How much practice is enough?

Do you see repetition as the enemy of a good math class? Or, do you see practice is the single biggest factor in learning? More practice might be better … it might be worse; however, repetition is not trivial in the learning process.

One reason I am thinking about repetition is the current emphasis with online homework systems, whether as part of redesigns like emporium or with modules or with ‘regular’ classes. Sometimes, these systems are marketed with an appeal to a high ‘mastery’ level (percent correct … not the same thing at all). To understand the impact of various practice arrangements, we need to review some cognitive psychology.

First, a lack of repetition normally places a high work load on short term memory; without repetition, the long-term memory (playing the role of ‘knowledge’ in this case) is anecdotal, like remembering the last web site you visited before leaving home. Without repetition, new knowledge does not become integrated with related knowledge. In the extreme, a contextualized math course has almost no repetition; each problem is a novel experience. In the science of cognition, this type of knowledge is called ‘declarative’ knowledge.

Second, the quality of the practice is a critical factor in how the information is stored. Much research has been done on factors that raise the quality of practice; in particular, ‘blocked’ (one type at a time) and ‘unblocked’ (mixed) both contribute to better learning. In my view, this is one of the major drawbacks of both online homework systems and modules … one objective at a time, practice on that, test and move on. (In cognitive science, ‘blocked’ is used strictly … same steps and knowledge used each time.)

Third, there is a connection between effective practice and math anxiety. As accuracy is established via repetition, anxiety can be lowered. [I am not claiming that practice, by itself, will lower anxiety. I am claiming that a lack of practice will reinforce the existing anxiety level.]

In the learning sciences, research talks about “automaticity” and “performance time”. Higher levels of automaticity are associated with faster performance time; both are factors in the brain’s efforts to organize information and ‘chunk’ material for easier recall.

Whatever class you are teaching, keep your practice consistent with your course goals. If you want students to organize knowledge, apply it to new situations, and improve attitudes, you should consider sufficient quantity and quality of practice.

Here are some references:

Cognitive Psychology and Instruction, 4th edition 2003 Bruning, Roger; Schraw, Gregory; Norby, Monica; Ronning, Royce (Pearson)

Beyond the Learning Curve: The Construction of the mind 2005 Speelman, Craig P and Kirsner, Ki (Oxford University Press)

Automaticity and the ACT* theory Anderson, John 1992 Available at http://act-r.psy.cmu.edu/publications/pubinfo.php?id=91

Radical Constructivism and Cognitive Psychology Anderson, John; Reder, Lynne; Simon, Herbert 1998 Available online at http://actr.psy.cmu.edu/~reder/98_jra_lmr_has.pdf

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cognition, Learning math | Jack Rotman | August 8, 2012 7:50 pm

1 Comment

By schremmer, August 9, 2012 @ 9:36 am

Practice is not the same as repetition.

Practice is often used in the sense of repetition as in “rehearse, run through, go over/through, work on/at; polish, perfect.” So, repetition has its points: it gets one used to a complex mechanism.

But, in learning, certainly in learning mathematics, one needs a space in which to “get acquainted” and in which to “experiment”, by which I do not mean some staggering attempts at new things, but rather small deviations, as when one tries one’s balance.

Inasmuch as “one learns from one’s mistakes”, the learner needs a space in which to make those “mistakes”: Why do I have to do this? One gets the answer by trying what happens when one does not do this.

Regards
–schremmer

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