Add With Me!

Operations—even basic operations such as addition and subtraction—have traditionally been reserved for school-age children. There are certainly still many who oppose any effort to engage preschool children in activities that involve number operations. However, this opposition is rapidly fading. Increasingly persuasive evidence now indicates that early concepts of basic addition and subtraction can become an important part of the young child's repertoire of mathematics understandings before they enter the kindergarten or the elementary school grades. In this article we begin by looking forward to the elementary grade expectations for children's understandings and abilities to perform number operations, then back to the types of early understandings we find evidenced in young learners (e.g., what and when certain abilities are developed), and finally what you can do to support and build upon these understandings.

Early expectations for understanding and performing mathematical operations

Analyses of national and state standards in the United States, learner outcomes in numerous provinces in Canada, and the National Curriculum of England reveal certain commonalities in student expectations for understanding and performing basic number operations at the elementary level. When taken as a whole, and distilled to their essence, these expectations provide a good picture of what we wish young children to understand or be fully prepared to understand as they enter the kindergarten and elementary school grades.

Figure 1 below illustrates a continuum of learning for three general learning expectations across two bands—Pre-K through grade 2, and grades 3 through 5. The bands are the same as those used in the U.S. document Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000).

Children's understandings progress systematically through all three continua. In the first continuum, children progress from understanding meaning and effect of the most basic operations of addition and subtraction to the meaning and effect of more complex operations of multiplication and division. In the second continuum, they progress from an understanding of relationships between addition and subtraction to relationships between multiplication and division. In the earlier grades, children begin to encounter situations that help them to become familiar with properties that are essential to learning mathematics in later years. The earliest involves the commutative property (e.g., 5+3 is the same as 3+5), which bears resemblance to the order irrelevance principle of early counting (see Count With Me!). At grades 3-5, the associative (Ex: (2+3)+5=2+(3+5)) then distributive (Ex: 3(2+4)=(3x2)+(3x4)) properties can begin to be addressed. In the third continuum, children progress in ability to perform operations, and in the complexity and variety of the strategies and tools (physical, visual, mental, pencil and paper, etc.) that they adopt and develop to address situations that entail those operations. They also mature in their ability to recognize and conceptualize strategies, and become more able to move between and among strategies as conditions warrant.

The three continua are mutually co-dependent. It is through exploring and using strategies (third continuum) that children develop the understandings described in the first two continua. Likewise, children's understandings of meaning and number relationships are a large factor in developing strategies that employ basic number combinations (Baroody, 1985). As we see in Figure 1, children are typically expected to begin to learn rational numbers (e.g., fractions and decimals) by cycling back into the process used for learning whole numbers. Later, in middle school, students will again cycle through the process when learning integers (negative and positive numbers). This cyclic learning progression is unique, and we find that it occurs on many levels from early preschool to high school and beyond.

Cyclic development of addition strategies: A flashpoint for learning

At the preschool and elementary grades especially, these small cycles of progressively complex strategies and number combinations devised and used by young learners offer the most powerful vehicle we have for encouraging growth in a child's understanding and ability to perform mathematical operations. It makes sense, both from what we see in individuals who are accomplished in performing operations, and from what we find to be natural tendencies—almost a type of inborn resourcefulness—on the part of very young learners still at pre-formal stages in their understandings. We see that "expert" children and adults naturally and spontaneously develop their own rules and procedures as new situations arise (Baroody, 1984, 1985; LeFevre, Sadesky, & Bisanz, 1996). We further find that most children from very early ages onward can and do devise and employ a wide variety of developmentally appropriate but very effective strategies when presented with new requirements and conditions, even when their counting range restricts them to physically "acting out" procedures with single-digit whole numbers, and that they continually progress to more efficient procedures as they grow and as the problems become more complex (Brownell, 1956/1987; Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993; Fuson, 1992; Kouba, 1989). These and other findings uncover the strategies that young children employ as well as the developmental levels at which these strategies become most apparent and appropriate. They also provide the key to identifying and designing effective instructional approaches that help young children to understand and perform basic operations.

Constructing trajectories for understanding and performing number operations

A learning trajectory is a sequence of understandings and abilities that students should master as their learning progresses. Though ages vary with individuals and circumstances, research evidence reveals certain consistencies in the pattern in which children grasp strategies for solving problems that involve operations. Figure 2 below illustrates three stages that are commonly observed—physical "acting out," non-visual counting, and general class procedures. The end result is some form of recall. We have known this for some time; it is for this reason that past programs often tried to stress memorization (e.g., the times tables). We know now, however, that when understanding is the goal, children cannot effectively jump to that stage. Rather, recall occurs for different children at various points throughout the process for different children and for a variety of different problem types.

