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:
- Ask each kid to compose a song to be played on their instrument
- Teach their song to a friend to play on their invented instrument
- The next day ask the kids to play the song they were taught yesterday from memory
- When they can’t remember how, you might ask each “composer” to write down the song so other players can remember it
- 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.
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 “viral” Goldieblox 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…
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…
- 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.
- Calling everything amazing or everyone a genius is lazy and counterproductive.
- 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.”
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.
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 firstname.lastname@example.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:
- 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.
- An article discovering an idea that millions of educators have known for decades, but this time with diminished expectations
- An ad for some test-prep snake oil or handful of magic beans
- 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!
- A terrible idea detrimental to teachers, students or public education
- 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.
* 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.
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.
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
- learn to value mathematics
- become confident in their ability to do mathematics
- become mathematical problem solvers
- learn to communicate mathematically
- 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.
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:
- drop a point on the screen (each point is a turtle and in MicroWorlds every turtle knows where it is in space)
- 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, 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:
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
ifelse even? :number [3n :number / 2] [3n (:number * 3) + 1]
to even? :number
output 0 = remainder :number 2
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:
- From order (a predictable set of rules), chaos emerges.
- 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.
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
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)
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.
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.
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.
- Abelson, H., & diSessa, A. (1981). Turtle Geometry. Cambridge, MA: MIT Press.
- Clayson, J. (1988). Visual Modeling with Logo. Cambridge, MA: MIT Press.
- Clements, D.H. (1991). Logo in Mathematics Education: Effects and Efficacy. Stevens Institute of Technology Conference Proceedings – Computer Integration in Pre-College Mathematics Education: Current Status and Future Possibilities, April 24, 1991. Hoboken, NJ: Stevens Institute of Technology/CIESE.
- Cuoco, A. (1990). Investigations in Algebra. Cambridge, MA: MIT Press.
- Goldenberg, E.P. (1993). Linguistics, Science, and Mathematics for Pre-college Students: A Computational Modeling Approach.Revised from Proceedings, NECC ‘89 National Educational Computing Conference, Boston, MA. June 20-22, pp. 87 -93. Newton, MA: Educational Development Center.
- Goldenberg, E.P. (1989). “Seeing Beauty in Mathematics: Using Fractal Geometry to Build a Spirit of Mathematical Inquiry.” Journal of Mathematical Behavior, Volume 8. pages 169-204.
- Goldenberg, E.P., & Feurzeig, W. (1987). Exploring Language with Logo Cambridge, MA: MIT Press.
- Harel, I. (1991). Children Designers: Interdisciplinary Constructions for Learning and Knowing Mathematics in a Computer-Rich School. Norwood, NJ: Ablex Publishing Corporation.
- Harel, I. & Papert, S. (editors) (1991). Constructionism. Norwood, NJ: Ablex Publishing Corporation.
- Harvey, B. (1982). Why Logo? Byte, Vol. 7, No.8, August 1982, 163-193.
- Harvey, B. (1985-87). Computer Science Logo Style, Volumes 1-3. Cambridge, MA: MIT Press.
- Kafai, Y. (1995) Minds in Play – Computer Design as a Context for Children’s Learning. Hillsdale, NJ: Lawrence Erlbaum and Associates.
- Lewis, P. (1990). Approaching Precalculus Mathematics Discretely. Cambridge, MA: MIT Press.
- National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
- Papert, S. (1980). Mindstorms: Children, Computers, and Powerful Ideas. (Second Edition, 1993) New York: Basic Books.
- Peterson, I. (1988). The Mathematical Tourist – Snapshots of Modern Mathematics. NY: W.H. Freeman and Company.
- Poundstone, W. (1985). The Recursive Universe… Chicago: Contemporary Books.
- Resnick, M. (1995). Turtles, Termites and Traffic Jams – Explorations in Massively Powerful MicroWorlds. Cambridge, MA: MIT Press.
