immediacy of concrete feedback makes debugging a skill that will serve learners for a lifetime.

Computing rewards habits of mind such as persistence, curiosity and perspective
Computers mediate a conversation with self in which constant feedback and incremental success propels learners to achieve beyond their expectations.


References

Cavallo, D. (1999) “Project Lighthouse in Thailand: Guiding Pathways to Powerful Learning.” In Logo Philosophy and Implementation. Montreal, Canada: LCSI.

Duckworth, E. (1996) The Having of Wonderful Ideas and Other Essays on Teaching and Learning. NY: Teachers College Press.

Ferris, T. (2002) Seeing in the Dark: How Backyard Stargazers Are Probing Deep Space and Guarding Earth from Interplanetary Peril. NY: Simon and Schuster.

Harel, I., and Papert, S., eds. (1991) Constructionism. Norwood, NJ: Ablex Publishing.

Kafai, Y., and Resnick, M., eds. (1996) Constructionism in Practice: Designing, Thinking, and Learning in a Digital World. Mahwah, NJ: Lawrence Erlbaum.

Levy, S. (2002) The Man Who Cracked the Code to Everything.Wired Magazine. Volume 10, Issue 6. June 2002.

Papert, S. (1980) Mindstorms: Children, Computers, and Powerful Ideas. New York: Basic Books.

Papert, S. (1990) “A Critique of Technocentrism in Thinking About the School of the Future,” MIT Epistemology and Learning Memo No. 2. Cambridge, Massachusetts: Massachusetts Institute of Technology Media Laboratory.

Papert, S. (1991) “Situating Constructionism.” In Constructionism, in  Harel, I., and Papert, S., eds. Norwood, NJ: Ablex Publishing.

Papert, S. (1993) The Children’s Machine: Rethinking School in the Age of the Computer. New York: Basic Books.

Papert, S. (1996) The Connected Family. Atlanta: Longstreet Publishing.

Papert, S. (1999) “The Eight Big Ideas of the Constructionist Learning Laboratory.” Unpublished internal document. South Portland, Maine.

Papert, S. (1999) “What is Logo? Who Needs it?” In Logo Philosophy and Implementation. Montreal, Canada: LCSI.

Papert, S. (2000) “What’s the Big Idea? Steps toward a pedagogy of idea power.” IBM Systems Journal, Vol. 39, Nos 3&4, 2000.

Resnick, M., and Ocko, S. (1991) “LEGO/Logo: Learning Through and About Design.” In Constructionism, in  Harel, I., and Papert, S., eds. Norwood, NJ: Ablex Publishing.

Stager, G. (2000) “Dream Bigger” in Little, J. and Dixon, B. (eds.) Transforming Learning: An Anthology of Miracles in Technology-Rich Classrooms. Melbourne, Australia: Kids Technology Foundation.

Stager, G. (2001) “Computationally-Rich Constructionism and At-Risk Learners.” Presented at the World Conference on Computers in Education. Copenhagen.

Stager, G. (2002) “Papertian Constructionism and At-Risk Learners.” Presented at the National Educational Computing Conference. San Antonio.

“The Dynabook Revisted” from the website, The Book and the Computer: exploring the future of the printed word in the digital age. (n.d.) Retrieved January 20, 2003 from http://www.honco.net/os/kay.html

Thornburg, D. (1984) Exploring Logo Without a Computer. Menlo Park, CA: Addison-Wesley.

Thornburg, D. (1986) Beyond Turtle Graphics: Further Explorations of Logo. Menlo Park, CA: Addison-Wesley.

Turkle, S. (1991) “Epistemological Pluralism and the Revaluation of the Concrete.” In Constructionism. Idit Harel and Seymour Papert (eds.), Norwood, NJ: Ablex Publishing.

Wolfram, S. (2002) A New Kind of Science. Champaign, IL: Wolfram Media, Inc.

“The Dynabook Revisted” from the website, The Book and the Computer: exploring the future of the printed word in the digital age. (n.d.) Retrieved January 20, 2003 from http://www.honco.net/os/kay.html.

ibid…

Following my presentation at the March ASCD National Conference, Sarah McKibben of ASCD interviewed me for an article, If You Build It: Tinkering with the Maker Mind-Set, published in the June 2014 issue of ASCD Education Update.

As is often the case, just a few of my comments made it into the final publication. Since I responded to a number of interview questions via email, I am publishing my full interview here.  The questions posed are in green.

How would you define making? I talked to Steve Davee at the Maker Education Initiative, and he says that making is more of a mind-set. “Where things that are created by people are recognized, celebrated, and there’s a common interdisciplinary thread.”  Would you agree?

I like to say that the best makerspace is between your ears. I agree that it’s a stance that prepares learners to solve problems their teachers could never have predicted with a strong sense of confidence and competence, even if only to discover that there is much more to learn.

Seymour Papert calls the learning theory underlying the current interest in “making,” constructionism. He asserts that learn best occurs when the learner is engaged in the process of constructing something shareable.

In our book, we argue that my friend and mentor Papert, is the father the maker movement as well as educational computing.

In a webinar on your website, Sylvia Martinez said that with making, assessment is intrinsic within the materials.” That it’s more “organic, formative, and internally motivated.” If you’re working with a material like cardboard, without any technology involved (and you can’t base success on something lighting up), how do you assess learning?

First of all, it would be best to take a deep breath and not worry about assessing everything. All assessment interrupts the learning process. Even just asking, “Hey, whatcha doing?” interrupts the learning process. It is up to reasonable adults to determine an acceptable degree of interruption. Perhaps building stuff out of cardboard is just fun.

The best problems and projects push up against the persistence of reality. One could observe a student’s habits of mind. Speak with them about her goals and what she has accomplished. One could imagine thinking about the understanding of physics involved in building a structure, understanding of history in their cardboard Trojan horse, or storytelling ability.

There isn’t anything magical about technology when it comes to a teacher understanding the thinking of each student. That said, we find over and over again that in productive learning environments, kids may combine media, like cardboard, lights, and microcontrollers in interesting and unpredictable ways. The computer is part of an expansive continuum of constructive material.

It seems that there’s a wide gamut of materials in making. From cardboard to Arduinos to expensive laser cutters. You mentioned in a presentation, something about “low threshold, high-ceiling materials.” Can you describe what you mean?

Sure, Tinkering and engineering requires a dialogue with materials in which it is possible for young or inexperienced users to enjoy immediate feedback so they continue to grow as fluency increases. Think of paint and brushes in that context or programming languages, such as Scratch or MicroWorlds. Like with LEGO, simple elements or tools may be used to create infinite complexity and expressiveness.

Can you give me an example of how, for instance, a high school English teacher might bring making into the classroom?

Making real things that matter with a real potential audience. Kids should write plays, poems, newspaper articles, petitions, manuals, plus make films, compose music, etc…  We need to stop forcing kids to make PowerPoint presentations on topics they don’t care about for audiences they will never encounter. Kids have stories to tell. They should act, write, sing, dance, film those stories AND learn to write the sort of scientific, technical and persuasive writing that nearly every career demands.

