I once heard former President Clinton say, “every problem in education has been solved somewhere.” Educators stand on the shoulders of giants and should be fluent in the literature of their chosen field.  We should be reading all of the time, but summer is definitely an opportunity to “catch-up.”

Regrettably too many “summer reading lists for educators” are better suited for those concerned with get-rich quick schemes than enriching the lives of children. Case-in-point, the President of the National Association of Independent Schools published “What to Read this Summer,” a list containing not a single book about teaching, learning, or even educational leadership. Over the past few years, I offered a canon for those interested in educational leadership and a large collection of suggested books for creative educators and parents.

When I suggested that everyone employed at my most recent school read at least one book over the summer, the principal suggested I provide options. Therefore, I chose a selection of books that would appeal to teachers of different grade levels and interests, but support and inspire the school’s desire to be more progressive, creative, child-centered, authentic, and project-based.

Gandini, Lella et al… (2015) In the Spirit of the Studio: Learning from the Atelier of Reggio Emilia, Second Edition.
Aimed at early childhood education, but equally applicable at any grade level.  Illustrates how to honor the “hundred languages of children.”




Little, Tom and Katherine Ellison. (2015) Loving Learning: How Progressive Education Can Save America’s Schools
A spectacular case made for progressive education in the face of the nonsense masquerading as school “reform” these days.




Littky, Dennis. (2004) The Big Picture: Education is Everyone’s Business.
Aimed at secondary education, but with powerful ideas applicable at any level. Students spend 40% each week in authentic internship settings and the remaining school time is focused on developing skills for the internship. This may be the best book written about high school reform in decades. 

Papert, Seymour. (1993) The Children’s Machine: Rethinking School in the Age of the Computer.
A seminal book that situates the maker movement and coding in a long progressive tradition. This is arguably the most important education book of the past quarter century.  Papert worked with Piaget, co-invented Logo, and is the major force behind educational computing, robotics, and the Maker Movement.

Perkins, David. (2010) Making Learning Whole: How Seven Principles of Teaching Can Transform Education.
A clear and concise book on how to teach in a learner-centered fashion by a leader at Harvard’s Project Zero. 


Tunstall, Tricia. (2013) Changing Lives: Gustavo Dudamel, El Sistema, and the Transformative Power of Music.
“One of the finest books about teaching and learning I’ve read in the past decade.” (Gary Stager) Tells the story of how hundreds of thousands of students in Venezuela are taught to play classical music at a high level. LA Philharmonic Conductor Gustavo Dudamel is a graduate of “El Sistema.” The lessons in this book are applicable across all subject areas. 

Check out the CMK Press collection of books on learning-by-making by educators for educators!

I became a pre-k through 8th grade teacher in the mid-1980s. I was literally in the last teacher education cohort who was expected to learn how to teach science, music, art, physical education, special education, make puppets out of Pop-Tart boxes, create math manipulatives, and fill a classroom with interdisciplinary projects. Teacher preparation was equal parts art and science. Then around 1985, a couple of years after A Nation at Risklegislatures around the world declared, “Teaching ain’t nothin’,” and replaced rich and varied teacher education curricula with Animal Control and Curriculum Delivery.

Today, anyone who has ever been a billionaire or 7-11 night manager can run the US Department of Education or be a superintendent of schools, while well-prepared and experienced educators are met with suspicion and derision. We say that, “we stand on the shoulders of giants,” but ask a room full of school leaders how many of the authors in this reading list they have read and prepare to be stunned by the blank stares. Suggest any teaching practice not sold by Pearson and you’re likely to have a school principal reply, “Oh! You mean like Montessori?” Quite simply, unqualified is the new qualified.

Elementary teaching has been narrowed and departmentalized in ways that make it as ineffective as high school. Truly getting to know each child and to engage them in meaning making through interdisciplinary projects has been the first casualty of the assault on the art of teaching. As teacher agency has eroded through mistrust, prescriptive curriculum, and standardized testing, teachers become less, not more, thoughtful in their practice. When you mechanize teaching and place it under constant surveillance, teaching quality becomes less human, rewarding, joyful, creative, and more compliant.

