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In Teaching Math, the Concrete Cannot Take the Place of the Abstract
In Teaching Math, the Concrete Cannot Take the Place of the Abstract

In Teaching Math, the Concrete Cannot Take the Place of the Abstract

And now we move on to a study — on the same Website, Science Daily, as yesterday’s post — that seems to contradict the “Do the Math Dance” study in yesterday’s post! In “Students Learn Better When The Numbers Don’t Talk And Dance,”  ScienceDaily.com says:
ScienceDaily (Oct. 11, 2005) — COLUMBUS , Ohio – Most teachers believe that students learn better when abstract concepts are taught using concrete materials or examples — but a new study suggests they may be wrong. Researchers found that when college students were taught an artificial form of mathematics and physics, they learned it better when it was presented using simple, abstract symbols – such as plain stars and raindrops — rather than more visually engaging and concrete 3-D objects that moved dynamically on a computer screen. Copyright © 1995-2009 ScienceDaily LLC  —  All rights reserved
Interesting that people don’t get this. Why do they think we have language in the first place? Why are concepts and language so powerful? Why do we have language, but all the other animals don’t? Why are we so much more successful at life than they? Why do we do so much more than babies? — It is more than physical strength: apes are stronger than us, and people in comas are stronger than babies.
The students were also more successful in applying what they learned to new situations when they were taught with abstract symbols rather than concrete objects, said Vladimir Sloutsky, co-author of the study and professor and director of the Center for Cognitive Science at Ohio State University.
Interesting. Maybe that relates to one important aspect of abstractions: they are “open-ended.” …But I’d think that it would also (or instead) have to do with the facts that our minds are finite and limited, and that things have identity, so we need to have a concrete (word or symbol) that stands for an abstraction only, that does not also have to stand for an actual concrete thing in our minds. A symbol might be confusing if it stood for both “sword” (or “weapon”) and for an actual sword; or if a picture stood for both “fish” and an actual dolphin. –But what, then, of Egyptian hieroglyphics? The Egyptians did not have the symbols at root of modern English nor did they have Arabic numerals, yet they were able to achieve, intellectually, a great deal. But they were held back in what they could achieve, and they did not achieve the level of abstraction and conceptualization of the Greeks. What’s more, language is the system(s) of shapes and sounds it is for a reason. Same with Arabic numerals.
The results of this study suggest that teachers may need to rethink one of the most widely accepted truisms of their profession, said Sloutsky, who is also associate dean of research at the university’s College of Human Ecology. “Many teachers believe that concrete materials make learning more fun for students, and that will increase their motivation and help them understand the concepts,” he said. “While this may be true, in many cases, the concrete materials also interfere with what they are trying to learn.” … The belief in the value of making the abstract concrete is widespread, however. For example, 84 percent of secondary school mathematics teachers in one survey said they believed concrete materials in their classes help students learn.
May need to rethink?” More like: are overdue to rethink. But to rethink this “truism” would entail rethinking their whole philosophy of pragmatism — which they just won’t do. Their philosophy blinds them to reality and to the need for change, just as the Scholastics’ basic Platonic ideas blinded them. The Scholastics (a school of philosophy during the Middle Ages) studied Aristotle, who said that nothing was so certain as the evidence of the senses — yet they stuck to the claim that an object falls with a speed in proportion to its weight, in the face of Galileo’s reasoning from the evidence of the senses that an object falls with a velocity that increases linearly with time; and they stuck to the claim that there was no change in the heavens, in the face of Galileo’s evidence that things were in the heavens as they were on earth, forever and ever. It is part and parcel of pragmatism to deny the importance and efficacy of concepts and abstractions. The ultimate culprit is the philosophy of John Dewey. (And, through Dewey, Immanuel Kant.) John Dewey leads people to have false, destructive ideas about learning, reasoning, thought, and concept formation. To teach math properly, you must get the right balance of concrete and abstract, and let each thing do what it is intended to do — sounds easy, but how often, as we see in our school system and in human history, do people go wrong! Good ideas about mathematics and how to teach it come from the practice of Archimedes, Newton, and Gauss; the educational theory of Maria Montessori; and the philosophies of mathematics of Ayn Rand and Aristotle. To read more about the study and the experiments done, refer to the Science Daily article, or look up the study itself, about which Science Daily says:
Sloutsky conducted the study with Jennifer Kaminski, graduate student at Ohio State and Andrew Heckler, assistant professor of physics at Ohio State . Their results were published in a recent issue of the journal Psychonomic Bulletin & Review.

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