At our Constructing Modern Knowledge summer institute, middle school humanities teacher, Kate Tabor of Chicago, used MicroWorlds to “make” the computer generate random Elizabethan insults. Teachers have used versions of Logo for decades to explore grammatical structure and conjugation rules by writing computer programs to generate random poetry or create the plural possessive form of a word.

Steve Davee also mentioned that a key to successful making in schools is to empower students to become the experts–to learn how to use a 3d printer on their own, for example, and to share that knowledge with others. He said that when a teacher has to be involved with a technology or material, it creates a “creative bottleneck.” On the other hand, you’ve mentioned that teachers need to tap into their own expertise to guide students. Can these two approaches coexist peacefully?

Kids are competent. I believe that teachers are competent too. I find it unfortunate that so many educators behave as if teachers are incapable of adapting to modernity.

There is a fundamental difference in stance between assuming that as a teacher I know everything as a fountain of knowledge and that the kids are smarter than me. There may be a “creative bottleneck,” but giving up on teachers or schools is an unacceptable capitulation.

Great things are possible when the teacher gets out of the way, but even greater possibilities exist when the teacher is knowledgeable and has experience they can call upon to help a kid solve a tough problem, connect with an expert, or toss in a well-timed obstacle that will cause the student encounter a powerful idea at just the right teachable moment.

Each year, teachers at Constructing Modern Knowledge construct projects that two years ago would have earned them a TED Talk and five years ago, a Ph.D. in engineering, and yet so much teacher PD is focused on compliance, textbook page turning or learning to “use the Google.”

How does making align with Piaget’s understanding, as you’ve mentioned, that knowledge is a consequence of experience?

Piaget said that knowledge is a consequence of experience. Papert said, “If you can make things with computers, then you can make a lot more interesting things and you can learn more by making them.” Both ideas serve as strong justification for making.

In a webinar, Sylvia Martinez mentioned that instead of looking at standards and creating projects around them, teachers might work backward by creating an educational experience, then filling in the standards. Do you agree with this approach? How would this look with making?

I agree with Papert that at best school teaches a billionth of a percent of the knowledge in the universe yet our entire educational system is hell-bent on arguing endlessly over which 1 billionth of a percent is important. As an educator, my primary responsibility is create a productive context for learning that democratizes access to experience and expertise while doing everything I can to make private thinking public in order to ready the environment for the student’s next intellectual development. Making is wholly consistent with this view.

As we have mechanized and standardized teaching over the past generation, teachers have been deprived of experience in thinking about thinking. Their agency has been robbed by scripted curricula, test-prep, the Common Core, and other nonsense I believe to be on the wrong side of history. As a result, they can’t help but become less thoughtful in their practice. My work is concerned with creating experiences during which teachers become reacquainted with learning in order to become more sensitive to the individual needs, passions, talents, and expertise of each student. The emerging tools of the Maker Movement provide an exciting basis for such experiences.

As I said at ASCD, you can’t teach 21st Century learners, if you haven’t learned this century.

The future viability of public education is dependent on a system of creative competent educators trusted to provide rich learning experiences for children.

All of my friends know I have serious reservations about smarmy self-important libertarianism of TED and loathe speaking in the format – essentially a television program without any of the accoutrements of a television studio. That said, I’ve now performed three of them.

My first TEDx Talk made me ill for months before and weeks following the talk. The pressure was unbearable. You see, I wanted to go viral and become a millionaire – an overnight sensation like that guy who has taken such a courageous stance for creativity. The clock got me and I left half of my prepared thoughts on the cutting room floor. That said, people seem to like the talk anyway. For that I am grateful.

My first TED experience was so unpleasant that I sought an opportunity to try it again. This time, I promised myself that I would not stress out or over plan. That strategy paid off and the experience was a lot less traumatic. The only problem is that the venue audio was a disaster and I’m yelling through the entire talk. Don’t worry. I won’t be yelling when I publish a print anthology of these performances.

In March, I was invited by my longtime client, The American School of Bombay, to do another TEDx Talk. I assembled my vast team of advisors and brainstormed how I could turn this talk into riches beyond my wildest dreams. I quickly abandoned that idea and decided to use the occasion to honor my dear friend, mentor, and colleague, Dr. Seymour Papert in a talk I called, “Seymour Papert – Inventor of Everything*

I hope you enjoy it (or at least learn something before I lose another game of Beat the Clock)! Please share, tweet, reload the page 24/7! I have not yet given up on becoming an overnight sensation.

2014 – Seymour Papert – Inventor of Everything*

2013 – We Know What to Do

2011 – Reform™

 

I am always looking for ways to help teachers be more intentional and create deeper learning experiences for their students. Today, through the haze of Bombay Belly, I had an epiphany that may help you in similar learning situations.

Authentic project-based learning is in my humble opinion incompatible with curricular tricks like, Understanding by Design, where an adult determines what a children should know or do and then gives the illusion of freedom while kids strive to match the curriculum author’s expectation.

I view curriculum as the buoy, not the boat and find that a good idea is worth 1,000 benchmarks and standards.

Whether you agree with me or not, please consider my new strategy for encouraging richer classroom learning. I call it, “…and then?”

It goes something like this. Whenever a teacher asks a kid or group of kids to participate in some activity or engage in a project, ask, “..and then?” Try asking yourself, “..and then?” while you teach.

For example, when the kindergarten teacher has every child make a paper turkey or a cardboard clock, ask, “…and then?” This is like an improvisational game that encourages/requires teachers to extend the activity “that much” further.

You ask first graders to invent musical instrument. Rather than being content with the inventions, ask, “…and then?” You might then decide to:

  1. Ask each kid to compose a song to be played on their instrument
  2. Teach their song to a friend to play on their invented instrument
  3. The next day ask the kids to play the song they were taught yesterday from memory
  4. When they can’t remember how, you might ask each “composer” to write down the song so other players can remember it
  5. This leads to the invention of notational forms which can be compared and contrasted for efficacy or efficiency. This invention of notation leads to powerful ideas across multiple disciplines.

I think, “…and then?,” has application at any age and across any subject area.

Try it for yourself and let me know what you think!

Candidly, I have not been enthusiastic about teaching “computational thinking” to kids. In nearly every case, computational thinking seemed to be a dodge intended to avoid computing, specifically computer programming.

“There is no expedient to which a man will not resort to avoid the real labor of thinking.”

(Sir Joshua Reynolds)

Programming is an incredibly powerful context for learning mathematics while engaged in being a mathematician. If mathematics is a way of making sense of the world, computing is a great way to make mathematics.

Most of the examples of computational thinking I’ve come across seemed like a cross between “Computer Appreciation” and “Math Appreciation.” However, since smart people were taking “computational thinking” more seriously, I spent a great deal of time thinking about a legitimate case for it in the education of young people.

Here it is…

Computational thinking is useful when modeling a system or complex problem is possible, but the programming is too difficult.

Examples will be shared in other venues.

“Young people have a remarkable capacity for intensity….”

Those words, uttered by one of America’s leading public intellectuals, Dr. Leon Botstein, President of Bard College, has driven my work for the past six or seven years. It is incumbent on every educator, parent, and citizen to build upon each kid’s capacity for intensity otherwise it manifests itself as boredom, misbehavior, ennui, or perhaps worst of all, wasted potential.

Schools need to raise the intensity level of their classrooms!

However, intensity is NOT the same as chaos. Schools don’t need any help with chaos. That they’ve cornered the market on.

capacity500
Anyone who has seen me speak is familiar with this photograph (above). It was taken around 1992 or 1993 at Glamorgan (now Toorak) the primary school campus of Geolong Grammar school in Melbourne, Australia. The kids were using their laptops to program in LogoWriter, a predecessor to MicroWorlds or Scratch.

I love this photo because in the time that elapsed between hitting the space bar and awaiting the result to appear on the screen, every ounce of the kid’s being was mobilized in anticipation of the result. He was literally shaking,

Moments after that image was captured, something occurred that has been repeated innumerable times ever since. Almost without exception, when a kid I’m teaching demonstrates a magnificent fireball of intensity, a teacher takes me aside to whisper some variation of, “that kid isn’t really good at school.”

No kidding? Could that possibly be due to an intensity mismatch between the eager clever child and her classroom?

I enjoy the great privilege of working in classrooms PK-12 all over the world on a regular basis. This allows me observe patterns, identify trends, and form hypotheses like the one about a mismatch in intensity. The purpose of my work in classrooms is to model for teachers what’s possible. When they see through the eyes, hands, and sometimes screens of their students, they may gain fresh perspectives on how things need not be as they seem.

Over four days last month, I taught more than 500 kids I never met before to program in Turtle Art and MicroWorlds EX. I enter each classroom conveying a message of, “I’m Gary. We’ve got stuff to do.” I greet each kid with an open heart and belief in their competence, unencumbered by their cumulative file, IEP, social status, or popularity. In every single instance, kids became lost in their work often for several times longer than a standard class period, without direct instruction, or a single  disciplinary incident. No shushing, yelling, time-outs, threats, rewards, or other behavioral management are needed. I have long maintained that classroom management techniques are only necessary if you feel compelled to manage a classroom.

In nearly every class I work with – anywhere, teachers take me aside to remark about how at least one kid shone brilliantly despite being a difficult or at-risk student. This no longer surprises me.

In one particular class, a kid quickly caught my eye due to his enthusiasm for programming. The kid took my two minute introduction to the programming language and set himself a challenge instantly. I then suggested a more complex variation. He followed with another idea before commandeering the computer on the teacher’s desk and connected to the projector in order to give an impromptu tutorial for classmates struggling with an elusive concept he observed while working on his own project. He was a fine teacher.

Then the fifth grader sat back down at his desk to continue his work. A colleague suggested that he write a program to draw concentric circles. A nifty bit of geometric and algebraic thinking followed. When I kicked things up a notch by writing my own even more complex program on the projected computer and named it, “Gary Defeats Derrick.” The kid laughed and read my program in an attempt to understand my use of global variables, conditionals, and iteration. Later in the day, the same kid chased me down the hall to tell me about what he had discovered since I left his classroom that morning.

Oh yeah, I later learned that the very same terrific kid is being drummed out of school  for not being their type of student.

I learned long ago. If a school does not have bad children, it will make them.

 

Student voice is good. We should take the needs, interests, concerns, talent, curiosity, discomfort, and joy of children seriously. (pretty courageous statement, eh?)

However, if one is truly committed to making the world better for kids, “voice,” is nice, but inadequate. “Voice” absent of power is often little more than propaganda or exploitation.

While I’ve been on a brief social media “skunk at the garden party” hiatus, Dean Shareski has generously filled-in by sharing his queasiness over the “viralGoldieblox video being passed around the Web. Señor Shareski set his BS detector  on high and has provided evidence that the “amazing” Rube Goldberg machine “made by girls” is merely a commercial for a new toy called, Goldieblox.

I am shocked! Shocked!

Anyone who knows me knows that I love toys. I find buying them irresistible. I’ve been seeing Goldieblox at Maker Faires for more than a year, but have not bought a set because I think they lack extended play value (a term LEGO uses internally). I’m not one to get all outraged that a toy for girls is pink. Goldieblox just hasn’t seemed very interesting to me or the girls I work with. It’s not part of my workshop road show sweeping the globe, “Invent To Learn.”

It just doesn’t seem that Goldieblox has any chance of measuring up to the self-promotion and hype of its creator that her box of ribbon and spools is “building women engineers.” I applaud the sentiment, but if we are truly serious about improving the education of girls, it will take a lot more work than a trip to Toys R Us.

I could be wrong. I’ve recently been upgrading my initial assessment of littleBits, based on my observations of children playing with the new toy/electronics construction kit. So, perhaps I will soon fall in love with Goldieblox, but I doubt it.

Back to Monsignor Shareski…

In his post critical of the Goldidblox video, Fake and Real Student Voice, Professori Shareski awakened several repressed social media memories I had long forgotten.

I took a lot of “brown porridge” when I called BS on the very same videos of yesteryear.

There was Dalton Sherman, the “amazing” 5th grader who was coached all summer-long to give a condescending speech, written by the Dallas Schools PR department  to Dallas teachers, right before laying off 400 of them.  I smelled a rat the second I saw the video. Was called a big fat poo-poo head by teachers on social media and was right. BTW: Dalton Sherman seems to have disappeared just like those teacher jobs. So much for being the voice of school reform.

Then there was Michael Wesch (who is an important scholar) made famous by the hostage film he created in which college students decried the state of education.

Fantastic. A college class with far too many students in it (200) attempts to revolutionize the educational system by whining in a five minute web video.

I’m sorry, but count me unimpressed!

Perhaps a student should hold up a sign saying, “My professor is wasting my time and money by making me participate in a piece of exploitative propaganda in which I get to insult either my generation or the one before me just to get on YouTube.”

How did bashing our own profession become such a popular sport? What possible value could demeaning educators have in a professional development setting? Are we desperate for moving pictures or are they merely a substitute for actual ideas?

From Hey Mom! Look What I Made in College (November 2007)

Aside from their lack of authenticity, what these three AMAZING viral videos of is how children and claims of “student voice” exploit children for propaganda purposes. The Goldieblox video is a commercial selling a toy. We don’t tweet Sir Grapefellow commercials (my preferred boyhood breakfast treat) as AMAZING examples of student voice, so why the wishful thinking about Goldieblox?

Señor Shareski rightfully cites my colleague Super-Awesome Sylvia (read Super-Awesome Sylvia in the Not So Awesome Land of Schooling) as a counter example to the fake Goldieblox commercial. I have worked closely with Sylvia over the past couple of years and made her part of the Constructing Modern Knowledge faculty, not because she is cute (she is), but because she is accomplished. She knows stuff. She has skills. She has a great work ethic and  is a terrific teacher (at 12).

