Mindstorms is so rich of powerful ideas that it is really hard to choose and pick one or two…. Reading the book is like taking a walk among almost every interesting concept I learned in my life, or, said in other words, meeting old friends again like Galileo, Aristoteles,Tarski, Poincare, Piaget and Bruner, all bringing new source of inspiration thanks to the powerful interpretation of their theory given by Papert. I’m particularly fascinated by Papert’s contributions about epistemology and its relationship with learning. It’s something related to my personal experience as primary school teacher with a background in theory of knowledge.

I’ve studied philosophy. During 5 years I used to spend my time enjoying the study of epistemology and logic of natural  language. It was wonderful passing the day in the library, reading, connecting abstract concept, playing with theories, comparing paradigms, analyzing my own process of learning and using language.

At the same time I was starting to feel a strong interest for education as political practice.  I discovered critical pedagogy and I got involved in several groups promoting child’s rights and new forms of  teaching/learning. I decided to convert this political passion into a job mixing my theoretical background with new practical pedagogical knowledge, and taking the challenge of promoting educational change in low income public school.

At that time I didn’t know that my epistemological background and my desire for educational change were strongly related and they were  part of my teacher’s “tool box”. At that time I was considering epistemology as something platonic, I mean idealistic, far from the ground,  not  a tool to use in education,  but just a tool to understand the history and development of knowledge.

I was convinced that the change needed in education was more about participation, human relationships, respect, freedom, and cooperation than about the structure of knowledge. I didn’t understand yet that how we think about knowledge affects how we think about us.

The teaching practice made me discover a new reality, more complex, less ideological and more epistemological. I started to enjoy using models in order to help kids become conscious of their own mental process, or physical activity in order to work on logical topics. The separation between my theoretical background and the practice was gone forever..epistemology, mental experiments, scientific paradigms, modularity, cognitive science,  history of science were everyday by my side, helping me making kids happy and comfortable about the knowledge they were building.

Below some Mindstorms ideas that I found very helpful for every teacher, not just math’s, to become conscious of the empowering effect of epistemology in learning and also to  limit the production of “cultural toxins” and  related damages on people.

• We need to promote a new sense of the child as a thinker, even as an epistemologist with the notion of the power of powerful ideas.
• It is necessary to encourage the child to become expert in recognizing and choosing among varying styles of thoughts, to acquire  multiple epistemology in order to build his own knowledge.
• Procedural thinking is a powerful mental tool needed for multiple epistemology.
• A domain of knowledge is a community of powerful ideas.
• To know a domain of knowledge is much like coming in a new community of people. The teacher plays the role of a mutual friend who provides introduction.
• How we think about knowledge affects how we think about us.
• The discrepancy between our experience of ourselves and our idealization of  knowledge has an effect: it intimidates us, it lessens the sense of our own competence and it leads us into counterproductive strategies for learning and thinking.
• The learner’s thinking is more like the mathematician’s than either is like the logical ideal.
• Powerful ideas help us grasping the world and have the capacity of helping us organizing our way to think about a particular class of problems. We don’t have to reorganize ourselves in order to use them. We put our skills and heuristic strategy into a kind of tool box- and while the interaction can, in the course of time, give rise to global changes, the act of learning is itself a local event.
• The importance of Tinkering: learning consist of building up a set of materials and tools that one can handle and manipulate.

Reading Papert’s Mindstorms: Children, Computers, and Powerful Ideas, my reactions and reflections went in several directions. As a historian, I was struck by some of the assumptions about the development of computers. In some ways, his predictions about the growth of the technology were right on, albeit a bit quaint sounding, “Readers who have never seen an interactive computer display might find it hard to imagine where this can lead. As a mental exercise they might like to imagine an electronic sketchpad, a computer graphic display of the not-too-distant future. This is a television screen that can display moving pictures in color. You can also ‘draw’ on it, giving it instructions, perhaps by typing, perhaps by speaking, or perhaps by pointing with a wand….” Sounds like an iPad to me. What struck me was his assumption that using such devices would continue to require students to learn how they worked, and be, at some basic level, masters of the technology. In fact, the opposite is true. Just like other technologies in our lives, we know less about what is making the computer work, about the code behind the graphical interface, than we do about our cars, our refrigerators, or our television sets. So does this amazing technological progress reflect any of the hope Papert had for computers as learning and teaching devices. I think my answer is maybe. When we use the computers to read, write, play video, record audio and video, or access the web, we are using a useful tool, we are perhaps opening up a gateway into a revolutionary new expanse of available knowledge, but we are not doing anything to help a child’s brain develop patterns for math, problem solving, or debugging.  However, when we use a few more creative tools in the classroom, movie making software and vector drawing programs come to mind from my own classroom experience, we do see students engaging in some of the same debugging activities that make LOGO so compelling.

