The DT approved by the DBE is freely downloadable on any mobile device such as tablets and mobile phones. The DT comprises 9 mathematics topics (sequences and series, functions, finance, trigonometry, polynomials, analytical geometry, Euclidean geometry, statistics, and probability) that are taught at Grade 12. The 9 topics are each written following the same format; thus, each topic begins with the revision of related concepts, followed by the content notes, a couple of worked out examples, and exercises at the end. The exercises have answers to enable learners to cross-check their solutions. In this article, we only discuss the topic of PCP.

Probability theory is a mathematical modelling (Blum et al., 2007, p. 4) of the phenomenon of chance or randomness. Randomness has a specific meaning in probability and statistics (Batanero et al., 1997; Batanero & Sanchez, 2013). Suppose T is a finite probability space. We assume that the physical characteristics of an experiment in T are such that the various outcomes of the experiment have equal chance of occurring. Such a probability space, where each point is assigned the same chance of outcome, is called a finite equiprobable space (Moore, Notz, & Fligner, 2013, p. 262; Spiegel, Schiller, & Srinivasan, 2013). However, events in T are far from being random. For example, a coin does not generate random numbers because a tossed coin obeys the laws of physics depending on the force used, angle of toss, and surface of the coin. So, why then do the results of tossing a coin look random? It is because the outcomes are extremely sensitive to the inputs, so that very small changes in the forces one applies when tossing a coin do change the outcomes, say from heads to tails and back again (Moore et al., 2013, p. 262).

Regarding counting principles, these are techniques for determining without direct enumeration, the number of possible outcomes of a particular experiment (May, Masson, & Hunter, 1990, p. 189). For example, if T defined as above, has k elements, then each point in T is assigned the probability 1k, and each event B ∈ T containing h points is assigned the probability hk. In other words, P(B)=n(B)n(T)=number of ways that the event B can occurnumber of ways that the sample T can occur=hk. We note that the formula for P(B) only applies to an equiprobable space T and not to a general space.

By multiple representations, we mean techniques of teaching PCP that include various objects such as graphs, diagrams, texts, and 3D visualisations to facilitate learners’ grasping of the underlying meaning of the concepts.

Probability and fundamental counting principles are relatively new topics in the South African mathematics syllabus (Zondo, Zewot & North, 2020), having been introduced for the first time in Grade 10 in 2012, and in Grade 11 in the following year. The topic was first examined in Grade 12 in 2014. Since their inclusion as compulsory topics in the South African mathematics syllabus, these principles have remained a challenge to many learners, with performance remaining generally poor over the years (DBE, 2015, 2016, 2017, 2018, 2019, 2020, 2021). For example, in 2013, the average score in the national examination was 30% (DBE, 2014). Although every year diagnostic reports are published stating specific concepts in PCP that learners show weakness in so that educators in schools can support them in overcoming such identified weaknesses, from 2014 to 2019, diagnostic reports strongly suggest that the same challenges faced by learners keep coming up in subsequent years.

There is also a lack of research to inform the teaching and learning of PCP at the school level. For example, a database search by the first author of articles in Pythagoras from 2016 to 2020 with probability as the keyword turned up only two outcomes, which were also not linked to the information on probability we were looking for. One article (Murray, 2017) sought to understand how the grades obtained at school for English and Mathematics affect the ‘probability’ of graduation at a university. Clearly, the context of said study was different from the current one. The second article (Prince & Frith, 2017) discussed quantitative literacy of South African school leavers who qualify for higher education. Again, this study is not related to PCP. These examples clearly confirm our assertion of little research in the topic of PCP at the secondary level. This study contributes to understanding mathematics teachers’ rating of the PCP learning tasks at Grade 12 in terms of the tasks’ cognitive demand levels. It may be that the underperformance in PCP at the national certificate is contributed to by the learning tasks that learners are prepared for on for the national examinations.

One of the proposals from the education authorities to try and reverse the poor performance at Grade 12 is the suggestion that the teaching and learning of PCP should incorporate multiple representations of tasks. Based on the mathematical task theoretical framework (Stein & Smith, 1998), we associate tasks with high cognitive demands with better prospects to enable learners to master PCP concepts, whereas tasks with low cognitive demand are associated with less chance of offering learners the opportunity to master PCP concepts. The objective is to understand the distribution of the cognitive demand levels of PCP learning tasks that are in the DT introduced earlier in the previous section of this study.

The Curriculum Assessment Policy Statements (CAPS) diagnostic reports (DBE, 2018, 2020) also recommend multiple representations of concepts as a strategy for teaching PCP at Grade 12. The effective learning of PCP at Grade 12 requires many resources. Teachers certainly play a vital role in supporting learners (Fennema & Franke, 1992).

