Adding It Up Helping Children Learn Mathematics (2001) / Chapter Skim
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6 Developing Proficiency with Whole Numbers
6 Developing Proficiency with Whole Numbers
Pages 181-230
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From page 181... ...
Even if children begin school with an unusually limited facility with number, intensive instructional activities can be designed to help them reach similar levels as their peers. Children's facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers.
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From page 182... ...
We use the term basic number combinations to emphasize that the knowledge is relational and need not be memorized mechanically. Adults and "expert" children use a variety of strategies, including automatic or semiautomatic rules and reasoning processes to efficiently produce the basic number combinations.2 Relational knowledge, such as knowledge of commutativity, not only promotes learning the basic number combinations but also may underlie or affect the mental representation of this basic knowledge.3 The domain of early number, including children's initial learning of singledigit arithmetic, is undoubtedly the most thoroughly investigated area of school mathematics.
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From page 183... ...
Furthermore, they can advance those counting skills as they solve more problems.6 In fact, it is in solving word problems that young children have opportunities to display their most advanced levels of counting performance and to build a repertoire of procedures for computation. Most children entering school can count to solve word problems that involve adding, subtracting, multiplying, and dividing.7 Their performance increases if the problems are phrased simply, use small numbers, and are accompanied by physical counters for the children to use.
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From page 184... ...
They invent a procedure that mirrors the actions or relationships described in the problem. This simple but powerful approach keeps procedural fluency closely connected to conceptual understanding and strategic competence.
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From page 185... ...
E; DEVELOP/NO PROF/C/ENCY WITH WHOLE NUMBERS 185 Table 6-1 A~l~lition anal Subtraction Problem Dupes Problem Type Join (Result Unknown) (Change Unknown)
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From page 186... ...
Additional types of multiplication and division problems are introduced later in the curriculum. These include rate problems, multiplicative comparison problems, array and area problems, and Cartesian products.9 As with addition and subtraction problems, children initially solve multiplication and division problems by modeling directly the action and relations in the problems.~° For the above multiplication problem with marbles, they form four piles with three in each and count the total to find the answer.
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From page 187... ...
Count Count on . Thinking strategies (forIarger numbers)
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From page 188... ...
Some of these procedures can be taught, which accelerates their use,~4 although direct teaching of these strategies must be done conceptually rather than simply by using imitation and repetition.~5 In some countries, children learn a general procedure known as "make a 10" (see Box 6-2~.~6 In this procedure the solver makes a 10 out of one addend by taking a number from the other addend. Educators in some countries that use this approach believe this first instance of regrouping by making a 10 provides a crucial foundation for later multidigit arithmetic.
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From page 189... ...
Other techniques that encourage students to use more efficient procedures are using large numbers in problems so that inefficient counting procedures cannot easily be used and hiding one of the sets to stimulate a new way of thinking about the problem. Intervention studies indicate that teaching counting-on procedures in a conceptual way makes all single-digit sums accessible to U.S.
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From page 190... ...
.22 Mathematical proficiency with respect to single-digit addition encompasses not only the fluent performance of the operation but also conceptual understanding and the ability to identify and accurately represent situations in which addition is required. Providing word problems as contexts for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students' adaptive reasoning and improving their understanding of addition processes.
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From page 191... ...
For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic.27 Examining the relationships between addition and subtraction and seeing subtraction as involving a known and an unknown addend are examples of adaptive reasoning. By providing experiences for young students to develop adaptive reasoning in addition and subtraction situations, teachers are also anticipating algebra as students begin to appreciate the inverse relationships between the two operations.28 Single-Digit Multiplication Much less research is available on single-digit multiplication and division than on single-digit addition and subtraction.
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From page 192... ...
Thus, treating multiplication learning as pattern finding both simplifies the task and uses a core mathematical idea. After children identify patterns, they still need much experience to produce skip-count lists and individual products rapidly.
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From page 193... ...
This practice can occur in many different contexts, including solving word problems.3~ Drill alone does not develop mastery of single-digit combinations.32 Practice that follows substantial initial experiences that support for understanding and emphasize "thinking strategies" has been shown to improve student achievement with single-digit calculations.33 This approach allows computation and understanding to develop together and facilitate each other. Explaining how procedures work and examining their benefits, as part of instruction, support retention and yield higher levels of performance.34 In this way, computation practice remains integrated with the other strands of proficiency such as strategic competence and adaptive reasoning.
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From page 194... ...
