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endnotes
Chapter 1: Open the Door
1 I’d like to point educators toward the book Redirect, by psychology professor Timothy Wilson, which describes the seminal importance of failure-to-success stories (Wilson 2011). Helping students change their inner narratives forms one of the important goals of this book. A leader in describing the importance of change and growth in mindset is Carol Dweck (Dweck 2006).
2 Sklar et al. 2012; Root-Bernstein and Root-Bernstein 1999, chap. 1.
Chapter 2: Easy Does It: Why Trying Too Hard Can Sometimes Be Part of the Problem
1 Default-mode network discussions: Andrews-Hanna 2012; Raichle and Snyder 2007; Takeuchi et al. 2011. More general discussion of resting states: Moussa et al. 2012. In a very different line of investigation, Bruce Mangan has noted that William James’s description of the fringe includes the following feature: “There is an ‘alternation’ of consciousness, such that the fringe briefly but frequently comes to the fore and is dominant over the nucleus of awareness” (Cook 2002, p. 237; Mangan 1993).
2 Immordino-Yang et al. 2012.
3 Edward de Bono is the grand master of creativity studies, and his vertical and lateral terminology is roughly analogous to my use of the terms focused and diffuse (de Bono 1970).
Astute readers will notice my mention that the diffuse mode seems to sometimes work in the background while the focused mode is active. However, research findings show that the default-mode network for example (which is just one of the many resting state networks), seems to go quiet when the focused mode is active. So which is it? My sense as an educator and a learner myself is that some nonfocused activities can continue in the background when focused work is taking place, as long as the focused attention is shifted away from the area of interest. In some sense, then, my use of the term diffuse mode might be thought of as “nonfocused mode activities directed toward learning” rather than simply “default-mode network.”
4 There are also a few tight links to more distant nodes of the brain, as we’ll explore later with the attentional octopus analogy.
5 The diffuse mode may also involve prefrontal areas, but it probably has more connections overall and less filtering out of seemingly irrelevant connections.
6 Psychologist Norman Cook has proposed that “the first elements in a central dogma for human psychology can be expressed as (1) the flow of information between the right and left hemispheres and (2) between the “dominant” [left hemisphere] and the peripheral effector mechanisms used for verbal communication” (Cook 1989, p. 15). But it should also be noted that hemispheric differences have been used to launch countless spurious overextrapolations and inane conclusions (Efron 1990).
7 According to the National Survey of Student Engagement (2012), engineering students spend the most time studying—senior engineering students spend eighteen hours on average per week preparing for class, while senior education students spend fifteen hours and senior social science and business students spend about fourteen hours. In a New York Times article titled “Why Science Majors Change Their Minds (It’s Just So Darn Hard),” emeritus engineering professor David E. Goldberg has noted that the heavy demands of calculus, physics, and chemistry can initiate the “math-science death march” as students wash out (Drew 2011).
8 For a discussion of evolutionary considerations in mathematical thinking, see Geary 2005, chap. 6.
Of course, many abstract terms aren’t related to mathematics. A surprising number of these types of abstract ideas, however, relate to emotions. We may not be able to see those terms, but we can feel them, or at least important aspects of them.
Terrence Deacon, author of The Symbolic Species, notes the inherent complexity of the encryption/decryption problem of mathematics:
“Imagine back when you were first encountering a novel kind of mathematical concept, like recursive subtraction (i.e., division). Most often this abstract concept is taught by simply having children learn a set of rules for manipulating characters for numbers and operations, then using these rules again and again with different numbers in hopes that this will help them ‘see’ how this parallels certain physical relationships. We often describe this as initially learning to do the manipulations ‘by rote’ (which is in my terms indexical learning) and then when this can be done almost mindlessly, we hope that they will see how this corresponds to a physical world process. At some point, if all goes well, kids ‘get’ the general abstract commonality that lies ‘behind’ these many individual symbol-to-symbol and formula-
to-formula operations. They thus reorganize what they already know by rote according to a higher-order mnemonic that is about these combinatorial possibilities and their abstract correspondence to thing manipulation. This abstraction step is often quite difficult for many kids. But now consider that this same transformation at a yet higher level of abstraction is required to understand calculus. Differentiation is effectively recursive division, and integration is effectively recursive multiplication, each carried out indefinitely, i.e., to infinitesimal values (which is possible because they depend on convergent series, which themselves are only known by inference, not direct inspection). This ability to project what an operation entails when carried out infinitely is what solves Zeno’s paradox, which seems impossible when stated in words. But in addition to this difficulty, the Leibnizian formalism we now use collapses this infinite recursion into a single character or the integral sign) because one can’t actually keep writing operations forever. This makes the character manipulation of calculus even less iconic of the corresponding physical referent.
“So the reference of an operation expressed in calculus is in effect doubly-encrypted. Yes, we’ve evolved mental capacities well-suited to the manipulation of physical objects, so of course this is difficult. But math is a form of ‘encryption,’ not merely representation, and decryption is an intrinsically difficult process because of the combinatorial challenges it presents. This is why encryption works to make the referential content of communications difficult to recover. My point is that this is intrinsic to what math is, irrespective of our evolved capacities. It is difficult for precisely the same reason that deciphering a coded message is difficult.
“What surprises me is that we all know that mathematical equations are encrypted messages, for which you need to know the key if you want to crack the code and know what is represented. Nevertheless, we wonder why higher math is difficult to teach, and often blame the educational system or bad teachers. I think that it is similarly a bit misplaced to blame evolution.” (Personal communication with the author, July 11, 2013.)
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