This case study investigated the Grade 12 mathematics teachers’ rating of learning tasks in a PCP chapter in a popular Grade 12 mathematics online textbook. Teacher ratings were interpreted following Stein and Smith’s (1998) cognitive demand levels of learning tasks. Two research questions guided the study: (1) What is Grade 12 mathematics teachers’ rating of the PCP learning tasks in one popular Grade 12 mathematics online textbook? (2) From the teachers’ rating of learning tasks, and from the secondary data available on Grade 12 learners’ performance in PCP over the years, what can be said about the two pieces of data – might there be a link between the achievement in probability at the national level by Grade 12 learners, and mean cognitive demand level of the learning tasks that the learners popularly use to prepare for the national examinations?

The idea of probability is empirical. That is, probability describes what happens in very many trials, and we must observe many trials to pin down a probability. In this article, we use the definition based on the notion of proportion or relative frequency. Relative frequency of a score is obtained by dividing the frequency of that score by the total number of scores. Similarly, the probability of an experiment yielding a particular result (e.g. a coin toss yielding heads) can be defined as the number of equally likely and mutually exclusive outcomes, divided by the total number of possible, equally likely, and mutually exclusive outcomes. By equally likely, we mean that in the long run each of the possible outcomes will occur with approximately equal frequency (May et al., 1990, p. 179; Moore et al., 2013, p. 260).

Probability knowledge for teaching (PKT) includes content knowledge of probability, and various ways of presenting this content to the learners so that learning takes place (Batanero, Chernoff, Engel, Lee, & Sanchez, 2016). Like mathematical knowledge for teaching, PKT can be divided into probability content knowledge (PCK) and probability pedagogical content knowledge (PPCK). Probability content knowledge requires teachers to have specialised training in probability beyond the content covered in the high school.

Probability pedagogical content knowledge is knowledge about presenting the concepts to the learners so that learners easily understand them (Kazima & Adler, 2006; Hill et al., 2008). For instance, how does a teacher present to the learners concepts such as variation and randomness, aware that many learners come into a probability class with deterministic (concrete) understanding of the world around them? Take, for example, a probability experiment such as tossing a coin. In such an experiment, the outcome is not predictable every time the coin is tossed. It is, thus, not a straightforward case to generalise the probability of events arising from the experiment unless the experiment is repeated very many times. Nevertheless, although an individual trial has an unpredictable outcome, there is a predictable pattern of outcomes that will be obtained over a long series of trials (Moore et al., 2013, p. 260). Hence, for a teacher who may be unaware of the foundation principles of probability, moving their learners from theoretical probability to the concrete results can result in misunderstanding by the learners.

In the mathematical task framework (MTF), a task is defined as a segment of classroom activity that is devoted to the development of a particular mathematical idea (Stein & Smith, 1998). A task can involve several related problems in each topic in mathematics, in this case PCP. Mathematical tasks used in the classroom, or used by learners in their homework, are the foundation for their learning (Doyle, 1988; Stein & Smith, 1998). Stein and Smith (1998) distinguish three phases through which tasks pass: the first phase includes tasks that are found in the instructional materials such as study guides, printed textbooks, and DTs. The second phase includes tasks that are prepared by the teacher, and the third phase involves tasks that the students engage with in the classroom or at home, as reflected in Figure 1.

This article will limit the discussion to phase 1 of Stein and Smith (1998) with a focus on PCP learning tasks in a DT. Learning tasks for PCP were chosen for two reasons. First, as noted earlier, PCP is a topic that learners show poor grades in at the matric level (see DBE, 2019, 2020). Second, through interaction with some Grade 12 mathematics teachers, we learned that the DT is widely used by learners and teachers. So, we wanted to understand the cognitive demand level of PCP tasks in the DT. This study contributes to providing research-based information on PCP at the matric level.

As shown in Figure 1, the three phases of mathematical tasks are: (1) curriculum tasks found in the learning materials such as textbooks, and other CAPS-compliant learning materials, (2) tasks that teachers select and use in their classroom teaching, and (3) tasks that students implement in their day-to-day learning (Stein, Smith, Henningsen, & Silver, 2000). The three phases are interrelated.

Stein et al. (1998) use the MTF to classify tasks into four levels of cognitive demand, namely: (1) memorisation, (2) procedures without connections, (3) procedures with connections, and (4) doing mathematics (see Table 1). According to the authors, tasks that promote memorisation and procedures without connections do not present any challenge to the learners since they do not require deep reflection to solve. Tasks that involve procedures but require other information that is not obvious in the tasks are classified as procedures with connections. Finally, doing mathematics is a level at which tasks are highly cognitively demanding (Stein, Grover, & Henningsen, 1996). Based on Stein et al.’s (1996) classification, we associate high cognitive demand tasks with tasks that require multiple representations to solve. By ‘doing mathematics’, we mean engaging students with PCP tasks that give them the opportunity to develop their thinking and reasoning skills thus leading them to meaningful mathematical understanding (Stein & Smith, 1998, p. 13).

