Dear Dr. Williams:

Thank you so much for being the first ISTE executive or board member to address the sad state of affairs expressed by my old friend and mentor David Thornburg. It is disappointing that David’s proposal was rejected. Dr. Thornburg is a pillar of educational computing.

I am grateful to David for bringing attention to ISTE’s non-existent response to the life and death of Seymour Papert. It is worth noting that the father of our field, Dr. Papert, was never invited to keynote ISTE or NECC; not after the publication of his three seminal books, not after the invention of robotics construction kits for children, not after 1:1 computing was borne in his image in Australia, not after Maine provided laptops statewide, not when One Laptop Per Child changed the world. This lack of grace implies a rejection of the ideas Papert advocated and the educators who had to fight even harder to bring them to life against the tacit hostility of our premiere membership organization.

One would imagine that a conference dedicated to linoleum installation would eventually have the inventor of linoleum to address its annual gathering. Last year (2015), ISTE rejected my proposal to lead a session commemorating the 35th anniversary of Papert’s book *Mindstorms * and the 45th anniversary of the paper he co-authored with Cynthia Solomon, “Twenty Things To Do with a Computer.” See the blog post I wrote at the time.

Such indifference was maddening, but the failure of the ISTE leadership to recognize the death of Dr. Papert this past July, even with a tweet, is frankly disgraceful. After Papert’s death, I was interviewed by NPR, the New York Times and countless other news outlets around the world. I was commissioned to write Papert’s official obituary for the prestigious international science journal *Nature. *Remarkably, unless I missed it, ISTE has failed to honor Dr. Papert in any way, shape, or form. I have begged your organization to do so in order to bring his powerful ideas to life for a new generation of educators. These actions should not be viewed as a grievance or form of attention seeking. ISTE’s respect for history and desire to provide a forum for the free exchange of disparate ideas are critical to its relevance and survival.

Dr. Papert himself might suggest that ISTE is idea averse. In its quest to feature new wares and checklists, it neglects to remind our community that we stand on the shoulders of giants. Earlier this year, I was successful in convincing NCWIT to honor Papert’s colleague, Dr. Cynthia Solomon, with its Pioneer Award. If only I could be so persuasive as to convince ISTE to honor the “mother of educational computing” before it’s too late. As we assert in our book, *Invent To Learn*, without Papert and Solomon there is no 1:1 computing, no Code.org, no CS4All, no school robotics, no maker movement.

In light of Papert’s recent passing, and the remarkable 50th anniversary of the Logo programming language in 2017, I submitted two relevant proposals for inclusion on the 2017 ISTE Conference Program.

- Papert Matters: Celebrating the Life of the Father of Educational Computing
- Logo at 50: Children, Computers, and Powerful Ideas

You guessed it. Both were rejected.

Anniversaries and deaths are critical milestones. They cause us to, pause, reflect, and take stock. In 2017, there are several major conferences, including one I am organizing, focused on commemorating Papert and the 50th birthday of Logo. Sadly, ISTE seems to be standing on the sidelines.

It is not that I have nothing to offer on these subjects or do not know how to 1) write conference proposals or 2) fill an auditorium. As someone who has worked to bring Papert’s powerful ideas to life in classrooms around the world for 35 years and who worked with Papert for more than two decades, I have standing. I edited ISTE’s journal dedicated to the work he began, was the principal investigator on Papert’s last major institutional project, gave a TEDx talk in India on his contributions, and am the curator of the Seymour Papert archives at dailypapert.com. I worked in classrooms alongside Seymour Papert. Last year, 30 *accepted* ISTE presentations cited my work in their bibliographies.

I am often asked why I don’t just give up on ISTE. The answer is because educational computing is my life’s work. I signed the ISTE charter and have spoken at 30 NECC/ISTE Conferences. It is quite possible that no one has presented more sessions than I. For several years, I was editor of ISTE’s Logo Exchange journal and founded ISTE’s SIGLogo before it was killed by the organization. I have been a critical friend for 25 years, not to harm ISTE, but to help it live up to its potential.

