Transcript of What's new – statistics and probability
Well, hello, and good afternoon, everyone. Welcome to our final session for the year in our series K-6 mathematics Adobe connect sessions. Thanks for joining us again. Today's session is on What's new: statistics and probability. I'm here with Chris Francis in the background, who will be answering questions in our chat today. Hopefully, she'll be able to answer your questions. If not, we'll get to them in a response email after our session today, so thanks for joining us. I know it's a busy time of the year for everyone, with report writing and presentation days coming up, so thanks for joining us today. So as I mentioned, today's session's on What's New: statistics and Probability.
So statistics and probability, our maths syllabus states, is developed initially in parallel, so they're sort of separate substrands that we teach throughout our K-6 syllabus, with the links between built progressively across the stages, so when it gets to Stage 2 and 3, they start to relate a lot more together as they develop into the statistics and probability strand that then extends into Stage 4 and 5. If we look at this on the organisation of content chart that we've been showing in our sessions, you can see statistics and probability sit over there in the orangey section. And we have Chance and data. So this is the same as what we currently have in our syllabus. It's just under a different strand title. So data used to be by itself and Chance was within Number, and now they've just been drawn back together under the title of Statistics and Probability. And you can see there where they develop into Stage 4 as well.
So our syllabus states that statistics is about collecting, organising, displaying and analysing data. And again, like in our other strand, it's based on real-life contexts using concrete materials first - that's for our early experiences. And we encourage our students to ask questions, so posing the questions themselves, and it's about interest, things that are interesting to them, and then we want them to investigate to then be able to collect the data. So it's about them developing the questions and investigating it for themselves, either by themselves or within a group.
The probability side of statistics and probability is chance, or that's where it starts out for us in Stage 1. And it's just about understanding the chance concepts at an early age. We want to build that understanding of chance situations that are further developed through experiments that we then produce data, so that's that link there, so they were parallel, and as we start to collect and represent data about the simple experiments, statistics and probability start to become an interrelated strand. So we want students to be able to make comparisons about the data they're collecting. And so in later stages, students see that chance concepts link to numerical probabilities, so when we start to look at those words like 'certain', 'uncertain', 'probable', 'probably', 'probably not' on a number line and how they relate to numbers.
I've just got a snippet there of language too that's from Stage 3, just to show you some of the language that's developing, and I'll go into that a little bit further now.
So in the previous sessions, we looked at some ... some progressions of learning, and in today's session, we really wanted to focus on language progression, statistics and probability and data and chance, as we sort of, I guess, currently know it, has a lot of language demands for our students and also for adults. A lot of adults don't understand the idea of chance and how it works. And it's quite a, you know, objective and subjective topic when we talk about it, so some people think about it as luck and some people see it as a much more mathematical term, so it's really good if we build these foundations upon the language and that students get a good understanding of that from the beginning. So in the data language... Chris has put them in different colours for us today. Just to try and show you the progression of how that language develops from Early Stage 1 to Stage 3, so there's some similarities there - you can see that...we talk about information that then becomes data. We talk about that they start by collecting, then it becomes gathering, then we create surveys, and then they tabulate the results. And you can see that they talk about groups, then categories, becomes categorical data, and then it delves into variables and numerical data, so you can see the language developing as we move through the stages. And then there's a whole lot of words that are just related around actually explaining what the data looks like and how we actually create the data into different forms. So there's a lot of language that needs to be developed around Data, some of it new, some of it we're quite familiar with already.
In Chance, the language progression, again, there's a whole lot of language that we're currently familiar with but not necessarily if you're a student who's from a non-English-speaking background. These kind of words can be quite problematic. So if you've got EAL/D learners in your school and a high number of those, then we need to focus on the language of Chance. And so this is a really nice link to literacy as one of the learning across the curriculum areas. So I've just highlighted a couple of words from each stage that I think are really important to get correct when we're talking with our students. So...that idea of 'possible' and 'impossible', a lot of students struggle with that, because often, 'impossible' is just something they've never seen before. So we really need to flesh that out a bit more for them. In Stage 2, there's some new terminologies like 'trials', like a set of trials, and also that idea of 'equally likely', so making sure they understand how that relates to fractions as well, which we'll get to a little bit later in the session today. And into Stage 3, we've really upped that ante with the language they're using about Chance and about 'likelihood' and 'probability'. And then we have sort of this new terminology around 'expected' and 'observed' probability, 'expected outcomes' and 'frequency' and being able to observe the frequencies of how they're happening within your Chance experiments. So there's a lot of language to take in within the new syllabus and also just in our current syllabus around statistics and probability.
