When we are learning math and science in the classroom, we are starting with some degree of expertise, and what we are expected to learn through the course of a semester is nothing like the vast jump in expertise experienced as a novice becomes a grand master at chess. When you are taking a class in some subject, you’re not going to see a dramatic neural difference occurring in one semester, similar to the dramatic difference between a novice and a grandmaster. But there is some indication that neural differences in how you process the material can show up even in a period of a few weeks (Guida et al. 2012). More specifically, Guida and colleagues note that experts preferentially make use of the temporal regions, which are crucial for long-term memory (2012, p. 239). In other words, when we steer students away from building structures in long-term memory, we are making it more difficult for them to acquire expertise. Of course, concentration on memorization alone without creative application is also a problem. Again—any teaching method alone can be misused; variety (not to mention competence) is the spice of life!
2 We’ve talked about interleaving the study of different techniques while you are studying a topic. But what about interleaving the study of completely different subjects? Unfortunately, there’s no research literature available on that as yet (Roediger and Pyc 2012, p. 244), so what I’m suggesting about varying what you are studying is simply common sense and common practice. This will be an interesting area to watch for future research.
3 Kalbfleisch 2004.
4 Guida and colleagues (2012, pp. 236–237) note that chunks in working memory and therefore in long-term memory (LTM) “get larger with practice and expertise . . . the chunks get also richer because more LTM knowledge is associated with each one of them. Moreover, several LTM chunks can become linked to knowledge. And eventually, if an individual becomes an expert, the presence of these links between several chunks can result in the creation of high-level hierarchical chunks. . . . For example, in the game of chess, templates can link to ‘. . . plans, moves, strategical and tactical concepts, as well as other templates’. . . . We suggest that the functional reorganization of the brain can be detected in expertise acquisition when LTM chunks and knowledge structures exist and are effective in the domain of expertise.”
5 Duke et al. 2009.
6 For a good review of the circumstances when deliberate practice is most effective, see Pachman et al. 2013.
7 Roediger and Karpicke 2006, p. 199.
8 Wan et al. 2011. This study sought to define the neural circuits responsible for rapid (within two seconds) intuitive generation of the best next move in spot games of shogi, an extraordinarily complex game of strategy. The part of the brain associated with quick, implicit, unconscious habit (the precuneus-caudate circuit) appeared central to the rapid generation of the best next move in professional players. See also McClain 2011.
9 Charness et al. 2005.
10 Karpicke et al. 2009; McDaniel and Callender 2008.
11 Fischer and Bidell 2006, pp. 363–370.
12 Roediger and Karpicke 2006, citing William James’s Principles of Psychology.
13 Beilock 2010, pp. 54–57.
14 Karpicke and Blunt, 2011b; Mastascusa et al. 2011, chap. 6; Pyc and Rawson 2010; Roediger and Karpicke 2006; Rohrer and Pashler 2010. John Dunlosky and colleagues, in their in-depth review of various learning techniques (2013), rate practice testing as having high utility because of its effectiveness, broad applicability, and ease of use. See also Pennebaker et al 2013.
15 Keresztes et al. 2013 provides evidence that testing promotes long-term learning via stabilizing activation patterns in a large network of brain areas.
16 Pashler et al. 2005.
17 Dunlosky et al. 2013, sec. 8; Karpicke and Roediger 2008; Roediger and Karpicke 2006.
Chapter 8: Tools, Tips, and Tricks
1 Allen 2001, pp. 85, 86.
2 Steel 2010, p. 182.
3 Beilock 2010, pp. 162–165; Chiesa and Serretti 2009; Lutz et al. 2008.
4 For those with an interest, please see the resources listed at the Association for Contemplative Mind in Higher Education, http://www.acmhe.org/.