We find that most children initially adopt and modify procedures that involve physically "acting out" a problem. For instance, when given two piles of blocks and asked to add the two, the child might physically transfer and count one block at a time until he comes to a solution. This is called "counting all." A similar but slightly more advanced approach might be to simply start with the number of blocks in one pile, and "count on" (some call it "add on") block by block from the second pile until she has the total sum of the blocks. Using either strategy, children are actively using representations to model their process, while simultaneously building upon and enhancing their grasp of one-to-one-correspondence, their fundamental understanding of number concept, and their understanding of basic counting principles. As teachers and parents, we can start the learning process by providing the materials (e.g., blocks, marbles, etc.), then observe the strategies the child initially adopts and help him/her to build from that point. It is important to use very small numbers first in order to not exceed the child's counting range, but the number of objects can be increased as the range grows. Eventually, and almost an intermediate step toward non-visual counting, children can begin to use their fingers and toes to perform these basic number combinations. Once the child has a rudimentary grasp of the meaning of number and numerals, he will be able to perform the operation mentally, or even with pencil and paper if he is able to form the symbols (e.g., 1,2,3, etc.). Finally, children will begin to adapt their strategies so that they function as general class procedures—strategies that work with a variety of similar operations or problems. As we find that children continuously move back and forth on the trajectory when presented with new situations, it is important to arrange new situations for them to encounter. Identifying the appropriate challenge, and the manner in which it should be presented, depends on the child. It may involve simply increasing the quantities being added, or not allowing the child to handle the physical objects. It may actually involve helping them to transfer and modify their strategies to work with a new operation (e.g., addition to subtraction). Regardless, an error is not costly—if you go too far and your child cannot solve the problem, just scale back again and try later. It is important to regularly present situations that press the limit of her present abilities. Throughout the entire process, use language. Talk with him about the entire process, and listen to his explanations. Ask questions, and encourage thinking aloud. This promotes awareness, and it plays an essential role in helping children (and adults as well) to understand what they are doing, why, and what they should do next. As we can see, there are a lot of changes that occur as a child progresses toward mastery of "simple" operations with whole numbers. It is important that teachers and parents recognize and utilize the changes that take place in the way children use, combine, and manipulate whole numbers as they grow in their understanding.

In short, we know now that children develop the ability to perform operations at a much earlier age than once thought. Evidence shows that most children age three (or younger) have the capacity to begin working in numerous productive ways with basic number combinations, and that some will even work out strategies that employ derived number combinations prior to the start of kindergarten (see sidebar). In addition to simply providing opportunities, there is a great deal we can do as teachers and parents to help promote young children's understandings and abilities to perform operations. Through arranging situations that are developmentally appropriate, encouraging children to explore these arrangements with both intuitive and taught approaches, and assisting and conversing with them during the process, children will not only improve in their abilities to compute but also improve in their understanding of meaning, effects, and abilities to use number relationships to solve problems.

Operationally speaking... Add With Me!

To ensure that children get the most benefit possible from addition exercises (or many other basic exercises involving operations), and that fundamental understanding of addition progresses along with recall, keep in mind certain considerations that will help keep your approaches flexible and child-centered. When helping young children learn to add:

- Use small numbers. If the numbers are not within the child's counting range, they will not be able to add or subtract even adjacent numbers.

- Begin with adjacent numbers. It is important to first establish the concept of addition and subtraction by working with numbers in which the computation itself is as easy as it can be (e.g., 2+1=3).

- Provide physical objects as "counters." Young children will work with the tools they have at their disposal, and at the earliest ages, one powerful tool is one-to-one correspondence with small whole numbers.

- Encourage "acting out" procedures. Acting out is a form of early mathematical modeling, and aside from the visual assistance that props provide (e.g., blocks, marbles, etc.), it also helps children at a very early age to begin to work through problems in a very logical and systematic manner, and helps to ensure that "understanding" progresses along with "procedure."

- Talk with children about their procedures. An important aspect of a child's development of the ability to add and subtract, and later to solve more complex problems, is the development of meta-cognition—the self-awareness of selecting procedures or strategies, trying them out, and making adjustments as necessary.

- Encourage use of specific counting procedures, such as "counting all" or "counting on" (often called "adding on" when performing addition).

- Encourage use of fingers and toes for early stages of addition through non-visual counting. There is nothing wrong with it—there is a reason we use the base-10 system.

- Gradually introduce situations that prevent children from using less advanced strategies. The need to develop and apply new strategies will help to move them to the next level.

- After children are able to use basic number combinations proficiently, begin to introduce methods that employ use of derived number combinations, such as "making tens" or "making doubles."

- Introduce word problems at a very early age. The very problems that cause middle school students the most problems are the perfect tool for teaching the youngest learners. Of course, you may need to help read the problem, or read it to them. Be sure as well to make the problem simple enough to understand what it is asking.

Baroody, A. (1984). The case of Felicia: A young child's strategies for reducing memory demands during mental addition. Cognition and Instruction, 1, 109-116.

Baroody, A. (1985). Mastery of the basic number combinations: Internalization of relationships or facts? Journal of Research in Mathematics Education, 16, 83-98.

Brownell, W. (1987). AT classic: Meaning and skill—maintaining the balance. Arithmetic Teacher, 34(8), 18-25. (Original work published 1956)

Carpenter, T., Ansell, E., Franke, M., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children's problem-solving processes. Journal for Research in Mathematics Education, 24, 428-441.

Fuson, K. (1992). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243-275). New York: Macmillan.

Kouba, V. (1989). Children's solution procedures for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20, 147-158.

LeFevre, J., Sadesky, G., & Bisanz, J. (1996). Selection of procedures in mental addition: Reassessing the problem-size effect in adults. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 216-230.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Verschaffel, L., & De Corte, E. (1993). A decade of research on word-problem solving in Leuven: Theoretical, methodological, and practical outcomes. Educational Psychology Review, 5(3), 1-18.