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Dig deeper into classroom artifacts using research-based learning progressions to enhance your analysis and response to student work, even when most students solve a problem correctly.

Instructional activities designed to encourage relational thinking in primary-grades classrooms can give students advantages when they reason about subtraction.

An exploration and development cycle emphasizing the use of concept maps for subtraction can expand preservice teachers' understanding.

Becoming fluent with basic facts is developmental. Use games infused with a focus on acquisition of strategies to help students in the early grades progress to computational fluency.

Appreciate the complexity of counting and adding skills by viewing them through the lens of an early numeracy progression.

By Henry Borenson A Balancing Act G lancing at Chandler’s Semiotics: The Basics (2007), I discovered that, generally speaking, signs have more than one meaning, depending on the context. Consider, for example, the meaning of the minus sign in the expression 2; it certainly does not mean subtraction. Similarly, the equal sign has various meanings, depending on the context in which it is used (Molina, Castro, and Castro 2009). The assumption that the equal sign can have only one meaning has led some educators to state that “it does not mean that the answer comes next.” The use of the equal sign to indicate the unique numerical result of the sequence of computations that precede it (the “calculator use” of the equal sign) is a valid and necessary use of this sign (Ginsburg 1996; Falkner, Levi, and Carpenter 1999; Seo and Ginsburg 2003). However, understanding the relational meaning of the equal sign also is essential for success in mathematics, and particularly in algebra. Whereas such equations as 3x + 5 = 26 can be understood knowing only the operational meaning of the equal sign (because 26 can be considered the result of the operations that appear to the left of the equal sign), examples such as 4 + 3 = __ + 2 and 4x + 2 = 2x + 6 cannot (Kieran 1992). In these examples, the equal sign indicates equivalence between two sets of expressions, each one of which includes one or more operations within it. Is the relational meaning of the equal sign self-evident to students? Do elementary school students intuitively understand the relational meaning of the equal sign? In a survey of 752 students in grades 1 –6, Falkner, Levi, and Carpenter (1999) found that only 5 percent of the students provided the correct response to the question 8 + 4 = __ + 5. As the authors noted, the equal sign usually is presented to elementary school students only “at the end of an equation, and only one number comes after it” (p. 233).

Fourth- and fifth-grade learners can use differentiated number sets within CGI problem structures to add and subtract fractions with unlike denominators.

Reflect and discuss: Everyday settings, like enjoying freshly baked cookies, can foster proficiency and enhance students� conceptual understanding

Word problems can play a prominent role in elementary school mathematics because they can provide practice with real-life problems and help students develop their creative, critical, and problem-solving abilities. However, word problems as currently presented in instruction and textbooks fail to accomplish these goals (Gerofsky 1996; Lave 1992). This failure is due, in part, to the unrealistic approach needed to solve them: the straightforward application of one arithmetic operation. Consequently, when faced with word problems in which context is critical to the solution, students fail to connect school mathematics with their real-world knowledge. Problems that cannot be solved by applying a straightforward arithmetic operation are called problematic. Several researchers have examined children's lack of use of their real-world knowledge to solve problematic word problems (Greer 1997; Reusser and Stebler 1997; Verschaffel and De Corte 1997).

Teaching the basic facts seemed like the logical thing to do. Wouldn't a study of the basic facts make mathematics computation much easier for my students in the future? How could I help my students memorize and internalize this seemingly rote information? How could I get rid of finger counting and move on to mental computation? As I embarked on my first year of teaching second grade following many years of teaching first grade, these questions rolled through my head.

Teachers usually present addition and subtraction at the same time to first and second graders. Many teachers have observed, however, that students find addition easier and more natural than subtraction. Children struggle with subtraction even when they learn “fact families” that ostensibly help them understand the relationship between addition and subtraction. Given that children continue to find subtraction difficult despite the use of time-honored practices, we suggest that teachers de-emphasize fluency in subtraction until their students become fluent in addition.