5 Boice 1996, p. 59.
6 Ferriss 2010, p. 485.
7 Ibid., p. 487.
8 Fiore 2007, p. 44.
9 Scullin and McDaniel 2010.
10 Newport 2012; Newport 2006.
11 Fiore 2007, p. 82.
12 Baddeley et al. 2009, pp. 378–379.
Chapter 9: Procrastination Zombie Wrap-Up
1 Johansson 2012, chap. 7.
2 Boice 1996, p. 120; Fiore 2007 chap. 6.
3 Ibid., p. 125.
4 Amabile et al. 2002; Baer and Oldham 2006; Boice 1996, p. 66.
5 Rohrer, et al. (in press).
6 Chi et al. 1981.
7 Noesner 2010.
8 Newport 2012, particularly chap. 1 (“Rule #1”).
9 Nakano et al. 2012.
10 Duhigg 2012, p. 137.
11 Newport 2012.
12 See Edelman 2012 for many such ideas.
Chapter 10: Enhancing Your Memory
1 Eleanor Maguire and colleagues (2003) studied individuals renowned for outstanding memory feats in forums such as the World Memory Championships. “Using neuropsychological measures, as well as structural and functional brain imaging,” they found “superior memory was not driven by exceptional intellectual ability or structural brain differences. Rather, [they] found that superior memorizers used a spatial learning strategy, engaging brain regions such as the hippocampus that are critical for memory and for spatial memory in particular.”
Tony Buzan has done much to bring the importance of memory techniques to the popular eye. His book Use Your Perfect Memory (Buzan, 1991) provides further information about some popular techniques.
2 Eleanor Maguire and colleagues (2003) note that memory techniques are often regarded as being too complicated to use, but some techniques, such as the memory palace, can indeed be very natural and helpful in allowing us to remember information that is important to us.
3 Cai et al. 2013; Foer 2011. Denise Cai and colleagues’ work indicates that specialization in one hemisphere (often the left) for language is accompanied by similar specialization in the other hemisphere for visuospatial capabilities. Specialization of a function in one hemisphere, in other words, appears to cause specialization of the other function in the other hemisphere.
4 Ross and Lawrence 1968.
5 Baddeley et al. 2009, pp. 363–365.
6 http://www.ted.com/talks/joshua_foer_feats_of_memory_anyone_can_do.html.
7 http://www.skillstoolbox.com/career-and-education-skills/learning-skills/memory-skills/mnemonics/applications-of-mnemonic-systems/how-to-memorize-formulas/.
8 A sense of the importance of spatial reasoning is provided in Kell et al. 2013.
Chapter 11: More Memory Tips
1 Two sources of information related to metaphor in late-nineteenth-century physics are Cat 2001 and Lützen 2005. For metaphor in chemistry and more broadly throughout science, see Rocke 2010, in particular chap. 11. See also Gentner and Jeziorski 1993. Imagery and visualization are beyond the scope of any single book—see, for example, the Journal of Mental Imagery.
2 As leading mathematical modeler Emanuel Derman notes: “Theories describe and deal with the world on its own terms and must stand on their own two feet. Models stand on someone else’s feet. They are metaphors that compare the object of their attention to something else that it resembles. Resemblance is always partial, and so models necessarily simplify things and reduce the dimensions of the world. . . . In a nutshell, theories tell you what something is; models tell you merely what something is like” (Derman 2011, p. 6).
3 Solomon 1994.
4 Rocke 2010, p. xvi.
5 Ibid., p. 287, citing Berichte der Durstigen Chemischen Gesellschaft (
1886), p. 3536. This was a mock issue of the nonexistent “durstigen” (thirsty) Chemical Society. The parody was sent to the subscribers of the Berichte der deutschen chemischen Gesellschaft and is virtually impossible to find today, since it was actually a spurious issue.
6 Rawson and Dunlosky 2011.
7 Dunlosky et al. 2013; Roediger and Pyc 2012. In a review of student flash card use, Kathryn Wissman and colleagues (2012, p. 568) observed: “students understand the benefits of practising to higher criterion levels (amount of practice) but do not typically implement or understand the benefits of practising with longer lags (timing of practice).”
8 Morris et al. 2005.
9 Baddeley et al. 2009, pp. 207–209.
10 In this book, you might think I’ve discussed all of the components of the SQ3R for study (sometimes SQ4R—for Survey, Question, Read, Recite, Review and wRite). So you might ask why I haven’t explored this method further in the text. The SQ3R was developed by psychologist Francis Pleasant Robinson as a general study tool. Central to the study of math and science is problem solving—the SQ3R approach simply doesn’t lend itself to this. I’m not the only one to notice. As physics professor Ronald Aaron and his son Robin Aaron note in Improve Your Physics Grade, ”. . . one Psychology text suggests studying by the SQ3R method. . . . For effective note taking in class it suggests the LISAN approach. . . . Do you believe that such approaches can help you? Do you believe in Santa Claus? The Easter Bunny?” (Aaron and Aaron 1984, p. 2).