When I started teaching digital storytelling in my 8th grade history classroom, I had to start with handouts, including screen shots, of how to use the basic features of iMovie. Today, when I assign a movie project, I don’t specify a program, and I don’t teach the software at all. The students, to a greater or lesser degree, know that part of the task is creating something, watching it, and seeing where the audio or the video did not work as they intended. Then they go back into the editing window and see what they can do to make it work.

All of this leads me on to reflect on the possibilities the computer presents, and the ways in which schools are, and are not, exploring the possibilities Papert points to as possibly fundamental in teaching and learning. It seems to me that we should have answers to some of his questions about how teaching programing (Turtle based, Scratch, whatever) have had any positive effects on students and “mathophobia.” I don’t know, however, if the presence of such evidence would be enough to move the behemoth that is American education in any useful direction.

I write this blog out of curiosity rather than authority. I do not think that I teach math. Even more, I decided in highschool that I was not “good at math,” and that decision prevented a potential career in a science lab at a scholarly institution. Instead, and one might argue for the better, I teach 5th and 6th grade science through the lens of making and problem solving. In the two years that I have been testing this curriculum, I have noticed that not only have my students (including the self-proclaimed “bad at math” students), but I too am developing a new love and appreciation of math through the work done in our fabrication lab, or FabLab. Having read Mindstorms; Children, Computers, and Powerful Ideas by Seymore Papert I was reminded of my new found “crush” on mathematics. Curiosity inevitably set in while reading Mindstorms and I wondered: Can we be learning math using tools that speak to the unconscious learning process? Can we build a sense of awe for mathematics using problem solving and making? And lastly, am I “teaching” math now, if the curriculum of problem solving and engineering is helping kids to see the beauty of math? My goal for this blog is to get a conversation going on how using Making in Education might get more students to love math for life, because doing so simply opens more doors for our students.

In the epilog of Mindstorms, Seymore Papert describes his “rejection of the dichotomy opposing a stereotypically ‘disembodied’ mathematics to activities engaging a full range of human sensitivities.” Here Papert is describing a very modern argument against the way mathematics is taught today.  In this essay, first published in the Wechsler edition of On Esthetics in Science, Papert presents the following questions in this essay:

1) Is the mathematical aesthetic really different from other systems or components of aesthetics (such as the arts)?

2) Does mathematical pleasure draw on its own pleasure principles or does it derive from those that animate other phases of human life (such as eating, procreating and exercise)?

3) Does mathematical intuition differ from common sense in nature and form, or only in content?

This last question is the one that relates to making in the classroom most, as it brings to mind the efforts of movement and manipulatives to teach math in more progressive settings, such as those seen in a Montessori classroom.

In the epilog, Papert goes on to share his understanding of distinguished mathematician Henri Poincaré’s view of mathematical education.  Poincaré stresses two ideas in his work, the first is the idea that seeing the beauty in mathematics, similar to that gained from the patterns in music or art, is fundamental to learning and doing math. “The distinguishing feature of the mathematical mind is not logical but aesthetic,” states Papert in response to Poincare’s viewpoint on what makes a mathematician. Poincaré’s attitude regarding beauty and mathematics is a living one, examined recently by authors such as Nathalie Sinclaire in a journal article entitled The Roles of the Aesthetic in Mathematical Inquiry (2009).

The second idea shown in Poincaré’s writing, and perhaps related to the first, was that mathematical creativity and understanding were the result of the unconscious, or intuition part of the brain, long before any non-anecdotal evidence existed to support that idea.  In “The Foundations of Science” by Henri Poincaré, published in 1908, Poincaré discusses the role of the unconscious in mathematical creativity and the understanding of a problem over time. Using his own success with mathematical creativity as a data set, he draws several conclusions about the role of the unconscious and problem solving.  Poincaré felt that “What the unconscious presents to the conscious mind is not a full and complete argument or proof, but rather ‘point of departure’ from which the conscious mind can work out the argument in detail. The conscious mind is capable of the strict discipline and logical thinking, of which the unconscious is incapable.” Support for these ideas is beginning to surface out of current research. In a study done in 2012 by Asael Y. Sklar. et.al, published in the Proceedings of the National Academy of Sciences, it was shown that “effortful arithmetic equations can be solved unconsciously.” If Poincaré and Papert are correct, then building and making from the earliest grades on can help to build a foundation for understanding more abstract mathematical concepts.

To learn more about this subject, I spoke to Diego Fonstad of the Bourne Idea Lab at the Castilleja School because he has been working on a concept using digital technologies to aid the production of analog, buildable objects for teaching math concepts. “I think that one element of concern (regarding math education) is that by the nature that we teach math we remove its beauty. We try to give it so much structure and logic that we’ve removed most people’s ability to see the inner beauty of it,” says Diego.  Diego is a firm believer that using making as a mode for learning, helps students gain an intuitive understanding of mathematical concepts. The hard part, Diego notes, is assessing these assumptions as educators.  “There so many externalities and other factors,” states Diego, “also (the efficacy of analog tools to teach math) would need to be measured longitudinally over many years.” Indeed, schools that offer maker style courses starting in pre-Kindergarten and Kindergarten would be ideal laboratories to study these ideas.  That is why Kickstarter projects such as Primo are so exciting, as they offer similar analog tools for teaching programming to kids. Projects such as Diego’s and Primo’s are offering a new kind of intuitive learning that educational settings such as Montessori style classrooms have already demonstrated a value for.

