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 BOX 5-5Accessible Algorithms In over a decade of working with a range of urban and suburban classrooms in the Children’s Math Worlds Project, we found that one multidigit addition method and one multidigit subtraction method were accessible to all students. The students easily learned, understood, and remembered these methods and learned to draw quantities for and explain them. Both methods are modifications of the usual U.S. methods. The addition method is the write-new-groups-below method, in which the new 1 ten or 1 hundred, etc., is written below the column on the line rather than above the column (see Jackie’s method in Figure 5-1). In the subtraction fix-everything-first method, every column in the top number that needs ungrouping is ungrouped (in any order), and then the subtracting in every column is done (in any order). Because this method can be done from either direction and is only a minor modification of the common U.S. methods, learning-disabled and special-needs students find it especially accessible. Both of these methods stimulate productive discussions in class because they are easily related to the usual U.S. methods that are likely to be brought to class by other students.

#### PRINCIPLE #3:A METACOGNITIVE APPROACH ENABLES STUDENT SELF-MONITORING

Learning about oneself as a learner, thinker, and problem solver is an important aspect of metacognition (see Chapter 1). In the area of mathematics, as noted earlier, many people who take mathematics courses “learn” that “they are not mathematical.” This is an unintended, highly unfortunate, consequence of some approaches to teaching mathematics. It is a consequence that can influence people for a lifetime because they continue to avoid anything mathematical, which in turn ensures that their belief about being “nonmathematical” is true.15

An article written in 1940 by Charles Gragg, entitled “Because Wisdom Can’t be Told,” is relevant to issues of metacognition and mathematics learning. Gragg begins with the following quotation from Balzac:

So he had grown rich at last, and thought to transmit to his only son all the cut-and-dried experience which he himself had purchased at the price of his lost illusions; a noble last illusion of age.

Except for the part about growing rich, Balzac’s ideas fit many peoples’ experiences quite well. In our roles as parents, friends, supervisors, and professional educators, we frequently attempt to prepare people for the future by imparting the wisdom gleaned from our own experiences. Some-

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