Student Reasoning About Chance

 

Developing Critical Numeracy Across the Curriculum

Student Reasoning About Chance

Over the early childhood and primary years students develop an understanding of chance in qualitative and descriptive settings that involve language usage.

  • Expressions reflect gross likelihood: impossible, certain, maybe

  • Examples given of life situations reflect levels of certainty:
    “It is certain the world will turn today.”
    “It might possibly rain today.”
    “It is impossible for me to fly to the moon.”

  • Expressions reflect finer degrees of likelihood: Buckley’s, unlikely, 50% chance, looking good, sure thing

  • Expressions interpret risk, e.g., can explain a “15% chance of getting a rash” from a medicine

Over the primary and middle school years students are likely to develop facility with measuring chance in a quantitative sense. The type of problems solved are likely to increase in complexity similar to the following sequence.

  • Working with a single event, e.g., an outcome from a die toss

  • Working with a single event but multiple characteristics, e.g., in a hat with 13 boys’ names and 16 girls’ names, what is the chance of drawing a girl’s name?

  • Comparing probabilities from two single events, stated in proportions

  • Producing distributions from repeated single events

  • Working with two independent events

  • Interpreting conjunction of events

  • Interpreting conditional events

  • Working with odds

As well as working quantitatively with chance events students develop understanding of the importance of meaningful explanations of the numerical answers they obtain. It is likely that justifications become more sophisticated in the following fashion.

  • Idiosyncratic reasoning: “it’s luck”, “because a 2 is easier than a 6”

  • “Anything can happen” reasoning:
    “It might be anything.”
    “You never know what you’ll throw.”

  • Reasoning based on physical considerations:
    “A die is a balanced cube.”
    “There is only 1 six on the die.”

  • Quantification: giving a numerical value (fraction, decimal, percent)

In the middle school years students should begin to develop the ability to make judgments about chance statements made in real-world contexts, looking for bias and questioning claims. This is likely to involve the following components.

  • Questioning very large or small numbers.

  • Appreciating risk statements and often their conditional nature.