A Mind For Numbers

by Barbara Oakley, PhD

  9 Bilalić et al. 2008.

  10 Geary 2011. See also the landmark documentary A Private Universe, available at http://www.learner.org/resources/series28.html?pop=yes&pid=9, which led to much research into misconceptions in understanding science.

  11 Alan Schoenfeld (1992) notes that in his collection of more than a hundred “videotapes of college and high school students working unfamiliar problems, roughly sixty percent of the solution attempts are of the ‘read, make a decision quickly, and pursue that direction come hell or high water’ variety.” You could characterize this as focused thinking at its worst.

  12 Goldacre 2010.

  13 Gerardi et al. 2013.

  14 Hemispheric differences may sometimes be important, but again, claims in this area should be taken with caution. Norman Cook says it best when he notes: “Many discussions in the 1970s went well beyond the facts—as hemisphere differences were invoked to explain, in one fell swoop, all of the puzzles of human psychology, including the subconscious mind, creativity, and parapsychological phenomena—but the inevitable backlash was also exaggerated” (Cook 2002, p. 9).

  15 Demaree et al. 2005; Gainotti 2012.

  16 McGilchrist 2010; Mihov et al. 2010.

  17 Nielsen et al. 2013.

  19 Immordino-Yang et al. 2012.

  18 A differing layout of this problem was provided in de Bono 1970—that was the inspiration for the problem outlined here. De Bono’s classic book contains a wealth of such insightful problems and is well worth reading.

  20 Although I’m speaking of lobbing between the focused and diffuse modes, there appears to be an analogous lobbing process of information back and forth between the hemispheres. We can get some sense of how information might flow back and forth between the hemispheres in humans by looking at studies of chicks. Learning not to peck a bitter bead involves a complex back and forth processing of the memory traces between the hemispheres over a number of hours (Güntürkün 2003).

  Anke Bouma observes, “An observed pattern of laterality does not mean that the same hemisphere is superior for all of the processing stages required by a particular task. There are indications that the [right hemisphere] may be dominant for one stage of processing, while the [left hemisphere] may be dominant for another processing stage. The relative difficulty of a particular processing stage seems to determine which hemisphere is superior for a particular task” (Bouma 1990, p. 86).

  21 Just move the coins as shown—do you see how the new triangle will point down?

  Chapter 3: Learning Is Creating: Lessons from Thomas Edison’s Frying Pan

  1 The cerebral distance model developed by Marcel Kinsbourne and Merrill Hiscock (1983) hypothesizes that concurrent tasks will interfere more with one another the closer together the two tasks are processed in the brain. Two simultaneous tasks using the same hemisphere and particularly the same area of the brain can really mess things up (Bouma 1990, p. 122). Perhaps the diffuse mode may be more capable of handling several tasks at once because of the unfocused nature of diffuse processes.

  2 Rocke 2010, p. 316, citing Gruber 1981.

  3 Ibid., pp. 3–4.

  4 Kaufman et al. 2010, in particular the disinhibition hypothesis on pp. 222–224; Takeuchi et al. 2012.

  5 In attempting to track down the provenance of this legend, I corresponded with Leonard DeGraaf, an archivist with the Thomas Edison National Historical Park. He noted, “I have heard the story of Edison and the ball bearings but have never seen any documentation that would confirm it. I’m also not sure about the story’s origin. This may be one of those anecdotes that had some basis in reality but became part of the Edison mythology.”

  6 Dalí 1948, p. 36.

  7 Gabora and Ranjan 2013, p. 19.

  8 Christopher Lee Niebauer and Garvey 2004. Niebauer refers to the distinction between object and meta-level thinking. The third, paradoxical error in the sentence, incidentally, is that there is no third error.

  9 Kapur and Bielczyc 2012, contains an excellent review on the importance of failure in problem solving.

  10 For a nice discussion of the many variations of what Edison actually might have said or written, see http://quoteinvestigator.com/2012/07/31/edison-lot-results/

  11 Andrews-Hanna 2012; Raichle and Snyder 2007.

  12 Doug Rohrer and Harold Pashler (2010, p. 406) note: ”. . . recent analysis of the temporal dynamics of learning show that learning is most durable when study time is distributed over much greater periods of time than is customary in educational settings.” How this relates to alternation between focused and resting state networks is an important topic for future research. See Immordino-Yang et al. 2012. In other words, what I’ve described is a reasonable supposition for what occurs while we learn, but needs to be borne out by further research.

