I’ve watched American Idol since its inception and am a fan. Months ago, I predicted that Angie would win this year. we will know for sure in a few weeks.

In the post-Simon Cowell years of American Idol, the quality of judging has become tedious, cloying and adoring of the young contestants. There has been little instructive teaching for the kids competing or the audience at home. That’s a shame because American Idol used to feature legendary artists every week as mentors who would perform a quickie masterclass for contestants (and audience) who otherwise would enjoy no such access to expertise. One of my favorite mentors a few years back was Harry Connick, Jr. It was also one of the lowest rated episodes of the season. Despite the relative (un)popularity of Mr. Connick, he taught the kids, played with them and wrote charts suited to their talents. He was a great mentor.

I was thrilled to see Harry back on Idol again this week and he ignited a firestorm when he refused to agree with the incredibly terrible advice being dispensed by an incredibly disingenuous Randy Jackson. You can the details of his awful advice in the well-written article linked below, but suffice to say that Mr. Jackson knows better. He may not have the talent and musical knowledge of Harry Connick, Jr., but he has enjoyed a great deal of success in the music business. If Randy Jackson had been paying for Kree’s studio time as a producer, his advice would have been exactly the same as that of Mr. Connick.

After Wednesday night’s show, an educator colleague of mine posted the following message on Facebook:

1. As Randy Jackson reminds us constantly, “this is a singing competition!” Singers should be able to sing anything.
2. The #1 album today is by Michael Bublé, a guy who sings the Great American Songbook. These classic songs are contemporary hits.

“The point Connick tried to make, which Jackson didn’t want to hear, was that the show’s contestants didn’t know these classic songs well enough to take liberties with their melodies and lyrics. In doing so, they were murdering the music.” – John Stark

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

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

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

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

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

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

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

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

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

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

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

Use testing stems as formative assessments and quizzes.

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

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

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

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

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

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

So, I watched the video…

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

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

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

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

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

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

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

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

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

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

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

Thank you

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

to double :number
output :number * 2
end

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

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

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

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

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

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

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

Dear Gary,

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

Interesting.
Yours,
Roger

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

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

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

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

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

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

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

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

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

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

References

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).

 

Recently, 5th and 6th grade girls in the school where I work came up to me in the hallway and volunteered, “I want to be an engineer.” While this is heartwarming, especially given the political rhetoric behind the importance of S.T.E.M. and the challenges of gender underrepresentation in the sciences, I would like to draw a totally different lesson for educators.

Anyone who knows anything about my teaching knows that I would never spend any time on “career education” with kids I teach. I create the context, conditions and projects   during which children are engaged in engineering. When building and programming robots, the kids are engineers – not contemplating a career for a dozen years later. The kids are smart enough to connect the dots and identify interest in a career related to their talent, interests or present mood, even if that interest is short-lived.

Time is the rarest of currencies in school. Therefore, time should be focused on authentic experiences, not meta experiences.

Affective qualities like collaboration, passion, curiosity, perseverance and teamwork are certainly desirable for teachers and students. However, these traits may be developed while engaged in real pursuits, even within the existing curriculum. All that is required is a meaningful project. This is why I question the use of “meta” activities like ropes courses, ice-breakers or trust-building exercises as a form of professional development or separate curriculum. Professional development resources are also scarce. Therefore, PD should be focused on learning to do or know. The affective skills should be byproducts of meaningful experiences intended to improve teaching.

Adults become better teachers when they enjoy firsthand learning adventures like they desire for their students. You can’t teach 21st Century Learners  if you haven’t learned this century. That is why I created Constructing Modern Knowledge.

Some educators have recognized that schools are too impersonal and that teachers should get to know their students. I could not agree more. However, the prescription is often to create advisory courses or extend homeroom to deal with pastoral care issues. The result is one teacher who gets to “know” students and time is borrowed from other courses where teachers should get to know their students formally and informally in the process of constructing knowledge together.