During this progression, individual children use a range of different procedures on different problems and even on the same problem encountered at different times.39 Even adults have been found to use a range of different procedures for simple addition problems.40 Further, it takes an extended period of time before new and better strategies replace previously used strategies.4~ Learning-disabled children and others having difficulty with mathematics do not use procedures that differ from this progression. They are just slower than others in moving through it.42 Instruction can help students progress.43 Counting on is accessible to first graders; it makes possible the rapid and accurate addition of all singledigit numbers.
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From page 195... ...
schools. Also, algorithms different from those taught in the United States today are currently being taught in other countries.46 Each algorithm has advantages Box 6-5 Heginning multiplication algorithm 752 X 23 6
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From page 196... ...
Research findings about learning algorithms for whole numbers can be summarized with seven important observations. First, the linkages among the strands of mathematical proficiency that are possible when children develop proficiency with single-digit arithmetic can be continued with multidigit arithmetic.
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From page 197... ...
This means that students in different classrooms and receiving different instruction might follow different learning progressions use different procedures.52 For single-digit addition and subtraction, the same learning progression occurs for many children in many countries regardless of the nature and extent of instruction.53 But multidigit procedures, even those for addition and subtraction, depend much more on what is taught. A fourth observation is that children can and do devise or invent algorithms for carrying out multidigit computations.54 Opportunities to construct their own procedures provide students with opportunities to make connections between the strands of proficiency.
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From page 198... ...
Research indicates that students' experiences using physical models to represent hundreds, tens, and ones can be effective if the materials help them think about how to combine quantities and, eventually, how these processes connect with written procedures. The models, however, are not automatically meaningful for students; the meaning must be constructed as they work with the materials.
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From page 199... ...
These intricacies are important because research has shown that it is difficult to develop procedural fluency with multidigit arithmetic without an understanding of the base-10 system.64 If such understanding is missing, students make many different errors in multidigit computations.65 This conclusion does not imply that students must master place value before they can begin computing with multidigit numbers. In fact, the evidence shows that students can develop an understanding of both the base-10 system and computation procedures when they have opportunities to explore how and why the procedures work.66 That should not be surprising; it simply confirms the thesis of this report and the claim we made near the beginning of this chapter.
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From page 200... ...
zoo ADDING IT UP Box 6-6 | A Third-Grade Class Finds 54 + 48 The students had worked on the problem at their desks for about 15 minutes and were sharing their procedures with the class. The teacher, Ms.
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From page 201... ...
Boxes 6-7 through 6-10 illustrate procedures for multidigit addition and subtraction. Method C in Box 6-7 captures, in written form, the thinking strategies that many students use as they continue constructing procedures for adding multidigit numbers.68 They usually begin by combining the larger units first and then combining the subtotals to find the sum.
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From page 202... ...
ZOO ADDING IT UP Box 6-7 Three Methods fear Multi~ligit A~l~lition A Common U.S. Algorithm Method A 1 1 568 876 1444 (a)
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From page 203... ...
, 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.7~ 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.
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From page 204... ...
Subtraction Algorithms 1 hundred 1 ten ~ 300~0, ~ 000 Students can construct multidigit subtraction procedures, though often these procedures are less similar to standard algorithms than is the case for addition. Still, as with addition, research has shown that students can learn a subtraction algorithm meaningfully if provided with appropriate experiences.
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From page 205... ...
question, "Can I subtract in this column? Is the top digit as big as or bigger than the bottom digit?
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From page 206... ...
than of addition and subtraction. Sample conceptual teaching lessons have been published for multiplication, and some alternative methods of instruction have been explored.74 A preliminary learning progression of multidigit procedures that fosters children's invention of
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From page 207... ...
Nevertheless, it is useful to examine algorithms students are expected to learn and to consider alternatives that might facilitate understanding. Standard multiplication and division algorithms used in the United States are complex procedures in which multiplying alternates with adding or subtracting (see Box 6-11~.
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From page 208... ...
They typically start at the right and multiply ones first. The expanded algorithm begins at the left, as students are naturally inclined to do.
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From page 209... ...
The expanded algorithm for these larger numbers is relatively easy to carry out because the necessary steps are visible, although the number of partial products more than doubles. Given the accessibility of calculators, it might not be wise for students to spend a great deal of valuable school learning time becoming efficient at multiplication with three-digit or larger numbers.
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From page 210... ...
Division Algorithms As we indicated earlier, relatively little research is available to shed light on how students think about multidigit division or what learning activities might be of most help to them. Sample teaching lessons have been proposed, and preliminary results suggest that students can construct their own procedures that, over time, approximate standard algorithms.77 As with multiplication, however, the best that educators can do at this point is to examine some alternative algorithms that are likely to support students' efforts to develop proficiency with multidigit division.