However, talent and achievement  did not made Sylvia immune from cynical exploitation by Rupert Murdoch and Joel Klein’s education cabal as documented in an article I wrote for the Huffington Post, Shameless Shape Shifters.

So the moral of our story is…

Three lessons…

  1. As a young blogger in 1971, The Brady Bunch taught me an important lesson relevant here, caveat emptor – buyer beware. Users of social media need to “follow the money,” have a highly-tuned BS Detector, and know when and what they are being sold.
  2. Calling everything amazing or everyone a genius is lazy and counterproductive.
  3. Student voice without what Seymour Papert calls “kid power” is worse than empty rhetoric, it is a lie. Escapism is not the same as freedom.  Too much of what is offered as “student voice” offers a false sense of agency, power, or freedom to the powerless. It is what Martin Luther King, Jr. called, “the intoxicating drug of gradualism.”

Papert circa 1999 enjoying the work of a middle schooler

I’ve been thinking a lot about my friend, colleague, and mentor Dr. Seymour Papert a lot lately. Our new book, “Invent to Learn: Making, Tinkering, and Engineering in the Classroom,” is dedicated to him and we tried our best to give him the credit he deserves for predicting, inventing, or laying the foundation for much of what we now celebrate as “the maker movement.” The popularity of the book and my non-stop travel schedule to bring the ideas of constructionism to classrooms all over the world is testament to Seymour’s vision and evidence that it took much of the world decades to catch up.

Jazz and Logo are two of my favorite things in life. They both make me feel bigger than myself and nurture me. Jazz and Logo provide epistemological lenses through which I view the world and appreciate the highest potential of mankind. Like jazz, Logo has been pronounced dead since its inception, but I KNOW how good it is for kids. I KNOW how it makes them feel intelligent and creative. I KNOW that Logo-like activities hold the potential to change the course of schooling. That’s why I have been teaching it to children and their teachers in one form or another for almost 32 years.

I’ve been teaching a lot of Logo lately, particularly a relatively new version called Turtle Art. Turtle Art is a real throwback to the days of one turtle focused on turtle geometry, but the interface has been simplified to allow block-based programming and the images resulting from mathematical ideas can be quite beautiful works of art. (you can see some examples in the image gallery at Turtleart.org)

Turtle Art was created by Brian Silverman, Artemis Papert (Seymour’s daughter) and their friend Paula Bonta. Turtle Art itself is a work of art that allows learners of all ages to begin programming, creating, solving problems, and engaging in hard fun within seconds of seeing it for the first time. Since an MIT undergraduate in the late 1970s, Brian Silverman has made Papert’s ideas live in products that often exceeded Papert’s expectations.

There aren’t many software environments or activities of any sort that engage 3rd graders, 6th graders, 10th graders and adults equally as Turtle Art. I wrote another blog post a year or so ago about how I wish I had video of the first time I introduced Turtle Art to a class of 3rd graders. Their “math class” looked like a rugby scrum, there was moving, and wiggling, and pointing, and sharing and hugging and high-fiving everywhere while the kids were BEING mathematicians.

Yesterday, I taught a sixth grade class in Mumbai to use Turtle Art for the first time. They worked for 90-minutes straight. Any casual observer could see the kids wriggle their bodies to determine the right orientation of the turtle, assist their peers, show-off their creations, and occasionally shriek with delight in a reflexive fashion when the result of their program surprised them or confirmed their hypothesis. As usual, a wide range of mathematical ability and learning styles were on display. Some kids get lost in one idea and tune out the entire world. This behavior is not just reserved to the loner or A student. It is often the kid you least expect.

Yesterday, while the rest of the class was creating and then modifying elaborate Turtle Art programs I provided, one sixth grader went “off the grid” to program the turtle to draw a house. The house has a long and checkered past in Logo history. In the early days of Turtle Graphics, lots of kids put triangles on top of squares to draw a house. Papert used the example in his seminal book, “Mindstorms: Children, Computers, and Powerful Ideas,” and was then horrified to discover that “making houses” had become de-facto curriculum in classrooms the world over. From then on, Papert refrained from sharing screen shots to avoid others concluding that they were scripture.

It sure was nice to see a kid make a house spontaneously, just like two generations of kids have done with the turtle. It reminded me of what the great jazz saxophonist and composer Jimmy Heath said at Constructing Modern Knowledge last summer, “What was good IS good.”

Love is all you need
This morning, I taught sixty 10th graders for three hours. We spend the first 75 minutes or so programming in Turtle Art.  Like the 6th graders, the 10th graders  had never seen Turtle Art before. After Turtle Art,  the kids could choose between experimenting with MaKey MaKeys, wearable computing, or Arduino programming. Seymour would have been delighted by the hard fun and engineering on display. I was trying to cram as many different experiences into a short period of time as possible so that the school’s teachers would have options to consider long after I leave.

After we divided into three work areas, something happened that Papert would have LOVED. He would have given speeches about this experience, written papers about it and chatted enthusiastically about it for months. Ninety minutes or so after everyone else had moved on to work with other materials, one young lady sat quietly by herself and continued programming in Turtle Art. She created many subprocedures in order to generate the image below.


Papert loved love and would have loved this expression of love created by “his turtle.” (Papert also loved wordplay and using terms like, “learning learning.” I’m sure he would be pleased with how many times I managed to use love in one sentence.) His life’s work was towards the creation of a Mathland where one could fall in love with mathematical thinking and become fluent in the same way a child born in France becomes fluent in French. Papert spoke often of creating a mathematics that children can love rather than wasting our energy teaching a math they hate. Papert was fond of saying, “Love is a better master than duty,” and delighted in having once submitted a proposal to the National Science Foundation with that title (it was rejected).

The fifteen or sixteen year old girl programming in Turtle Art for the first time could not possibly have been more intimately involved in the creation of her mathematical artifact. Her head, heart, body and soul were connected to her project.

The experience resonated with her and will stay with me forever. I sure wish my friend Seymour could have seen it.

Love,

 

 


Turtle Art is free for friends who ask for a copy, but is not open source. It’s educational efficacy is the result of a singular design vision unencumbered by a community adding features to the environment. Email contact@turtleart.org to request a copy for Mac, Windows or Linux.