So here's some of those important and critical skills we want our students to develop. So it's about calculating the probabilities, representing outcomes, analysing what they've found, exploring different methods of collecting the data, questioning, recognising misleading data, which is a really important skill, right into real-world context for our students. Making reasoned judgements when they compare data is also really important. So...all of these concepts and skills, they're based...and it depends upon students having a sound knowledge, understanding and use of the terminology. So we talked about those language focuses there for both data and Chance. And so students being able to have a good understanding of what those things really are is important to be able to be successful in using and applying these skills.
So statistics and probability, as I mentioned before, they're developed at first parallel, and then the links are built progressively, so Data is in the centre of that little honeycomb. So we start with the concrete materials. We're posing questions. We want them to investigate statistics in society, so those real-world contexts. We use electronic tools, things as simple as using Excel, collecting and organising their data and then displaying it. And there's also this sort of new terminology that's come down to K-6 around understanding numerical and categorical data and sort of data as variables, so we're going to go into that a little bit today as well. So we want to build a shared understanding of what these things are, so these meanings, these definitions, are straight from the glossary in our new syllabus that you can look up online or in your hard copy of your syllabus. So categorical data, so categories - just think of it as categories - and they're distinct groups. So maybe the shoes are sorted into groups according to the colour - black, brown, white. They can still be numbers, though, so we don't just want to get into our minds sort of thinking, "Oh, if it's got numbers, it's numerical," because things like postcodes could be distinct groups. And I'll go into a little bit of how that's different to numerical data with some pictures in a moment. So numerical data is expressed by numbers, and it's obtained by counting...and that 'discrete', in brackets, is what they delve into into Stage 4 - i.e. like the number of kittens in a litter - or by measuring, which is continuous data - e.g. temperature over time or over days. So that idea of discrete and continuous comes in in Stage 4, but we're talking about in our context around counting and measuring. And variable - this is a word that's probably new to K-6. It's pretty much a word they use to represent data in Stage 4. And what it means is that it's...something is measurable or observable that's expected to change over time or between observations, between the times in which you take the data and collect the data. So I've just got on the side there that variables are used when we compare two data sets, so we would use that term 'variables' when we're looking at two different parts of data. Maybe it's from two different classes that have collected the data. And just to let you know that categoricals, or the categories, looking at things in distinct groups, is the focus from Stage 1, 2... It does delve into Stage 3 as well. And that numerical data doesn't really come into our syllabus until Stage 3 onwards. So for Stage 1 and 2, you're just looking at categorical. When you're in Stage 3, you start to look at categorical and numerical data. So just to take it a little bit further with the sort of idea of data and variables, so the term 'categorical variable', it's used in our new syllabus. Sometimes the term 'variable' and 'data' are interchanged within our syllabus content, both for K-6 and 7-10, and we did speak to Nagla about this in 7-10, and she said that it...you know, they can be interchanged and they...they generally use the word 'variable', and that's because they're starting to compare a lot of their data. So you might see the wording 'categorical data' or you might see the wording 'categorical variable'. OK? So they can sort of be interchangeable. But just to note that students aren't required to use the term 'variable' until Stage 3, and it's not even defined as...you know, "Students use this term and can define it" until Stage 4, so it's fine to use the word 'data'. We just want you to be aware that that word 'variable' is now there. And we would talk about variables in Stage 3, when we're looking at two-way tables, so let's say you had information gathered about boys' interests and girls' interests, you could then compare that data, and that's comparing variables, OK? Hopefully, that's sort of clearing up what that means for people. So this is a little picture that Nagla created for us about variables, and that's sort of that numerical and categorical, and that's pretty much as far as we get by the end of Stage 3. Into Stage 4, they start to look at continuous versus discrete, and in Stage 5, they start to look at ordinal and nominal. Now, I'm not going to go into those today, but it really is a good idea that if you do teach across stages into Stage 3 and 4, if you're a central school, or if you just want to know where the learning continuum is going for statistics and probability, it might be a good idea to have a look at where these topics go into Stage 4 and 5 in the online syllabus if you don't have a hard copy for yourself. So categorical or numerical - OK, the easiest way that we've sort of had a discussion about, Chris and myself, the easiest way to remember is to ask yourself, "Can I find the average of this data?" So here's my data around favourite colours, so there's red, blue, green, orange, pink, yellow. Sorry they're all blue. It was easier to do it that way. So would you find the average favourite colour? No. I'm saying no, that you wouldn't do that. It doesn't make sense. You can't say, "Oh, the average favourite colour is green." You can work out what the favourite colour is, what someone's favourite colour is, and you can work out what the least popular colour was there, but you wouldn't try and find the average of that information to compare it together. So it's categorical. OK? So try and think about it in that way. So if I then look at something like a graph around daily temperature, so I've taken the temperature in our classroom on Monday, Tuesday, Wednesday, Thursday, Friday, would you, or could you, find the average of the daily temperature? Yes, you would do that. So it's numerical. OK, so I find that's a nice, easy way to remember if it's categorical or numerical. So feel free to go back over these slides later if you want a slowed-down version of looking at some of these areas. So the glossary in our syllabus has a really good little section on numerical variables, or numerical data. So it just says that they're generally numbers for which our arithmetic processes, such as adding and subtracting or calculating average, make sense. So before, I talked about postcodes. So if I worked out all the students that lived in particular postcodes in my classroom, I then can't add them all together, divide them and find what the average postcode number is, because postcodes don't work that way. So just be aware that you can have numbers as categories but that when we talk about numerical data, it's about... Think about average, think about adding and subtracting the values together to find some information. So that's just, hopefully, an easy way to think about the difference between the two, remembering that numerical data and variables don't enter our syllabus until Stage 3, OK? So looking at Chance. So Chance is introduced from Stage 1. That's the same as what our current syllabus is as well. So we're looking again at language, starting informally, and then moving on to understanding what some of those Chance words means. We're talking about that they later go into that numerical probability and fractions and decimals, using probabilities on a number line and also seeing probability as relating to fractions, and talking about equally likely, frequencies, probabilities, events. These are all the language that comes out. And I've just green... I've put in green that simple experiments to produce data. That's where that link is between Chance and data, so... As you do the Chance experiments, you collect data on it. There it is - statistics and Probability are now interrelated, so...you'll see that coming out in the syllabus. So probability: Frequency and Probability. So these might be new terms, apart from probability. Frequency probably isn't something we're as familiar with in K-6. So probability is the value used to describe a chance of an outcome occurring, so it's a general... It's between 0 and 1. OK? When you look at, numerically, what the probability is. Whereas frequency is the number of times it happens, the particular outcome happens, in a chance experiment, OK, so... That just means that when I have my event, my chance event, just because I should get a certain result doesn't mean I will. OK, so the frequency of it happening within my chance experiment might be different to what the probability states it should be. So probability is the likelihood of event, and it's a mathematical number between 0 and 1, and the students need to be able to identify if all the outcomes are possible - are they all equally likely? So you'd be familiar with that...with the idea of spinners. For example, in NAPLAN, you see questions around spinners. If all the segments of the circle are equal or if maybe there's three blue and two red and one white, then they're not equally likely to occur, and so that sum of probabilities of outcomes together, they equal to 1, and there's a very strong link to fractions. So here, I'm just... I'm sorry about the box moving in my presentation, by the way. So let's say I'm rolling a dice, so I have a 1-in-6 or a sixth of a chance of rolling a five. I also have a sixth of a chance of rolling a one, a two, a three, a four or a six. So when I add all those sixths together, it equals 1 - that idea of that they all have to equal to 1. So, but this might not actually happen in your trial, which is the frequency what I expected or observed happening. So on my little graph there, a little side-by-side column graph, and so the blue represents what the probability is if I roll it 20 times. They should all come out about three times, 3.3 times, but when I actually did the rolling for 20 rolls, that's not what I got. But in Stage 3, the students get to the point where they start to see that if I made the trial bigger, more rolls, and did it in a larger data set that they would get closer and closer and closer to what the probability states it should be. OK? So that's just to clear up how frequency and probability are different. So getting into some of that 'What's new'. So we'll go through each of the stages now to give some information on that. So there's no changes