Sit next to a student engaged in a science experiment and talk with them. Lead vigorous discussions or chat with a kid about the book they’re reading. You don’t need a class period set aside for asking “How was your weekend?” or for building trust. Join a group of students for lunch. Say, “hi,” while passing in the hallway. Dennis Littky tells the story of making Time Magazine because as a school principal he greeted students when they entered school in the morning. Have we lowered our expectations so much that knowing students is some sort of awesome systemic accomplishment? Humane, thoughtful, even casual interaction between teachers and students does not require an NSF grant or special class.

When educators create a productive context for learning, achievement improves, students feel more connected and behavioral problems evaporate. For three years, Seymour Papert, colleagues and I created a learner-centered, project-based alternative learning environment for at-risk learners inside of a troubled prison for teens. When the needs, interests, passions, talents and curiosity of our students were put ahead of a random list of stuff, they were not only capable of demonstrating remarkable competence, but there was not a single discipline incident in ever that required a kid to leave the classroom.

Students can develop self-esteem by engaging in satisfying work. Classroom management is not required when teachers don’t view themselves as managers. Kids can learn “digital citizenship” while learning to program, sharing code and interacting online. They can feel safe at school by forming relationships with each of their teachers. Study skills are best gained within a context of meaningful inquiry.

Learning is the best way to learn. Accept no substitutes!

CMK Founder Gary Stager, Ph.D. gave a presentation in November 2012 about the philosophy and practice of Constructing Modern Knowledge. The following video is a recording of that presentation about the institute.

Click here to register for Constructing Modern Knowledge 2013 today!

 

Constructing Modern Knowledge may be the most important work of my career. For five years, we have demonstrated the competence and creativity of educators who spend four days of their summer vacation learning to learn in the digital age. I marvel at the complexity, sophistication and ingenuity illustrated by the educator’s projects created at Constructing Modern Knowledge. It is not an exaggeration to say that several of the projects created at CMK 2012 would have earned the creator(s) a TED Talk two years ago and an MIT Ph.D. five years ago.

CMK remains committed to creating a space where educators remake themselves by engaging in personally meaningful projects and learn through firsthand experience. It is NOT a conference. It is a samba school, laboratory, playground, library, maker space, film studio, atelier or workshop filled with people and objects to think with.

Constructing Modern Knowledge is a reflection of each participant. Some alums will say that CMK is about being at the forefront of the Maker movement, or about the Reggio Emilia approach, or about creativity, or robotics or filmmaking, or history, or school reform, or about S.T.E.M., or music composition or collaboration or visiting the MIT Media Lab. CMK is all of those things and what each participant makes of the experience.

Our remarkable faculty supports the learning of each participant and our guest speakers share a daily dose of inspiration. Given the diversity of the participants and the enormous range of projects created, CMK means different things to different people. So, what is CMK about?

Constructing Modern Knowledge is about:

