Transcript of Mathematics Stage 2 diagnostic tasks
Ayesha Ali Khan: Hello all and thank you for taking the time to view this Adobe Connect recording. I am Ayesha Ali Khan the Mathematics Adviser K-6 and I have Linda De Marcellis the Numeracy Adviser K-6 here with me today.
Linda De Marcellis: Hi everyone
Ayesha Ali Khan: This recording is to familiarise and support you in using the Mathematics Stage 2 diagnostic tasks. You may like to have a copy of this resource to refer to throughout this recording and the Mathematics K-10 continuum of key ideas. You can find this in the files pod below. Please note hard copies of the Mathematics Stage 2 diagnostic tasks were distributed to schools during Term 3 of 2018.
During this Adobe Connect recording you will be provided with information about:
- the introduction section of this document
- how to use this resource
- ways to support students in using this resource
- task considerations
- how you can access this resource
- and other diagnostic tasks available.
Linda De Marcellis: In New South Wales students in Stage 2 work towards the achievement of outcomes from the New South Wales Mathematics K-10 syllabus created by the New South Wales Education Standards Authority. That's NESA. The Mathematics K-6 continuum of key ideas is designed to provide an overview of the sequence of learning from Early Stage 1 to Stage 3 in each of the sub-strands of the Mathematics K-10 syllabus. On the NESA website within the mathematics K-10 support materials and programming section, you can locate the mathematics K-10 continuum of key ideas under the heading ‘Additional support materials’ and we've also included a downloadable PDF on the file there, on the left. You can access that.
Ayesha Ali Khan: Each question in this resource focuses on the Stage 2 key ideas from sub-strands within the strands of the Mathematics K-10 syllabus. For example within the strand of number and algebra in K-6 there are 5 sub-strands. Within the sub-strand of whole numbers these are the key ideas identified which are on the right of your screen.
So part 1:
- count forwards and backwards by tens and hundreds from any starting point
- state the place value of digits and numbers of up to four digits
- read write and order numbers of up to four digits.
- state the place value of digits and numbers of up to five digits
- read write in order numbers of up to five digits
- record numbers of up to five digits using expanded notation.
The Mathematics Stage two diagnostic tasks resource has questions that focus on each of these key ideas and other key ideas throughout the document. As the information provided in NESA’s document titled ‘Introduction to the continuum of key ideas’, it states that these are not a substitute for the syllabus content. Teachers are expected to refer to the outcomes and content of the syllabus to determine the breadth and depth of the knowledge, skills and understanding expected in each Stage. When planning an appropriate learning experiences for your students you may choose to refer to the key ideas to assist you to:
- identify the Stage in which a particular key concept is introduced introduced
- identify the key concepts that precede or follow those introduced in a particular Stage of a sub-strand
- program units that build on and extend the key concepts developed in earlier Stages of the sub-strand
- and identifying compare when related concepts are introduced in other sub-strands.
Linda De Marcellis: The Mathematics Stage 2 diagnostic tasks can be used by teachers to generate a snapshot of student learning aligned to syllabus expectations at Stage two. This information may assist teachers in tailoring teaching and learning experiences to more effectively meet students’ needs. It may assist teachers to:
- make professional judgments of student progress
- tailor teaching and learning experiences
- support students transition in mathematics learning between Stages 2 and 3.
Ayesha Ali Khan: As you can see in the table of contents each sub-strand has a dedicated section in this resource. The contents page also refers to two other sections in this document the introduction and notes. The introduction states information about:
- using this resource
- supporting students
- and task considerations - which is also being detailed in this Adobe Connect recording.
The notes section allows you to record information that you may find useful in relation to this resource. You may like to take notes in this section whilst watching this recording. Tasks in this resource have been designed for flexible use. A sub-strand could form a pre-assessment task for example whole numbers or alternatively, teachers may decide to use particular questions to gather evidence about a student's or a group of students knowledge, skills and understanding of key ideas in mathematics. The page featured to the right is an example where by question 3 focuses on understanding place value and the key idea of - record numbers of up to five digits using expanded notation. This question also has a related key idea recorded from addition and subtraction - use and record a range of mental strategies for addition and subtraction of two, three, four and five digit numbers. Throughout this resource you will find related key ideas recorded for some tasks.
Linda De Marcellis: Okay, working mathematically components have been included in some questions. This facilitates the examination of students working out, thereby assisting teachers to assess the efficiency of strategies used by students as they apply their knowledge, and skills of mathematical concepts. Here is an example (page 8) from this resource that ask students to show their working out and you can see it says addition and subtraction strategy one working out and strategy two and there's sufficient amount of space there that you can see that students can use to write as well as draw or just show their working.
Ayesha Ali Khan: Teachers can also ask students to use technologies such as an iPad to recall how to record how they have solved, worked out or share their thinking to gather information about a student's knowledge, skills and understanding, in regards to a key idea of the syllabus and the working mathematically outcomes in Stage 2. This is an example of a final product video recording completed by a student using an application on an iPad whereby he's recorded his thinking as he solves the task. This student is completing the two-dimensional space task question 5 splitting 2D shapes on page 82.
Student: [silence][video] Hi today I am gonna make 2 ways to make a hexagon. So the first way you can add 2 triangles, add 2 triangles and another one over here. And then if you know if you add 2 squares you can make a rectangle. Like over here. It kinda looks like a ‘H’. That’s the funny thing about it. Now this one, you have to make 2 trapeziums together and tilt it, because if you just leave it it will look diagonally weird. So it really needs to actually be a hexagon so that’s actually it. Thank you.
Ayesha Ali Khan: So in this next video that we will play it is of the student actually completing the task.
Student: Hi today I am gonna make 2 ways to make a hexagon. So the first way you can add 2 triangles, add 2 triangles and another one over here. And then if you know if you add 2 squares you can make a rectangle. Like over here. It kinda looks like a ‘H’. That’s the funny thing about it. Now this one, you have to make 2 trapeziums together and tilt it, because if you just leave it it will look diagonally weird. So it really needs to actually be a hexagon so that’s actually it. Thank you.
Ayesha Ali Khan: So you can see how using technology can really further support teachers to gather information about a student's understanding and thinking when solving problems. You may like to pause this recording now and discuss what information you can find out or note down in regards to Ali’s thinking and about the key idea of splitting common shapes into other shapes and recording the result.
[Pause video] So from observing this video clip that Ali has presented us we can gather more information about his understanding of the orientation of two-dimensional shapes and how he saw he could see triangles in a different orientation but when he was explaining about the hexagon divided into two trapeziums, he was saying that he had to see it on the side for him to recognize the trapezium.
Linda De Marcellis: Another way you can use this resource is also engaging students with technology and we can do that through the area task on question 4 - comparing areas on page 54. The students use technology here to complete the task by taking photos and then they can use those photos to show their understanding.
Ayesha Ali Khan: So we really want to have a look at what students are sharing with us to identify the next point in learning. This is a teacher’s observation which I have recorded and the student is completing the length task so question 8 - calculating perimeter on page 43.
The key idea noted for this task is estimate and measure perimeters of two-dimensional shapes. You may also like to pause this recording again and reflect on what information about um her learning you could gather from this observation.
Student: [video] [silence] First I did 32 centimetres divided by 4 because there are 4 sides in a square equals 8 so I worked out this was 8 and this was 8 too. And then I did 8 divided by 2 because this side cut in half equals that means it’s in 2 parts so then I divided it and then it equals 4. Then I did 4 plus 4 plus 8 plus 8 which is 24.
Linda De Marcellis:I think the great thing when we were observing that, that Ayesha and I realised, is how the student was talking about you know her knowledge also about square and that all the slides were equal and I really think you don't always get that understanding even though it is a length task it's also really important to know what other information students are bringing in when they're working out a problem.
Ayesha Ali Khan: So I wanted to find out more information, about how she gained 24 centimetres as the perimeter of one rectangle so I've asked her a question and we've recorded that.
Ayesha Ali Khan: So how did you do your adding to find the perimeter can you explain that?
Student: First I did 4 plus 4 equals 8 with 8 times 3 because there are 3 8’s equal 24.
Ayesha Ali Khan: So by asking a question about how she solved the problem, I've gained a lot more information not just around the key ideas mentioned in relation to that task. So we can see here that Fiona has demonstrated an understanding of fractions and division as well as multiplication.
So here we have a work sample of an addition and subtraction task - so question 1 which focuses on addition, and the key idea noted for this question is use and record a range mental strategies for addition and subtraction of 2, 3, 4 and 5 digit numbers. So I've I have observed Fiona completing section B which asks 16 plus 8 plus 4 equals and Fiona has completed this task and I observed her constructing a line for strategy 2 and then crossing it out and then she proceeded to record 16 plus 4 equals 20 and then 20 plus 8 equals 28. So I wanted to find out more about Fiona's thinking. So I interviewed her and recorded some anecdotal notes
I asked her to explain how she solved this problem. Prior to me sharing this with you, you may like to pause this recording and reflect on what you think she may share just viewing what she's written.
