Practice

• Practice Makes Perfect -- But Only If You Practice Beyond the Point of Perfection by Daniel T. Willingham, American Educator, American Federation of Teachers (AFT), Spring 2004. "It is difficult to overstate the value of practice. For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary. This column summarizes why practice is so important and reviews the different effects of intense short-term practice versus sustained, long-term practice. That students would benefit from practice might be deemed unsurprising. After all, doesn't practice make perfect? The unexpected finding from cognitive science is that practice does not make perfect. Practice until you are perfect and you will be perfect only briefly. What's necessary is sustained practice. By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year). This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.

• Learning May Unify Distant Brain Regions by B. Bower, Science News, March 6, 1999. A tantalizing bit of evidence in favor of repetition and practice: "In studies of many animal species, the brain shows signs of exerting less effort as individuals learn to perform simple tasks or to recognize relationships between repeatedly presented items. ... Neural activity slackened in both pathways on later trials, as individuals demonstrated better knowledge ... pathways increasingly pool their efforts during learning trials." (Emphasis added)

• "Applications and Misapplications of Cognitive Psychology to Mathematics Education", by John R. Anderson, Lynne M. Reder and Herbert A. Simon, Department of Psychology, Carnegie Mellon University. Excerpt: "This criticism of practice (called 'drill and kill,' as if this phrase constituted empirical evaluation) is prominent in constructivist writings. Nothing flies more in the face of the last 20 years of research than the assertion that practice is bad. All evidence, from the laboratory and from extensive case studies of professionals, indicates that real competence only comes with extensive practice (e.g., Hayes, 1985; Ericsson, Krampe, Tesche-Romer, 1993). In denying the critical role of practice one is denying children the very thing they need to achieve real competence. The instructional task is not to 'kill' motivation by demanding drill, but to find tasks that provide practice while at the same time sustaining interest. Substantial evidence shows that there are a number of ways to do this; 'learning-from-examples,' a method we will discuss presently, is one such procedure that has been extensively and successfully tested in school situations."

• Why is it that practice makes perfect? Researchers begin to understand how the brain changes to accommodate new learning and hone motor skills. by Beth Azar, APA Monitor, American Psychological Association, January 1996. Practice may not always make you perfect, but it is likely to make a lasting impression on your brain. Two new brain studies have found that repetitive motor sequences can trigger changes in the parts of the brain that accept sensory information and control motor function. These changes may explain why coordination and ability improve with practice on tasks such as typing and playing an instrument. This research adds another dimension to a growing body of work on brain plasticity--the brain's ability to alter its circuitry."

• In Defense of "Mindless Rote" by Ethan Akin, professor of mathematics, City College of New York. What is the role of "learning the basics" in mathematical operations and fundamental algorithms? How do repeated practice and mastery of computation lead to "higher-order thinking" later? Here's a wonderful essay that gets to the heart of these questions!

• The Expert Mind by Philip E. Ross, Scientific American, July 24, 2006. "[M]uch of the chess master's advantage over the novice derives from the first few seconds of thought. This rapid, knowledge-guided perception, sometimes called apperception, can be seen in experts in other fields as well. Just as a master can recall all the moves in a game he has played, so can an accomplished musician often reconstruct the score to a sonata heard just once. And just as the chess master often finds the best move in a flash, an expert physician can sometimes make an accurate diagnosis within moments of laying eyes on a patient. But how do the experts in these various subjects acquire their extraordinary skills? ...
       "So, too, experienced physicists may on occasion examine more possibilities than physics students do. Yet in both cases, the expert relies not so much on an intrinsically stronger power of analysis as on a store of structured knowledge. ...
       "The conclusion that [chess] experts rely more on structured knowledge than on analysis is supported by a rare case study of an initially weak chess player, identified only by the initials D.H., who over the course of nine years rose to become one of Canada's leading masters by 1987. Neil Charness, professor of psychology at Florida State University, showed that despite the increase in the player's strength, he analyzed chess positions no more extensively than he had earlier, relying instead on a vastly improved knowledge of chess positions and associated strategies."

• Discovery Learning in Math: Exercises Versus Problems by Barry Garelick, Nonpartisan Education Review, Vol.5, No.2, 2009. Ed school theorists promote the notion that traditional math courses provide merely mechanical and algorithmic approaches that do not lead to "off the script" thinking. This article takes a close look at how so-called "exercises" do indeed lead to discovery and to unscriptlike thinking. Excerpt:
       "Whether in driving, math, or any other undertaking that requires knowledge and skill, the more expertise one accumulates, the more one can depart from the script and successfully take on novel problems. It's essential that at each step, students have the tools, guidance, and opportunities to practice what they learn. It is also essential that problems be well posed. Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems. As a result, the challenge is in figuring out what they mean -- not in figuring out the math. Well-posed problems that push students to apply their knowledge to novel situations would do much more to develop their mathematical thinking."
       "Students given well-defined problems that draw upon prior knowledge, as described in this article, are doing much more than simply memorizing algorithmic procedures. They are developing the procedural fluency and understanding that are so essential to mathematics; and they are developing the habits of mind that will continue to serve them well in more advanced, college level mathematics courses. Poorly-posed problems with multiple 'right' answers turn mathematics into a frustrating guessing game. Similarly, problems for which students are expected to discover what they need to know in the process of solving it do little more than confuse. But well-posed problems that lead students in manageable steps not only provide them the confidence and ability to succeed in math, they also reveal the logical, hierarchical nature of this powerful and rewarding discipline."