• Jamming on a cupcake
• Looking up
• Looking in
• Cool tools
• Floating above the classroom
• Bringing Edison back to life
• Reinventing yourself
• Painting a piano
• Programming random Shakespearean insults
• Giving Lego a ukulele lesson
• Teaching a robot to use Twitter
• Becoming the next great YouTube filmmaker
• Getting lost in the flow
• Learning to solder
• Scoring a cartoon
• Snapping lots of photos
• Creating an animation
• Having lunch with your hero
• Sneaking around the MIT media lab
• Feeling smart
• Time lapse photography
• Laughing really hard
• Charging your iPhone by peddling a bike
• Tinkering
• Being a historian
• Working alone
• Working in teams
• Cool tools
• Aluminum foil
• Understanding astrophysics through dance
• Being silly
• Being serious
• A digital butler keeping your beer cold
• Engineering
• Secret ice cream
• Measuring your whiffle bat swing
• Manch Vegas
• Brightening a Rwandan child’s day
• Flow
• Fixing the future with air-curing rubber
• Makey Makey
• Conquering the geometry of islamic tiles
• Conductive paint
• Mathematical thinking
• Designing a video game
• Making friends
• Expanding your personal learning network
• Feeling smart
• Feeling foolish
• Confusion
• Finding science in your art and electronics in your peanut butter
• Satisfaction
• Scratch
• Learning to learn
• Bursting balloons
• The Reggio Emilia Approach
• Clarity
• Turning trash into treasure
• Reading
• MicroWorlds
• Constructionism
• Computer graphics
• Storytelling
• The 100 languages of children
• Chatting with Marvin Minsky
• Ingenuity
• Choreographed t-shirts
• Turtle Art
• Coffee with a legend
• Writing
• Progressive education
• Creativity unleashed
• Computing
• An amazing faculty
• Powerful ideas
• Changing the world
• A smile-controlled robot
• Exploring linguistic patterns of the 1940s
• Challenging yourself
• Sounding like Eleanor Roosevelt
• Brazilian churascaria
• Wearable computing
• Whimsy
• Never finding the pool
• Raising standards
• Blowing your mind
• MIDI
• Conversation
• Re-imagining education
• Expanding your comfort zone
• Being super awesome
• Taking off your teacher hat
• Putting on your learner hat
• Action!

Join the learning adventure with us July 9-12, 2013 in Manchester, NH!

Register today!

Download a printable brochure for Constructing Modern Knowledge 2013

 

 

Larry Ferlazzo invited me to share a vision of computers in education for inclusion in his Classroom Q&A Feature in Education Week. The text of that article is below.

You may also enjoy two articles I published in 2008:

1. What’s a Computer For? Part 1 – It all depends on your educational philosophy
2. What’s a Computer For? Part 2 – Computer science is the new basic skill

Technology is Not Neutral
Educational computing requires a clear and consistent stance
Gary S. Stager, Ph.D.
constructingmodernknowledge.com

There are three competing visions of educational computing. Each bestows agency on an actor in the educational enterprise. We can use classroom computers to benefit the system, the teacher or the student. Data collection, drill-and-practice test-prep, computerized assessment or monitoring Common Core compliance are examples of the computer benefitting the system. “Interactive” white boards, presenting information or managing whole-class simulations are examples of computing for the teacher. In this scenario, the teacher is the actor, the classroom a theatre, the students the audience and the computer is a prop.

The third vision is a progressive one. The personal computer is used to amplify human potential. It is an intellectual laboratory and vehicle for self-expression that allows each child to not only learn what we’ve always taught, perhaps with greater efficacy, efficiency or comprehension. The computer makes it possible for students to learn and do in ways unimaginable just a few years ago. This vision of computing democratizes educational opportunity and supports what Papert and Turkle call epistemological pluralism. The learner is at the center of the educational experience and learns in their own way.

Too many educators make the mistake of assuming a false equivalence between “technology” and its use. Technology is not neutral. It is always designed to influence behavior. Sure, you might point to an anecdote in which a clever teacher figures out a way to use a white board in a learner-centered fashion or a teacher finds the diagnostic data collected by the management system useful. These are the exception to the rule.

While flexible high-quality hardware is critical, educational computing is about software because software determines what you can do and what you do determines what you can learn. In my opinion the lowest ROI comes from granting agency to the system and the most from empowering each learner. You might think of the a continuum that runs from drill/testing at the bottom; through information access, productivity, simulation and modeling; with the computer as a computational material for knowledge construction representing not only the greatest ROI, but the most potential benefit for the learner.

Piaget reminds us ,“To understand is to invent,” while our mutual colleague Seymour Papert said, “If you can use technology to make things, you can make more interesting things and you can learn a lot more by making them.”

Some people view the computer as a way of increasing efficiency. Heck, there are schools with fancy-sounding names popping-up where you put 200 kids in a room with computer terminals and an armed security guard. The computer quizzes kids endlessly on prior knowledge and generates a tsunami of data for the system. This may be cheap and efficient, but it does little to empower the learner or take advantage of the computer’s potential as the protean device for knowledge construction.