Over the past thirty years, educators have lost control, freedom, and memory of classic pedagogical practices. During my work in classrooms around the world, I am often struck by how teachers are unaware of teaching practices I have long taken for granted. For example, I just assumed that every teacher knew about classroom centers, could defend their use, and make them a staple of each learning environment. I was wrong. That’s one of the reasons I wrote “Thoughts on Classroom Centers,” although I would still love to find the seminal work(s) on the topic.

Choice Time

While mentioning this lingering question to one of my heroes, Deborah Meier, she suggested I ask Renée Dinerstein. (I intend to) Ms. Dinnerstein is the author of a fine new book, Choice Time – How to Deepen Learning Through Inquiry and Play, PreK-2. The book focuses on the critical element of student choice and what they do during learner-centered classroom time. Classroom centers are the magic carpet of choice time.

I just purchased the book and cannot recommend it highly enough. It is a beautiful guide filled with clear and practical advice for teachers without being condescending or treating its readers like imbeciles. The book is not 500 pages of jargon and reproducibles, but rather 165 pages of inspiration intended to rekindle creative teaching in order to create more productive contexts for learning by children. It also helps teachers observe and understand the thinking of each child.

Although it says that the book’s wiscom is intended for PK-2nd grade, I would recomment the book to teachers at any grade level.

The author maintains a web site, investigatingchoicetime.com, intended to extend the inspiration shared in the book.


Since I am known as a man of impeccable taste and endless fascination, I humbly share a collection of the books I have recently purchased. Happy reading!
Isogawa, Yoshihito
Isogawa, Yoshihito
Martin, Laura C.
Pratt, Caroline
Curtis, D. & Carter, M

Making Sense of Algebra
Goldbenberg, E. Paul


Mathematical Mindsets – Unleashing Students’ Potential Through Creative Math, Inspiring Messages, and Innovative Teaching
Bowler, Jo

Schank, Roger C.
Naval Education And Training Program

Random stuff Amazon’s robots think you might enjoy…

Unlike most media outlets, The Huffington Post actually pretends to take an interest in education. However, I continue to believe that their Education section was created to be an advertising platform for the truly awful film, “Waiting for Superman,” remembered as the Howard the Duck of education documentaries by the three other schmucks and I who paid to see it.

Regardless of their motives, The Huffington Post, is a frequent mouthpiece for the charter school movement and unofficial stenographer for corporations trying to make a quick buck off the misery of teachers and students.

The Huffington Post recently featured an article, “The Most Popular Books For Students Right Now,” authored by their Education Editor Rebecca Klein. I clicked on the headline with interest, because I’m a fan of books and reading (I know a truly radical view for an educator). What I found was quite disappointing.

Aside from the fact that six books were the favorite across twelve grade levels, the books fell into two obvious camps; books kids like and books they were required to read by a teacher.

Nonetheless, data is data and Web users like lists.

What I do not like is when basic tenets of journalism, like “follow the money,” are ignored in order to mislead readers. The source for the “independent reading habits of nearly 10 million readers“ is Renaissance Learning, described by The Huffington Post as “an educational software company that helps teachers track the independent reading practices of nearly 10 million students.”

That’s like saying ISIS is a magazine publisher Donald Trump, owner of an ice cream parlor. While factually true, this is what Sarah Palin might call putting lipstick on a pig.

Renaissance Learning is a wildly profitable company that sells Accelerated Reader, a major prophylactic device for children who might otherwise enjoy reading. The product is purchased by dystopian bean counters who view small children as cogs in a Dickensian system of education where nothing matters more than data or achievement.

Their product creates online multiple-choice tests that schools pay for in order to quantify each child’s “independent” reading. If the school doesn’t own the test for a particular book a kid reads, they receive no credit. Kids routinely dumb down their reading in order to score better on the quizzes. Accelerated Reader rewards compliance and speed by turning reading into a blood sport in which winners will be rewarded and their classroom combatants, punished.

