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(II) The Role of Implicit and Explicit Learning in Mathematics

"We Learn By Doing

Not many years ago I began to play the cello. Most people would say that what I am doing is 'learning to play' the cello. But these words carry into our minds the strange idea that there exists two very different processes: (1) learning to play the cello; and (2) playing the cello. They imply that I will do the first until I have completed it, at which point I will stop the first process and begin the second. In short, I will go on 'learning to play' until I have learned to play' and then I will begin to play. Of course, this is nonsense. There are not two processes, but one. We learn to do something by doing it. There is no other way."


- John Holt, taken from
Chicken Soup for the Soul
(Canfield and Hansen, 1993, p. 129)

"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 left-hand 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 rules-oriented 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 second-nature, then more complicated rules to be learned which build on the simple ones, and more practice to merge it all until it becomes second-nature, and on and on and on it goes. In fact, it goes on beyond what most naturally-inclined 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 second-nature 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 ill-suited 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 over-learned (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 sub-skills 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 higher-order thinking to creatively manipulate those rules to fit the new context (May, Rabinowitz and Mantyka, 2002).


- taken from The Math Plague, 2007, pp. 14-17.

For more detail, please see May, Rabinowitz and Mantyka, 2002, pp. 18-19.



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