for Early Stage 1 in data. Just a note that there's a descriptor that organise objects into simple data displays. It has no AC number next to it, and there are a few descriptors like that in our syllabus. And that's because they were created by the Board of Studies for continuity. So we currently have a data outcome in Early Stage 1, and we wanted to keep it for the new syllabus, so that's why it's there, because we feel that students in Early Stage 1 can produce data and can create picture graphs, for instance, so we wanted to keep it in there. And so with Chance, it's the same as our current syllabus, and we're focusing on the students themselves and their environment. So Chris made this little activity for us here about "What type of container do you bring your lunch in to school?" So you get the students to take a photo of their lunch container. They then have a look at the photos and determine some characteristics to group the photos together. You can then show them this prepared display, which is in the file pod today, and we can have a look and discuss, "Well, how have the lunch containers been grouped?" So...here, they've grouped them if they're lunch bags or if they were lunch boxes. And then you discuss the number of containers in each row. And then we get on to this idea of misleading data. So we want to discuss why the row with four objects looks longer than the row with five objects and how that can actually mislead people to think which group is larger, so we want to have these discussions from very early on about misleading data, and we want the students to then create a display and compare the two data displays. So this is straight from the syllabus that matches that activity there. A really strong link to reasoning, so students need to give reasons why they think those rows might look bigger than the other one when they actually have less objects in it and then interpret the information from the displays. What's new for Stage 1 - again, not a large amount of difference there. We want the students to pose their own questions to gather and sort data, and they're using lists and tables, and they record their observations. Now, just a note there that in data 2, in the AC content descriptor, it uses the term 'one categorical variable'. Now, we're not saying that students in Stage 1 need to be able to use that terminology but just to let you know that you can see that language is coming in from Stage 1 about categories and actually referring to it not just as data but as what the type of data that it is, that it's categorical data that we're looking at. So for Stage 1, there's a little activity as well that Chris made up. So we need to discuss and record the features of a picture graph, and she's made a lovely little rotation there that talks about the real fundamentals of creating a graph and creating the picture graph in particular for this one, but it applies to a lot of graphing as well, so...that you need to have a baseline. Are all those objects aligned together? Is there equal spacing? Are they the same-sized symbol? Is there a key? Are there labels on the graph? So all of these features of a graph that our students need to be exploring in Stage 1. So we observe the data in the display. Some of those features are missing. So how could that be misleading? Again, that idea of looking at making sure our graph actually displays the information correctly. And that focus from our syllabus there on one-to-one correspondence. And, again, focus on reasoning. What's new for Stage 2? So as we move into Stage 2, we start to up the ante with that language, so they're going to conduct a set of number trials in chance experiments, so more than one - they're going to compare their results from before and they're going to start to recognise the variations. So that variation, variable, variety, all those words - it's a really nice opportunity to focus on the metalanguage of mathematics and where those words come from and why we use that word. They're then going to compare their chance events. They're using the term 'outcome' and they're predicting. I've just put a couple of examples there of the AC descriptors that are from Stage 2 that are quite wordy. It's about looking at events where one can't happen if the other happens and identifying events where the chance of one occurring will not be affected by the occurrence of another. Now, it seems quite wordy, but when you actually get down to the content, there's a lot there, but basically what it's looking at is when you have chance experiments, how does it affect the next go? So I've got a picture there of some coins, so if you're tossing a coin, when I toss it, doesn't matter how many times I've tossed it before, those previous turns are not going to affect the result of me tossing it again, whereas if I had a deck of cards and I drew a card out and then didn't replace it back in the pack before we did the activity again, it would actually affect the probability, the results that could happen from drawing a card from that deck, so it's just delving into some of those issues and some of those different results that happen from chance experiments. So for Stage 2, there's an activity around collecting data. A lot of questioning involved with data. If you're familiar with using here, hidden and head questions in literacy or English, they're a great way to get into mathematics as well. In this case, we actually want the students in small groups to create a question they would like to