11 Curiously, it appears very little work has been done in this area—what little is available seems to simply affirm that writing things out by hand helps us assimilate information better than typing. See Rivard and Straw 2000; Smoker et al. 2009; Velay and Longcamp 2012.
12 Cassilhas et al. 2012; Nagamatsu et al. 2013; van Praag et al. 1999.
13 Guida et al. 2012, p. 230; Leutner et al. 2009.
14 Levin et al. 1992 describes how students who use mnemonics outperform students who apply contextual and free learning styles.
15 Guida et al. 2012 points out that training in memory techniques can speed up the process of acquiring chunks and knowledge structures, thus helping people become experts more rapidly by allowing them to use part of their long-term memory as working memory.
16 Baddeley et al. 2009, pp. 376–377, citing research by Helga and Tony Noice (2007).
Chapter 12: Learning to Appreciate Your Talent
1 Jin et al. 2014.
2 Partnoy 2012, p. 73. Partnoy goes on to note: “Sometimes having an understanding of precisely what we are doing unconsciously can kill the natural spontaneity. If we are too self-conscious, we will impede our instincts when we need them. Yet if we aren’t self-conscious at all, we will never improve on our instincts. The challenge during a period of seconds is to be aware of the factors that go into our decisions . . . but not to be so aware of them that they are stilted and ineffectual” (Partnoy 2012, p. 111).
3 Partnoy 2012, p. 72, citing Klein 1999.
4 Klein 1999, p. 150, citing Klein and Klein 1981. But note the small sample size in Klein and Klein 1981.
5 Mauro Pesenti and colleagues (2001, p. 103) note, “We demonstrated that calculation expertise was not due to increased activity of processes that exist in non-experts; rather, the expert and the non-experts used different brain areas for calculation. We found that the expert could switch between short-term effort-requiring storage strategies and highly efficient episodic memory encoding and retrieval, a process that was sustained by right prefrontal and medial temporal areas.”
Already in 1899 brilliant psychologist William James wrote, in his classic Talks to Teachers on Psychology: “You now see why ‘cramming’ must be so poor a mode of study. Cramming seeks to stamp things in by intense application immediately before the ordeal. But a thing thus learned can form but few associations. On the other hand, the same thing recurring on different days, in different contexts, read, recited on, referred to again and again, related to other things and reviewed, gets well wrought into the mental structure. This is the reason why you should enforce on your pupils habits of continuous application” (William 2008, [1899], p. 73).
6 In a classic study, William Chase and Herbert Simon (1973) found that the intuitive generation of next moves by chess experts is based on the superior, quick perception of patterns that has been achieved through practice. Fernand Gobet and colleagues (2001, p. 236) define a chunk as “a collection of elements having strong associations with one another, but weak associations with elements within other chunks.”
7 Amidzic et al. 2001; Elo 1978; Simon 1974. A figure of 300,000 chunks was cited by Gobet and Simon 2000.
8 Gobet 2005. Gobet goes on to note that expertise in one domain doesn’t transfer to another. That’s true—certainly if you learned Spanish, it’s not going to help you when you go to order sauerkraut in Germany. But the metaskills are important. If you learn how to learn a language, you can pick up a second language more easily.
That, again, is where developing an expertise in something like chess can be quite valuable—it provides a set of neural structures that are similar to those you need when learning math and science. Even if the neural structures are as simple as you need to internalize the rules of the game—that’s a valuable insight.
9 Beilock 2010, pp. 77–78; White and Shah 2006.
10 Indeed, there is modest support for this type of finding in the research literature. See Simonton 2009.
11 Carson et al. 2003; Ellenbogen et al. 2007; White and Shah 2011.
12 Merim Bilalić and colleagues (2007) point out that some players with an IQ of between 108 and 116 fell into the elite player group by virtue of their extra practice. The elite group had an average IQ of 130. See also Duckworth and Seligman 2005.
