Page 244 - A Mind For Numbers: How to Excel at Math and Science

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 on
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.