For decades, David Thornburg and I have spoken at ISTE/NECC at our own expense. This is just one way in which I know that we are both committed to what ISTE can and should be. I have also written for ISTE’s *Learning and Leading with Technology*.

It would be my pleasure to discuss constructive ways to move forward.

Happy holidays,

*Gary*

Gary S. Stager, Ph.D.

CEO: Constructing Modern Knowledge

Co-author: Invent To Learn – Making, Tinkering, and Engineering in the Classroom

PS: Might I humbly suggest that ISTE hire or appoint a historian?

*Logo Exchange*. For more than two decades, Logo Exchange supported computer science in schools by igniting the curiosity and competence of teachers while using programming as a vehicle for powerful ideas. I had the great honor of serving as this important publication’s final editor.

*Logo Exchange*here. You would be surprised how much of

*Logo Exchange’s*wisdom is still useful in the classroom.

Join Dr. Gary Stager in a free Twitter Chat about computer programming in schools December 7, 2016. Learn more here.

*Creative Computing Magazine*had 400,000 subscribers in 1984. Thirty years ago, my friend and Constructing Modern Knowledge faculty member, Dr. Dan Watt, sold more than 100,000 books of Learning with Logo. Tens of thousands of educators taught children to program in the 1980s and then again after laptops were introduced in the 1990s. This was not for an hour, but over sufficient time to develop fluency.

Sphero is hardly the first programmable robot. My friend Steve Ocko developed Big Trak for Milton Bradley in the late 1970s. Papert, Resnick, Ocko, Silverman, et al developed LEGO TC Logo, the first programmable LEGO building system in 1987. (Watch Seymour Papert explain the educational benefits in 1987)

**Related articles:**

- The Invent To Learn Guide to Block Programming
- Programmable toys for controlling with the iPad
- The Secret Key to Girls and Computer Science
- Legislators Finally Admit the Obvious
- President Obama Discovers Coding – Yippee!

**Professional learning opportunities for educators:**

Constructing Modern Knowledge offers world-class hands-on workshops across the globe, at schools, conferences, and museums. During these workshops, teachers learn to learn and teach via making, tinkering, and engineering. Computer programming (coding) and learning-by-making with a variety of materials, including Sphero and Tickle. For more information, click **here**.

Hate to be a killjoy, but I just looked at one of the Code.org activities for programming turtle graphics in App Lab.

As someone who has taught various dialects of Logo to kids and teachers for 34+ years, I was horrified by the missed learning opportunities and design of the activity. My concerns are in lesson/interface design and lost learning opportunities.

https://studio.code.org/s/cspunit3/stage/2/puzzle/1

First of all, you connect any blocks and then hit Next. It doesn’t matter if you solve the actual problem posed or not.

Second and MUCH more importantly, ALL of the power and intellectual nutritional value of turtle geometry is sacrificed in order to teach a much simpler lesson in snapping blocks together in service of “efficiency.”

The power of turtle geometry is well – geometry, also measurement, and number. There are no numerical inputs to the turtle geometry blocks and all of the turns are in 90 degree increments.

The use of Javascript (presumably the blocks were added to the environment for this exercise and are not actual primitives) adds needless and confusing punctuation to the command structure WITHOUT the benefit of allowing users to change the input to FD or LT. Therefore, any opportunity to explore powerful mathematical ideas

As we approach the 50th anniversary of Logo and are celebrating the 35th anniversary of the publication of Mindstorms – Children, Computers, and Powerful Ideas, it sure would be nice if Code.org would learn some fundamental lessons of children, computers, and powerful ideas instead of depriving kids of an opportunity to learn mathematics while learning computer science.

**Discussion:**

Since posting the above statement to a CS discussion forum on Facebook, Hadi Partook – founder of Code.org responded as follows.

