A project upon which we spent a lot of time was listing all the “awful” lines in the new mathematics Curriculum. Readers have not paid much attention, but that’s understandable, and readers were not really the point. Compiling the list has given us a clearer sense of the absurd nature of the Curriculum, the list has been and will be the basis for more specific posts, and the list is there ready for the future: next year, when the Curriculum kicks in and people start to realise just how bad it is, we’ll be ready with the “We told you so”.

Compiling the list was not just tiring, it was tricky. Determining what to include or exclude, and why, was difficult. We decided to aim for comprehensiveness: if a line seemed faulty for even a minor reason then it would be listed. As a consequence, many lines listed are not truly “awful”; they are merely “bad” or “wonky”. Given that standard, it resulted in a *lot* of lines being listed. The vast majority of curriculum lines are, at minimum, wonky.

Nonetheless, although we didn’t look to excuse bad lines, we also attempted to give proper respect to the OK lines, and the rare good lines. So, for each Year-Stream section, there were at least a few lines that seemed acceptable or better, and so were not listed. Except for one: Year 5 Probability.

For Year 5 Probability, we wound up listing both content descriptors and all twelve elaborations. We looked again, and again, and could not see how to do otherwise. We could not find a single line that didn’t make us cringe. Here it is, in full.

**Year 5 Probability**

*list the possible outcomes of chance experiments involving equally likely outcomes and compare to those which are not equally likely* (AC9M5P01) (21/11/22)

*discussing what it means for outcomes to be equally likely and comparing the number of possible and equally likely outcomes of chance events; for example, when drawing a card from a standard deck of cards there are 4 possible outcomes if you are interested in the suit, 2 possible outcomes if you are interested in the colour or 52 outcomes if you are interested in the exact card *(AC9M5P01) (21/11/22)

*discussing how chance experiments that have equally likely outcomes can be referred to as random chance events; for example, if all the names of students in a class are placed in a hat and one is drawn at random, each person has an equally likely chance of being drawn *(AC9M5P01) (21/11/22)

*commenting on the chance of winning games by considering the number of possible outcomes and the consequent chance of winning* (AC9M5P01) (21/11/22)

*investigating why some games are fair and others are not; for example, drawing a track game to resemble a running race and taking it in turns to roll 2 dice, where the first runner moves a square if the difference between the 2 dice is zero, one or 2 and the second runner moves a square if the difference is 3, 4 or 5; responding to the questions, “Is this game fair?”, “Are some differences more likely to come up than others?” and “How can you work that out?” *(AC9M5P01) (21/11/22)

*comparing the chance of a head or a tail when a coin is tossed, whether some numbers on a dice are more likely to be facing up when the dice is rolled, or the chance of getting a 1, 2 or 3 on a spinner with uneven regions for the numbers *(AC9M5P01) (21/11/22)

*discussing supermarket promotions such as collecting stickers or objects and whether there is an equal chance of getting each of them *(AC9M5P01) (21/11/22)

*conduct repeated chance experiments including those with and without equally likely outcomes, observe and record the results ; use frequency to compare outcomes and estimate their likelihoods *(AC9M5P02) (22/11/22)

*discussing and listing all the possible outcomes of an activity and conducting experiments to estimate the probabilities; for example, using coloured cards in a card game and experimenting with shuffling the deck and turning over one card at a time, recording and discussing the result *(AC9M5P02) (22/11/22)

*conducting experiments, recording the outcomes and the number of times the outcomes occur, describing the relative frequency of each outcome; for example, using “I threw the coin 10 times, and the results were 3 times for a head, so that is 3 out of 10, and 7 times for a tail, so that is 7 out of 10” *(AC9M5P02) (22/11/22)

*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) (22/11/22)

*comparing the results of experiments using a fair dice and one that has numbers represented on faces more than once, explaining how this affects the likelihood of outcomes *(AC9M5P02) (22/11/22)

*using spreadsheets to record the outcomes of an activity and calculate the total frequencies of different outcomes, representing these as a fraction; for example, using coloured balls in a bag, drawing one out at a time and recording the colour, replacing them in the bag after each draw *(AC9M5P02) (22/11/22)

*investigating First Nations Australian children’s instructive games; for example, Diyari koolchee from the Diyari Peoples near Lake Eyre in South Australia, to conduct repeated trials and explore predictable patterns, using digital tools where appropriate *(AC9M5P02) (22/11/22)

I might quibble with the wording of the first content descriptor – although, any sane person will get the gist of this line in the curriculum, and teach it appropriately. When it says “… and compare to those which are not equally likely”, is the intent to compare the *outcomes* of the experiments, or to compare the *experiments*, or to compare the *probabilities* of those outcomes? I guess to an extent all three should be compared, but I think its the last which is most relevant. ie. compare two chance experiments, both of which are the rolling of a 6-sided die labelled 1 to 6, but one of the dice is weighted towards 6.