  13 Baumeister and Tierney 2011.

  14 I want to make it clear that these are only my “best guess” ideas about what might promote diffuse-mode thinking, based on where people seem to get many of their most creative, “aha!” insights.

  15 Bilalić et al. 2008.

  16 Nakano et al. 2012.

  17 Kounios and Beeman 2009, p. 212.

  18 Dijksterhuis et al. 2006.

  19 Short-term memory is the activated information that is not actively rehearsed. Working memory is the subset of short-term memory information that is the focus of attention and active processing (Baddeley et al. 2009).

  20 Cowan 2001.

  21 If you’re interested in the neural geography underlying all of this, it looks like both long-term memory and working memory use overlapping regions in the frontal and parietal lobes. But the medial temporal lobe is used only for long-term memory—not working memory. See Guida et al. 2012, pp. 225–226, and Dudai 2004.

  22 Baddeley et al. 2009, pp. 71–73; Carpenter et al. 2012. Spaced repetition is also known as distributed practice. Dunlosky et al. 2013, sec. 9, provides an excellent review of distributed practice. Unfortunately, as noted in Rohrer and Pashler 2007, many educators, particularly in mathematics, believe overlearning is a good way to boost long-term retention—hence many similar problems are assigned that ultimately devolve to make-work with little long-term benefit.

  23 Xie et al. 2013.

  24 Stickgold and Ellenbogen 2008.

  25 Ji and Wilson 2006; Oudiette et al. 2011.

  26 Ellenbogen et al. 2007. The diffuse mode may also be related to low latent inhibition—that is, being rather absentminded and easily distractable (Carson et al. 2003). There’s creative hope for those of us who tend to switch thoughts in the middle of a sentence!

  27 Erlacher and Schredl 2010.

  28 Wamsley et al. 2010.

  Chapter 4: Chunking and Avoiding Illusions of Competence: The Keys to Becoming an “Equation Whisperer”

  1 Luria 1968.

  2 Beilock 2010, pp. 151–154.

  3 Children learn through focused attention, but they also use the diffuse mode, with little executive control, to learn even when they are not paying focused attention (Thompson-Schill et al. 2009). In other words, it seems that children don’t need to use the focused mode as much as adults do when learning a new language, which may be why it’s easier for young children to pick up a new language. But at least some focused learning appears necessary to acquire a new language beyond early childhood.

  4 Guida et al. 2012, sec. 8. Recently, Xin Jin, Fatuel Tecuapetla, and Rui Costa revealed how neurons in the basal ganglia play an important role in signaling the concatenation of individual elements into a behavioral sequence—the essence of chunking (Jin et al. 2014). Rui Costa has received a 2 million euro grant to study the mechanism of chunking—his unfolding research will bear watching.

  5 Brent and Felder 2012; Sweller et al. 2011, chap. 8.

  6 Alessandro Guida and colleagues (2012, p. 235) noted that chunk creation apparently relies initially o
n working memory, which is in the prefrontal areas, and results from focused attention, which helps binds chunks. These chunks also begin to reside, with developing expertise, in long-term memory related to the parietal regions. A very different aspect of memory involves neural oscillatory rhythms, which help bind perceptual and contextual information from many areas of the brain (Nyhus and Curran 2010). See Cho et al. 2012 for an imaging study of the development of retrieval fluency in arithmetic problem solving in children.

  7 Baddeley et al. 2009, chap. 6; Cree and McRae 2003.

  8 Baddeley et al. 2009, pp. 101–104.

  9 The “big picture” I’m referring to can be thought of as a cognitive template. See Guida et al. 2012, in particular sec. 3.1. Templates arising from the study of math and science would naturally tend to be more amorphous than those arising from the crisp outlines of chess. Chunks, Guida notes, can be built very quickly, but templates, which involve functional reorganization, take time—at least five weeks or more (Guida et al. 2012). See also the discussion of schemata in Cooper and Sweller 1987; Mastascusa et al. 2011, pp. 23–43. Also useful in understanding these ideas related to developing expertise is the discussion in Bransford et al. 2000, chap. 2. Prior knowledge can be helpful in learning something new and related—but prior knowledge can also act as a hindrance, as it can make it more difficult to make changes in schemata. This is particularly noticeable with students’ erroneous embedded beliefs about basic concepts in physics, which are notoriously resistant to change (Hake 1998; Halloun and Hestenes 1985). As Paul Pintrich and colleagues (1993, p. 170) note: “a paradox exists for the learner; on the one hand, current conceptions potentially constitute momentum that resists conceptual change, but they also provide frameworks that the learner can use to interpret and understand new, potentially conflicting information.”