The Problem My work with Ashleigh began when a neighbor found out that I like to “do mathematics” with elementary students. She asked me to work with her third-grade daughter, Ashleigh, who had “a problem with subtraction,” she said. On timed tests of addition and even multiplication facts, Ashleigh was receiving grades of 90 to 100. On tests of subtraction facts, however, she was consistently coming home with scores in the 50s or lower. Ashleigh’s mom said they had studied with flash cards many times, but Ashleigh still could not remember her subtraction facts. “Also, she does this thing with her fingers when she does her subtraction homework,” Ashleigh’s mom said. “Her fingers get to flying, counting up and down and backward. . . . You’ll have to watch her to see what I mean.” When I did watch Ashleigh, I realized that she was trying to use a prescribed touching technique and also count backward with her fingers to do subtraction. These techniques became complicated when she needed to solve a problem such as 17 – 8. Her use of fingers caused accuracy errors, her dependence on touching the numerals to subtract was slowing down her performance rate, and her backward counting sometimes became muddled and ended in wrong answers. I noticed another side effect of her dependence on only these two techniques: She was not thinking about the numbers and their relationships. She was thinking only about counting backward or about where to touch the numerals. When she wrote her solution, she was unaware of whether it “made sense”; it simply was the answer she got when she was finished. Her lack of awareness of number relationships showed again when she was playing card games. Ashleigh knew that a 4 and a 6 make 10 and could get this answer when she added on paper. But when she tried to solve 10 – 6 or 10 – 4

FEBRUARY 2003 R ichfield Elementary School is in the midst of its literacy campaign. Room 24 has read 674 books and Room 16 has read 328 books. The teacher, Meghan Steinmeyer, has asked her students in Room 16 to figure out the number of books they must read to catch up to Room 24. One student, John, works with the number 330 rather than 328 to make the subtraction easier (see fig. 1). He subtracts 330 from 674 and gets 344. Because he subtracted a number that he increased by 2, he compensates by adding 2 back to the 344 to get 346. Learning computation is an active problemsolving process and a collaborative one in this third-grade classroom. The students entered the classroom with some strategies for subtracting lesser numbers and now need to extend these methods and move toward fluency with greater numbers. Fluency means that students can flexibly choose computational strategies to solve problems, understand and explain their methods, and produce accurate answers efficiently (NCTM 2000; Russell 2000). The purpose of this article is to share what we have learned about supporting students as they move toward fluency with subtracting greater numbers. We believe that meaningful computation occurs at the intersection of number relationships, understanding of operations, and children’s ways of thinking. We begin with examples of children’s strategies that highlight relationships among numbers and operations. Then we examine whole-class discourse that promotes understanding of strategies and efficiency. Finally, we present a collection of common strategies and consider the connection between a teacher’s own fluency and teaching for computational fluency.

suggests that computationally fluent procedures are accurate, efficient, flexible, and consolidated. Students develop these characteristics over time as they explore different kinds of procedures, experiment with numbers of different sizes, and analyze one another’s procedures. At first, students’ procedures may include minor computation errors and many steps. Some students may choose only one method as their preferred procedure and rarely venture to use others. Over time, however, as teachers increase their expectations, students should become more accurate, efficient, and flexible. Students also will begin to consolidate ideas when they are no longer exploring alternative approaches but have a firm understanding of a few procedures and know when it is more appropriate to use them. The call to develop elementary students’ computational fluency poses a particular challenge to current and future teachers, who usually are not computationally fluent themselves (Ma 1999). Understandably, because of the instruction that they received, teachers often are wed to traditional, single-method paper calculation procedures, are calculator-dependent, know few meanings or only one meaning of an operation—such as subtraction only as “take away”—and are generally unfamiliar with alternative ways of using reasoning to calculate. Teachers must move away from these limitations and build their own computational fluency in order to foster such thinking in children. This article describes aspects of a program that was designed to develop teachers’ computational fluency. We share some of the experiences we have developed for work related to whole-number subtraction that we have used with both current and prospective teachers. We selected subtraction because it provides rich opportunities for developing computational fluency. It also is one of the more challenging whole-number operations, both in calculating and in justifying what happens when numbers are changed and adjustments are made. Understanding the reasons for different subtraction strategies is challenging for both children and teachers. Although children first encounter subtraction in the primary grades, they continue to struggle with it through the upper grades.