Linda De Marcellis:For strategy 1 and also for strategy 2.
Ayesha Ali Khan: So, this is what I gathered from what Fiona shared with me and I took these notes.
So for strategy one, she started to record 16 plus 8. So going across the question and realized that she knew a known fact that 2 times 8 is in 16 so therefore, she's used that knowledge and understanding and realised therefore 3 times 8 equals 24.
She's then proceeded to add 24 plus 4 equals 28 and renamed that.
Linda De Marcellis: And when Ayesha and I were talking about this, she had the note there, that even though she didn't write 3 times 8 equals 24, she actually explained it as 3 times 8. So it was really interesting of course as you know it's an addition and subtraction task. So she's sort of gone along the way of showing her understanding through addition but in actual fact her flexibility has given her entry into multiplication as well, and really seen that double 8 is 16, and then associated that to make it 3 times 8 with the 24. I think it's really important to have time to talk to students about their working out because it clearly shows us that Fiona was doing an additive way of demonstrating her strategy but in actual fact her thinking was through multiplication for most of the part of that solving.
Ayesha Ali Khan: So for strategy 2, this is what made me curious, she started with the number line and crossed it out. she explained that she was drawing a number line but she figured that there was a quicker way. She explained to me that she knew her friends of 10 so 6 plus 4, and used this understanding for her friends of 20. So 16 plus 4 is what she's written which equals 20, and then she has written down 20 plus 8 equals 28.
Linda De Marcellis: And so when Ayesha and I when we were talking you know we think of Cathy Forsnot and she says that mathematicians when they're looking at problems there don't look at the problem and then work out which what they're going to do - they actually look at the numbers and they look at what what the numbers are sort of telling them and what they can do with the numbers. So even though the number here the problem was 16 plus 8 plus 4 you can see Fiona is actually seeing the numbers doing other things for her and a real mathematician really looks at that and says which is sort of the best way to work that problem out.
Ayesha Ali Khan: So this really highlights that by giving students diagnostic tasks to complete independently and then collecting them to mark will provide you with some valuable information. However by observing and delving into it more by asking them questions you find out more information about what they're thinking and that adds to their recording. If I wasn't observing her I may not have realised that she constructed the number line first and I would have definitely not known that she had used and that she knew that she had to use addition in her working out - but she actually applied multiplicative thinking. So I also gathered information about her understanding of combinations to 10 and how she uses this knowledge and recognises the relationships with combinations to 20 .
Linda De Marcellis: Um so yeah we sort of want to make sure that when you're using this resource obviously students are doing this independently if you want to or you want to see how they're working as a group but, it really is important to try and have that opportunity to either have one-on-one discussion with them, or if you're unable to do that maybe do that recording like we did with the first student, and get them to record their thinking. Also it's important that you know obviously Fiona's done a really good explanation of writing with her working out. But an actual fact there's probably some teaching here that we'd like to look into for her to further explore so that we can see that she was doing a multiplicative thinking that she can also record that as well .
Ayesha Ali Khan: And I guess this resource is really there to support us with finding out where we can go next in order to support students.
Linda De Marcellis: Yes and where the gaps are and how we can extend but it also shows us that students are working a lot more flexibly than what their sort of writing and the writing shows us.
Ayesha Ali Khan: So we have another work sample (page 23) here. Prior to me sharing this with you, you may like to pause this recording and reflect on what you think Zakey has demonstrated to solve this problem
So from observing him I've taken this anecdotal note. So he started by creating these circles which we presume represented counters. In column 1, he's recorded 7 and in column 2 he's recorded 8 um circles.
And what he did was he attempted to use those 8 counters to keep tracks of 7, of counts of 7 so he went to 7, 14, 21 and then referred back to those 7 circles and countered by 1. So while he's counting by ones he's lost track of the count and somehow arrived at 72 instead of stopping at 56.
Linda De Marcellis: And you can see clear marking that he's marked 1, 2, 3, 4, 5 so he's sort of keeping, trying to keep some sort of track of his count but somewhere along the line counting by ones isn't really an efficient strategy and and we know that and that's why he somehow came to 56 at 72.
Ayesha Ali Khan: That's correct because as I was observing him he said 7,14, 21 and then he started you could see him looking at each of the seven circles.
Linda De Marcellis: Mm-hm
Ayesha Ali Khan: And trying to count them - placing a dot next to the counter 4 in that column of 8 recounting 7 again and adding on and then reloading so somewhere along the line he missed the count.
