| Fun with Probability!The Probable Pen in the Cereal BoxRelationship to NCTM Standards |
This document describes how the extended Fun with Probability! lesson relates to the NCTM Curriculum Standards Document published by the National Council of Teachers of Mathematics. Please refer to the official site (linked above) for the most recent updates.
The Fun with Probability! project has foundation in three process (1-4) and four content (5-13) NCTM standards in the 5-8 grade range. They are listed below, along with links back to the NCTM site. Following that is a brief discussion for each of the standards addressed.
CURRICULUM STANDARDS FOR GRADES 5-8
STANDARD 1. MATHEMATICS AS PROBLEM SOLVING Discussion
STANDARD 2. MATHEMATICS AS COMMUNICATION Discussion
STANDARD 4. MATHEMATICAL CONNECTIONS Discussion
STANDARD 5. NUMBER AND NUMBER RELATIONSHIPS Discussion
STANDARD 7. COMPUTATION AND ESTIMATION Discussion
STANDARD 9. ALGEBRA Discussion
STANDARD 10. STATISTICS Discussion
STANDARD 11. PROBABILITY Discussion
1. MATHEMATICS AS PROBLEM SOLVING
(1.1, 1.2, 1.3, 1.4, 1.5, 1.6) As discussed in this standard, it is important that computers be used as problem-solving tools. It is also critical that students have the opportunity to solve problems in a cooperative environment, use technology, and address interesting mathematical ideas. They should also use multiple problem-solving techniques and be allowed ample time to explore, discuss and experiment, all attributes of this project. Finally, students are encouraged to generalize solutions to other situations, a step reached on the final day.
2. MATHEMATICS AS COMMUNICATION
(2.1, 2.2, 2.4, 2.5) The class participation/discussion activities of Day 1 and Day 5, as well as the group interaction activities of Day 2 and Day 3 provide ample opportunity for the collaborative goals of the second standard. Students will be using oral (discussion), written (tabulating data), concrete (simulating the experiment), and graphical modes of communication. The lesson also relies heavily on the ability of the students to listen and interpret results as well as discuss their thoughts and conjectures about what is happening over the course of the lesson.
(4.1, 4.4, 4.5) As part of the discussion, the teacher is encouraged to continually relate the simulation and model back to the real world. On the final day, the class might calculate the cost of the cereal required to get each of the pens, comparing it to the cost of a set of pens bought at an art supply store. Seeing how the mathematical nature of the problem helps them to interpret the real world is the whole point of this standard.
5. NUMBER AND NUMBER RELATIONSHIPS
(5.3, 5.5) It is difficult to imagine any mathematics lesson which doesn't incorporate at least some of standard 5. Many manipulations of numbers will eventually lead to fractional or percentage computations. This lesson also incorporates graphing of one-dimensional graphs.
(7.1, 7.4, 7.5) One new focus of this standard is the focus on technology as a vehicle for performing repeated and routine computations. This in fact underpins the entire lesson -- it would take hours to manually calculate (with or without a calculator) the averages and pool the values for histogram purposes. A corollary of this standard is to develop skills to translate from the computer model back into the problem. I feel that this lesson does just that.
(9.2) The second part of the algebra standard supports representing information in a variety of formats or modes. This lesson supports tabular, graphic and oral interpretations of the data.
(10.1, 10.2, 10.3, 10.4, 10.5) This lesson is based on the ability to generate and gather data in an organized form. The student will have to create and interpret graphs, and create general inferences about the data. Finally, the students will compare their experimental results to the actual expected value, and will have to justify/explain the differences in some way.
(11.1, 11.2, 11.3, 11.4, 11.5) I was surprised to see the Fun with Probability! project concept listed as an example in the probability standard as one of several "experiments ... that students can determine probabilities inherent in more complex situations using simple methods." Obviously, this being a probability project, we would expect the probability content standard would be addressed. Three of the objectives that I set out to reach when creating the project (before I knew the standards existed) were: