• Research
• Deaf and Hard of Hearing Research

One of the organizing tenets of the research that went into To Market, To Market was an emphasis on “core difficulties” as outlined by Robbie Case, Sharon Griffin¹, and others. For students to be successful, it is imperative that they begin their academic careers with a strong intuitive feel for the nature of numbers. Case has researched the identification of certain universal stumbling blocks that trip up students who do not have this background in number sense. Countering these difficulties by providing a foundational understanding of numbers was one of the key goals of this project. Awareness of these difficulties, as well as a clear understanding of what “good” number sense is, helped the researchers implement a design aimed toward that end.

In “Re-Thinking the Elementary Mathematics Curriculum,” Case suggests that it is important to understand not only students’ failures, but also their successes. “It seems particularly important to understand those successes that depend on intuition and creativity, rather than just on the mastery of explicitly defined concepts and algorithms.”² Case goes on to outline the common abilities of students with good number sense:

• moving seamlessly between the “real” world of quantities and the mathematical world of numbers, symbols, and operations
• inventing their own procedures for conducting mathematical operations: procedures that are flexible and intuitive, not completely tied to an algorithm that has been learned in school
• recognizing and using number patterns that reveal something about the structure of the number system (e.g., 21+10=31; 22+10=32; 23+10=33; etc.)
• recognizing gross errors, that is, errors that are off by an order of magnitude
• representing a single number in multiple ways, depending on the context and purpose of the mathematical activity (e.g., 9=5+4 or 10-1)

To Market, To Market combines the research from cognitive scientists like Case and mathematics educators such as Klein, Cobb, Treffers, Kaput, and others in a technology program for primary grade students. The product is also based on extensive research about children’s responses to the proposed environment:

• the “child appeal” of contexts and mathematical objects including the use of real-world mathematical applications, the use of models, including the number line, and how children select and implement them
• the opportunity for students to create their own productions
• the incentives that are successful in getting students to communicate their mathematical thinking
• understanding of students’ beginning intuition on number and multiple perspectives on number

Some of this research includes:

Real World

Freudenthal (1987)³ describes children’s mathematization in activities in which they seem to move back and forth between the real world and the world of symbols as they build their understanding. Realistic Mathematics Education has, at its central focus, the use of context problem situations in which students are able to construct their own knowledge of mathematics. Problems which have meaning to the children must be identified. Freudenthal’s work in the “reinvention” of mathematics made great strides in providing a model for activities that foster the formation of the concepts from within “the phenomena by which the concepts appear in reality.” (De Lange, 1988)⁴

Models

In addition to the importance of problems in realistic contexts, another focus of the research is the use of models. Models can act as a bridge for raising students’ understanding from one level to the next. One way the research has described these three levels of understanding is as follows:

• Level 1: describing and solving concrete problems through informal mathematizing
• Level 2: solving similar problems with mathematical tools that build upon the informal strategies
• Level 3: gradually evolving strategies into simplified and formalized mathematics

Streefland points out that in the first level of mathematizing, computing is a concrete procedure and not yet divorced from the context. Later, a model such as the empty number line may be used so that reference to the concrete still exists, though it is probably no longer needed to apply notations. And finally, on the third or abstract level, when the procedures have been formalized, understanding will transfer from the illustrative context and exist completely within the formal number system. It is through the use of such models as the empty number line applied to realistic concrete situations that students develop and invent their own informal strategies for problem solving. “...The empty number line is such a tool, which functions as a bridge between the concrete and the formal level.” (1990)⁵

Own Productions

Traditional mathematics instruction has emphasized the avoidance of mistakes. In recent literature, children’s “own productions” are also valued, as well as their incorrect solutions to equations, and their informal attempts at reasoning through problems. The idea supporting the encouragement of informal strategies and providing contexts in which students can create their own productions is that children will learn from their first attempts to solve problems and describe mathematical situations on their own, even if these attempts are, at first, flawed.

In addition, technologies such as animation may help to expand students’ understanding of numbers.

The goal of To Market, To Market is to synthesize four important elements in a single environment:

1. students’ intuitive feel for numbers and multiple perspectives on numbers
2. use of sound to transition from acoustic counting to quantity counting and to determine number
3. the provision of multiple tools to encourage students’ own strategies and a move toward more formal mathematizing
4. the opportunity to use mathematics creatively

1. Griffin, S. A., Case, R., & Siegler, R. S. “Rightstart: Providing the Central Conceptual Prerequisites for First Formal Learning of Arithmetic to Students at Risk for School Failure.” In K. McGilly (Ed.) Classroom Lessons: Integrating Cognitive Theory and Classroom Practice. Cambridge, Mass: M.I.T. Press, pp. 25-50. 1994.
2. Case, Robbie. “Re-Thinking the Elementary School Mathematics Curriculum.” Background paper prepared for a colloquium with Joan Moss, presented at the Centre for Applied Cognitive Science. November, 1995.
3. Freudenthal, H. Theorievorming bij het Wiskundeonderwijs-geraamte en Gereedschap. Panamapost 5, 4-15. 1987.
4. De Lange, Jan. “Curriculum Contents and Their Evolution” in Senior Secondary Mathematics Education. Utrecht, the Netherlands: Research Group at Educational Computer Centre. 1988.
5. Streefland, Leen. “Realistic Mathematics Education. What Does it Mean?” in Context Free Productions Tests and Geometry in Realistic Mathematics Education. Research Group for Mathematical and Educational Computer Centre; State University of Utrecht, Netherlands. 1990.
6. Cobb, P. “Information-Processing Psychology and Mathematics Education-A Constructivist Perspective.” The Journal of Mathematical Behavior. 6(1), 4-40. 1987.
7. Van den Heuvel-Panhuizen, Marja. “Student Generated Problems: Easy and Difficult Problems on Percentage.” For the Learning of Mathematics: An International Journal of Mathematics Education. Montreal, Canada. Issue 15 (3) November, 1995; or Issue 16 (1) February, 1996.