The development of number sense and computational fluency should be an integral part of the mathematics curriculum. Because other areas of the curriculum such as data and measurement are closely related to and sometimes dependent on these skills, students must have a firm foundation in number. Teachers should provide activities and experiences that develop a conceptual understanding of number and operations, instead of focusing on the memorization of rules and procedures. Meaningful mathematical learning then can occur. When left to use strategies that are natural for them, children are wonderful problem solvers and are able to make sense of numbers in their world. This article focuses on the development of number sense in the primary grades using the ideas of Kathy Richardson (1999).

During the last twenty-five years, arithmetic curricula have included more hands-on activities with an increasing emphasis on problem solving. According to the Representation Standard in Principles and Standards, “instructional programs from prekindergarten through grade 12 should enable all students to use representations to model and interpret physical, social, and mathematical phenomena” (NCTM 2000, p. 136). Representations, such as manipulatives, help children develop understanding by building a foundation for the later use of symbols. When given a choice of how to solve a problem, children gravitate toward manipulatives because they can act out the situation or relationships in the problem. When children are free to choose strategies for problem solving in single-digit arithmetic, they typically progress from using concrete strategies to more abstract strategies. For example, consider the following problem: “Lisa had 8 seashells in her collection. Hal gave her 5 more seashells. How many seashells did Lisa have then?” To solve this problem, children in the beginning stage of development will make a set of 8 objects and another set of 5 objects, join the two sets, and count all the objects. As they progress in their development, children do not need to build the first set but can count on from 8 to get the answer. Eventually, children can solve the problem abstractly without requiring a concrete model of either quantity in the problem (see Carpenter et al. [1999] for a complete discussion of these developmental stages). This progression of strategies is quite natural, and even five-year-olds move from counting-all strategies to counting-on strategies without instruction (Groen and Resnick 1977). Some children progress along the same kind of developmental path for multidigit addition and subtraction, initially using concrete strategies in which they manipulate the quantities in the prob

At the beginning of the school year, Mrs. Underwood's second graders could work with groups of tens or groups of ones, but they did not consider the relationship between the numerals in the tens and ones places. For example, to add 25 + 19, children added the numerals in the tens place, then counted on by ones. First, they would add 20 and 10 to get 30. Next, they would count on by ones to add 14, instead of adding 5 and 9 to make 14, then partitioning 14 into 10 and 4, and adding all the tens and all the ones. To subtract 35 – 17, not surprisingly, students explained that 3 – 1 was 2 and that 5 – 7 was 2 to derive an answer of 22.

In its Principles and Standards for School Mathematics, the NCTM suggests that fluency with basic addition and subtraction number combinations is a goal in teaching whole-number computation (NCTM 2000, p. 84). A mastery of lower-order skills instills confidence in students and facilitates higher-order thinking. The ability to automatically recall facts strengthens mathematical ability, mental mathematics, and higher-order mathematical learning. Without this automation, students have difficulty performing advanced operations.

Research studies and experiences with young children reveal that many of them have difficulty learning and applying place-value concepts in mathematics.

Addition and subtraction with regrouping and across zeros can be difficult concepts for second and third graders. Although the traditional algorithms are not the only successful processes that students may use, they are commonly used in mathematics textbooks and, especially when zeros are involved, can be difficult processes to teach. I have found that several days spent with the Three Pigs can go a long way to increase students' understanding of the traditional algorithms.

A friend, who e adult mathematical skills enable him to balance his checkbook and decide which brand of coffee is a better buy, still remembers the trauma of first grade. “There I was, happily adding away,” he says. “Then one day they hit me with subtraction.”

Without having been taught a method for solving subtraction problems that require regrouping, a first-grader applies his natural knowledge of numbers and operations to solve problems containing two- and three-digit numbers. This conversation features his reasoning as he formulates an explanation of the computational strategy he developed. Online extra: http://www.nctm.org/tcm010

This teacher explored how young students build understanding of number operations and what strategies they naturally choose--without prompts or modeling--to recognize distinctions and adjust solution methods among different problem types involving addition and subtraction; joining problems, result unknown; and compared set, difference unknown. Online extra: http://nctm.org/tcm005

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