• How I Rewired My Brain to Become Fluent in Math: Sorry, education reformers, it's still memorization and repetition we need. by Barbara Oakley, Nautilus, Issue 17, October 2, 2014. "In the current educational climate, memorization and repetition ... are often seen as demeaning and a waste of time for students and teachers alike. ... Imparting a conceptual understanding reigns supreme, especially during precious class time. The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don't. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence."

• Math Discoveries Catch Kids Unawares by B. Bower, Science News, January 2, 1999. Many of those who have careers that depend on math generally feel they learned it not by talking about "ways" of solving problems, but rather by doing math, with lots and lots of practice. This article from Science News reports on an interesting study that suggests exactly that!

  Three amazing conclusions from this study:

  1. kids deduced an important mathematical algorithm simply from doing computation problems,
  2. nearly all of the kids in the study eventually did this, and
  3. most of the kids were not able to explain in words what they were doing, thus suggesting that being forced to convert mathematical understanding into words was a skill quite distinct from mathematical competency.

  "This study shows that conceptual insights emerge unconsciously during practice"

  Here are some excerpts: TEMPORAL "Many educators and scientists assume that conscious knowledge is the engine that drives learning. A new study suggests instead that, at least among grade-schoolers, unconscious problem-solving insights often set the stage for academic advances. Second-graders who practice solving inversion problems -- such as 8+10-10 = 8 -- start out by computing the answers but frequently turn to a more efficient strategy unconsciously. ... However, after becoming aware of the shortcut, kids employ it only part of the time, returning at other times to more time-consuming calculations. In the long run, the child's nurturing of an array of problem-solving tactics allows for adjustments in tougher math challenges. ... 'This study shows that conceptual insights emerge unconsciously during practice,' remarks psychologist David C. Geary of the University of Missouri in Columbia."

• How Can Learning Facts Make Thinking More Enjoyable -- and More Effective?
  by Daniel T. Willingham, American Educator, American Federation of Teachers, Spring 2009.
       "Data from the last 30 years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that's true not simply because you need something to think about. The very processes that teachers care about most -- critical thinking processes like reasoning and problem solving -- are intimately intertwined with factual knowledge that is in long-term memory (not just in the environment). ...
       "Take two algebra students -- one is still a little shaky on the distributive property, whereas the other knows it cold. When the first student is trying to solve a problem and sees a(b + c), he's unsure whether that's the same as ab + c or b + ac or ab + ac. So he stops working on the problem, and substitutes small numbers into a(b + c) to be sure that he's got it right. The second student recognizes a(b + c), and doesn't need to stop and occupy space in working memory with this subcomponent of the problem. Clearly, the second student is more likely to successfully complete the problem."

• "Most mathematics textbooks follow precisely the approach that our studies find so ineffective"

  Temporal Spacing and Learning by Hal Pashler, Doug Rohrer, and Nicholas J. Cepeda, The Observer, Association for Psychological Science (APS), March 2006. "Studies going back a century and more have found that spacing learning episodes across time sometimes enhances memory. The so-called spacing effect is the topic of hundreds of articles, and one might assume that we know all we need to know about it. However, the subtitle of an article on spacing effects that Frank Dempster published in American Psychologist in 1988 -- "A case study in the failure to apply the results of psychological research" -- remains appropriate now. Whether one looks at classrooms, instructional design texts, or language learning software, there is little sign that people are paying attention to temporal spacing of learning.
       "[O]ur team ... has been teaching students abstract mathematics skills ... In a study ..., students learned to solve a type of permutation problem, and then worked two sets of practice problems. One-week spacing separating the practice sets drastically improved final test performance (which involved problems not previously encountered). In fact, when the two practice sets were back-to-back, final performance was scarcely better than if the second study session was deleted altogether. This fits with other research from our team showing that benefits of over-learning decline sharply with time ...
       "Interestingly, most mathematics textbooks follow precisely the approach that our studies find so ineffective: a brief lesson on a topic is followed by a practice set containing virtually every problem in the book relating to this topic. Far more useful, we suspect, is to intersperse problems related to older topics covered over past weeks and months."

• "The ideal textbook also has a tremendous number of practice problems because practice, practice and more practice is demanded from each student. Finding enough practice problems has always been difficult, thus I am currently developing my own texts and practice workbooks for class and homework use that are consistent with the lesson plans of the program."
  -- legendary math teacher Jaime Escalante

• "Exercise in repeatedly recalling a thing strengthens the memory."
  -- Aristotle

• "Don't practice until you get something right, practice until you can't do it wrong."
  -- unknown