Gary, the goal of this course isn’t to teach turtle programming. Most of the students in our course sequence would have done that years earlier. This is a high school course to introduce students to JavaScript (including the syntax) and making apps. It begins with a few turtle stages because turtle programming would be familiar to these students as a concept fully explored in our CS Fundamentals courses – including all the geometric glory you mention, and problems that tell you whether you solved them or not. In our high school course the theme isn’t “solving puzzles” because it’s about “making apps,” with just a few turtle examples to carry forward from something students already know.

- Low engagement
- Limits on student creativity, exploration, and tinkering
- A missed opportunity for students to learn/use mathematical ideas
*while*learning Javascript

More than 20 years ago, a graduate student of mine, named Beth, (surname escapes me, but she had triplets and is a very fine high school math teacher) used an early version of MicroWorlds to program her own version of a toolkit similar to Geometer’s Sketchpad. Over time, I ran a similar activity with kids as young as 7th grade. I’ve done my best to piece together various artifacts from my archives into a coherent starting point for this potentially expansive activity. Hopefully, you’ll be able to figure out how to use the tools provided and improve or expand upon them.

Students (middle and high school) will use MicroWorlds EX create their own tool for exploring two-dimensional geometry similar to Geometers’ Sketchpad, Cabri, or GeoGebra. [1]

As students build functionality (via programming) into a tool for creating and measuring geometric constructions, they reinforce their understanding of important geometric concepts. As the tool gets more sophisticated, students learn more geometry, which in turn leads to a desire to explore more complex geometric issues. This is an ecological approach to programming. The tool gets better as you learn more and you learn more as the tool becomes more sophisticated.

Along the way, students become better programmers while using variables, list processing, and recursion in their Logo procedures. They will also engage in user interface design.

### Resources:

- Teacher and student project instructions
- MicroWorlds EX Geometry Toolkit starter template file
- An example of a more elaborate Geometry Toolkit created by Beth

[1] I would not show commercial models of the software to students until after they have programmed some new functionality into their own tools.

In November, I had a the great honor of working with my colleagues at the Omar Dengo Foundation, Costa Rica’s NGO responsible for computers in schools. For the past quarter century, the Fundacion Omar Dengo has led the world in the constructionist use of computers in education – and they do it at a national level!

While there, I delivered the organization’s annual lecture in the Jean Piaget Auditorium. The first two speakers in this annual series were Seymour Papert and Nicholas Negroponte.

The first video is over an hour in length and is followed but the audience Q & A. The second portion of the event gave me the opportunity to tie a bow on the longer address and to explore topics I forgot to speak about.

I hope these videos inspire some thought and discussion.

**Gary Stager “This is Our Moment “ – Conferencia Anual 2014 Fundación Omar Dengo **(Costa Rica)

San José, Costa Rica. November 2014

.

**Gary Stager – Questions and Answers Section – Annual Lecture 2014** (Costa Rica)

San José, Costa Rica. November 2014

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**Love is all you need**

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

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

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

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

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

Love,

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

**Computationally-Rich Activities for the Construction of Mathematical Knowledge – No Squares Allowed**

©1998 Gary S. Stager with Terry Cannings

This paper was published in the proceedings of the 1998 National Educational Computing Conference in San Diego

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

**Abstract
**The NCTM Standards state that

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

**Introduction**

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

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

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

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

- 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.

**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:

- 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
**“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

It’s odd

It’s 3 * 3 * 3 (an opportunity to introduce the concept of cubed numbers)

The sum of the digits = 9

The number is greater than 25

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

to 3n :number

print :number

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

end

to even? :number

output 0 = remainder :number 2

end

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

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

Fractal Geometry and Chaos Theory

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

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

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

- 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.

**Animation**

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

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

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

to double :number

output :number * 2

end

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

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

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

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

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

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

Date: Sun, 06 Nov 1994 09:52:44 -0400 (EDT)

Subject: fractions

To: gstager@pepperdine.edu

Dear Gary,

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

Interesting.

Yours,

Roger

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

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

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

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

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

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

Conclusion

Conclusion

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

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

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

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

**References**

- 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).

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.

Click here to register for Constructing Modern Knowledge 2013 today!