Introduction
Probability is an important mathematical concept that supports critical thinking through inquiry learning and “applying logic and reasoning” (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2016, para. ACMSP067). This review will discuss the concept of probability and the related big ideas. Then, misconceptions and issues with probability will be discussed. An overview of effective pedagogies will precede a critical review of the curriculum in regards to probability.
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Key understandings
The concept of probability and data representation is evident in the everyday lives of children and adults (Siemon et al., 2015). In order for children and adults to “make informed decisions” (Reys et al., 2012, p. 432) they require an understanding of statistics and probability. For example, in advertising they might claim that 3 out of 4 people approve a product but the sample size was only 40 people. Children have an “intuitive understanding” (Frykholm, 2001, p. 112) of probability from a young age which should be developed throughout their schooling, and it is beneficial to the development of other mathematical concepts. It is also important to note that probability activities “serve as a foundation for data analysis” (Tarr, 2002, p. 485) when data is collected from trials. Probability also provides an important link to the concepts of fractions and percentages, ratio and proportion (Van de Walle, Karp & Bay-Williams, 2014)
Probability is a form of measurement, measurement of the likelihood of a particular event occurring (Tarr, 2002). The language associated with probability needs to be taught to students so they understand there is a probability continuum of how likely or unlikely it is an event will occur (Frykholm, 2001). Students also need to understand that “chance has no memory” (Van de Walle, Karp & Bay-Williams, 2014, p.482) and that one trial has no impact on another. This is also referred to as randomness which occurs when there is no pattern to the results (Reys et al., 2012).
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Lastly, there are two types of probability: experimental probability and theoretical probability (Van de Walle, Karp & Bay-Williams, 2014). Theoretical probability can be determined by analysing “the chance of events happening under ideal circumstances” (Siemon et al., 2015, p. 706). It relies on the students’ ability to identify the all possible outcomes of an event, known as the sample space (Van de Walle, Karp & Bay-Williams, 2014). While experimental probability uses the relative frequency to indicate the likelihood of an event (Haylock, 2014). It is linked to “number sense … percent, decimals, ratios and fractions” (Andrew, 2009, p. 35) which shows the importance of teaching with simulations and experiments. Activities that rely on experimental probability provide an understanding of the “fundamental principles of chance” (Andrew, 2009, p. 36).
Possible misconceptions or issues
Probability is a complex concept that results in numerous misconceptions being experienced by children. Children’s view of probability can be influenced by their prior intuitive knowledge of fairness, uncertainty and luck which is evident with “subjective responses” (Frykholm, 2001, p. 113); Siemon et al., 2015). They can base predictions on their preference (Reys et al., 2012). Students can also be “reluctant to acknowledge that an event can occur” (Tarr, 2002, p. 483) which impacts their ability to identify sample space. Additionally, students may not understand that probability estimates need to add to 1 or 100 percent (Van de Walle, Karp & Bay-Williams, 2014). It is essential that enough time is allowed to explore activities as estimating probabilities with a small number of trials “can be misleading” (Tarr, 2002, p. 484). Additionally, using a range of hands-on activities will improve their ability to make inferences from data and “apply basic concepts of probability” (Reys et al., 2012, p. 455).
Effective pedagogies
Probability should be developed through hands-on experiences with concrete materials that initially relate to student language, then materials language (Frykholm, 2001; Jamieson-Proctor and Larkin, n.d.). It is beneficial for students to simulate situations in order to find the likelihood of a real event occurring (Van de Walle, Karp & Bay-Williams, 2014). Additionally, activities should involve group work to develop students’ understanding that chance is objective and not related to a particular person or object (Andrew, 2009). Informal activities based on intuitive knowledge are also important to “foster the beginning of experimental probability” (Frykholm, 2001, p. 113). Task involvement is essential, however it is just as important to follow up by reflecting on the tasks and to “debate the findings” (Booker, Bond, Sparrow & Swan, 2010, p.524) with the purpose of identifying misconceptions. Furthermore, discussions about thinking will develop students’ reasoning skills (Reys et al., 2012).
Probability tasks can be extended with the introduction of virtual manipulatives. They can provide visual representation and support at the mathematics language stage, particularly for a large number of trials (Leavy & Hourigan, 2014; Jamieson-Proctor and Larkin, n.d.). Additionally, the results of trials can be entered into spreadsheets to compare data between groups or between the experimental probability and theoretical probability (Leavy & Hourigan, 2014). This incorporates symbolic language (Jamieson-Proctor and Larkin, n.d.). Continuing tasks to a large number of trials can develop their understanding of the ‘law of large numbers’ and it “strengthens the meaning of theoretical probability (Andrew, 2009, p. 34; Leavy & Hourigan, 2014).
Probability tasks can be extended with the introduction of virtual manipulatives. They can provide visual representation and support at the mathematics language stage, particularly for a large number of trials (Leavy & Hourigan, 2014; Jamieson-Proctor and Larkin, n.d.). Additionally, the results of trials can be entered into spreadsheets to compare data between groups or between the experimental probability and theoretical probability (Leavy & Hourigan, 2014). This incorporates symbolic language (Jamieson-Proctor and Larkin, n.d.). Continuing tasks to a large number of trials can develop their understanding of the ‘law of large numbers’ and it “strengthens the meaning of theoretical probability (Andrew, 2009, p. 34; Leavy & Hourigan, 2014).
Critique of curriculum
The Australian curriculum features ‘chance’ from Year 1. However, Frykholm (2001) suggests that probability should feature from the beginning of schooling as children bring intuitive knowledge of fairness, uncertainty and luck which can be developed on. Year 1 and 2 content is in line with Frykholm’s (2001) recommendations that children start with identify and describing likelihood of an event occurring with specific language (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2016). The language becomes more specific within the Year 2 curriculum but continues through to Year 4 (ACARA, 2016). Andrew (2009) suggests that experimental probability be introduced to provide an understanding of the “fundamental principles of chance” (p. 36) which from Year 3 onwards (ACARA, 2016). From there more complex concepts are developed. Sample space is introduced in Year 5 as students begin to represent probability as fractions. However, it is not until Year 6 that students are required to understand that “probability of an event occuring is between 0 and 1” (Van de Walle, Karp & Bay-Williams, 2014, p.482). The curriculum does not however explicitly mention the big idea of ‘randomness’ which underpins “all learning in probability” (Reys et al., 2012, p. 458).
Conclusion
Probability is a mathematical concept that can be taken beyond a mathematics lesson as it is experienced in the everyday lives of children. For example, language relating to probability can be developed throughout the day by asking questions that require students to think about the likelihood of an event occurring, such as whether the lunch bell will sound today. Probability has a number of key concepts that together develop probabilistic reasoning, and misconceptions that can be overcome with adequate practice and class discussion. Hands-on group activities are essential to students developing understanding of the concepts.