Librarika: Smart Shopping Math (21st Century Lifeskills: Mathematics)
We teach mathematics symbolically because, for many centuries, symbolic representation has been the most effective way to record mathematics and pass on mathematical knowledge to others. Still, given the comparison with music, can't we somehow manage to break free of that historical legacy? Well, things are not quite so simple. Like all analogies, the comparison of mathematics with music, while helpful, only takes you so far. Although mathematical thinking is a mental activity, for the most part the human brain can do it only when supported by symbolic representations.
In short, the symbolic representation seems far more crucial to doing mathematics than is musical notation for performing music. We are all aware of successful musicians who cannot read or write a musical score. In fact, much of mathematics - including all advanced mathematics - deals with symbolically defined, abstract entities. Without the symbols, there would be no entities to reason about.
Learning Math: The Symbol Barrier
The one exception, where the brain does not require the aid of symbolic representations and where the comparison with music holds well is what for several years now I have been calling "everyday mathematics. In contrast to the mathematics required for science, engineering, economics, advanced finance, and many parts of business, where fluency with symbolic expressions is essential. Roughly speaking, everyday mathematics comprises counting, arithmetic, proportional reasoning, numerical estimation, elementary geometry and trigonometry, elementary algebra, basic probability and statistics, logical thinking, algorithm use, problem formation modeling , problem solving, and sound calculator use.
Yes, even elementary algebra belongs in that list. The symbols are not essential. For much of its roughly fifteen-hundred-year history, algebra was not written down symbolically, rather was recorded, described, and taught using ordinary language, with terms like "the unknown" where today we would write an"x".
True, people sometimes scribble symbols when they do everyday math in a real-life context. But for the most part, what they write down are the facts needed to start with, perhaps the intermediate results along the way and, if they get far enough, the final answer at the end. But the doing math part is primarily a thinking process - something that takes place primarily in your head. Even when people are asked to "show all their work," the collection of symbolic expressions that they write down is not necessarily the same as the process that goes on in their heads when they do math correctly.
In fact, people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations. It is with everyday mathematics that the symbol barrier emerges. With concealed tape recorders, they posed as ordinary market shoppers, seeking out stalls being staffed by young children between 8 and 14 years of age.
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At each stall, they presented the young stallholder with a transaction designed to test a particular arithmetical skill. The purpose of the research was to compare traditional instruction which all the young market traders had received in school since the age of six with learned practices in context. In many cases, they made purchases that presented the children with problems of considerable complexity.
There was more to the study.
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Posing as shoppers and recording the transactions was only the first part. About a week after they had "tested" the children at their stalls, the three researchers went back to the subjects and asked each of them to take a pencil-and-paper test that included exactly the same arithmetic problems that had been presented to them in the context of purchases the week before, but expressed in the familiar classroom form, using symbols.
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The investigators were careful to give this second test in as non-threatening a way as possible. It was administered in a one-on-one setting, either at the original location or in the subject's home, and the questions were presented in written form and verbally. The subjects were provided with paper and pencil, and were asked to write their answer and whatever working they wished to put down.
They were also asked to speak their reasoning aloud as they went along. The children were absolute number wizards when they were at their market stalls, but virtual dunces when presented with the same arithmetic problems presented in a typical school format. The researchers were so impressed - and intrigued - by the children's market stall performances that they gave it a special name: As you might imagine, when the three scholars published their findings in the book Street Mathematics and School Mathematics, Cambridge University Press, Cambridge, UK, , it created a considerable stir.
Many other teams of researchers around the world carried out similar investigations, with target groups of adults as well as children, and obtained comparable results.
Smart Shopping Math (Electronic book text, Teacher ed.)
It is simply not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math. In fact, for most people, it's not accurate to say that the problems they are presented in paper-and-pencil format are "the same as" the ones they solve fluently in a real life setting. When you read the transcripts of the ways they solve the problems in the two settings, you realize that they are doing completely different things. I present some of those transcripts in my book.
Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being "the same. It's huge and it is pervasive. For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces in order to store and distribute mathematical knowledge, that barrier has prevented millions of people from becoming proficient in a cognitive skill-set of evident major importance in today's world, on a par with the ability to read and write. With video games, we can circumvent the barrier.
Read series, "How to design video games that support good math learning," at profkeithdevlin. Related posts The Daily Riff: Something is missing from that idea, however, so we offer a corollary: Different people, when presented with exactly the same information in exactly the same way, will learn different things.
So the next time you fret that your child is on a mindless video game of course, depending which one , take a chill pill. It may not be as bad as you think.
It's another play on Goyte's hugely viral video. The Symbol Barrier by Dr. Behavior Screening and Autism Spectrum. Semantics, Vocabulary and Word Finding. Speech, Language and Literacy. Visual Perception and Motor Skills. Common Core Language Arts Resources. Common Core Mathematics Resources.
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Professional Development for Mathematics. Professional Development for multiple subjects. Fall Special Needs Catalog Items displaying this icon are digital and available for immediate download after purchase. Description Build your students' confidence and enhance the math skills they use in everyday situations! Each of these six pp reproducible books presents topics students are likely to encounter in everyday life.
Problems involve estimation, equations, mental math, calculators, and critical thinking and use additional concept-specific skills such as graphing, averages, statistics, ratios, and measurement. The associated Enhanced eBook on CD is an electronic version of the book that can be loaded onto Read full description Build your students' confidence and enhance the math skills they use in everyday situations!
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