We made it. We have read ACARA’s entire idiotic mathematics curriculum, and we’ve completed the Complete Awfullest Works. Meaning, soon, we’ll be on to Stage Two. But first, following on from Number and Algebra and Measurement and Space and Statistics, we have the post for the awfullest Probability lines.

The Probability stream is, of course, awful, but the awfulness is notably different in character from that of the Statistics. Whereas the statistics is a homogeneous gruel of “data”, with the only mathematical substance being the tiny, tasteless raisins of “median” and its kin, the probability stream has just enough substance to be consistently, solidly stupid.

Note that the Probability stream only begins in Year 3 since, unlike Algebra, the kids need to know a little before considering such concepts.

### YEAR 3

*identify practical activities and everyday events involving chance; describe possible outcomes and events as ‘likely’ or ‘unlikely’ and identify some events as ‘certain’ or ‘impossible’ explaining reasoning* (AC9M3P01)

*classifying a list of everyday events or sorting a set of event cards according to how likely they are to happen, using the language of chance and giving reasons for classifications; discussing how impossible outcomes cannot ever happen, uncertain outcomes are affected by chance as they may or may not happen whereas certain events must always happen, so they are not affected by chance *(AC9M3P01)

### YEAR 4

*conduct repeated chance experiments to observe relationships between outcomes; identify and describe the variation in results* (AC9M4P02)

*playing games such as Noughts and Crosses or First to 20 and deciding if it makes a difference who goes first and whether you can use a particular strategy to increase your chances of winning* (AC9M4P02)

### YEAR 5

*list the possible outcomes of chance experiments involving equally likely outcomes and compare to those which are not equally likely* (AC9M5P01)

*experimenting with and comparing the outcomes of spinners with equal-coloured regions compared to unequal regions; responding to questions such as “How does this spinner differ to one where each of the colours has an equal chance of occurring?”, giving reasons *(AC9M5P02)

### YEAR 6

*listing the different possible outcomes for rolling a dice and using a scale to locate the relative probability by considering the chance of more or less than for each possible event ; for example, the probability of getting a number greater than 4* (AC9M6P01)

*conduct repeated chance experiments and run simulations with an increasing number of trials using digital tools; compare observations with expected results and discuss the effect on variation of increasing the number of trials* (AC9M6P02)

### YEAR 7

*identify the sample space for single-stage events; assign probabilities to the outcomes of these events and predict relative frequencies for related events* (AC9M7P01)

*exploring and observing First Nations Australian children’s instructive games; for example, Koara from the Jawi and Bardi Peoples of Sunday Island in Western Australia, to investigate probability, predicting outcomes for an event and comparing with increasingly larger numbers of trials, and between observed and expected results *(AC9M7P02)

### YEAR 8

*recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts* (AC9M8P01)

*describing events using language of “at least”, exclusive “or” (A or B but not both), inclusive “or” (A or B or both) and “and” *(AC9M8P02)

### YEAR 9

*list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes* (AC9M9P01)

*discussing two-step chance experiments, such as the game of Heads and tails, describing the different outcomes and their related probabilities* (AC9M9P01)

### YEAR 10

*design and conduct repeated chance experiments and simulations using digital tools to model conditional probability and interpret results* (AC9M10P02)

*recognising that an event can be dependent on another event and that this will affect the way its probability is calculated *(AC9M10P02)

### YEAR 10 OPTIONAL

*counting principles, and factorial notation as a representation that provides efficient counting in multiplicative contexts, including calculations of probabilities*

*performing calculations on numbers expressed in factorial form, such as to evaluate the number of possible arrangements of n objects in a row, r of which are identical, for example 5 objects, 3 of which are identical, can be arranged in a row in *

*different ways*

> playing games such as Noughts and Crosses or First to 20 and deciding if it makes a difference who goes first and whether you can use a particular strategy to increase your chances of winning (AC9M4P02)

These games are both deterministic. The only chance-based aspect here is whether your opponent doesn’t play optimally in Noughts and Crosses, and whether that’s modellable with chance is another matter entirely.

> identify the sample space for single-stage events;

“listing the different possible outcomes for rolling a dice” from Year 6 is already an example of identifying the sample space for a multiple-stage event. These two points appear to be in the wrong order.

> exploring and observing First Nations Australian children’s instructive games; for example, Koara from the Jawi and Bardi Peoples of Sunday Island in Western Australia, to investigate probability

Using First Nations games to investigate probability is not necessarily a bad idea in principle IMO, but given all the complaints on this blog about there not being useful teaching resources for a bunch of the rest of the curriculum, I don’t expect there to be good teaching resources for this either.

Actually, having just Googled Koara, it seems like this refers to games involving a propeller toy (“how accurately can you make it land” or “how high can you make it fly”, among others)? It’s very unclear what sort of probability one can learn from this without doing hundreds of trials to collect experimental data first (and that falls squarely under Statistics). Seems like exactly the sort of activity that will bore Year 7s to death while not actually teaching anything.

I’d love to see what ACARA had in mind while writing that curriculum line. It feels like they just put that in as token representation.

> recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts (AC9M8P01)

If we’re already teaching students about sample spaces in Year 6, then we should already be touching on complementary events and addition of probabilities of mutually exclusive events then, instead of waiting two years. Part of the curriculum you didn’t quote talks about:

> using the relation Pr(A and B) + Pr(A and not B) + Pr(not A and B) + Pr(not A and not B) = 1 to calculate probabilities, including the special case of mutually exclusive events where Pr(A and B) = 0

but it doesn’t at all mention how to calculate P(A or B) when A and B are mutually exclusive; I genuinely can’t think of any way you would teach this as some “special case” of the given “relation” (“identity”?).

> describing events using language of “at least”, exclusive “or” (A or B but not both),

I’m not convinced it’s useful to explicitly introduce “exclusive or”. Just tell the students that in mathematics, “or” means “at least one of the two is true”, or informally “A or B or both”, and don’t ever use “or” exclusively in a question without explicitly saying “but not both”. Unless, of course, the exams are expected to use “or” in both an inclusive and an exclusive manner, in which case I really feel sorry for the poor students who are forced to deal with yet another source of confusion.

Also, surely “at least” can already be introduced in Year 6 alongside the “listing the different possible outcomes for rolling a dice” thing? If you’re already asking about “the probability of getting a number greater than 4”, why not also ask about “the probability of getting a number that’s at least 5”?

> recognising that an event can be dependent on another event and that this will affect the way its probability is calculated

AC9M9P01 already talks about “list all outcomes for compound events both with and without replacement”. Surely students should already understand at Year 9 that e.g. when drawing cards from a deck without replacement, events relating to the second card are by-and-large dependent on events relating to the first card?

What a mess. So much of this material is either introduced in the wrong order or dragged out unnecessarily.

There are many games that involve randomness. Noughts and crosses is not one of them.

In the 17th century, John Milton described schooling in England thus.

“[W]e do amiss to spend seven or eight years meerly in scraping together so much miserable Latine and Greek, as might be learnt otherwise easily and delightfully in one year.”

I often think about this sentiment when reading extracts provided above.