School concoctions like information literacy, digital citizenship or making PowerPoint presentations represent at best a form of “Computer Appreciation.” The Conservative UK Government just abandoned their national ICT curriculum on the basis of it being “harmful and dull” and is calling for computer science to be taught K-12. I could not agree more.

My work with children, teachers and computers over the past thirty years has been focused on increasing opportunity and replacing “quick and easy” with deep and meaningful experiences. When I began working with schools where every student had a laptop in 1990, project-based learning was supercharged and Dewey’s theories were realized in ways he had only imagined. The computer was a radical instrument for school reform, not a way of enforcing the top-down status quo.

Now, kindergarteners could build, program and choreograph their own robot ballerinas by utilizing mathematical concepts and engineering principles never before accessible to young children. Kids express themselves through filmmaking, animation, music composition and collaborations with peers or experts across the globe. 5^th graders write computer programs to represent fractions in a variety of ways while understanding not only fractions, but also a host of other mathematics and computer science concepts used in service of that understanding. An incarcerated 17 year-old dropout saddled with a host of learning disabilities is able to use computer programming and robotics to create “gopher-cam,” an intelligent vehicle for exploring beneath the earth, or launch his own probe into space for aerial reconnaissance. Little boys and girls can now make and program wearable computers with circuitry sewn with conductive thread while 10^th grade English students can bring Lady Macbeth to life by composing a symphony. Soon, you be able to email and print a bicycle. Computing as a verb is the game-changer.

Used well, the computer extends the breadth, depth and complexity of potential projects. This in turn affords kids with the opportunity to, in the words of David Perkins, “play the whole game.” Thanks to the computer, children today have the opportunity to be mathematicians, novelists, engineers, composers, geneticists, composers, filmmakers, etc… But, only if our vision of computing is sufficiently imaginative.

Three recommendations:

1) Kids need real computers capable of programming, video editing, music composition and controlling external peripherals, such as probes or robotics. Since the lifespan of school computers is long, they need to do all of the things adults expect today and support ingenuity for years to come.

2) Look for ways to use computers to provide experiences not addressed by the curriculum. Writing, communicating and looking stuff up are obvious uses that require little instruction and few resources.

3) Every student deserves computer science experiences during their K-12 education. Educators would be wise to consider programming environments designed to support learning and progressive education such as MicroWorlds EX and Scratch.

I recently heard that a conference speaker told his audience, “We need fewer teachers and more facilitators.” My first reaction was, “1986 called and would like its keynote back.” My second thought was that the speaker is dead wrong!

The use of terms like “facilitator” always makes me queasy. The desire to rebrand teaching as facilitation results more from the low self-esteem of educators than either public opinion or a serious commitment to pedagogical progress. Regardless of the speaker’s intent, “teacher as facilitator” is a cliché that makes teaching sound more mechanistic and impersonal, not more. Modern medicine evolves and changes constantly, yet we still call its practitioners doctors. The invention of Viagra didn’t cause the public to make erector appointments. They call their doctor.

If one truly wants to improve the educational experience of children, then we need more teachers and fewer facilitators.

A popular parlor game among educators is debating the precise moment when “education went bad.” (Whether or not you believe there is a crisis in education.) A Nation at Risk, No Child Left Behind, Race-to-the-Top are often cited as the tipping point in the decline of K-12 education. I don’t blame a specific piece of legislation or blue-ribbon report entirely for the challenges faced by educators on a daily basis.

In my humble opinion, classrooms became less productive contexts for learning when teacher education became more concerned with training facilitators than creating teachers. The die was cast when professional educators accepted such dystopian rebranding as “facilitator.”

