Chapter 6. Master copy of Probability discovery cycle
Chapter 1. This booklet contains information on how to introduce probability to year 9 mathematics classes so that they can dispel some of their misconceptions about probability or chance as well as enjoy a topic that is relatively new to many of them.
Many pupils do not have a very good understanding of many of the ideas taught in traditional probability courses and this is highlighted in external examination results (see chapter 2). The way that probability is currently taught in a lot of schools is simply not working. It is hoped that teachers using this set of lessons will be able to provide an interesting and fun module which aims to dispel some heuristic thinking and fallacies.
This module is relatively short, focusing on just 7 lessons as well as a diagnostic and summative assessment task. Much of the language may be unfamiliar to Mathematics teachers, especially if they were pure mathematics rather than statistically trained and therefore I would recommend that teachers using this set of class lessons read chapter 2 before using these lessons in their classrooms.
There is very little theoretical probability in these lessons so students of all abilities should be able to engage in the tasks. I am hoping to develop a Year 10 Probability module in 2008.
Suggested equipment (to cover all lessons)
Computer and Data-show (projector) – Used in all lessons
Large dice or OHP “6 sided die” (not essential but good visual)
Class set of dice – Lesson 1
Big card board coin (not essential but good visual)
Toy coins (could provide real money if needed) – Lesson 2
Coloured counters/beans and opaque containers – Lesson 3
Education is in a constant evolutionary process with the aim of improving students understanding through better teaching. Many of these evolutionary steps are quite small and un-noticed by most, a school Mathematics Department redefining its course structure is an example of a small evolutionary step. Other steps are moderate, such as the government driven projects such as the Numeracy Project that was introduced into Primary and Intermediate schools in 2000, the Literacy Project, introduced in 1998 and more recently, the Numeracy Project that was launched into Secondary schools in 2005. These projects may or may not affect a particular school as often a school must apply to receive assistance. Large steps also occur in the evolutionary ladder and can divide the education community. The NCEA and its associated assessment schedules and the proposed Curriculum are examples of large steps. In this essay a moderate evolutionary step in the teaching of probability is proposed in order to improve students’ probabilistic understanding. The misuse or misunderstanding of probability and probabilistic language will be explored and preconceived notions about probability in their reasoning will be investigated. Finally, changes will be suggested that could help teachers and students recognise pitfalls to enable them to understand this branch of the curriculum better.
Despite probability having been taught in the New Zealand curriculum for the same length of time as statistics, probability, in my experience, is still seen by many teachers as statistic’s little unwanted cousin and in many junior secondary school Mathematics courses, probability is often simply taught at the end of the statistics topic, and only if there is time. By the time students face their first national examination for probability, a significant amount of the participants may have only received about 5 to 6 weeks of probability teaching prior to this examination at secondary school. According to Shaughnessy and Bergman (1993, p. 178) “Many students will not have the opportunity to take it (probability) and teachers may even be tempted to skip it altogether.”
Results from the 2006 NCEA Level 1 Mathematics and Statistics Examinations (table 1) show that the probability strand had the greatest number of students failing with 15,701 of the 34,432 participants receiving a Not Achieved Grade (45.6%). Probability also received the dubious distinction of having the smallest percentage of Achieved with Excellence with-in the Mathematics and Statistics strand with just 3.2% of the 34,432 students attaining this grade.
By year 13, New Zealand’s final year of secondary school, the students sit their University Entrance equivalent examination, NCEA Level 3. The pass rates at this level make for even bleaker reading with over half the participants failing the “solve straightforward problems involving probability” examination with the carry on effect of pass rates being significantly lower than all the other papers at the Achieved and Merit level.
Why are questions involving straightforward probability being misunderstood or simply not answered correctly at so much higher rates than other statistical and probability topics? Distributions such as the Poisson and Binomial are currently set at Level 8 of the current national curriculum. The topic of Confidence Intervals is also set at Level 8 of the curriculum, yet students of Probability and Statistics are passing these topics in far greater numbers than straightforward probability, a strand of the curriculum that is supposedly introduced to them at Level 1 of the national curriculum, that is, in their first year of school. While it is arguable that the NCEA does not assess what students need to know about probability and its place in everyday life, it does highlight that many students do not understand the basics of the topic.
Heuristics, bias and fallacies
It is naïve to believe that students arrive at secondary school with a tabula rasa, and have no exposure to probability as is it after all, prescribed in the curriculum. Despite the fact that they have had a lot or no classroom exposure to probability they have certainly talked, seen, used and informally calculated probabilities in their day to day life (Watson, 2005). Students and adults have their own set of probabilistic beliefs and biases, which can be of either benefit or hindrance to their conclusions about a probabilistic outcome (Tversky & Kahneman, 1982; Watson, 1995). Many of these preconceived notions of probability fall under the title of bias or fallacy, they can often be the precursor to the heuristic concepts, representativeness, availability and anchoring that will lead to an incorrect conclusion (Mckean, 1985).
There is a need for students to be exposed to their heuristic beliefs and biases so that they can become aware of the flaws in their reasoning and make better judgments in the real world. However, how can teachers break down these barriers unless they are aware of these heuristics and biases in the first place? Other misconceptions that students bring to their lessons are the notions of luck, fairness, random behavior and equiprobability bias (Watson, 2005; Shaughnessy, 2003). Fallacies include the conjunction fallacy, the gambler’s fallacy and Post hoc ergo propter hoc which is a logical fallacy (Watson & Moritz, 2002; Tversky & Kahneman, 1982). While it is not enough to change probabilistic thinking simply based upon awareness of biases, fallacies and heuristics (Shaughnessy, 1977), it is a starting point and with other techniques such as modeling real life situations and more thoughtful use of experiments and simulations it may lead onto more informed decision making processes under uncertainty and better cognitive processes.
A heuristic, derived from the Greek word "heurisko" (εὑρίσκω), which means “I find”, is a procedure or method that people may use when problem solving. It often involves mental shortcuts that may work well or could lead to incorrect conclusions. Students will have an intuitive feel for question posed to them and is important that they have an appropriate schema for the situation. While the teaching and highlighting of potential pitfalls may not make students necessarily change their methodology, it is hoped that it will make them aware of beliefs or assumptions that may cloud their judgment. The three main heuristics are representativeness, availability and anchoring.
1. The representative heuristic is a heuristic where it is assumed there is a link between events or objects. The stronger an event can represent an event