  10 Geary et al. 2008, pages 4-6 through 4-7; Karpicke 2012; Karpicke et al. 2009; Karpicke and Grimaldi 2012; Kornell et al. 2009; Roediger and Karpicke 2006. For reviews, see McDaniel and Callender 2008; Roediger and Butler 2011.

  11 Karpicke et al. 2009, p. 471. See also the Dunning-Kruger effect, where incompetent people mistakenly note their ability higher than they should. Dunning et al. 2003; Kruger and Dunning 1999; Ehrlinger et al. 2008; Bursonet et al. 2006.

  12 Baddeley et al. 2009, p. 111.

  13 Dunlosky et al. 2013, sec. 4.

  14 Longcamp et al. 2008.

  15 Dunlosky et al. 2013, sec. 7.

  16 See in particular Guida et al. 2012, which notes how experts learn to use long-term memory to expand their working memory. See also Geary et al. 2008, 4-5, which observes, “Working-memory capacity limits mathematical performance, but practice can overcome this limitation by achieving automaticity.”

  17 The solution to the anagram is “Madame Curie.” Attributed to Meyran Kraus, http://www.fun-with-words.com/anag_names.html.

  18 Jeffrey Karpicke and colleagues (2009) suggested the relationship between illusions of competence in learning and the difficulty of anagrams when you see the solution as opposed to when you don’t see the solution.

  19 Henry Roediger and Mary Pyc (2012, p. 243) note: “Professors in schools of education and teachers often worry about creativity in students, a laudable goal. The techniques we advocate show improvements in basic learning and retention of concepts and facts, and some people have criticized this approach as emphasizing “rote learning” or “pure memorization” rather than creative synthesis. Shouldn’t education be about fostering a sense of wonder, discovery, and creativity in children? The answer to the question is yes, of course, but we would argue that a strong knowledge base is a prerequisite to being creative in a particular domain. A student is unlikely to make creative discoveries in any subject without a comprehensive set of facts and concepts at his or her command. There is no necessary conflict in learning concepts and facts and in thinking creatively; the two are symbiotic.”

  20 Geary 2005, chap. 6; Johnson 2010.

  21 Johnson 2010, p. 123.

  22 Simonton 2004, p. 112.

  23 This is my own rephrasing of a common sentiment in science. Santiago Ramón y Cajal cited Duclaux in noting, “Chance smiles not on those who want it, but rather on those who deserve it.” Cajal went on to note, “In science as in the lottery, luck favors he who wagers the most—that is, by another analogy, the one who is tilling constantly the ground in his garden” (Ramón y Cajal 1999, pp. 67-68). Louis Pasteur noted, “In the fields of observation chance favors only the prepared mind.” Related expressions include the Latin-based proverb “Fortune favors the bold” and the British Special Air Service motto: “Who dares wins.”

  24 Kounios and Beeman 2009 [1897]; Ramón y Cajal 1999, p. 5.

  25 Rocke 2010.

  26 Thurston, 1990, p. 846–847.

  27 See the foundational work of Karl Anders Ericsson on development of expertise (e.g., Ericsson 2009). For insightful popular approaches related to the development of talent, see Coyle 2009; Greene 2012; Leonard 1991.

  28 Karpicke and Blunt 2011a; Karpicke and Blunt 2011b. For further information, see also Guida et al. 2012, p. 239.

  29 Of interest is that left hemisphere prefrontal regions appear active during the encoding phase of memorization, while right hemisphere regions are activated during retrieval. This has been reported by many groups using a great variety of imaging techniques (Cook 2002, p. 37). Is it possible that retrieving memorized materials creates the beginnings of diffuse-mode concept mapping-like connections? See also Geary et al. 2008, 4-6 to 4-7.

