None of the topics below are likely to be part of the content an elementary school teacher is expected to teach. Nevertheless, probability is important for at least two reasons. Probability, particularly the law of large numbers and the normal distribution, provides the foundation for statistical inference, an area that teachers of all levels will use when interpreting test results and research outcomes. In addition, topics such as expected value and Baye's theorem are important to understand to be an educated citizen and consumer. We recognize that in many settings elementary education majors will encounter these topics in a the context of a course intended for students from a variety of majors. We believe all students will benefit from an inquiry-based approach to these topics, as recommended in the resources below.
D5.a. Understand that probability provides a way to describe the ‘long-run’ random behavior of an outcome occurring and recognize how to use simulation to approximate probabilities and distributions.
D5.b. Explain why theoretical and experimental probabilities may differ for a given event in a particular experimental situation.
NOTES:
Recommended text: Beckmann, Sybilla, Mathematics for Elementary Teachers with Activities, 5th Ed., chapter 16.
Recommended materials: dice (possibly include 4-sided, 8-sided, etc), cards, coins, beads of at least two different colors:
Recommend: Use surveys of class for data when possible
Recommended Websites and Apps:
random.org (random number generator, simulations for coin-flipping, die-rolling, card-shuffling)
phone apps for die-rolling, coin-flipping, card-selecting (Do a search; be aware some card-shuffling apps allow repeats while others do not. Test to see.)
For Android, "Dice Roller" by Pilot Student Studios works well.
Basic Probabilities, Law of Large Numbers, Empirical vs Theoretical Probability:
These experiences illustrate the tendency of long-run experimental probabilities to center at the mean, while also showing the variability that occurs. This background knowledge is important for understanding inference in statistics.
And, Or, Complement, Addition Rules, Mutually Exclusive:
These examples give opportunities to focus on precision in mathematical language (SMP 6) and correctly representing given information (SMP 4).
Multi-stage independent events (introduce tree diagrams):
These activities give opportunities to reinforce the concept of "independence" and address the "gambler's fallacy" that a long string of one outcome means another outcome is "due."
Probability Distributions:
Creating histograms to summarize probabilities is a stepping stone to understanding the normal distribution and the concept of area as probability. It links the more concrete notion of "likelihood" of an outcome to the more abstract representation of area in a graph (SMP 2).
Conditional Probability in Two-Way Tables:
Emphasize how the denominator changes in conditional probability to reflect the "given" or "known" information.
Conditional Probability in Multi-Stage Events (use tree diagrams):
The probabilities on the second level of branching in tree diagrams will reflect the "given" condition (e.g., if a red marble has already been chosen there are fewer red marbles, and fewer total marbles, in the bag).
Tree diagrams are a powerful and useful model when conditional events occur (SMP 4).
Reversing the Conditioning (using tree diagram):
Without giving a formula or even naming Baye's Theorem, students can calculate probabilities like "given the test result is positive, what is the probability the person really does have the medical condition?" This is an example of useful real-life knowledge. The Post article cited gives examples of medical professionals mis-interpreting this type of outcome. Students have the opportunity to critique the informal mistaken reasoning cited in the article and construct a coherent explanation for the correct interpretation (SMP 3).
Expected Value:
Expected Value can connect probability to practical situations such as a lottery or insurance. This is another opportunity to emphasize the Law of Large Numbers, also: even if a person gains money at a single casino visit, the negative expected value means if they continue to play they will eventually lose money overall (SMP 3, 4).
Counting Principles (multiplication rule, permutations, combinations):
Rules for counting become complex very quickly. Depending on the course goals, this unit may be brief or eliminated altogether. However the basic multiplication principle is another opportunity for practical applications. Challenging contexts give opportunities to practice perseverance and modeling (SMP 1, 4).
Probability Misconceptions:
These can be addressed throughout the unit, and then reviewed at the end.
Basic Probabilities, Law of Large Numbers, Empirical vs Theoretical Probability:
D5.a. Activity "Randomness, Probability, and Simulation"
D5.b. Activities from Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed :
16C Empirical vs Theoretical Probability: Picking Cubes from a Bag
16E If You Flip 10 Pennies, Should Half Come Up Heads?
D5.a. "Ratios" Activity in Navigating Through Probability in Grades 6 - 8, pp 26-28, 95-98.
D5.a. Article describing activity: "Is the Last Banana Game Fair?" Mathematics Teacher: Learning and Teaching PK - 12, January 2020.
D5.a. Applet to quickly take samples to illustrate distribution of a proportion:
And, Or, Complement, Addition Rules, Mutually Exclusive:
D5. Activity. "Two Way Tables and Venn Diagrams":
Multi-stage independent events (introduce tree diagrams):
D5. Activities from Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed.
16L Using the Meaning of Fraction Multiplication to Calculate a Probability
16H Number Cube Rolling Game
16M Using Fraction Multiplication and Addition to Calculate a Probability
16J Critique Probability Reasoning About Compound Events
D5. Activity. "Probability Bingo"
D5. Activity. "Analyzing Games of Chance"
D5. Activity. Probability with Spinners
Probability Distributions:
D5. b Book with Activities. Navigating Through Probability in Grades 6 - 8, pp 29-50
Conditional Probability in Two-Way Tables:
D5. Activity. "Do You Prefer English or Math?"
D5. Book with Activities. Navigating Through Probability in Grades 9 - 12, pp 27-35 and associated activities
Conditional Probability in Multi-Stage Events (use tree diagrams):
D5. Activity. "Picking Two Marbles from a Bag of 1 Black and 3 Red Marbles" 16I in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed
D5. Activity. "General Multiplication Rule: Can You Get a Pair of Aces or a Pair of Kings?"
D5. Activity. "Representing Conditional Probabilities 1"
D5. Activity. "Representing Conditional Probabilities 2"
Reversing the Conditioning (using tree diagram):
D5. Activity "Medical Testing"
D5. Activity. "The Surprising Case of Spot, the Drug-Sniffing Dog," Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed. pp 746-747 (recommend giving students the information in the first four paragraphs and asking groups to a) guess, b) model with a tree diagram and calculate).
D5. Article. "What the Tests Don't Show"
Expected Value:
D5. Activity. "Expected Earnings from the Fall Festival," 16K in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed.
D5. Book with Activities. Navigating Through Probability in Grades 9 - 12, pp 49-58 and associated activities
D5. Activity. "Designing a Game of Chance"
Counting Principles (multiplication rule, permutations, combinations):
D5. Activity. "Too Many Choices Too Early in the Morning"
D5. Activities in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed:
16F How Many Keys Are There?
16G Counting Outcomes: Independent vs Dependent
D5. Activity. "How Many Ways Can You Order a Hamburger?"
D5. Article. "How Can We Make Stronger Passwords?"
Probability Misconceptions:
D5. Activity. "Evaluating Statements about Probability"