collect data about, so, again, it's about context for them. Now, I've just suggested one there. "What's the most common number of children in a family?" We want the students to choose a question, though. So they collect their data. They represent their data as a column graph. They can use technology or not. And then, I've sort of separated this out, because this is where it gets really important, that we want them to write three interesting facts about the information in their graph and reflect upon the questions they chose and possible improvements, so now they're actually starting to analyse their own graphing work, their own data collection, and I think that's a really important skill from Stage 2 onwards. They need to present their findings, and then they can compare the effectiveness of the graphs they've produced, and that's communicating the data. So that's straight from the syllabus, and that's how it links in there about being able to represent the data, use one-to-one correspondence with and without technology. So Chris has adapted this lesson from Stage 3's teaching data: Dot plots Building capacity resource. If you've not seen that, if you've been to the... This is all hyperlinked, by the way, these images, if you want to have a look. The New South Wales syllabuses for the Australian Curriculum - mouthful, but there it is. There's a link on the intranet page to that. There's these great Building Capacity resources that the team at DEC have created, and this one's about Teaching Data Stage 3: Dot plots, so... Chris changed that lesson a bit to be about column graphs and for Stage 2, but this is a great resource for teaching data around dot plots, so if you're not familiar with them, I would go here at some point and download that resource that you can use. So, what's new for Stage 3? So, again, comparing. We talked about comparing last week with Measurement and Geometry, and, again, it comes out in the statistics and Probability strand. So we want to compare a range of data and then choose appropriate displays. They're drawing conclusions. They're identifying bias in representations. And newspapers are a great source of data that is biased. So please use things that the kids are exposed to every day. We're also getting them on to assigning values, that idea of using numerical values for probability, and seeing the connection with fractions, like I talked about that, you know, rolling a five has a one-sixth of a chance when you're having a roll of each of...of a dice six times. So...the new content is that terminology around categorical and numerical data. So it's just drawing out the specific types of data we're looking at. Dot plots is new. Frequencies and equally likely events. And also looking at data representations in digital media and critically evaluating it, and that's a really important skill as well. Just a sidenote as well that sector graphs and divided bar graphs, so graphs around using percentages, is now located in Stage 4, so we bumped that up to Stage 4 so that we can focus on the numerical data and the categorical data without delving into percentages and creating sector graphs. So an example for Stage 3. So just talking about what dot plots are. OK, if you've not seen one before. They're a visual representation of whole number counts of data. They can be used as an alternative to a column graph. They sort and plot the data along an axis, only one axis, and usually only for small collections of data, because there is a dot for each single picture, so there's a symbol either...it might be a cross or a dot that represents each single piece of data. So that Syllabus BITES, there is a hyperlink as well. That's currently sitting in TaLE. And they're a fantastic resource as well. That one is about what dot plots are and two-way tables, so... Feel free to have a look at that as well at some point, because those two aspects of statistics is quite new for Stage 3. OK, then you've just got the outcome at the top there that talks about dot plots and line graphs and two-way tables - this is just helping our students organise and display their data. So the features of a stacked dot plot... So you can see that one there. It's a dot plot of 5K's school shoe size. And, again, I've just put the little pictures of the circles up there because that idea of baseline, equal spacing, same-sized symbols, keys and labels all fits in with dot plots just like it fitted in with picture graphs. So it's usually a horizontal axis and it's labelled. As I said, it's a dot or an x would represent one piece of data. And when there's more than one piece of data related to the same category, the dots are stacked on top of one another, as you can see there, so... This is a nice way to use a different type of graph than a column graph. There'd be no point using a column graph, because when you use the axes on a column graph, you always start with 0, and you'll have all this information from 0 that you don't really need because no-one's going to be a shoe size 0. And so on a dot plot, the extremities of the axis normally cover the smallest and the largest numbers, so if you're doing height as well, you don't need to have from 0. For, you know, students in Year 6, you probably only need to have from, you know, a metre, or a bit more than that, up. OK, so you're sort of getting rid of that wasted space that you might have in a column graph if you were going to use that, so... It's a nice little visual representation, and it's really nice to see two of