Ironically, I wrote about Accelerated Reader in The Huffington Post back in 2012. (Read Mission Accomplished)

When you look at the “favorite” book list featured in The Huffington Post, please consider that kids read The Giver and The Crucible because they are standard parts of the curriculum. This tells us nothing about what kids at grades 7, 8, or 11 actually like to read. Seeing Green Eggs and Ham as the first grade winner should make you sad. Can you imagine taking a comprehension test on this classic??? How vulgar!

The Grade 2 favorite is also likely assigned by teachers, Click, Clack, Moo: Cows That Type. The mind reels when I try to imagine the test measuring comprehension of the comic book/graphic novel, named favorite book by 3rd, 4th, 5th, AND 6th graders, Diary of a Wimpy Kid: The Long Haul. First of all, we should be alarmed that this simple book tops the charts for four years, but don’t forget that kids will be tested by a computer on their comprehension of this delightful comic book.

“Nothing forced can ever be beautiful.” – Xenophone

Caveat emptor!

This time of year, schools scramble to select a book for their entire faculty to read over the summer. Although it would be nice if everyone read the same book as a basis for common dialogue and for teachers to read more than one book about learning each year, I just assembled a list for the (DK-8) school where I serve as the Special Assistant to the Head of School for Innovation. Based on our overarching goals of action, reflective practice, progressive education, learning-by-making, energetic classroom centers, creativity, and collegiality, I recommended the following books for this summer. If a school community was to read one book (besides Invent To Learn – Making, Tinkering, and Engineering in the Classroom) , I would recommend David Perkins’ book, Making Learning Whole.

If you wish to give your faculty (K-12 in any configuration), a list of selections to choose from, I recommend the following in no particular order.
  1. Perkins, David. (2010) Making Learning Whole: How Seven Principles of Teaching Can Transform EducationA clear and concise book on how to teach in a learner-centered fashion. 
  2. Gandini, Lella et al… (2015) In the Spirit of the Studio: Learning from the Atelier of Reggio Emilia, Second EditionA beautiful and practical book aimed at early childhood education, but equally applicable at any grade level. 
  3. Littky, Dennis. (2004) The Big Picture: Education is Everyone’s BusinessAimed at secondary education, but with powerful ideas applicable at any level. This may be the best book written about high school reform in decades. 
  4. Tunstall, Tricia. (2013) Changing Lives: Gustavo Dudamel, El Sistema, and the Transformative Power of MusicOne of the finest books about teaching and learning I’ve read in the past decade. This lessons in this book are applicable across all subject areas. 
  5. Papert, Seymour. (1993) The Children’s Machine: Rethinking School in the Age of the ComputerA seminal book that situates the maker movement and coding in a long progressive tradition. This is arguably the most important education book of the past quarter century. 
  6. Little, Tom and Katherine Ellison. (2015) Loving Learning: How Progressive Education Can Save America’s Schools  A spectacular case made for progressive education in the face of the nonsense masquerading as school “reform” these days. 

You could also indulge yourself in the richest professional learning event of your life by participating in Constructing Modern Knowledge 2016. Limited space is still available.

The Best Invention and Tinkering Books, plus other cool stuff – including toys and kits

Available Now!

Available Now!

I’m enormously pleased that our publishing company, Constructing Modern Knowledge Press, has just released The Invent To Learn Guide to 3D Printing in the Classroom – Recipes for Success. The book is currently available in print and Kindle formats from Amazon.com.

The following is the text of the Foreword I wrote for the book. I hope you enjoy it.

3D printers are hot. They’re so hot that even schools are buying them. Although, schools are thought to be late adopters of emerging technology, I’ve been pleasantly surprised by how many already own 3D printers.

Investing in a school’s first 3D printer may be a down payment on the future of education; a future in which learning to learn with one’s head, heart, and hands will be equally critical. Making things is a great way to learn and an ability to make the things you need is an important 21st Century skill. The confidence and competence required to solve problems that the school curriculum or your teachers never anticipated will be the mark of a life well lived.

That said, once a school gets their 3D printer working reliably enough for each seventh grader to print an identical Yoda keychain, many educators are at a loss for next steps. That’s where this book comes in. David, Norma, and Sara share 18 projects designed to help teachers teach 3D design and enrich multiple curricular subjects.

Once you get the hang of 3D printing, you will realize how simple the hardware is. The real revolution may not be the printer as much as it is the democratization of design and the Z-axis.