While earning my BA in teacher education during the early to mid-1980s, I was in the last class required to learn to play the piano a little bit, teach physical education, make puppets out of pop-tart boxes, create math manipulatives, design science experiments and setup a convivial classroom environment. When teaching was viewed as equal parts art and science, teacher education reflected that balance.

Around 1985, legislatures across the nation concluded that “teaching ain’t nuthin’” and changed credentialing requirements to ensure that teachers studied something “real” instead of education courses. Today, Teach-for-America spends five weeks preparing college grads to be teachers – less than half the time required for Marine Corps basic training and exponentially less time than I spent becoming an elementary school teacher. Educators know well that when elementary teacher preparation is less child-centered, secondary education gets even worse.

Today, new teachers truly are facilitators. They are “trained” to manage classrooms and deliver the curriculum handed to them. That’s about it.

This is great news for policy-makers and ideologues. Teachers are more compliant and less questioning than ever before. Flip the classroom? Sure! Tie teacher pay to standardized testing? Why not? Abandon labor protections secured by unionization? You betcha!

I remember being taught explicitly how to justify playing Scrabble for days or putting on a puppet show as educationally efficacious. This wasn’t just a “cover-your-ass in the plan book strategy,” but a way of understanding and articulating what your students were learning. The deafening calls for “accountability” are partially the result of teachers incapable of making learning visible. The less teachers have to think about their students’ thinking, the less thinking they do generally. Teaching needs to be thoughtful.

I have been stunned to observe the complete and utter return to whole class instruction in nearly every school I visit (public, private, rich, poor, urban, suburban and rural) everywhere in the world. New teachers have little or no experience with classroom centers, independent work, student projects and the sorts of agency that allow children to enjoy the “flow” experiences that build upon their obsessions and lead to understanding. Even when teachers are not lecturing from bell-to-bell, the classroom agenda is top-down and leaves little chance for serendipity or student initiative.

The most generous rationale for the Common Core Content Standards is that teachers lack a personal compass for what students should know and do. Teachers expert in inspiring long-term, personally meaningful and interdisciplinary projects or thematic instruction regularly exceed the standards, but that realization is lost on facilitators.

Great teachers know their students in deeper ways than any data can provide. They ask kids about their weekends. They chat about what kids are reading and console them when their hamster dies. Teachers spend thirty minutes per month in Toys R Us on the lookout for cool stuff to use in the classroom and as a means to learning about the culture of the children they serve. They learn continuously for themselves and their students. Teachers share their love of reading and are patrons of the arts. They are active citizens and engage students in current events. Outstanding teachers are not afraid to appear silly or create a whimsical classroom environment. They play in the snow with kindergarteners like Maria Knee.

The best thing we can do for children is to have them spend as much time with possible with interesting adults. So, great teachers need to be passionate, competent and interesting humans beyond the scope and sequence of the curriculum.

If we truly wish to make the world a better place for children, then we need many more teachers and a lot fewer facilitators!

While waiting for the 5th grade class to settle  down between recess and their holiday party, I wrote this project starter for creating arithmetic flashcard software in MicroWorlds. While the “math” isn’t particularly interesting or open-ended, there are plenty of opportunities for the students to improve and augment the software.

Bad drill and practice doesn’t become good because it is programmed in Logo, or by kids. However, the person who learns the most from “educational” software is the person who made it.

I thought of doing this because “practice multiplication facts” has been written on the classroom board for months. If the kids “write the software, perhaps they’ll think about multiplication a bit.

This is also an opportunity for introducing concepts, like percent, in order to create a cumulative score.

Download the PDF project starter by clicking the link below:

 A “Math” Game Only A Mother Could Love (PDF)

Treat yourself or the other makers in your life to these incredible new (or old favorite) materials and sources of inspiration for future learning adventures.

Be sure to click on the links at the bottom of this list for additional materials you’ll want under the tree.

All of the recommended products are affordable and may be purchased online with one-click!

Read out latest newsletter for creative educators. There you will find other book reviews and recommendations for stimulating learning adventures!

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