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# Mathematics (page 3)

By Pearson Allyn Bacon Prentice Hall
Updated on Jul 20, 2010

#### Examples of Common Counting Strategies for Addition

 Strategy Sample Problem Use of Strategy Counting manipulatives 2 + 5 Child counts two manipulatives (e.g., blocks, coins, candies), then counts five more manipulatives, then counts all seven objects. Counting fingers 2 + 5 Child raises two fingers on one hand, then five on the other hand. Child then counts all seven fingers one at a time, moving each one as it is counted. Counting all (sum) 2 + 5 Child counts the first number aloud: "One, two." Child then continues counting on by the second number "three, four, five, six, seven"—and then states the answer as "seven." Counting on first 2 + 5 Child states the first number, "two"; counts on by the second number—"three, four, five, six, seven"; then states the answer as "seven." Counting on larger (minimum, or min) 2 + 5 Child states the biggest number, regardless of whether it is first or second. Here, child says "five," then counts on by the smaller number ("six, seven"), then states the answer as "seven." Decomposition (deriving a fact) 2 + 5 Child decomposes the 6 into 3 + 3, adds 7 + 3 and gets 10, adds the other 3 to 10, and gives the answer: 13. (The child could choose to decompose the other number, 7, into 4 + 3, add 4 + 6 to get 10, then add the remaining 3 to 10.) Fact retrieval 7 + 6 Child retrieves the answer from long-term memory and states the answer as "seven." Regrouping 16 + 23 Child decomposes both numbers into tens and units (16 = 10 + 6; 23 = 20 + 3), sums the tens and units separately (10 + 20 = 30; 6 + 3 = 9), adds the subtotals (30 + 9 = 39), and gives the answer: 39.

Source: Adapted from Geary (1994)

### Mathematical Skills during Older Childhood and Adolescence

Children continue to increase their understanding of basic mathematical principles throughout their school years, although some principles take some time to develop (Geary, 2006). Sometimes misunderstandings about fundamental math principles lead to bugs, or systematic errors in children's problem-solving procedures. The table below presents examples of common bugs in multidigit subtraction (Brown & Burton, 1978; Brown & VanLehn, 1982). Buggy procedures are common in all arithmetic operations, and they can persist well into the late elementary school years.

As children learn about more complex math topics such as fractions, geometry, and algebra, they consistently attempt to build new understanding on what they already know. At times this leads to misunderstandings of new principles; but when students recognize their mistakes, they begin to debug their math knowledge and understand the accurate procedures. Instruction that builds on prior knowledge is typically more effective in helping students understand new mathematical concepts. In contrast, instruction that focuses on memorization of facts and rules gives students less opportunity to work out their buggy procedures. As a result, it can take longer for the child to gain correct understanding of a concept (Clement, 1982).

#### Subtraction Bugs

 Problems and Correct Answers 307 – 49 = 258 286 – 68 = 218 293 – 171 = 122 Bug: Borrow across zero Child decrements first nonzero number correctly but does not alter the zero. 248 (combination of "borrow across zero" and "0 – N = N" bugs) 218 122 Bug: Blank instead of borrow When borrowing is needed, child skips that column and goes to the next. 3 22 122 Bug: Smaller from larger Child always subtracts the smaller number from the larger number. 342 222 122 Bug: Smaller from larger instead of borrow from zero Child does not borrow from zero but instead subtracts the smaller from the larger number. 262 218 122 Bug: Borrow, no decrement Child borrows but does not decrement the number being borrowed from. 368 228 122 Bug: Borrow from zero Child changes zero to a nine but does not decrement the next digit to the left. 358 218 122 Bug: Zero minus N equals N If top digit is a zero, child writes down the bottom number as the answer. 348 (combination of "borrow, no decrement" and "0 – N = N" bugs) 218 122

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