Also a comment on the 4th elaboration for the first content descriptor: fairness is a slightly tricky concept for probability neophytes, because we also want to consider in the “utility” or “value” or “goodness” of an outcome, not just its probability. I guess in the game described, the only options are move one square or move no squares, so it’s not complicated. But a simple extension of the game (which one can imagine a teacher being tempted to run) is that the number of squares moved = difference between numbers on the dice, and it’s definitely more complicated to determine if this is fair (at least, I would imagine so for 9/10 year olds).

This is a question about probability. I hope that it is in the right place.

Is a percentage a number? Tony Gardiner thinks not (Extension Mathematics: alpha, p. 28). AC:M suggests that percentages are numbers; e.g. students in Year 5 should recognise that 60% is 10% more than 50%. To be fair, AC:M avoids talking about equality; e.g. 0.5 “represents” 50%, but a teacher might be forgiven for reading equality into this. I ask because I have noticed that students seem to prefer expressing probabilities as percentages. Toss a coin, then P(H) = 50%. What do others think?

Very good question, and one I’ve never properly thought about. I always felt queasy when reading ACARA stuff seeming to imply that percentages were numbers, but I never stopped to analyse my queasiness. I definitely don’t like the idea of expressing probabilities as percentages (and I’m not thrilled with decimals).

Fractions and decimals have their own advantages; decimals would be used to express P(Z 1.5) where Z is a standard Normal rv; expressing a probability as a fraction m/n indicates the student’s working; m successes in n trials.

There is research in the cancer literature about how to best express, to the patient, the probability of the cancer returning.

If we’re talking about students getting comfortable the ideas of (discrete) probability, I can see no advantages to decimals.

According to the glossary of AC:M v. 8.4 (is there a glossary for v.9?) the definition of probability is this.

“The probability of an event is a number between 0 and 1 that indicates the chance of that event happening; for example, the probability that the sun will come up tomorrow is 1, the probability that a fair coin will come up ‘heads’ when tossed is 0.5, while the probability of someone being physically present in Adelaide and Brisbane at exactly the same time is zero.”

What does “chance” mean? The glossary does not define chance.

Why is it that “the probability that the sun will come up tomorrow is 1”? Laplace came up with a different estimate.

What is a “fair” coin? Is it a coin where the probability of heads = probability of tails, and hence = 0.5? This is defining fairness in terms of probability.

The Brisbane/Adelaide example seems to be saying only that the probability of an impossible event is 0. Perhaps this is defining the probability of an impossible event to be 0. But how do we define probability of a possible event?

So what is meant by probability? And how do we introduce it to our students?

Hi Terry,

The v9 Glossary is here (Word, idiots).

I’m not sure of the point that you’re making. Is that the glossary entry sucks? Or that “probability” is difficult to define?

Thanks; helpful as always.

It is difficult to define probability, especially for high school students, and there are issues with the definition in the glossary. Let me concentrate in the latter.

In version 9, the definition of probability is:

“The chance of something happening shown on a scale from 0 and 1 (inclusive), e.g. the probability that a fair coin toss will come up ‘heads’ is 0.5.”

So, what is “chance”? Chance is not defined in the glossary. I have noticed that the word “chance” is often used instead of “probability”. This definition suggests that they mean the same thing.

What is a “fair” coin? Is it a coin where the probability of heads = probability of tails, and hence = 0.5? This is defining fairness in terms of probability.

BTW, the definition of a random sample is given as “A subset of the population chosen such that every element of the population has an equal chance of being selected.” This is wrong.

The latest Victoria curriculum has the same glossary at AC:M v.8.4 with the explanation of probability that I gave above, and has the same comments that I indicated above apply.

Except you didn’t comment; you raised questions. I’m still not sure of the target of your questions.

On probability, odds, and gambling.

When looking at online gambling websites in Australia I noticed that odds are expressed as decimals. Suppose that the odds are 6.2. This means that if I bet a dollar on that outcome, and it occurs, then I will get back 6.2 dollars making a profit of 5.2 dollars.

Websites tell me that this means that the probability of that outcome is 1/6.2 = 0.1612.

Now when I look at cricket matches, odds are given for team A to win, team B to win, and a draw. If I calculate the corresponding probabilities, they do not add up to one. Why is it so?

Um, maybe because the “websites” are written by idiots?

“The odds on display never reflect the true probability or chance of an event occurring.” The probabilities calculated as 1/odds are called “implied probabilities”.

https://www.investopedia.com/articles/dictionary/042215/understand-math-behind-betting-odds-gambling.asp

Are you conversing, or talking to yourself? If the latter, I’ll leave you be.

Initially I was conversing; I thought that someone on this blog might know more about gambling than I do; but the last post was simply saying that I found an answer to my initial question. Feel free to ignore me.

OK, fine. The answer was so obvious, I took your question to be rhetorical.