The MTF is used to classify PCP tasks found in a DT in terms of either high or low cognitive demand levels. Tasks that are set at a high cognitive demand level require multiple strategies to solve (Stein et al., 1996). Low cognitive demand tasks occupy learners with reproducing known facts. Tasks promoting memorisation (level 1) and procedures without connections (level 2) require less reflection to solve and are categorised as low cognitive demand level (Stein et al., 1996). For example:

This task does not demand much more than using a formula and substituting in the respective values, then solving for the unknown. Let T represent tea, and C represent coffee, P(T) is the probability that Jabu likes tea and P(C) is the probability that Jabu likes coffee. Then, P(T∪C) = P(T) + P(C) – P(T∩C).

Tasks of high cognitive demand level require some thinking and reasoning to solve (Stein et al., 1996). Take an example adapted from Moore et al. (2013):

One reason is that the two events are not necessarily independent, because not all 10% of adult full-time students are above 50 years of age. Moreover, it is reasonable to expect that younger adults are more likely than older adults to be college students. Hence, P(college student|over 50 years) < 0.10. This example fits in level 3 or level 4 of Stein et al.’s (1996) categorisation of learning tasks, namely procedures with connection, or doing mathematics.

The PCP tasks used in this article are obtained from a digital Grade 12 mathematics textbook. The textbook is endorsed by the DBE in South Africa. The book is freely available to South African users. Users are free to download and read the book on their mobile devices or print and read offline. The only restriction is for users to keep the book’s cover, title, contents, and short-codes unchanged.

We chose the digital book from among other books for three reasons. First, the book is used by many Grade 12 mathematics teachers and learners in South Africa, so it is a popular learning resource. Second, the book is freely available. Third, the book covers all mathematics topics taught in Grade 12 in South Africa. In this article, we focus only on the topic of PCP.

The PCP section is divided into eight sub-topics. For the purposes of this article, we limited our discussion to only four sub-topics, namely: the fundamental counting principles, factorial notation, tasks involving the application of counting principles, and tasks involving application of probability. We picked a total of 48 different learning tasks and asked five senior mathematics teachers at Grade 12 to rate the tasks according to the four levels of cognitive demand developed by Stein et al. (1996). Details of the study design are contained in the methodology section.

This study is a case study taking a descriptive statistical approach. This approach enables us to transform qualitative data into quantifiable form and use it to make sense of the cognitive demand levels of learning tasks in PCP.

Participants in the study are five secondary mathematics and probability teachers, pseudo-named A, B, C, D, and E to ensure anonymity. Initially, seven secondary mathematics teachers (six male and one female) were contacted by email to take part in the study. A questionnaire with clear instructions was emailed to all the seven teachers to complete. However, only five teachers, all male, from four secondary schools in two provinces in South Africa (Gauteng and KwaZulu-Natal) returned the questionnaire. The five questionnaires were entered into a spreadsheet (Table 2) for analysis.

Table 2 has 48 rows and 8 columns. Each row represents one task taken from the DT. The first column provides the serial number of the task for identification during analysis. The second column gives the location of the task in the DT. An ‘exercise’ is a collection of tasks. For example, 10.4 (1) represents task 1 found under exercise 10.4 in the DT. It can also be observed from Table 2 that exercise 10.4 has a total of seven different tasks. The remaining 47 tasks are presented in a similar format. For instance, exercise 10.5 has a total of three PCP tasks. The next five columns after column 2 are the five participant teachers who independently rated the 48 PCP tasks according to the four cognitive demand levels on a scale from 1 to 4 for each task. The last column shows the mean rating for each task.

The mean cognitive demand ratings (MCDR) by five senior PCP teachers from four schools in two provinces in South Africa were received by email by both authors of this article. Each teacher rated the 48 tasks independently of the other teachers. The first author entered the original data received from all teachers in Table 2 and the second author corroborated the entries with the original submissions. Table 2 was again cross-checked by the first author to ensure accuracy and consistency of measurements. In reporting the findings, we have rounded off the cognitive demand ratings of PCP tasks to the nearest digits.

The study was carried out during the restrictions due to COVID-19, where physical contacts were restricted as recommended by the health authorities to keep individuals and the public safe from contracting the disease. Communication during the data gathering process depended mainly on email with an attached questionnaire for the teachers, and follow-up phone calls. We contacted seven PCP teachers, but in the end only five teachers returned the questionnaire. Although only five teachers responded, percentage wise, it still represented a reasonable percentage considering that our initial target was seven senior teachers of probability at Grade 12. We were not able to observe the teaching of PCP in the classrooms for the same reason explained above. Finally, this study focused only on the learning tasks that are available to learners in the online textbook, so we missed the teaching tasks and the nuances that the teachers incorporate in their actual lessons. Obviously, we also missed observing the tasks that learners implement in their learning in the classrooms (Stein et al., 1996). Future empirical studies should consider these uncovered areas with respect to PCP.