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From page 211... ...
DEVELOP/NO PROF/C/ENCY WITH WHOLE NUMBERS Z11 Box 6-14 A Common Algorithm f:ar Multidigit Division 68 46 J 3 1 29 -276 369 368 Box 6-15 | Expanded Algorithm and Model for M"ltidigit Division Accessible Division Algorithm: Abbreviated Model: Take away copies of 46 until no more remain Build up copies of 46 40 + 6 46) 31 29 —2 3 0 0 50 (5s are easy: take 8 2 9 half of 10 x 46)
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From page 212... ...
children can understand and explain procedures for calculating with multidigit numbers rather than just executing them mechanically. This conclusion, which is especially well established for addition and subtraction,79 means that mathematical proficiency with multidigit arithmetic is achievable by students even at early grades.
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From page 213... ...
Some students invent their own methods for performing multidigit computations, and some learn by listening to others another student or the teacher explain a method. Whatever avenue students take, their procedural fluency is intertwined with their conceptual understanding and adaptive reasoning.
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From page 214... ...
Mental Arithmetic and Estimation Written procedures for adding, subtracting, multiplying, and dividing are the major focus of mathematics in the elementary school curriculum, and we have discussed how they can be integrated into the other strands of children's developing mathematical proficiency. We end this chapter by considering two other kinds of calculation methods and the roles they can play in fostering the development of mathematical proficiency.
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From page 215... ...
There is evidence, though, that some instruction on mental arithmetic in upper elementary grades, if it is focused on understanding and uses number and operation properties, can move students away from the clumsy and error-prone mental use of written algorithms toward use of a variety of mental procedures better adapted to particular number combinations.89 Beyond its many practical uses in the modern world, mental arithmetic can promote mathematical proficiency by bringing together the various strands. Mental arithmetic should be taught to encourage children to reason about the problem situation and the numbers involved, to take advantage of their conceptual understanding of the properties and rules of arithmetic, and to strategically select and adapt procedures to simplify a computation and calculate the answer.
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From page 216... ...
It also requires recognizing that the appropriateness of an estimate is related to the problem and its context.9~ Estimation requires a flexibility of calculation that emphasizes adaptive reasoning and strategic competence, guided by children's conceptual understanding of both the problem situation and the mathematics underlying the calculation. Research on estimation shows how difficult it is for students who receive conventional instruction, with its frequent overemphasis on routine paperand-pencil calculation, to move from calculating exact answers to estimating wisely.
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From page 217... ...
These methods are more sophisticated mathematically, use structural properties such as commutativity, and use the place-value symbolic notation in productive ways. As students begin multidigit arithmetic, it is vital that teachers and classrooms provide support for all to build understanding of
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From page 218... ...
Basic number combinations may be represented not as a table of facts but as a network of facts and interconnecting relations (e.g., Baroody, 1985, 1987b, 1992~. This idea is consistent with research in cognitive science, which says that expert knowledge is organized and connected (Bransford, Brown, and Cocking, 1999~.
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From page 219... ...
49. For a synthesis on the relationship between conceptual and procedural knowledge for multidigit addition and subtraction, see Rittle-tohnson and Siegler, 1998.
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From page 220... ...
67. Student-invented procedures are sometimes not really algorithms because the steps are not precisely specified but instead follow a path that emerges through the process and that path may be slightly different if the same problem is posed again.
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From page 221... ...
For a similar discussion about estimation, see Buchanan, 1978. Beishuizen, 1993, discusses students connecting mental arithmetic procedures to using base-10 blocks and hundreds squares.
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From page 222... ...
InL.~.Morrow&M.~.Kenney(Eds.) , The teaching andlearningafalgorithmsin school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp.
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From page 223... ...
InL.~.Morrow&M.~.Kenney(Eds.) , The teaching andlearningafalgorithmsin school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp.
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From page 224... ...
(1993~. Group case studies of second graders inventing multidigit addition procedures for base-ten blocks and written marks.
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~ 1997~. Children's conceptual structures for multidigit numbers and methods of multidigit addition and subtraction.
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From page 226... ...
, The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp.
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From page 227... ...
(1998~. The teaching and learning of algorithms in school mathematics ~ 1998 Yearbook of the National Council of Teachers of Mathematics)
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From page 228... ...
, New directionsfor elementary school mathematics ~ 1989 Yearbook of the National Council of Teachers of Mathematics, pp.
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From page 229... ...
(1978~. Emphasizing thinking strategies in basic fact instruction.
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