Counting the Pinecones: Children's Addition and Subtraction Strategies

As mathematics educators, we have found Maria Montessori's work to be innovative and ahead of her time. Instead of mandating a curriculum she felt children should learn, she spent time observing children to find out what children could learn and wanted to learn. She collected data regarding children's reactions and understandings of her presentations and materials in order to revise her approach to a concept, if necessary. In the language of current education, this is called "action research." Montessori intended this knowledge to be shared with other teachers, increasing the Montessori community's understanding of children's thinking. A group of Montessorians has even tried to formalize this process with a program called Teachers' Research Network (ChattinMcNichols & Loeffler, 1989). Similarly, our intent is to share mathematics education research and practices. Specifically, we would like to suggest the use of word problems to help children build a more abstract understanding of addition and subtraction.

In mathematics education, researchers are examining how children invent arithmetic operations in a program called Cognitively Guided Instruction (CGI). Notice that the title implies an approach to teaching that is similar to Montessori's method of using research of children's thinking to guide instructional decisions. Recognizing that our symbolic squiggles do not have inherent meaning to children, Carpenter et al. (1999) used word problems as a medium to contextualize addition, subtraction, multiplication, and division. By observing how children pursue word problems, the researchers were able to assess what the children understood about operations, looking beyond whether or not they could perform the simple arithmetic calculations. Through quantitative and qualitative analyses, they discovered that children were indeed capable of solving complex word problems, including problems that involved more than one operation, in a variety of ways. Carpenter et al. shared this information with teachers, creating a new level of research, an investigation of how teachers use their knowledge of children's mathematical learning to make informed, instructional decisions. To begin the journey of datadriven decision making, teachers in CGI programs are first shown the types of word problems that promote children's algorithmic inventions and then discuss children's typical solution strategies.

Many teachers are surprised when they learn about children's methods of problem solving. For instance, children and adults respond to the following word problem in different ways.

Ricardo and Stephanie are searching their school playground for pinecones. Stephanie had some pinecones. Ricardo gave her 6 more. Now she has 14 pinecones. How many pinecones did Stephanie start with?

While adults would generally recognize this as a subtraction problem and subtract the 6 from the 14 to get the answer of 8, children think of the action involved in the situation. A common strategy for a child solving this type of problem would be to grab a handful of counters, add 6 to the pile, and then check to see if there are 14 altogether. Then they would repeat the process pulling out larger or smaller start piles. After solving many problems, a child begins to realize the relationship between addition and subtraction and will not need to mimic the action of the problem with concrete objects. Only then will a child begin to understand the symbols involved in mathematical sentences and be able to recognize arithmetic categories, such as the missing addend problem shown above. CGI researchers looked at children's ways of thinking and organized their findings around how children thought about the problems, not what most adults would consider the most likely operation to be used.

Types of Situations-Joining and Separating

According to CGI, the pinecones problem above would be classified as a Join Start Unknown (JSU). The problem indicates the physical merging (joining) of two distinct sets (Stephanie's pinecones and Ricardo's pinecones), thus the label Join. Because we do not know how many pinecones Stephanie had to begin with, it is further labeled as a Start Unknown. Thus, rather than distinguishing between addition and subtraction, CGI organizes problem types according to their implied actions. The table on this page shows how any of the three numbers in a problem (the starting number, the number indicating a physical change, and the resulting quantity following the change) can be used as an unknown.

Join Result Unknown (JRU) and Separate Result Unknown (SRU) are the types of problems found most often in textbooks. Usually children add or subtract the numbers in a word problem based upon what operation they have just studied in class. This is not true problem solving. It is important for teachers to provide a variety of problem types and for students to create their own methods of solution, thus leading to their own deep understanding of the arithmetic operations. Including Change Unknown and Start Unknown problems in addition to the simpler Result Unknown problems provides opportunities for examining all possible situations involving joining and separating. Within this rich mathematical environment, children begin to view addition and subtraction as inverse operations. The key is for children to know how to decode and use information to solve problems, not just how to compute.

Levels of Understanding

Similar to the Montessori tradition of "following the child," CGI teachers believe that children's own solution methods are the most empowering for them. Instead of insisting upon a specific (traditional) approach where a child mimics an adult's solution methods, teachers in CGI classrooms learn to spend time understanding children's methods. Because the children own the teaching and learning in this type of classroom, they gain confidence in their ability to understand and perform mathematics. By studying children's methods of solution, CGI researchers noticed that children attacked problems in a variety of ways. In general, students begin by directly modeling the situation presented in the problem and progressively move toward applying abstract number facts.

Consider the following Separate Start Unknown (SSU) problem:

Stephanie had some pinecones. She gave 6 to Ricardo. Now she has 8 pinecones. How many pinecones did Stephanie start with?

Needing materials to solve a problem, a child using direct modeling would pull out a large pile of counters and remove 6. After counting the remainders, if there were more or fewer than 8, the child would push all the counters together again and adjust the size of the starting pile accordingly. Then he would repeat the actions of removing, counting, checking, and adjusting until there were 8 counters remaining. The child must recognize the need to count the 6 removed and the 8 remaining counters to get an answer of 14.

Once children gain confidence in not needing to physically represent the problem situation, they frequently use counting strategies to solve problems. A child at this level might solve the above problem by starting at 8 and counting up 6 more to get to 14, possibly using her fingers to count.

At advanced levels of understanding, children know an easier way to solve the problem using addition or subtraction facts. A child at these levels immediately knows that adding 6 and 8 can solve the problem. If a child knows "6+6=12" then he can add 2 to that total to reach 14 (derived facts). However, the most expedient solution method would be to automatically know that 6 added to 8 are 14 (known facts).

In CGI classrooms, teachers observe children as they solve problems using these varieties of strategies. Using this observational data, the teacher can scaffold to more challenging problems, and encourage each student to move toward more abstract thinking at his or her own pace. In addition, classroom discussions about the word problems help children accept and adopt a variety of solution strategies presented by their peers. If a child understands another child's more advanced solution methods, he or she will readily embrace the "new and improved" method.

Throughout this article, we have described the aspects of CGI that are similar to the Montessori tradition. Children use a variety of materials and strategies to solve problems. The role of the teacher is to modify the environment (using a variety of problem types and difficulties) to learn about each child's understanding. The teacher's new understanding of the children's mathematical thinking is then used to vary the types of problems given in order to help children become more abstract thinkers. Once you can present a variety of problems, your skills as an action-researcher in your own classroom are enhanced.

We have only scratched the surface of the findings of CGI researchers. Many more problem types exist for all four operations. There are also research findings of children's understanding of other mathematical topics (see appendix). We encourage you to investigate these for use in your classrooms.

References

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L. & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Chattin-McNichols, J. & Loeffler, M. H. (1989). Teachers as researchers: The first cycle of the teachers' research network. Young Children, 44(5), 20-27.

DR. MICHELLE K. REED is an assistant professor with a joint appointment to the Departments of Mathematics & Statistics and Teacher Education at Wright State University in Dayton, OH.

DR. JEFFREY P. SMITH is an associate professor in the Department of Mathematics at Otterbein College in Westeruille, OH.

Copyright American Montessori Society Spring 2005
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