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# Mindful of Symbols [Preview]

On the way to learning that one thing can represent another, young children often conflate the real item and its symbol. These errors show how difficult it is to start thinking symbolically

About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a small-scale model of a room that was part of my lab. The real space was furnished like a standard living room, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory to figure out where to find the toy in the large room.

The three-year-olds were very successful. After they observed the small toy being placed behind the miniature couch, they ran into the real room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the big room, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.

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1. 1. Borenson 09:52 PM 1/1/08

Judy Deloach makes the observation, regarding the use of manipulatives in mathematics, that "If children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive."

What can we say, however, if student do understand the relationship between the manipulatives and what they represent? Current research on a manipulatives-based instructional system known as Hands-On Equations shows that the use of manipulatives, when students understand the relationship between the manipulatives and what they represent, and when they are used as an integral component of a well-structured sequence of concept development, has the potential to [u]significantly[/u] increase student achievement in mathematics.

In particular, when the first six lessons of the Hands-On Equations program were presented to 413 students consisting of 4th, 6th, and 8th graders, under the same instructional and testing conditions, it was found that the 4th graders achieved at the same level as the 6th and 8th graders, with all three grade groups obtaining a significant pre- to post-test gain. This study, â€œ[url http://www.borenson.com/Validation/ComparisonReportHOENov2107.pdf]A comparison of Algebra Achievement by 4th, 6th and 8th Graders[/url],â€ (Barber and Borenson, 2007) also showed that all groups scored above 85% on the post-test.

In a study recently completed ( Barber and Borenson, 2008) â€œ[url http://www.borenson.com/Validation/BrowardCountyStudy131MA%20Final.pdf]The Effect of Hands-On Equations on the Learning of Algebra by 4th and 5th Graders of the Broward County Public Schools[/url],â€ it was found that 80% of the 195 students in the regular classroom study were able to successfully solve the equations 3x = x + 12 and 4x + 3= 3x + 6 after seven lessons of instruction, and again three weeks later on a retention test. (The percentage of these students who solved these equations on the [u]pre-test[/u] was less than 10%).

These kinds of results are not available, as of this writing, by any means other than the use of manipulatives, and the use of Hands-On Equations in particular. The understanding that is possible through visual and hands-on means can be as significant in mathematics as in any other field of endeavor.

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Edited by Borenson at 03/31/2008 3:23 AM

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