Almost daily, a colleague I respect posts a link to some amazing tale of classroom innovation, stupendous new education product or article intended to improve teaching practice. Perhaps it is naive to assume that the content has been vetted. However, once I click on the Twitter or Facebook link, I am met by one of the following:

  1. A gee-whiz tale of a teacher doing something obvious once, accompanied by breathless commentary about their personal courage/discovery/innovation/genius and followed by a steam of comments applauding the teacher’s courage/discovery/innovation/genius. Even when the activity is fine, it is often the sort of thing taught to first-semester student teachers.
  2. An article discovering an idea that millions of educators have known for decades, but this time with diminished expectations
  3. An ad for some test-prep snake oil or handful of magic beans
  4. An “app” designed for kids to perform some trivial task, because “it’s so much fun, they won’t know they’re learning.” Thanks to sites like Kickstarter we can now invest in the development of bad software too!
  5. A terrible idea detrimental to teachers, students or public education
  6. An attempt to redefine a sound progressive education idea in order to justify the status quo

I don’t just click on a random link from a stranger, I follow the directions set by a trusted colleague – often a person in a position of authority. When I ask them, “Did you read that article you posted the link to?” the answer is often, “I just re-read it and you’re right. It’s not good.” Or “I’m not endorsing the content at the end of the link, “I’m just passing it along to my PLN.”

First of all, when you tell me to look at something, that is an endorsement. Second, you are responsible for the quality, veracity and ideological bias of the information you distribute. Third, if you arenot taking responsibility for the information you pass along, your PLN is really just a gossip mill.

If you provide a link accompanied by a message, “Look at the revolutionary work my students/colleagues/I did,” the work should be good and in a reasonable state of completion. If not, warn me before I click. Don’t throw around terms like genius, transformative or revolutionary when you’re linking to a kid burping into Voicethread!! If you do waste my time looking at terrible work, don’t blame me for pointing out that the emperor has no clothes.

Just today, two pieces of dreck were shared with me by people I respect.

1) Before a number of my Facebook friends shared this article, I had already read it in the ASCD daily “Smart” Brief. Several colleagues posted or tweeted links to the article because they yearn for schools to be better – more learner-centered, engaging and meaningful.

One means to those ends is project-based learning.  I’ve been studying, teaching and speaking about project-based learning for 31 years. I’m a fan. I too would like to help every teacher on the planet create the context for kids to engage in personally meaningful projects.

However, sharing the article, Busting myths about project-based learning, will NOT improve education or make classrooms more project-based. In fact, this article so completely perverts project-based learning that it spreads ignorance and will make classroom learning worse, not better.

This hideous article uses PBL, which the author lectures us isn’t just about projects (meaningless word soup), as a compliment to direct instruction, worksheets and tricking students into test-prep they won’t mind as much. That’s right. PBL is best friends with standardized testing and worksheets (perhaps on Planet Dummy). There is no need to abandon the terrible practices that squeeze authentic learning out of the school day. We can just pretend to bring relevance to the classroom by appropriating the once-proud term, project-based learning.

Embedding test-prep into projects as the author suggests demonstrates that the author really has no idea what he is talking about. Forcing distractions into a student’s project work robs them of agency and reduces the activity’s learning potential. The author is also pretty slippery in his use of the term, “scaffolding.” Some of the article doesn’t even make grammatical sense.

Use testing stems as formative assessments and quizzes.

The  article was written by a gentleman who leads professional development for the Buck Institute, an organization that touts itself as a champion of project-based learning, as long as those projects work backwards from dubious testing requirements. This article does not represent innovation. It is a Potemkin Village preserving the status quo while allowing educators to delude themselves into feeling they are doing the right thing.

ASCD should be ashamed of themselves for publishing such trash. My colleagues, many with advanced degrees and in positions where they teach project-based learning, should know better!

If you are interested in effective project-based learning, I’m happy to share these five articles with you.

2) Another colleague urged all of their STEM and computer science-interested friends to explore a site raising money to develop “Fun and Creative Computer Science Curriculum.” Whenever you see fun and creative in the title of an education product, run for the hills! The site is a fund-raising venture to get kids interested in computer science. This is something I advocate every day. What could be so bad?

Thinkersmith teaches computer science with passion and creativity. Right now, we have 20 lessons created, but only 3 packaged. Help us finish by summer!

My experience in education suggests that once you package something, it dies. Ok Stager, I know you’re suspicious of the site and the product searching for micro-investors, but watch the video they produced. It has cute kids in it!

So, I watched the video…

Guess what? Thinkersmith teaches computer science with passion and creativity – and best of all? YOU DON’T EVEN NEED A COMPUTER!!!!!!

Fantastic! Computer science instruction without computers! This is like piano lessons with a piano worksheet. Yes siree ladies and gentleman, there will be no computing in this computer science instruction.

A visitor to the site also has no idea who is writing this groundbreaking fake curriculum or their qualifications to waste kids’ time.

Here we take one of the jewels of human ingenuity, computer science, a field impacting every other discipline and rather than make a serious attempt to bring it to children with the time and attention it deserves, chuckleheads create cup stacking activities and simplistic games.

There are any number of new “apps” on the market promising to teach kids about computer science and programming while we should be teaching children to be computer scientists and programmers.

At the root of this anti-intellectualism is a deep-seated belief that teachers are lazy or incompetent. Yet, I have taught thousands of teachers to teach programming to children and in the 1980s, perhaps a million teachers taught programming in some form to children. The software is better. The hardware is more abundant, reliable and accessible. And yet, the best we can do is sing songs, stack cups and color in 2013?

What really makes me want to scream is that the folks cooking up all of these “amazing” ideas seem incapable of using the Google or reading a book. There is a great deal of collected wisdom on teaching computer science to children, created by committed experts and rooted in decades worth of experience.

If you want to learn how to teach computer science to children, ask me, attend my institute, take a course. I’ll gladly provide advice, share resources, recommend expert colleagues and even help debug student programs. If you put forth some effort, I’m happy to match it.

There is no expedient to which a man will not resort to avoid the real labor of thinking.
-Sir Joshua Reynolds

Don’t lecture me about the power of social media, the genius of your PLN, the imperative for media literacy or information curation if you are unwilling to edit what you share. I share plenty of terrible articles via Twitter and Facebook, but I always make clear that I am doing so for purposes or warning or parody. The junk is always clearly labeled.

Please filter the impurities out of your social media stream.You have a responsibility to your audience.

Thank you


* Let the hysterical flaming begin! Comments are now open.

Computationally-Rich Activities for the Construction of Mathematical Knowledge – No Squares Allowed
©1998 Gary S. Stager with Terry Cannings
This paper was published in the proceedings of the 1998 National Educational Computing Conference in San Diego

Based on a book chapter: Stager, G. S. (1997). Logo and Learning Mathematics-No Room for Squares. Logo: A Retrospective. D. L. Johnson and C. D. Maddux. Philadelphia, The Haworth Press: 153-169.

Abstract
The NCTM Standards state that fifty percent of all mathematics has been invented since World War II. (National Council of Teachers of Mathematics, 1989) Few if any of these branches of mathematical inquiry have found their way into the K-12 curriculum. This is most unfortunate since topics such as number theory, chaos, topology, cellular automata and fractal geometry may appeal to students unsuccessful in traditional math classes. These new mathematical topics tend to be more contextual, visual, playful and fascinating than adding columns of numbers or factoring quadratic equations. Logo provides a powerful medium for rich mathematical explorations and problem solving while providing a context in which students may fall in love with the beauty of mathematics. The examples in this paper are intended to spark the imaginations of teachers and explore several mathematical areas ripe for Logo-based investigations.

Introduction
While it may seem obvious to assert that computers are powerful computational devices, their impact on K-12 mathematics education has been minimal. (Suydam, 1990) More than a decade after microcomputers began entering schools, 84% of American tenth graders said they never used a computer in math class.(National Center for Educational Statistics, 1984) Computers provide a vehicle for “messing about” with mathematics in unprecedented learner-centered ways. “Whole language” is possible because we live in a world surrounded by words we can manipulate, analyze and combine in infinite ways. The same constructionist spirit is possible with “whole math” because of the computer. In rich Logo projects the computer becomes an object to think with – a partner in one’s thinking that mediates an ongoing conversation with self.

Many educators equate Logo with old-fashioned turtle graphics or suggest that Logo is for the youngest of children. Neither of these beliefs is true. Although traditional turtle graphics continues to be a rich laboratory in which students construct geometric knowledge, Logo is flexible enough to explore the entire mathematical spectrum. Logo continues to satisfy the claim that it has no threshold and no ceiling. (Harvey, 1982) Best of all, Logo provides a context in which children are motivated to solve problems and express themselves.

The National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics recognizes Logo as a software environment that can assist schools in meeting the goals for the improvement of mathematics education. In fact, Logo is the only computer software specifically named in the document.

The Goals of the NCTM (1984) Standards for All Students

  1. learn to value mathematics
  2. become confident in their ability to do mathematics
  3. become mathematical problem solvers
  4. learn to communicate mathematically
  5. learn to reason mathematically

The NCTM Standards state that fifty percent of all mathematics has been invented since World War II. (National Council of Teachers of Mathematics, 1989) Few if any of these branches of mathematical inquiry have found their way into the K-12 curriculum. This is most unfortunate since topics such as number theory, chaos, topology, cellular automata and fractal geometry may appeal to students unsuccessful in traditional math classes. These new mathematical topics tend to be more contextual, visual, playful and fascinating than adding columns of numbers or factoring quadratic equations. Logo provides a powerful medium for rich mathematical explorations and problem solving while providing a context in which students may fall in love with the beauty of mathematics.

Computer microworlds such as Logo turtle graphics and the topics of constructions and loci provide opportunities for a great deal of student involvement, In particular, the first two contexts serve as excellent vehicles for students to develop, compare and apply algorithms. (National Council of Teachers of Mathematics, 1989, p. 159)

The examples in this paper are intended to spark the imaginations of teachers and explore several mathematical areas ripe for Logo-based investigations. The project ideas use MicroWorlds, the latest generation of Logo software designed by Seymour Papert and Logo Computer Systems, Inc. MicroWorlds extends the Logo programming environment through the addition of an improved user interface, multiple turtles, buttons, text boxes, paint tools, multimedia objects, sliders and parallelism.

Parallelism allows the computer to perform more than one function at a time. Most computer-users have never experienced parallelism or the emergent problem solving strategies it affords. MicroWorlds makes this powerful computer science concept concrete and usable by five year-olds. The parallelism of MicroWorlds makes it possible to explore some mathematical and scientific phenomena for the first time. Parallelism also allows more conventional problems to be approached in new ways.

Euclidian Geometry
One source of inspiration for student Logo projects is commercial software. Progressive math educators have found software like The Geometric Supposer and the more robust Geometers’ Sketchpad to be useful tools for exploring Euclidian geometry and performing geometric constructions. I noticed that while teachers may use these tools as extremely flexible blackboards, kids can pull down a menu and request a perpendicular bisector to be drawn without any deeper understanding than if the problem was solved with pencil and paper.

Could middle or high school students design collaboratively their own such tools? If so, they would gain a more intimate understanding of the related math concepts because of the need to “teach” the computer to perform constructions and measurements. Throughout this process, teams of students are asked to brainstorm questions, share what they know and define paths for further inquiry. Students as young as seventh grade have developed their own geometry toolkits in MicroWorlds.

Much of learning mathematics involves naming actions and relationships. Logo programming enhances the construction of mathematical knowledge through the process of defining and debugging Logo procedures. The personal geometry toolkits designed by students are used to construct geometric knowledge and questions worthy of further investigation. As understanding emerges the tool can be enhanced in order to investigate more advanced problems.

At the beginning of this project students are given a few tool procedures to start with. These procedures are designed to:

  1. drop a point on the screen (each point is a turtle and in MicroWorlds every turtle knows where it is in space)
  2. compute the distance between two points


With these two sets of tool procedures students can create tools necessary for generating geometric constructions, measuring constructions and comparing figures. MicroWorlds’ paint tools may be used to color-in figures and to draw freehand shapes. The procedural nature of Logo allows for higher level functions to be built upon previous procedures. Figures 1a, 1b & 1c are screen shots of one student’s geometry toolkit.

Probability and Chance
Children use MicroWorlds to explore probability via traditional data collection problems involving coin or dice tosses and in projects of their own design. Logo’s easy to use RANDOM function appears in the video games, races, board games and sound effects of many students.

Perhaps the best use of probability I have encountered in a MicroWorlds project is in a project I like to call, “Sim-Middle Ages.” In this project a student satisfied the requirements for the unit on medieval life in a quite imaginative fashion. Her project allows the user to specify the number of plots of land, number of seeds to plant and the number of mouths to feed. MicroWorlds then randomly determines the amount of plague, pestilence, rainfall and rate of taxation to be encountered by the farmer.

On the next page there are two buttons. One button announces if you live or die in the middle ages and the other tells why, based on the user-determined and random variables. You may then go back and adjust any of the values in an attempt to survive. (figures 2a, 2b and 2c)




Things happen in the commercial simulations, but users often don’t understand the causality. In student-created simulations, students use mathematics in a very powerful way. They develop their own algorithms to model historical or scientific phenomena. This type of project can connect mathematics with history, economics, physical science and life science in very powerful ways.

Number Theory
“Number theory, at one time considered the purest of pure mathematics is simply the study of whole numbers, including prime numbers. This abstract field, once a playground for a few mathematicians fascinated by the curious properties of numbers, now has considerable practical value… in fields like cryptography.”(Peterson, 1988) Software environments, such as MicroWorlds, provide a concrete environment in which students may experiment with number theory. “Experimental math” projects benefit from Logo’s ability to control experiments, easily adjust a variable and collect data. Kids control all of the variables in an experiment and can swim around in the beaker with the molecules. Intellectual immersion in large pools of numbers is possible due to computer access. The scientific method comes alive through mathematical experimentation.

A fascinating experimental math problem to explore with students is known as the 3N problem. The problem is also known by several other names, including: Ulam’s conjecture, the Hailstone problem, the Syracuse problem, Kakutani’s problem, Hasse’s algorithm, and the Collatz problem. The 3N problem has a simple set of rules. Put a number in a “machine” (Logo procedure) and if it is even, cut in half – if it is odd, multiply it by 3 and add 1. Then put the new value back through the machine. For example, 5 becomes 16, 16 becomes 8, becomes 4, 4 becomes 2, 2 becomes 1, and 1 becomes 4. Mathematicians have observed that any number placed into the machine will eventually be reduced to a repeating pattern of 4…2…1…

While this is an interesting pattern, what can children explore? Well, it seems that some numbers take a long time to get to 4…2…1… I call each of the numbers that appear before 4, a “generation.” I often expose students to this problem by trying a few starting numbers and leading a discussion. Typing SHOW 3N 1 takes 1 generation to get to 4. Students may then predict that the number 2 will take two generations and they would be correct. They may then hypothesize that the number entered will equal the number of generations required to get to 4. However, 3N 3 takes 5 generations! I then ask, “how can we modify our hypothesis to save face or make it look like we were at least partially right?” Kids then suggest that the higher the number tried, the longer it will take to get to 4…2…1… They may even construct tables of the previous data and make numerous predictions for how the number 4 will behave only to find that 4 takes zero generations (for obvious reason that it is 4).

I then tell the class that they should find a number that takes a long time to get to 4…2…1… I do not specify what I mean by a “long time” in order to let the young mathematicians agree on their own limits. The notion of limits is a powerful mathematical concept which helps focus inquiry and provides the building blocks of calculus. Students often test huge numbers before realizing that they need to be more deliberate in their experimentation. The working definition of “long time” changes as the experiment continues. Eleven generations may seem like a long time until a group of kids test the number 27. Gasps and a chorus of wows can be heard when 27 takes 109 generations. Then I ask the class to tell me some of the characteristics of 27. Students often list some of the following hypotheses:

Its factors are 1, 3, 9, 27
It’s odd
It’s 3 * 3 * 3 (an opportunity to introduce the concept of cubed numbers)
The sum of the digits = 9
The number is greater than 25
 

We then test each of the hypotheses and discard most of them. The cubed number hypothesis is worthy of further investigation. If we test the next cubed number, 4, with SHOW 3N 4 * 4 * 4 we find that it does not take long to get to 4. One student may suggest that only odd perfect cubes take a long time. I then suggest that the other students find a way to disprove this hypothesis by finding either an odd perfect cube that doesn’t take a long time or an even cube that does. Both exist.

to 3n :number
print :number
ifelse even? :number [3n :number / 2] [3n (:number * 3) + 1]
end

to even? :number
output 0 = remainder :number 2
end

A simple tool procedure may be added to count the number of generations for the “researcher.” The more you play with this problem, the more questions emerge. A bit more programming allows you to ask the computer to graph the experimental data or keep track of numbers that take longer than X generations to reach 4…2…1… Running such experiments overnight may lead to other interesting discoveries, like the numbers 54 and 55 each take 110 generations. What can adjacent numbers have in common? 108, 109 and 110 each take 111 generations. Could this pattern have something to do with place value? How could you find out? (see figures 4a & 4b)

The joy in this problem for kids and mathematicians is connected to the sense that every time you think you know something, it may be disproven. This playfulness can motivate students to view mathematics as a living discipline, not as columns of numbers on a worksheet. For many students, problems like 3N provide a first opportunity to think about the behavior of numbers. “For the most part, school math and science becomes the acquisition of facts that have been found by people who call themselves scientists.” (Goldenberg, 1993) Logo and experimental math provides another opportunity to provide children with authentic mathematical experiences.

Fractal Geometry and Chaos Theory
The contemporary fields of fractal geometry and chaos theory are the result of modern computation. Many learners find the visual nature of fractal geometry and the unpredictability of chaos fascinating. Logo’s turtle graphics and recursion make fractal explorations possible. The randomness, procedural nature and parallelism of MicroWorlds brings chaos theory within the reach of students.

Fractals are self-similar shapes with finite area and infinite perimeter. Fractals contain structures nested within one another with each smaller structure a miniature version of the larger form. Many natural forms can be represented as fractions, including ferns, mountains and coastlines.

Chaos theory suggests that systems governed by physical laws can undergo transitions to a highly irregular form of behavior. Although chaotic behavior appears random, it is governed by strict mathematical conditions. Chaos theory causes us to reexamine many of the ways in which we understand the world and predict natural phenomena. Two simple principles can be used to describe Chaos theory:

  1. From order (a predictable set of rules), chaos emerges.
  2. From a random set of rules, order emerges.

MicroWorlds may be used to explore both chaos and fractal geometry simultaneously. Figure 3shows two similar fractals called the Sierpinski Gasket. The fractal on the left is created by a complex recursive procedure. The fractal on the right is generated by a seemingly random algorithm discovered by Michael Barnsley of Georgia Institute of Technology. The Barnsley Fractal is created by placing three dots on the screen and then randomly choosing one of three points, going half way towards it and putting another dot. This process is repeated infinitely and a Sierpinski Gasket emerges. In fact, if you grab the turtle from the “chaos fractal” and move it somewhere else on the screen, it immediately finds its way back into the “triangle” and never leaves again. The multiple turtles and parallelism of MicroWorlds makes it possible to explore the two different ways of generating a similar fractal simultaneously. Experimental changes can always be made to the procedures and the results may be immediately observed.

Animation
One of the most attractive aspects of MicroWorlds is its ability to create animations. Students are excited by the ease with which they can create even complex animations. MicroWorlds animations require the same mathematical and reasoning skills as turtle graphics. The difference is that the turtle’s pen is up instead of down and the physics of motion comes into play. Multiple turtles and “flip-book” style animation enhance planning and sequencing skills. Even the youngest students use Cartesian coordinates and compass headings routinely when positioning turtles and drawing elaborate pictures.

Perhaps the best part of MicroWorlds animation is that the student-created animation and related mathematics are often employed in the service of interdisciplinary projects. Using animation to navigate a boat down the ancient Nile, simulate planetary orbits, design a video game or energize a book report provides a meaningful context for using and learning mathematics.

Functions and Variables
Logo’s procedural inputs and mathematical reporters give kids concrete practice with variables. Functions/reporters/operations are easy to create in MicroWorlds and can even be the input to another function. For example, the expression SHOW DOUBLE DOUBLE DOUBLE 5 or REPEAT DOUBLE 2 [fd DOUBLE DOUBLE 20 RT DOUBLE 45] are possible by writing a simple procedure, such as:

to double :number
output :number * 2
end

Many teachers are unaware of Logo’s ability to perform calculations (up through trigonometric functions) in the command center or in procedures. SHOW 3 * 17 typed in the command center will display 51 and REPEAT 8 [fd 50 rt 360 / 8] will properly draw an eight-sided regular polygon.

A favorite project I like to conduct with fifth and sixth graders creates a fraction calculator. First we decide to represent fractions as a (Logo) list containing a numerator and a denominator. Then we write procedures to report the numerator and denominator of a fraction. From there, the class can easily collaborate to write a procedure which adds two fractions. Some kids can even make the procedure add fractions with different denominators. From there, all of the standard fraction operations can be written as Logo procedures by groups of children. The next challenge the kids typically tackle is the subtraction of fractions.