For decades, CAD/CAM (computer-aided design/computer-aided manufacturing) software was too complicated and expensive for more than a few students to use. It was relegated to drafting classes and vocational settings. Now affordable and accessible software like Tinkercad make design child’s play. The ease of use associated with this new generation software does not mean that the design process has become any less rigorous. Design is where the mathematical reasoning, artistic sensibility, and engineering processes come to the fore.

We were all taught about the X- and Y-axes in school math class. Some of us may even use that coordinate system from time-to-time. However, with the exception of the occasional SAT question about the volume of a cylinder, you might conclude that we live in a 2D world. 3D printing and its design software bring us the Z-axis and provide an authentic context for using and understanding three-dimensional space. This book makes the conscious pedagogical decision to transition from 2D design to 3D artifact.

A common trope in educational discussions is, “Technology changes constantly.” Oh, if only that were true. If your school has spent two decades teaching kids to make PowerPoint presentations on subjects they don’t care about for an audience that doesn’t exist, then “technology” hasn’t changed much for you or your students (in school) since Alf went off the air.

In rare instances, there are revolutionary advances in technology that impact classroom practice. The technologies most closely associated with the maker movement, including: laser cutters, open-source microcontrollers like Arduino, and new ways to embed circuitry in everyday objects may indeed represent a paradigm shift in educational technology.

Since affordable and accessible 3D design is in its infancy, the authors provide you with experience exploring a variety of different software environments. You will also need to adapt instructions for the proclivities of your specific printer. Through this experience, you should be able to decide which software best meets the needs of you and your students. The hardware and software will change. Some of the companies producing your favorite software or printer may not last a school year. As a pioneer, you will need to remain flexible and on the lookout for better solutions. Once you find a software solution (or two) that works for you, use it. You don’t need to jump on every bandwagon or pretend that your students are learning something valuable because you keep changing software. Understanding which tools you choose to use and why is important.

In 1985, I flew cross-country to attend one of my first educational computing conferences. At the opening reception, I stumbled upon two gentlemen engaged in a mind-blowing discussion of Ada Lovelace’s work. One of the combatants was Brian Silverman and the other, David Thornburg. Over the past four decades, Brian and David have contributed as much as anyone in the world to what children are able to do with computers.

As I eavesdropped on the fascinating conversation, I silently vowed to spend the rest of my life in the company of smart people like the Lady Ada fans at that party. Fortunately for me, both men have become great friends and close colleagues. Prior to meeting David, I was familiar with his work through his many articles and the fantastic Logo books he authored. I had also taught with the Koala Pad, an affordable and reliable drawing tablet he had designed. David was already an accomplished mathematician, computer scientist, engineer, and designer with Xerox PARC and Stanford on his CV by the time I met him. Since then, David has been a great friend, collaborator, and trusted advisor.

David Thornburg has a knack for anticipating hot trends and getting educators excited about the future just around the corner. His presentations and countless books have inspired two genera- tions of teachers to use technology in a playful, deep, and constructive fashion.

3D printing and David fit each other like a hand and virtual reality glove. David is a renaissance man – part mathematician, part computer scientist, part engineer, part educator, part designer, part musician, part humorist, and full-time tinkerer.

My longtime colleagues in the Thornburg Center, Norma Thornburg and Sara Armstrong joined David in bringing this volume to life. They too have made indelible contributions to the field of education.

It seemed natural that Constructing Modern Knowledge Press would publish a book by David, Norma and Sara, which situates the 3D printing revolution in a classroom context. I commend you, brave pioneer, as you and your students design the future together.

— Gary Stager, PhD
Publisher, Constructing Modern Knowledge Press

Buy the book now!

©2014 Constructing Modern Knowledge Press - All rights reserved.

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.




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.

A boyhood dream has come true. I was interviewed by California School Business Magazine!

I certainly sized the opportunity to pull no punches. I left no myth behind.  Perhaps a few school business administrators will think differently about some of their decisions in the future.

A PDF of the article is linked below. I hope you enjoy the interview and share it widely!

Edtech Expert Discusses the Revolution in Computing

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

  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]

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:

  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.