Nobel Prize winner Richard Feynman liked to tout his relatively low IQ score of 125 as evidence that you could go pretty far whatever tests might indicate about your intelligence. Feynman clearly had natural smarts, but even as a youngster he was practicing obsessively in developing his mathematical and physical knowledge and intuition (Gleick 1992).
13 Klingberg 2008.
14 Silverman 2012.
15 Felder 1988. See also Justin Kruger and David Dunning (1999), who note “the miscalibration of the incompetent stems from an error about the self, where the miscalibration of the highly competent stems from an error about others.”
Chapter 13: Sculpting Your Brain
1 DeFelipe 2002.
2 Ramón y Cajal 1937, 309.
3 Ramón y Cajal 1999 [1897], pp. xv–xvi; Ramón y Cajal 1937, p. 278.
4 Ramón y Cajal 1937, 154.
5 Fields 2008; Giedd 2004; Spear 2013.
6 Ramón y Cajal 1999 [1897].
7 Bengtsson et al. 2005; Spear 2013.
8 Cajal could clearly plan well—witness his construction of the cannon. But he couldn’t seem to make the connection with the bigger picture consequences of his actions. Taken up with the exciting task of blowing up a neighbor’s gate, for example, he couldn’t make the obvious prediction that he would be in deep trouble as a consequence. See Shannon et al. 2011, with their intriguing finding that functional connectivity in troubled teens connects the dorsolateral premotor cortex to the default-mode network (“a constellation of brain areas associated with spontaneous, unconstrained, self-referential cognition” p. 11241). As troubled teens mature and their behavior improves, the dorsolateral premotor cortex instead appears to begin connecting with the attention and control networks.
9 Bengtsson et al. 2005; Spear 2013; Thomas and Baker 2013. As Cibu Thomas and colleagues note (p. 226), “the evidence from animal studies suggests that the large-scale organization of axons and dendrites is very stable and experience-dependent structural plasticity in the adult brain occurs locally and is transient.” In other words, we can make modest changes in our brain, but we can’t indulge in whol
esale rewiring. This is all commonsense stuff. For a terrific popular book on brain plasticity, see Doidge 2007. The best technical approach to this topic is Shaw and McEachern 2001. It is fitting that Cajal’s own work is now gaining recognition as foundational in our understanding of brain plasticity (DeFelipe 2006).
10 Ramón y Cajal 1937, p. 58.
11 Ibid., pp. 58, 131. The ability to grasp the key ideas—the gist of the problems—appears to be more important than verbatim ability to memorize. Verbatim as opposed to “gist” memories seem to be encoded differently. See Geary et al. 2008, 4–9.
12 DeFelipe 2002.
13 Ramón y Cajal 1937, p. 59.
14 Root-Bernstein and Root-Bernstein 1999, pp. 88–89.
15 Bransford et al. 2000, chap. 3; Mastascusa et al. 2011, chaps. 9–10.
16 Fauconnier and Turner 2002.
17 Mastascusa et al. 2011, p. 165.
18 Gentner and Jeziorski 1993.
Chapter 14: Developing the Mind’s Eye through Equation Poems
1 Plath 1971, p. 34.
2 Feynman 2001, p. 54.
3 Feynman 1965, 2010.
4 This section is based on the wonderful paper by Prentis (1996).
5 Excerpts from the song “Mandelbrot Set,” © Jonathan Coulton, by kind permission of Jonathan Coulton. Lyrics excerpted from song fully given at http://www.jonathancoulton.com/wiki/Mandelbrot_Set/Lyrics.
6 Prentis 1996.
7 Cannon 1949, p. xiii; Ramón y Cajal 1937, p. 363. In a related vein, see Javier DeFelipe’s extraordinary Butterflies of the Soul, which contains some of the beautiful illustrations produced in the early days of research in neuroscience (DeFelipe 2010).
8 Mastascusa et al. 2011, p. 165.
9 Keller 1984, p. 117.
10 See discussions of elaborative interrogation and self-explanation in Dunlosky et al. 2013.
A Mind For Numbers Page 26