  30 There are, of course, caveats here. For example, what if a student is asked to recall material to determine what belongs on a concept map? There are also undoubtedly disciplinary differences. Some subjects, such as those involving communication processes in biological cells, inherently lend themselves more readily to “concept map” approaches in understanding key ideas.

  31 Brown et al. 1989.

  32 Johnson 2010, p. 110.

  33 Baddeley et al. 2009, chap. 8.

  34 Ken Koedinger, a professor of human-computer interaction and psychology at Carnegie Mellon University, notes, “To maximize retention of material, it’s best to start out by exposing the student to the information at short intervals, gradually lengthening the amount of time between encounters. Different types of information—abstract concepts versus concrete facts, for example—require different schedules of exposure” (quoted in Paul 2012).

  35 Dunlosky et al. 2013, sec. 10; Roediger and Pyc 2012; Taylor and Rohrer 2010.

  36 Rohrer and Pashler 2007.

  37 It appears that “mass practice” techniques of presenting the material provide an illusion of competence in teaching. Students appear to learn quickly, but as studies have shown, they forget quickly as well. Roediger and Pyc (2012, p. 244) note: “These outcomes show why teachers and students can be fooled into using strategies that are inefficient in the long run. When we learn we are so focused on how we are learning, we like to adopt strategies that make learning easy and quick. Blocked or massed practice does this. For better retention in the long run, however, we should use spaced and interleaved practice, but while we are learning this procedure seems more arduous. Interleaving makes initial learning more difficult, but is more desirable because long term retention is better.”

  38 Rohrer et al. 2013.

  39 Doug Rohrer and Harold Pashler (2010, p. 406) observe: “. . . the interleaving of different types of practice problems (which is quite rare in math and science texts) markedly improves learning.”

  40 Personal communication with the author, August 20, 2013. See also Carey 2012.

  41 Longcamp et al. 2008.

  42 For examples, see http://usefulshortcuts.com/alt-codes.

  Chapter 5: Preventing Procrastination: Enlisting Your Habits (“Zombies”) as Helpers

  1 Emsley 2005, p. 103.

  2 Chu and Choi 2005; Graham 2005;
Partnoy 2012.

  3 Steel (2007, p. 65) notes: “Estimates indicate that 80%–95% of college students engage in procrastination . . . approximately 75% consider themselves procrastinators . . . and almost 50% procrastinate consistently and problematically. The absolute amount of procrastination is considerable, with students reporting that it typically occupies over one third of their daily activities, often enacted through sleeping, playing, or TV watching . . . Furthermore, these percentages appear to be on the rise . . . In addition to being endemic during college, procrastination is also widespread in the general population, chronically affecting some 15%–20% of adults.”

  4 Ainslie and Haslam 1992; Steel 2007.

  5 Lyons and Beilock 2012.

  6 Emmett 2000.

  7 See extensive discussion in Duhigg 2012, which in turn cites Weick 1984.

  8 Robert Boice (1996, p. 155) noted that procrastination appears to involve a narrowing of the field of consciousness. See also pp. 118–119.

  9 Boice 1996, p. 176.

  10 Tice and Baumeister 1997.

  11 Boice 1996, p. 131.

  Chapter 6: Zombies Everywhere: Digging Deeper to Understand the Habit of Procrastination

  1 McClain 2011; Wan et al. 2011.

  2 Duhigg 2012, p. 274.

  3 Steel 2010, p. 190, citing Oaten and Cheng 2006 and Oaten and Cheng 2007.

  4 Baumeister and Tierney 2011, pp. 43–51.

  5 Steel 2010, citing the original work of Robert Eisenberger, 1992, and others.

  6 Ibid., p. 128-130, referring in turn to the work of Gabriele Oettingen.

  7 Beilock 2010, pp. 34-35.

  8 Ericsson et al. 2007.

  9 Boice 1996, pp. 18–22.

  10 Paul 2013.

  Chapter 7: Chunking versus Choking: How to Increase Your Expertise and Reduce Anxiety

  1 One important point is that much of the literature on experts involves individuals who have trained for years to attain their level of expertise. But there are differing levels of experts and expertise. For example, if you know the acronyms FBI and IBM, it’s easy to remember the sequence as a chunk of two rather than a disparate grouping of six letters. But this easy chunking presumes that you are already an expert, not only with the meaning of FBI and IBM, but with the Roman alphabet itself. Imagine how much more difficult it would be to memorize a Tibetan sequence like this: