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David Watson

Hello everyone and welcome to this workshop titled, Assessing Working mathematically, prepared by the Mathematics Curriculum team of the New South Wales Department of Education.

We'd like to begin by acknowledging the traditional Custodians of the many lands you may be joining us from today and of the lands from which we are recording, the Darug Nation here in Parramatta. We pay respect to Elders past, present and emerging, and extend that respect to any Aboriginal and Torres Strait Islander people joining us today.

The learning intentions and success criteria for this session are designed to empower you to create summative and formative assessment opportunities that allow students to demonstrate what our syllabus values.

The session is divided into 3 parts. First, we will look at assessment requirements before considering the implications on summative assessment. This will be followed by an activity for you to do with your colleagues before returning to the recording for our session on formative assessment. We'll complete the professional learning by engaging in a second activity with your colleagues.

Let's begin by looking into policies and related guidelines that impact school-based decisions with assessments.

NESA's website includes descriptions of assessment, what it entails and its purpose. You can see in these statements that much of the focus of assessment is meant to be about helping the students who are completing the assessment and this should inform our decisions about what assessment looks like.

For the new Mathematics K–10 Syllabus, there is one overarching Working mathematically outcome.

The outcome is the same for all newly published mathematics syllabuses. You'll notice an emphasis on connections, reasoning and deep understanding. It's important to highlight that Working mathematically is an outcome and when we see the word 'outcome' in NESA's advice, we need to therefore consider working mathematically.

The common grade scale hasn't changed. The benefit and drawback of a common grade scale is that it can be applied to all year levels in all schools, in all subjects. That means that if you're assigning a grade to report on a student's understanding in mathematics, you would have to determine the difference between an extensive and a thorough knowledge, as well as other descriptors.

We ran an activity in a statewide staffroom, where we asked teachers to suggest what each of these descriptions might look like in maths.

You wouldn't be surprised to hear that teachers really struggled to describe the difference and used similar words to describe both terms.

So, if the common grade scale only gives us a general idea of where students should be at, we're lucky, then, to have the course performance descriptors, which are subject-specific.

Let's look at a sample of what we've been working with in our current syllabus. Nine grades from E2 to A10, focusing on C6. There are 2 components to each grade.

At the top, we see a description of Working mathematically processes that tend to come through with students performing at this grade level.

Underneath that is a list of content-specific skills that students at this grade typically demonstrate.

In the new course performance descriptors, there are 5 grades instead of 9. It's a simple A to E, focusing here on Grade C.

You'll notice that there's no mention of content in this grade description of a C grade and that's the same for each of the A to E grades. Instead, students are to be assessed solely on what used to be the top paragraph, the Working mathematically skills and processes.

It's important to highlight that because there is no mention of content, a student does not have to have done particular Path content, or indeed any Path content, to achieve an A grade. The challenge we will see is that we need to make sure we are giving students tasks and opportunities to demonstrate these skills by embedding Working mathematically into our day-to-day lessons and assessment tasks.

It's also important to note that the course performances descriptors are used to assign grades at the end of Stage 5, but we need to be constantly exposing them and building them up from the beginning of Year 7.

As we're talking about the course performance descriptors, we thought it would be worth referring to the PISA descriptors, as they share a lot of similarities. You'll notice that there is no mention of content in these descriptors either. This gives us an indication of the skills that are valued globally.

The small image on the right of screen shows the different levels, and we've pulled out and zoomed in on Level 4 to compare with our course performance descriptors. At Level 4, students are required to select and integrate different representations, connect to real-world situations, and construct and communicate explanations.

So now let's have a look at the impact these priorities will have on assessments within schools.

NESA defines summative assessment as it compares to assessment of learning with a key purpose often being to determine a result that can be communicated to the wider community, including parents, educators or students themselves.

The definition is also clear that the effectiveness of any assessment task depends on how the task is designed, with a few points on the screen and particular attention being paid to validity. Does a task actually assess what is then being graded and reported on?

Before we begin, we need to note that in every focus area, including this one, the Working mathematically outcome is included, along with the content outcome, and how students are able to apply Working mathematically processes, while engaging with the content outcome, needs to be assessed and valued.

We're going to use Algebraic techniques A, a core focus area from Stage 5, as an example to consider how writing a valid test might work. Specifically, we're going to be looking at operations with algebraic fractions.

So a sample set of test questions for this focus area could look like those on the screen. As a matter of fact, these 4 problems are listed in the syllabus for us to consider in the examples.

A student's work could look like what we see on the screen here. As a matter of fact, this student seems to have performed exceptionally well. It might be difficult to perceive a student performing better in this test.

This leads us to the question of what grade we should be assigning to their performance. We might tend to think that such work would deserve an A grade.