Lastly, in David and Goliath, by Malcom Gladwell, Gladwell spends a chapter explaining the drawbacks of attending an Ivy League school from the perspective of graduating students in STEM fields that eventually become employed in a STEM career.  His argument is that people do much better when they do not feel they occupy the bottom rungs of intellectual tiers, i.e. small fish in a big pond. Rather, students thrive in less renowned institutions when they are big fish in a little pond. Maybe the math classrooms of today that use only abstract learning methods, set up similar tier systems of who is “good at math,” hindering some students from seeing themselves as good math students, which leads to a drop in interest in STEM related careers. Maybe, the value of a FabLab style classroom, and potentially art studios as more art departments incorporate these digital technologies, is that math is made accessible through a wider range of learning styles.

Please share your ideas on this topic.  

References:

1. Asael Y. Sklara, Nir Levya, Ariel Goldsteinb, Roi Mandela, Anat Marila, and Ran R. Hassina. 2012. “Reading and doing arithmetic nonconsciously.” Psychology Department, Cognitive Science Department and Center for the Study of Rationality, Hebrew University, Jerusalem 91905, Israel

2. Sinclaire, Nathalie. 2004. “The Role of the Aesthetic in Mathematical Inquiry.” Mathematical Thinking and Learning Volume 6, Issue 3, pages 261-2843.

3. Papert, Seymore. 1978. “Poincaré and the Mathematical Unconscious” in Wechsler, J. (Ed.) On esthetics in science. Cambridge, MA: MIT Press.

4. Papert, Seymour. 1980. Mindstorms: children, computers, and powerful ideas. New York: Basic Books.

5. Lillard, Angeline and Else-Quest, Nicole. 2006. Evaluating Montessori Education. Science Magazine. Vol. 313 http://www.dmaball.org/ScienceLillard060929.pdf

6. Lopata, Christopher, Wallace, Nancy V.  and Finn, Kristin V. 2005. “Comparison of Academic Achievement Between Montessori and Traditional Education Programs.” Journal of Research in Childhood Education, Vol. 20 No. 1 http://www.pearweb.org/teaching/pdfs/Schools/Cambridge%20Montessori%20Elementary-Middle%20School/Articles/Montessori%20article.PDF

7. The Unconscious Mind According to Poincaré

I like when Seymour explain about the process to develop his writing as the analogy to new perspective to look at learning. The first ‘unacceptable draft’ that leads to revision with ‘critical  eyes’ and kind of self-assessment and develop work from feedback into presentable form, these steps made me look at the learning process in the different way. Looking at the mistakes as the opportunity to learn and develop not just for marking as failures  is really the key change.

Education is filled with acronyms and buzzwords, some invented by educators and others borrowed from industry and psychology and even popular culture memes. Why is it that an experience as basic as learning has been so sliced and diced into so many pieces that it has become unrecognizable? School vision statements are peppered with the buzz words of the day, false testament that these things are occurring on a regular basis within the walls of the school.

These buzzwords and acronyms hold teachers captive and somewhat chained to the ideas and decisions that too often come from above and from non-teachers. Actions speak louder than words, but in so many cases it seems that the words are all that matters.

This past fall I have attended and participated in a a number of really great workshops for educators. Smart and creative people pushing the boundaries and thinking of new ways to bring experiences and learning to their students. Teachers so excited about being immersed in hands on activities. Seeing some amazing prototypes and ideas being produced by teachers and educators who make it clear that there are some really really great teachers working in public and private schools across all grade levels.

But there is one common thread that I just don’t understand. The wrap ups, the conclusions, the “how would I use this in my classroom” is far too often, “I can’t”.

What are the reasons given for this?

• There isn’t a good rubric for it.
• It isn’t really aligned to the Common Core
• How would I be able to measure my students learning?
• I can’t have students doing different projects.
• What would the quiz look like?
• There wouldn’t be enough time to do this.

So then teachers sit around and begin discussing ways that they could force-fit what started as exciting and engaging work into the more formal and rigid boxes we like to call school.

• The idea would have to be modified, the rubric would need to be aligned to the Common Core
• The instruction would have to be differentiated and adhere to the Danielson Framework.
• The idea must “look” like the ideas that the policy makers and non-educators who decide upon what passes for good education in school.

and so on….

How do teachers beat this system of constraint? How can they teach outside the system, without fear of poor evaluations by administrators? Why must teachers feel so constrained by methodologies and structures they often do not believe to be effective?

I am interested in how I can measure learning and the success of my students and how I can use data to represent what is happening in my classroom, but only if the measurements and methods are in synch with the pedagogy that I hope takes place in my classroom.