**Teaching is a skill, and like any other skill it takes both practice and proper training to excel. I think that what sets me apart from other tutors is the amount of time and energy that I have put into seeking out the most effective methods and learning how to apply them. It is notoriously difficult to introspect on one’s methods and explain them, imagine trying to explain to someone all of the motions and coordinations that go into swinging a golf club or surfing. And it is also notoriously difficult to take abstract psychological theories and apply them to the real world. What I have tried to do here is bridge the gap. **

I’ve put the effort into it for many reasons. First, I find this interesting and I think that it has vastly improved my abilities as a teacher and as a student. Second, I know that your or your child’s education is of the highest importance to you and you want to get to know what your tutor’s principles are. In the following I try to explain the guidelines I use when teaching without the jargon, if you have any questions about any of these feel free to contact me.

• METAPHOR – Many of the difficulties that students encounter when learning mathematics stem from their inability to decipher the metaphorical properties of the language in which concepts are cast (Danesi 2007). Further, metaphor seems to play a central role in inferential reasoning from everyday reasoning to scientific and mathematical reasoning (Lakoff and Johnson 1980).

• “MIND – BASED MATHEMATICS” - Mathematics was not created in a vacuum. Rather, it is based on everyday cognitive mechanisms and once a student is able to understand this, she has overcome an initial, psychological barrier to success in mathematics.

PERSONAL RELAVENCE - People are much better at remembering information when it relates to them personally. I try to find out what individual student's interests are and make connections to the material at hand. Each one of my students is a special case to me. I spend a lot of time thinking about each individual student and how I can make the material appealing to them.

**• EMOTIONAL SALIENCE - Students learn better when they are into the material. A fundamental question that all students ask when they are asked to do school work is "why does this matter, who cares?" I think that this question deserves a well-thought answer, and I hope mine will help students become emotionally engaged in the material.**

• CONCEPTUAL REDEPLOYMENT - Often learning is a matter of redeploying a concrete, well-known concept to more general, abstract phenomena, rather than forming a completely new concept. A famous example of this is Newton's insight that the orbit of the moon around the Earth is just like the parabolic motion one observes when throwing a stone on the Earth. There is a school of thought called "Embodied Philosophy" (I'm a big fan) that says that all abstract concepts are cognitively rooted in lower functions. For example, our geometrical concepts (points, lines, parallel ect.) are rooted in our motor systems. So instead of telling students to memorize lists of theorems, I encourage students to think about problems in terms of their common experiences.

• FALSIFICATION - The only way to tell if a student has actually learned something is to test them on it. Many students don't understand the difference between passively understanding something and actively being able to produce it. I typically leave my students with short tests to do before the next session without any help to see where they're at. I am also crazy about flashcards, I carry around stacks of them for my own classes and I encourage students to do the same. I called this principle falsification because my teaching strategy is highly influenced by the work of Karl Popper (Conjectures and Refutations), who suggested that scientists work by boldly jumping to conjectures and then trying as hard as they can to falsify those conjectures. When problems arise I encourage students to make hypothesis about the solution and then come up with ways to test their hypothesis, rather than just giving them the correct answer.

• PLATONIC FALLACIES - Particularly in mathematics, students are presented with concepts as if they came from some Heaven in the sky. Ideas are presented in their polished form as if they are clever tricks of a Math Magician, who has some supernatural understanding of the way thing really are. This illusion causes many students to feel that there must be something wrong with them because they just "can't get it." In my view mathematics is a tool for describing reality that was invented by humans, and anyone who is willing to put enough effort into it can acquire the skills to use mathematics effectively.

• "HAS SEEN 1000 TREES, BUT NEVER A FORREST" - Again, my program is a BIG PICTURE program. I've found that the curriculum in public schools now is driven by a need to meet standards. Teachers want their students to perform well on standardized tests so that the school will receive more funding. Fine, but this causes many students to completely miss the essential links that ties everything together. I know that not everyone finds historical and philosophical questions fascinating, but sometimes in order to understand something like how to graph on the co-ordinate plane, one must ask for just a minute "what is a co-ordinate plane?"

• HISTORICAL PROGRESSION - The way information is often presented in schools makes students think that the concepts are dead, set in stone, immutable. This just isn't the case, everything that we think we know is always up for modification reinterpretation. I encourage students to see themselves as active participants in our living, breathing intellectual culture. I can't stress enough the importance of seeing oneself as an active agent in our historical progression.

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Bazzini, L. (2001). From grounding metaphors to technological devices. Eduacational Studies in Mathematics, 47: 259-271.

Danesi, M. (2007). A conceptual metaphor framework for the teaching of mathematics. Studies in the Philosophy of Education, 26: 225-236.

Danesi, M. (2003). Conceptual metaphor theory and the teaching of mathematics: Findings of a pilot project. Semiotica 145: 71-83.

Fauconnier, G., & Turner, M., (2002). The Way we Think: Conceptual Blending and the Mind’s Hidden Complexities, Basic Books, New York.

Frege, G. (1884, 1999). The Foundations of Arithmetic, Evanston: Northwestern University Press

Gold, B. (2001). Where mathematics comes from: How the embodied mind brings mathematics into being, Read this! The MAA Online book review column. Availible online at http://www.maa.org/reviews/wheremath.html

Lakoff, G. & Johnson, M. (1980). The Metaphors we Live by, The University of Chicago Press, Chicago.

Lakoff, G. Metaphors Homepage. http://cogsci.berkeley.edu/lakoff/metaphor

Lakoff, G. Nunez, R. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York.

Nunez, R., Edwards, L., & Matos, J. F. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Eduacational Studies in Mathematics, 39: 45-65.

Schoenfeld, A. (1987). Cognitive science and mathematics education: An Overview. Cognitive Science and Mathematics Education. Lawrence Erlbaum Associates. Hillsdale, New Jersey.

Schoenfeld, A. (2002a). A highly interactive discourse structure. In J. Brophy (Ed.), Social Constructivist Teaching: Its Affordances and Constraints (Volume 9 of the series Advances in Research on Teaching) 131-170. Amsterdam: JAI Press.

**Schedule a Session! Embodied Mathematics**