What We Know

The probability of an event is the chance of it occurring. The concept of sample space is fundamental to all probabilistic reasoning. Because calculating the probability of events is not necessarily an intuitive process and may be overly influenced by our personal experiences, instruction in probability is important for students (Principles and Standards for School Mathematics, 2000). Instruction is also important because of common misunderstandings in probability. Green (1983a, 1983b) found that students lack understanding of some concepts basic to success in probability, including ratio (needed to predict probabilities); the language of probability (such as "at least" or "certain"); and concepts of randomness, stability of frequencies and inference.

Probabilistic thinking begins when children differentiate between whether an event will always happen, sometimes happen or never happen. When a glass is knocked off a table, it will always fall toward the floor. Sometimes it will break. It will never fall toward the ceiling. Yet while young children are aware of a range of outcomes to events, they often cannot distinguish between definite and possible events. Many think likely events will always happen and unlikely events will never happen (Shaughnessy, 1992).

Research shows that a majority of middle-grade students incorrectly answer questions involving independent events. Most students think an event will affect the possibility of a future outcome (such as if a coin flip results in heads, tails are more likely on the next event) (Shaughnessy, 1992). Studies have shown, though, that instruction can increase student understanding. In one study, middle-grade students who engaged in calculating probabilities and predicting events performed better on a measure of probability than did students who did not receive this instruction (Fischbein & Gazit, 1984).

Jones, Thornton, Langrall and Tarr (1999) describe the progression of students' probabilistic reasoning through the middle grades as follows:

• Level 1 Subjective: Students seldom present a complete list of possible outcomes. They focus on what a small-sample or personal experiences tell them are most likely to occur rather than focusing on what is possible?
• Level 2 Transitional: Students identify a complete set of outcomes for a one-stage experiment. They lack full understanding of sample space and probability and may identify the most likely outcome subjectively.
• Level 3 Informal Quantitative: Students list outcomes for a two-stage experiment. They begin to understand differences between experimental and theoretical probabilities, fair and unfair probability situations and independent and dependent events.
• Level 4 Numerical: Students employ systematic strategies to generate possible outcomes of experiments and determine numerical probabilities.