them side by side that you can have a look at and compare data. Maybe you want to do the shoe size of boys and girls in your class. So side-by-side column graphs, so... This is also something that's a little bit new for us, but I don't think it's too difficult to understand. So I got this activity straight from CensusAtSchool, from the Australian Bureau of statistics. If you've not been there before, it's a great site. You can join in and add to the collection of statistics from your class at your school. They have a whole lot of prepared samples and infographics that you can use, but they also have some teacher-submitted activities, and this one was about eye colour comparison, and so... Regency Park had their data on there about eye colour. I just created some data from my class around eye colour. And then the idea is that the students create them into a two-way table, OK, so I've used Excel there to create a little two-way table about the eye colour in my school and Regency Park data. And then they need to interpret it, so I then used that in Excel to create a side-by-side column graph, and you can see it down there - it's not particularly easy to see on my presentation today. But you can see the difference and you can compare the two schools. And within this activity, which, the activity's hyperlinked, and all the activities for this are also in the pod today. You can have a look at other data on the CensusAtSchool site about, "Is your class a typical Australian class when it comes to eye colour?" You could even do it within your own school - "Is your class a typical class in your school with eye colour?" Or you could do it about anything you like. But it's really nice for students to use data with other students from other schools. They find it really a nice connection, and, look, you could do your own data collection over Connected Classroom if you want. You don't have to use the data at CensusAtSchool. But there's a lot there, and it's all generated around what students in primary and secondary would be interested in knowing, and the infographics, by the way, are excellent. They're lovely pictures around some general commonalities they've found in their data when they were collecting it over the years, so you can look at each year separately as well, which is really helpful. So that's just how you would use side-by-side column graphs in the classroom. Just some other resources. We don't have a lot of resources on data. And we haven't sort of delved into that yet. Obviously, now, we probably will, because there's quite a strong focus on it in the new syllabus, so I've just got a couple of images there. I've put the Shaping statistics, Building capacity... These are all hyperlinked, apart from the one on the right, top right. The Shaping statistics in Stage 4. I know I'm going overtime, so just... I'm nearly finished. Shaping statistics in Stage 4 - it's great, again, if you're a central school or you want to know where statistics goes into Stage 4, have a look at that resource. There's 'The Development of Graph Understanding in Mathematics'. It's a reading, but it's really helpful to understand the different types of graphs and why we use them. I've also got a link to a numeracy wrap there - 'What Is the Chance?' That's just on TaLE. It's got some lovely activities and notebooks there on Chance. And then I've popped in a picture of the 'problem Solving Through Chance and data'. Now, that book is from 1993, and as you can see, I had it when I was at the University of New South Wales, and it's a really good resource, and it has some lovely activities on Chance and data. Now, you cannot... No, you can no longer buy it. I couldn't find it on the Board of Studies website. But if you have it in your school, go hunting for it, because it might be a great opportunity now to bring it back out. OK? But they're just some extra resources if you need any more. So thanks for joining us. There's our details. Hopefully, people have got those down by now, our email addresses and phone numbers. We're happy to answer further questions. We will again look at the chat. Although I haven't heard Chris typing too much today, which is good. Hopefully, that means I've been clear with my message. There's the files in the file pod for today, if you'd like to use any of those, including this presentation, if you'd like to use it for your school. Just to let you know that next year, we'll be running some more Adobe Connect series. It'll actually be every two weeks, and it will be on a Tuesday, the same day every two weeks for the whole term. The first one will be on Tuesday 11 February. So we have some suggestions already. Thank you for people that added some into the chat today. If you've got any more suggestions, please add them in to the chat before you leave. We do have some ideas of our own as well that we'll be sharing next year, so...thank you so much for being committed to mathematics, and we wish you all the best for implementing the syllabus into your school for the rest of this year and into next year, and we look forward to catching up with you next year. We will be posting up the enrolment dates, obviously, on SchoolBiz, early next year, but we'll also have the links in here, so if you visit in this room probably the last couple of weeks of the school holidays, the links will be in here in the landing page of our Adobe Connect room. So please keep coming back in here and having a look to see when we have it ready. But thanks for coming today. 'Bye.