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Adding It Up: Helping Children Learn Mathematics (2001) Center for Education (CFE)

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. "6 Developing Proficiency with Whole Numbers." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

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Adding + It Up: Helping Children Learn Mathematics

Method B is a variation of Method A that addresses two of these three problems (it also moves from right to left). Method B is taught in China and has been invented by students in the United States.70 In this method the new 1 or regrouped 10 (or new hundred) is recorded on the line separating the problem from the answer. This arrangement makes it easier to see the 14 that generated the regrouped 10 than when the 1 is written above the problem. Because the new 1 sits below in the answer space, it does not change the top number. Adding is easy: You just add the two numbers you see and then increase that total by one.

Methods A and B both require that children understand what to do when they get 10 or more in a given column. Because they can only write 9 or less of a given grouping in a column, they must make a group of 10 ones (or tens or hundreds, etc.) and give that group to the next left place. This conceptual trouble spot for students is called carrying or regrouping or trading. Method C, reflecting more closely many students’ invented procedures, reduces the problem by writing the total for each kind of unit on a new line. The carrying-regrouping-trading is done as part of the adding of each kind of unit. Also, Method C can be done in either direction (Box 6–7 shows the left-to-right version). Because you write out the whole value of each partial sum (e.g., 500 + 800 = 1300), this method also facilitates children’s thinking about and explaining how and what they are adding. Accessibility studies indicate that young children can solve multidigit addition problems using methods like B and C and some other methods also.71

Drawings like that in Box 6–8 can be used to support children’s understanding of the quantities in the problem and how those quantities are grouped to make new tens, hundreds, or thousands. Such drawings can be used with any of the three methods (or with other methods). Whether drawings or objects are used to support understanding of an addition method, it is vital that they be linked to the numbers in the algorithm until the student can perform it with understanding. If the drawings (or physical models like base-10 blocks) are used simply to calculate answers, they lose their ability to help connect understanding to procedures. The benefits of using the materials come from seeing that the actions performed on the drawings or objects to get answers are the very actions that are used in carrying out the algorithm. Learning the algorithm then becomes a matter of students recording with numbers on paper the actions and thinking they did with the drawings or objects. This linking process takes time. Asking students to explain their procedure as if the numbers were the drawings or physical models can facilitate the linking process.

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 Front Matter (R1-R20) Executive Summary (1-14) 1 Looking at Mathematics and Learning (15-30) 2 The State of School Mathematics in the United States (31-70) 3 Number: What Is There to Know? (71-114) 4 The Strands of Mathematical Proficiency (115-156) 5 The Mathematical Knowledge Children Bring to School (157-180) 6 Developing Proficiency with Whole Numbers (181-230) 7 Developing Proficiency with Other Numbers (231-254) 8 Developing Mathematical Proficiency Beyond Number (255-312) 9 Teaching for Mathematical Proficiency (313-368) 10 Developing Proficiency in Teaching Mathematics (369-406) 11 Conclusions and Recommendations (407-432) Biographical Sketches (433-440) Index (441-454)