"As with other beautiful and useful areas of mathematics, probability has in practice only a limited place in even secondary school instruction" (Moore, 1990, p. 119). The development of students' mathematical reasoning through the study of probability is essential in daily life. Probability represents real-life mathematics. Probability also connects many areas of mathematics, particularly counting and geometry (NCTM, 1989). "Research in medicine and the social sciences can often be understood only through statistical methods that have grown out of probability theory" (Huff, 1959, p. 11). Moore (1990) stated,
Probability is the branch of mathematics that describes randomness. The
conflict between probability theory and students' view of the world is due at
least in part to students' limited contact with randomness. We must therefore
prepare the way for the study of chance by providing experience with random
behavior early in the mathematics curriculum.
An understanding of probability theory is essential to understand such things as politics, weather
reports, genetics, sports, and insurance policies. As shown Table 1, the list of questions presented
in the different areas shows the need for experimental probability. Experimental probability is
the actual results of an experiment or trial. These questions require considerations of
probabilities and what they mean (Huff, 1959).
The inclusion of activities dealing with experimental probability in the elementary school enhances children's problem-solving skills and provides variety and challenges for children in a mathematics program (Kennedy & Tipps, 1994). Current and past recommendations for the mathematics curriculum identify experimental probability as one of several critical basic skill areas that should occupy a more prominent place in the school curricula than in the past (National Council for Supervisor of Mathematics (NCSM), 1989; Mathematical Sciences Education Board [MSEB], 1990; Willoughby, 1990; NCTM, 2000). Unfortunately, many schools fail to introduce probability until the end of the school year or not at all.
From a historical perspective, members of the Cambridge Conference on School Mathematics (1963) also acknowledged the role probability and statistics played in our society. The Cambridge Conference was an informal discussion of the condition of the mathematics curriculum in the United States at the elementary and secondary level. The members of the conference addressed revisions to the mathematics curriculum. They recommended that probability and statistics not only be included as part of the modern mathematics of that day, but these were also recommendations that they considered for 1990 and 2000. Other researchers have suggested that elements of statistics and probability be introduced in the secondary school curriculum and possibly at the elementary level as part of the basic literacy in mathematics that all citizens in society should have (Schaeffer, 1984; Swift, 1982).
Changes are being made today to introduce probability into the elementary school curriculum (NCTM, 2000). Experience with probability can contribute to students' conceptual knowledge of working with data and chance (Pugalee, 1999). This experience involves two types of probability--theoretical and experimental. There may be a need for students to be exposed to more theoretical models involving probability. Theoretical models organize the possible outcomes of a simple experiment. Some examples of theoretical models may include making charts, tree diagrams, a list, or using simple counting procedures. For example, when asked to determine how many times an even number will appear on a die rolled 20 times, students can list the ways of getting an even number on a die (2, 4, 6) and may conclude that one should expect an even number one-half of the time when a die is rolled. Then, students can roll the die 20 times, record their actual results and make conclusions based on their experiment. Another example that involves experimental modeling is the following:
If you are making a batch of 6 cookies from a mix into which you randomly
drop 10 chocolate chips, what is the probability that you will get a cookie with
at least 3 chips? Students can simulate which cookies get chips by rolling a
die 10 times. Each roll of the die determines which cookie gets a chip.
The most important use of probability is to help us make decisions as we go through life
(Newman, Obremski, & Schaeffer, 1987). For example, in issues of fairness, students may pose
a question based on claims of a commercial product, such as which brand of batteries last longer
than another (NCTM, 2000).
The study of statistics and probability stress the importance of questioning, conjecturing, and searching for relationships when formulating and solving real-world problems (NCTM, 2000). We live in a society where probabilistic skills are necessary in order to function. Probability describes the world in which we live. Many everyday skills depend on knowing and understanding probability. Milton (1975) suggested the following reasons for introducing probability as early as the primary level:
1. The basic role which probability theory plays in modern society both in the daily lives of the public at large, and the professional activities of groups within the society, e.g. in the sciences (natural and social), medicine and technology.
2. Probability theory calls upon many mathematical ideas and skills developed in other areas of school coursed, e.g. set, mapping, number, counting, and graphs.
3. Students are able to work in a branch of mathematics, which is relevant to current activities in life.
There is much support on children learning concepts of probability as early as elementary school. Many important documents, such as Principle and Standards (2000), have been revised to include the teaching and learning of probability as early as kindergarten.Although few studies exist at the elementary level, researchers recognize the importance of introducing probability at an earlier age (Shaughnessy, 1992; Vahey, 1998). The discussion in this study indicates that probability is a very useful concept that needs to be introduced early in the development of children and continuously revisited. It may not be necessary to wait until basic skills are mastered before exposing children to probability concepts. Because fractions are needed in demonstrating and explaining probability to students, why not include probability at this point, instead of the end of the school year? There is no reason to postpone studying concepts of probability until after fraction skills have been mastered. The study of probability could even possibly serve as an introduction to fractions. It may be that a spiral curriculum, designed to increase student interest and motivation in a particular subject area, would be useful for students when studying probability. With the repetition of fundamentals and the integration of subject content, the spiral curriculum should increase retention of basic skills and concepts. To accomplish the spiral curriculum as teaching moves upward, constantly circle back to build upon previous understandings to help students develop to their full potential in mathematics education.
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