(II) The Role of Implicit and Explicit Learning in Mathematics
"We Learn By Doing


"Cognitive psychology is the branch of
psychology concerned with the functioning of the brain, especially
in regards to mental processes operating in response to stimuli,
such as how well a student learns in response to different teaching
methods" (Collins and
Harding, 2004, p. 9). In the literature of cognitive
psychology, the "learning by doing" process described in the
quotation on the lefthand page is called implicit learning. This
is the way most humans learn their first language. This is also the
way most humans learn to play a sport or a musical instrument, or
learn how to dance. However, contrary to Mr. Holt’s
contention in his quote, it is not the only way humans learn. There
is another process called explicit learning, and it is this process
which is more common and more effective in the learning of
mathematics.
Implicit learning is characterized by imitating behavior without
necessarily acknowledging the underlying rules that determine this
behavior. For example, children learn to speak their first language
employing correct rules of grammar without being able to name those
rules of grammar. On the other hand, explicit learning is rules
oriented. Rules are learned, practiced in drills, and then
implemented in problem situations. This is the most common way that
adults learn a second language. They memorize the alphabet of the
new language, they learn some vocabulary, they learn the
rudimentary rules of grammar, and then they attempt to string words
together according to the rules of grammar to generate simple
sentences. With enough practice, this leads to an ability to
converse in that language.
The rulesoriented approach to the teaching of mathematics is
immediately recognizable by all. As children we are taught to
memorize addition facts and then multiplication facts. Then we are
taught algorithms for using these facts, and simple but abstract
relationships between addition and subtraction and multiplication
and division to enable us to subtract and divide whole numbers. In
junior high school we are taught to employ these facts and these
rules in more complex procedures for dealing with all the analogous
arithmetic operations for ratios of whole numbers, fractions. Facts
to be memorized, rules to be learned, practice to merge the two
until they both become secondnature, then more complicated rules
to be learned which build on the simple ones, and more practice to
merge it all until it becomes secondnature, and on and on and on
it goes. In fact, it goes on beyond what most naturallyinclined
implicit learners can tolerate after the age of twelve. That is
when most implicit learners stop acquiring new knowledge of
mathematics.
If this is so, why can’t mathematics be taught to those
individuals implicitly? To a point, it can. In fact, once rules of
mathematics have become secondnature as described in the previous
paragraph, then invocation of those rules is more in keeping with
the process of implicit learning. That is, correct rules are
employed without explicit acknowledgement of that rule at that
moment. At this ideal stage of learning mathematics, this process
is really a combination of the two. Unfortunately, without a lot of
drill and practice of all the basic skills in mathematics, the
learner will not achieve this ideal.
Another element of implicit learning that makes it illsuited to
the learning of mathematics is the lack of transference. In
general, implicit learning is associative and context bound, while
explicit learning is based on hypothesis testing and transfers
across contexts (Berry &
Dienes, 1993). Many basic mathematics skills can be learned
associatively, but skills learned this way will be context bound
and of limited value without an explicit understanding. For
example, a child who cannot describe the conceptual meaning of
multiplication but is merely drilled on multiplication tables, has
in effect learned multiplication facts associatively. As a result,
this child might be able to quickly state the answer to 7 x 9 but
would fail to use multiplication when asked about the number of
cups on a tray with seven rows, each with nine cups in it.
Automaticity of a skill is about making an explicitly learned skill
also available implicitly. Thus, in the context or contexts in
which the skill is highly overlearned (that is, is second nature)
it will be appropriately and quickly retrieved with limited
investment of mental resources.
Initially, it is essential that mathematical skills be learned
explicitly if those skills are to be useful across different
contexts without learning and practicing them in all potentially
relevant situations. This explicit accessibility facilitates the
use of that skill in new contexts.
Driving an automobile is a skill most of us acquire explicitly
which eventually becomes automatic. This skill is ordinarily
executed with so little demand on resources that most of us fail to
remember what we did and what traffic signs we saw on the way to
work. On the other hand, we can explicitly retrieve and describe
the relevant subskills when teaching someone to drive.
Furthermore, we do explicitly use and monitor these skills when
road conditions are poor.
This is the ideal for mathematical skills to be useful in problem
solving. When routine situations are encountered, we want
invocation of mathematical procedures to be automatic so that
working memory can attend to higher order thinking. On the other
hand, when unfamiliar problem situations in mathematics confront
us, we want to be able to retrieve explicit rules and use our
higherorder thinking to creatively manipulate those rules to fit
the new context (May,
Rabinowitz and Mantyka, 2002).
 taken from The
Math Plague, 2007, pp. 1417.
For more detail, please see May, Rabinowitz and
Mantyka, 2002, pp. 1819.