One day, a fifth grader, Billy, made an interesting discovery while testing his subtraction “machine.” Billy typed, SHOW SUBTRACT [1 3] [2 3] (meaning 1/3 – 2/3), and -1 3 appeared in the command center. I noticed the negative fraction and mentioned that when I was in school we were taught that fractions had to be positive. Therefore, there is no such thing as a negative fraction.

Billy exclaimed, “Of course there is! The computer gave one to us!” This provoked a discussion about “garbage in – garbage out,” the importance of debugging and the need for conventions agreed upon by mathematicians and scientists. We even discussed the difference between symbols and numbers. Billy listened to this discussion impatiently and announced, “That’s ridiculous because I can give you an example of a negative fraction in real-life.”

Billy said, “I have a birthday cake divided into six slices and eight people arrive at my party. I’m short two sixths of a cake – negative 2/6!” He went on to say, “If the computer can give us a negative fraction and I can provide a real-life example of one, then there must be negative fractions.” The hazy memory of my math education diminished the confidence required to argue with this budding mathematician. Instead, I agreed to do some research.

I looked in mathematics dictionaries, but found more ambiguity than clarity. I also spent several weeks consulting with math teachers. Most of these people either dismissed the question of negative fractions as silly or complained that they lacked the time to adequately deal with Billy’s dilemma. After a bit more time, I ran into a university mathematician at a friend’s birthday party. Roger did not dismiss Billy’s question. Instead he asked for my email address. The next morning the following email message awaited me.

Date: Sun, 06 Nov 1994 09:52:44 -0400 (EDT)
Subject: fractions
To: gstager@pepperdine.edu

Dear Gary,

It was fun to have a chat at Ihor’s party. This morning I got out my all time favorite source of information on things worthwhile, the Ninth Edition of the Encyclopedia Britannica. (With its articles by James Clerk Maxwell et al.) It is very clear. Fractions come about by dividing unity into parts, and are thus by definition positive.

Interesting.
Yours,
Roger


Now what should a teacher tell Billy? In the past, you might hope that he forgot the matter. Today, Billy can post his discovery on the Internet and engage in serious conversation – perhaps even research with other mathematicians. Access to computers and software environments like MicroWorlds makes it possible for children to make discoveries that may be of interest to mathematicians and scientists. It is plausible that kids can contribute to the construction of knowledge deemed important by adults.

New Data Structures
MicroWorlds has two new data structures that contribute to mathematical learning. With the click of the mouse, sliders and text boxes can be dropped on the screen. As input devices, sliders are visual controls that adjust variables. Each slider has a name and a range of numbers assigned to it. Like a control on a mixing board the slider can be set to a number in that range. The slider’s value can then be sent to a turtle whose speed or orientation is linked to the value of the slider. The slider can also be used to set the values of variables used in a simulation.

Sliders may also be used as output devices. A procedure can change the value of a slider to indicate an experimental result. If a slider named, counter, is in a MicroWorlds project then the command, SETCOUNTER COUNTER + 1, can be used to display the results of incrementing the counter.

MicroWorlds text boxes also function as both input and output devices. A text box is like a little word processor drawn on the MicroWorlds page to hold text. Text boxes also have names that when evoked report their contents. If a user types the number 7 in a text box named FOO, then typing SHOW FOO * 3 will display 21 in the command center. FD FOO * 10 will move the turtle forward 70 steps. The command, SETFOO 123 will replace the contents of the text box, FOO, with 123. Therefore, text boxes may be used as experimental monitors or calculator displays. Constructing a garden-variety calculator with a text box and MicroWorlds buttons or turtles is deceptively simple, but provides one illustration of how text boxes could be used in a mathematical context.

A basic spreadsheet can be built in MicroWorlds with just one line of Logo code. If three text boxes are named, cell1, cell2 and total, then a button with the instruction, SETTOTAL CELL1 + CELL2, will put the sum of the first two cells in the third. Making the button run many times will cause the “spreadsheet” to perform automatic calculations. A bit more programming will allow you to check for calculation efforts, graph data or cause a turtle to change its behavior based on the result of a calculation. Building a model spreadsheet helps students understand how a commercial spreadsheet works, develop computation skills and add automatic calculation to their Logo toolbox.

Instructional Software Design
Children can use Logo as a design environment for teaching others mathematical concepts. Idit Harel’s award-winning research (Harel, 1991) and the subsequent research by her colleague, Yasmin Kafai (Kafai, 1995), demonstrated that when students were asked to design software (in LogoWriter or MicroWorlds) to teach other kids about “fractions” they gained a deeper understanding of fractions than children who were taught fractions and Logo in a traditional manner. These students also learn a great deal about design, Logo programming, communication, marketing and problem solving. Harel and Kafai have confirmed that children learn best by making connections and when actively engaged in constructing something meaningful. Their research provides additional evidence of Logo’s potential as an environment for the construction of mathematical knowledge.

Conclusion

Increased access to computers and imaginative teachers will open up an infinite world of possibilities for Logo learning. Software environments, such as MicroWorlds provide children with an intellectual laboratory and vehicle for self-expression. MicroWorlds inspires serendipitous connections to powerful mathematical ideas when drawing, creating animations, building mathematical tools or constructing simulations.

Excursions into the worlds of number theory, fractal geometry, chaos and probability rely on MicroWorlds’ ability to act as lab assistant and manager. Paul Goldenberg suggests that it is difficult to test out ideas unless one has a slave stupid enough not to help. (Goldenberg, 1993) The computer plays the role of lab assistant splendidly, yet the student still must do all of the thinking. MicroWorlds makes it possible to manage large bodies of data by running tedious experimental trials millions of times if necessary, collecting data and displaying it in numerical or graphical form. The procedural nature of MicroWorlds makes it possible to make small changes to an experiment without having to start from scratch.

MicroWorlds provides schools with a powerful software package flexible enough to grow with students. In days of tight school budgets it is practical to embrace a software environment with which students can address the demands of numerous subject areas. The sophistication with which students confront intellectual challenges improves along with their fluency in MicroWorlds.

Seymour Papert was horrified at how the simple example of commanding a turtle to draw a house, depicted in Mindstorms, became “official Logo curriculum” in classrooms around the world. However, providing students with a rich “mathland” in which to construct mathematical knowledge has always been one of the goals in the design and implementation of Logo. This paper attempts to provide simple examples of how MicroWorlds may be used to explore a number of mathematical concepts in a constructionist fashion. Those interested in additional ideas should read (Abelson & diSessa, 1981), (Cuoco, 1990), (Clayson, 1988), (Goldenberg & Feurzeig ,1987), (Lewis, 1990) and (Resnick, 1995). More detailed examples and teacher materials related to this paper are available on my World-Wide-Web site at: http://moon.pepperdine.edu/~gstager/home.html.

References

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