- Silverman, B. (1987). The Phantom Fishtank: An Ecology of Mind. Montreal: Logo Computer Systems, Inc. (book with software for Apple II or MS-DOS)
- Stager, G. (October, 1988). “A Microful of Monkeys.” The Logo Exchange .
- Stager, G. (1990). “Developing Scientific Thought in a Logo-based Environment.” Proceedings of the World Conference on Computers in Education. Sydney, Australia: IFIP.
- Stager, G. (1991). “Becoming a Scientist in a Logo-based Environment.” Proceedings of the Fifth International Logo Conference. San José, Costa Rica: Fundacion Omar Dengo.
- Suydam, M. N. (1990). Curriculum and Evaluation Standards for Mathematics Education. (ERIC/SMEAC Mathematics Education Digest No. 1, 1990) Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education. (ERIC Document Reproduction Service No. ED319630 90).
Recently, 5th and 6th grade girls in the school where I work came up to me in the hallway and volunteered, “I want to be an engineer.” While this is heartwarming, especially given the political rhetoric behind the importance of S.T.E.M. and the challenges of gender underrepresentation in the sciences, I would like to draw a totally different lesson for educators.
Anyone who knows anything about my teaching knows that I would never spend any time on “career education” with kids I teach. I create the context, conditions and projects during which children are engaged in engineering. When building and programming robots, the kids are engineers – not contemplating a career for a dozen years later. The kids are smart enough to connect the dots and identify interest in a career related to their talent, interests or present mood, even if that interest is short-lived.
Time is the rarest of currencies in school. Therefore, time should be focused on authentic experiences, not meta experiences.
Affective qualities like collaboration, passion, curiosity, perseverance and teamwork are certainly desirable for teachers and students. However, these traits may be developed while engaged in real pursuits, even within the existing curriculum. All that is required is a meaningful project. This is why I question the use of “meta” activities like ropes courses, ice-breakers or trust-building exercises as a form of professional development or separate curriculum. Professional development resources are also scarce. Therefore, PD should be focused on learning to do or know. The affective skills should be byproducts of meaningful experiences intended to improve teaching.
Adults become better teachers when they enjoy firsthand learning adventures like they desire for their students. You can’t teach 21st Century Learners if you haven’t learned this century. That is why I created Constructing Modern Knowledge.
Some educators have recognized that schools are too impersonal and that teachers should get to know their students. I could not agree more. However, the prescription is often to create advisory courses or extend homeroom to deal with pastoral care issues. The result is one teacher who gets to “know” students and time is borrowed from other courses where teachers should get to know their students formally and informally in the process of constructing knowledge together.
Sit next to a student engaged in a science experiment and talk with them. Lead vigorous discussions or chat with a kid about the book they’re reading. You don’t need a class period set aside for asking “How was your weekend?” or for building trust. Join a group of students for lunch. Say, “hi,” while passing in the hallway. Dennis Littky tells the story of making Time Magazine because as a school principal he greeted students when they entered school in the morning. Have we lowered our expectations so much that knowing students is some sort of awesome systemic accomplishment? Humane, thoughtful, even casual interaction between teachers and students does not require an NSF grant or special class.
When educators create a productive context for learning, achievement improves, students feel more connected and behavioral problems evaporate. For three years, Seymour Papert, colleagues and I created a learner-centered, project-based alternative learning environment for at-risk learners inside of a troubled prison for teens. When the needs, interests, passions, talents and curiosity of our students were put ahead of a random list of stuff, they were not only capable of demonstrating remarkable competence, but there was not a single discipline incident in ever that required a kid to leave the classroom.
Students can develop self-esteem by engaging in satisfying work. Classroom management is not required when teachers don’t view themselves as managers. Kids can learn “digital citizenship” while learning to program, sharing code and interacting online. They can feel safe at school by forming relationships with each of their teachers. Study skills are best gained within a context of meaningful inquiry.
Learning is the best way to learn. Accept no substitutes!
CMK Founder Gary Stager, Ph.D. gave a presentation in November 2012 about the philosophy and practice of Constructing Modern Knowledge. The following video is a recording of that presentation about the institute.