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)
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.


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.


  1. Abelson, H., & diSessa, A. (1981). Turtle Geometry. Cambridge, MA: MIT Press.
  2. Clayson, J. (1988). Visual Modeling with Logo. Cambridge, MA: MIT Press.
  3. 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.
  4. Cuoco, A. (1990). Investigations in Algebra. Cambridge, MA: MIT Press.
  5. 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.
  6. 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.
  7. Goldenberg, E.P., & Feurzeig, W. (1987). Exploring Language with Logo Cambridge, MA: MIT Press.
  8. Harel, I. (1991). Children Designers: Interdisciplinary Constructions for Learning and Knowing Mathematics in a Computer-Rich School. Norwood, NJ: Ablex Publishing Corporation.
  9. Harel, I. & Papert, S. (editors) (1991). Constructionism. Norwood, NJ: Ablex Publishing Corporation.
  10. Harvey, B. (1982). Why Logo? Byte, Vol. 7, No.8, August 1982, 163-193.
  11. Harvey, B. (1985-87). Computer Science Logo Style, Volumes 1-3. Cambridge, MA: MIT Press.
  12. Kafai, Y. (1995) Minds in Play – Computer Design as a Context for Children’s Learning. Hillsdale, NJ: Lawrence Erlbaum and Associates.
  13. Lewis, P. (1990). Approaching Precalculus Mathematics Discretely. Cambridge, MA: MIT Press.
  14. National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
  15. Papert, S. (1980). Mindstorms: Children, Computers, and Powerful Ideas. (Second Edition, 1993) New York: Basic Books.
  16. Peterson, I. (1988). The Mathematical Tourist – Snapshots of Modern Mathematics. NY: W.H. Freeman and Company.
  17. Poundstone, W. (1985). The Recursive Universe… Chicago: Contemporary Books.
  18. Resnick, M. (1995). Turtles, Termites and Traffic Jams – Explorations in Massively Powerful MicroWorlds. Cambridge, MA: MIT Press.
  19. Silverman, B. (1987). The Phantom Fishtank: An Ecology of Mind. Montreal: Logo Computer Systems, Inc. (book with software for Apple II or MS-DOS)
  20. Stager, G. (October, 1988). “A Microful of Monkeys.” The Logo Exchange .
  21. Stager, G. (1990). “Developing Scientific Thought in a Logo-based Environment.” Proceedings of the World Conference on Computers in Education. Sydney, Australia: IFIP.
  22. 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.
  23. 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).


Few authors, activists, intellectuals or teachers move me like Jonathan Kozol. For nearly a half century, Kozol has given voice to the optimistic, playful, scared, sad and hungry children in our society. He spends time with the children most of us never think about and confronts us with our spiritual beliefs and the policies that most acutely affect the least of us in society. To meet a man with the greatness, humility, decency and literary genius of Kozol would be a miracle. To be able to work with him is a rare gift. To have him introduce me at Constructing Modern Knowledge 2011 as “one of my oldest friends in education” was a blessing I will never forget. Watch his CMK11 talk.

After far too long of a hiatus, Jonathan’s latest book, “Fire in the Ashes: Twenty-Five Years Among the Poorest Children in America,” is out today! I have read the galleys and the book is riveting, profound, tragic, hopeful and beautifully written. You should read it AND buy a copy for a friend or colleague. Click to buy from Amazon.com.

Jonathan Kozol & Gary Stager at CMK 2011

This school year, Constructing Modern Knowledge will expand beyond its unique summer institute (July 9-12, 2013 – Manchester, NH) to offer some exciting new learning opportunities for learners and parents. The first event by Constructing Modern Knowledge Productions is in collaboration with my colleagues at the Willows Community School in Culver City, California.

On September 10th at 7:00 PM, The Willows Community School will host An Evening with Jonathan Kozol, Acclaimed Author and Educational Activist. Due to the generosity and public mindedness of the school, the event is free and open to the public! Reservations are required via the web site.

At this event, Kozol will speak and sign his new book, Fire in the Ashes: Twenty-Five Years Among the Poorest Children in America. I hope you will join us for this very special evening!