Linda De Marcellis: And you know I looked at that and we sort of talked about, Ayesha and I, if he got to 21 quite quickly and Ayesha does remember him getting back to getting to 21 really quickly. It would have been interesting to see if we said look if you double 21 which I'm pretty sure that he would have been able to do and get to 42, that would have taken him straight to 6, and then he would have known that just adding another 2 more which is you know the 14 then he would have arrived at 56 probably, easier and quicker than counting and then obviously getting the incorrect answer of 72.
Ayesha Ali Khan: So it's really important that we use the information that we're gathering from these, and question students completing these questions in this resource to guide our teaching and our focus. So you may also be having a look at other strategies and that we can discuss with Zakey or have students share with Zakey - perhaps you know using his understanding of the 5 times tables and going 7 times 5 is 35, and then we have an understanding that Zakey knows and 7, 14 and 21 or 3 times ...
Linda De Marcellis: Yep and 3 times 7.
Ayesha Ali Khan: 7 and then combining those to get 56.
Linda De Marcellis: And this is a hard thing that students do find - is moving out of that additive thinking and going into multiplicative thinking where they really are trying to move away from just counting and repeated counting - repeated addition. I think you know it'd be really interesting to stop this and have a talk to amongst the teachers if you are there or even if you're on your own just to actually reflect and think how else would best suit to sort of support Zakey and his thinking as well.
So also with the mathematics Stage 2 diagnostic tasks resource, it is not intended then to provide a comprehensive assessment, but it's giving additional evidence that we should also be drawing from other sources. Such as like we said teachers observations which we'd like to think that you would do this throughout your investigation within and also involvement within the task, in the diagnostic. But there's other teacher observations that you're going to observe throughout the day as well. As well as interviews and using student work samples and also some anecdotal records that you would be taking. But all together these will give a good evidence of support for teachers in making professional judgments about students progress towards achievement in Stage 2 syllabus outcomes.
Ayesha Ali Khan: So please note adjustments can be made with students with individual learning needs these may include:
- the use of large print papers
- extra time
- rest breaks
- small group group or individual supervision
- use of a reader or writer
- and the use of computer or assistive technologies.
Additional support should also be provided to students who are learning English as an additional language. On the PDF version online, where possible, Web Content Accessibility Guidelines 2.0 AA have been met.
Linda De Marcellis: To support the teachers ease of use the following considerations should be noted so:
- when you see a hand icon it identifies that there's a practical task so it really does require extra material sometimes so you have to sort of have that all ready
- and also a ruler icon identifies tasks requiring the use of a ruler.
So here are some examples of a practical task. On the top left is an area task question for comparing areas on page 54, the top right is a two-dimensional space question 4 combining 2D shapes on page 82, the bottom left a chance question 3 chance experiments on page 103 and the bottom right is also a chance question 4 and that's also on page 104. So you can see students here are utilizing different equipment - there's rulers and 2D shapes also dice and coin . So when you're using the task make sure if there are hands on you have those equipment ready for the students. Also too remember when printing this resource, do not scale worksheets instead print them at 100% on the A4 paper. Now this is important to note as it will affect some length and position tasks like the examples that we've used on page 40 and 92 which require a ruler so both of those tasks.
Ayesha Ali Khan: So 4 copies of the Mathematics Stage 2 diagnostic tasks were sent to all primary schools to support the continued implementation of the New South Wales K-10 mathematics syllabus. This resource is also available on the mathematics curriculum website located on the education.nsw.gov.au site. Please follow the steps above to locate it online.
Follow these steps:
- Teaching and Learning
- Key learning areas
- Early Stage 1 to Stage 3
- Mathematics Stage 2 Diagnostic Tasks, (PDF 2284.31KB)
The direct link is also in the links pod below.
Linda De Marcellis: Please note that there are also Stage 3, 4. 5 diagnostic tasks resources available and they're also available through the mathematics curriculum website mathematics Early Stage 1 to Stage 3 resource page which had the links on the previous slide. And we just like to say you know a thank you for coming this afternoon and if you start using these tasks or have been using these tasks and you'd like to share some really great discoveries, please share them with us you can email them to us. We'd love to hear how they're working and what your students are doing and show how flexible they are. You know if you've got photos please do that. We'd love to see what's happening with the diagnostics.
Ayesha Ali Khan: So I like to take this opportunity to thank the principals, teachers and especially the students in trialing this resource prior to publication. It was greatly appreciated and I hope you find this resource valuable. If you have any further questions about this resource please feel free to contact either of us.
Linda De Marcellis: Otherwise happily enjoy the diagnostic tasks and hopefully we get to hear and see your lovely stories and learning with the students
Ayesha Ali Khan: Thank you
Linda De Marcellis: Thank you
End of transcript