The course performance descriptors focus on Working mathematically processes and skills and give us an indication of what we should be valuing when we look at student responses.

Has this student considered any relationships between mathematical concepts, moved between any representations, solved any non-routine problem, or produced any reasoning or explanations? Now, while some of this may have happened, the evidence isn't really in the responses by the student.

The student has done some things that are highlighted as being important in the course performance descriptors. Namely, they have used multiple steps to simplify these familiar expressions. They've also used and moved between abstract representations of an algebraic fraction.

Not only have they created equivalent fractions in most questions, but they've also recognised that dividing by a fraction is the equivalent of multiplying by its inverse.

It's easy to think that the best way to challenge students to demonstrate more A-level skills and understanding is to add further, more difficult questions. However, it's difficult to identify anything from the course performance descriptors that we just saw that students will need to draw on if the questions are just more challenging.

It's not necessary to elicit all descriptors in any one question or even in any one assessment. However, we can include questions that provide further opportunities for students to demonstrate these skills and hence provide greater evidence of their grade level.

Some considerations we suggest undertaking before making modifications include thinking about what the key concepts and principles are that we want students to understand deeply.

For example, or in our example, it is possible that students are able to follow shown procedures without actually understanding that each pronumeral represents a number and that the expression represents numerical possibilities.

We should give thought to what it would look like for students to be presented with something that's unfamiliar or non-routine.

We should also think about what representations might be useful for students in this focus area, including concrete, such as algebra tiles, and abstracts, such as those shown in the original examples.

Let's have a look at how we can modify these questions to give students further opportunities to demonstrate the skills that we want to see.

So, what if we ask students to consider the result of one of the operations and what it implies? The expression within this question is one from the earlier test. By asking them to make a comparison of size, students need to consider the expression as having a value, and best responses will display an understanding of how this actually works.

A problem like this could potentially encourage a student to use their calculation to support a conclusion, and in doing so, they can demonstrate a deeper understanding of what their solution actually is.

In answering this problem, students have the opportunity to display their understanding of the relationship between the algebraic expression and the numerical possibilities that it's generalising.

They're asked to explain their solution in greater detail and give reasoning to support how they are interpreting this simplified expression. This student has produced a concise argument to justify their result.

They're still asked to provide multiple-step solutions to a problem that could be considered routine and familiar.

There's also an opportunity for students to use representations beyond the abstracts that we saw earlier.

The same response from a student can be improved using a bar model, demonstrating that even without knowing the value of A, we can know how the expression compares with A.

To make this likely in a student response, we could take certain measures. This includes ensuring we teach with representations and aim to empower students to use them.

This use of a bar model can be introduced while teaching operations with numeric fractions, so that its use in this topic becomes a familiar extension. Finally, pay attention to the end of the question just now.

So, we can modify the question we ask to specifically request a visual representation of what's happening in the expression.

In this second modified question, we're asking students to compare the given algebraic expression to a numeric value.

While students might be able to examine the original expression, a response where a student chooses to simplify demonstrates their understanding of the usefulness of this skill of simplification.

Students are then also encouraged by the question to include some sort of explanation in their response.

Once again, this type of question encourages students to show an understanding of the algebraic expression.

Students are challenged to effectively communicate their reasoning and explain their solution. And again, a visual representation can be used as an aid.

Using bar models effectively could potentially support students to understand the problem for themselves. They can also be used to aid their communication and provide the teacher with evidence of understanding of key concepts, such as the constant size of X in these 3 bars.

In this final problem, we've modified one of the questions to ask students to further consider the solution and offer an opinion supporting this position with adequate reasoning.

A sample response could look something like this. The student has again simplified, applying appropriate algebraic techniques, but has taken the time to interpret some meaning behind this particular simplified expression.

If students genuinely understand the connection between the algebraic expression and the underlying concept, there are multiple approaches they could take to a successful solution.

Again, a visual representation using a bar model could assist students, both to understand why the relationship between the 2 algebraic terms in the expression is a constant, that one is one-sixth of the other, and potentially to convey this reasoning.

So, it's now time for you to try modifying some questions. Ideally, we would like you to have an existing assessment task for your school that you can examine one, or some of the questions from, with a group of your colleagues.

The task does not need to be a test. It could be an investigation task or any other type of assessment that you use in your school where students are asked to respond to a question in some way.

We want you to highlight any opportunities that already exist for students to demonstrate the course performance descriptors.

Consider any modifications you could make to questions, or questions being asked, to provide greater opportunities for students to demonstrate these descriptors at higher levels, and then ask yourselves, how likely is it that any students would respond in the ways that you hope? Does this require modifications to how the mathematics is taught, perhaps?

