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.
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