Pause the recording and complete this task, sharing your group's result with all that are present.

[mellow bright music]

Welcome back. You will have noticed that in our presentation, that certain course performance descriptors were more frequently able to be drawn out from the test question responses than others.

While we were able to create some examples of questions that could benefit from students producing or using visual representations, they're not always natural in a response to a test question and higher grade levels ask students to do far more than just interpret a representation.

Unfamiliar situations may be challenging to define for classes of students as maybe non-routine problems.

This is why it's so important to ensure that a variety of assessment types are used to collect evidence of student skills and understanding.

The resources published by the New South Wales Department of Education to support the 7–10 section of the Mathematic K–10 Syllabus will be supported by a variety of summative sample assessment tasks similar to the example shown on the screen for Years 7 and 9.

When we talk about summative assessment, we are essentially limiting the purpose of our work to telling students how they have performed against the standard set by the syllabus outcomes, including the overarching Working mathematically outcome.

Formative assessment is where we use what we learn from assessment to modify teaching and help students to make further progress.

If we follow a traditional cycle of teach for a set period of time, then assess and give a grade, students aren't given the opportunity to learn from assessment and improve.

Going back to one of our problems from our modified test, the response displayed on the screen would likely be considered correct. Unfortunately, the student has not taken the opportunity to demonstrate some of the Working mathematically processes that we value such as producing an explanation for their solution.

While a class test may generally be considered a summative assessment, if teachers use student responses to inform decisions about future teaching of this group, then it can serve both purposes.

Often, we know our students well enough that we know if they're capable of more than what they've written and this knowledge of our students has to come from somewhere.

What we'll be talking about now are strategies that we can use to gather evidence within our lessons to build a more complete picture of our students, and become more effective at filling in any potential skill or knowledge gaps as a result.

Finger votes are great for assessing multiple choice with students. It can be a simple question that leads to a concept, perhaps based on an initial opinion, or it can be a full problem that requires calculation. Students hold up one, 2, or 3 or 4 fingers to vote for either A, B, C or D.

Advantages of this strategy include teachers gathering immediate feedback across the class and for each individual, as well as students naturally looking around and considering the responses of their peers.

A similar strategy is using mini whiteboards in class. Students can hold up their work with responses but also have the opportunity to record working and work in small groups. The ability to erase work has been shown to increase students' willingness to take a risk and attempt a response to questions.

Taking this one step further is using a vertical non-permanent surface, such as a plastic pocket stuck on the wall or whiteboards around the room if this is available to you, and this has been shown to enhance thinking in the classroom. Students' work is displayed for teachers to review and give immediate feedback, as well as to take notes to consider implications for future teaching.

A Think-Pair-Share strategy involves allowing time for students to think independently about a prompt or problem before pairing up to discuss and share with the class. Advantages include time to individually reflect, as well as being able to share the opinion or knowledge of a partner, which can feel less of a risk for students.

Given we are suggesting students be asked to explain their responses in assessment tasks, Think-Pair-Share can be a great activity to develop this skill while also being an opportunity for teachers to gather evidence from overhearing individual discussions and leading the whole-class sharing.

An exit ticket is a formative assessment tool that requires students to respond to a few key questions or prompts at the end of a lesson.

In the example on the screen, students are asked to simply complete a calculation. This could be satisfactory, but as we discussed earlier in the session, modifying questions to focus on the underlying concept can draw out more of the skills that we value in students.

The lessons published by the New South Wales Department of Education highlight the way that assessment should be considered while planning and reflecting on teaching.

In each lesson, opportunities for formative assessment are embedded and called out in the suggested opportunities for assessment section. Assessment should be frequent and ongoing, allowing students to demonstrate growth.

For your final activity, please use page 8 and 9 of your participant booklet to select a formative assessment strategy, such as those that we've just displayed, or another strategy you may be familiar with.

Review the week ahead and consider opportunities to incorporate this formative assessment strategy in your class. Then plan how you will implement this strategy. Remember that this plan involves considering how to run the activity, possible student responses and how to act on these responses.

How significantly do you feel you need to be ready to adjust your teaching plan for following lessons based on the results of your assessment strategy?

Pause the recording to complete this task.

[mellow bright music]

Welcome back. We hope you enjoyed that activity and that you've come up with some great plans for your classes moving into the rest of this term.

We hope that moving forward, if you need any assistance with anything that you've learned from today's session, that you can contact the Curriculum team by joining the statewide staffroom and reaching out through there, or by contacting us on the mathematics 7–12 email.

You can give us feedback via the evaluation link that you can see. We look forward to hearing from you with all your experiences with Assessing Working mathematically.

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