Using formative assessment rubrics in Mathematics · Web viewFormative assessment is any assessment...

Using formative assessment rubrics in Mathematics

Strategies for multiplication Levels 1–3

Authorised and published by the Victorian Curriculum and Assessment AuthorityLevel 7, 2 Lonsdale StreetMelbourne VIC 3000

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Using formative assessment rubrics in Mathematics – Strategies for multiplication, Levels 1–3

ContentsWhat is formative assessment?.........................................................................................................5

Using formative assessment rubrics in schools..............................................................................5

The formative assessment rubric.......................................................................................................6

Links to the Victorian Curriculum F–10...........................................................................................6

The formative assessment task.........................................................................................................8

Description of the task (administration guidelines).........................................................................8

Evidence collected from this task...................................................................................................9

Interpreting evidence of student learning.........................................................................................10

Setting the scene..........................................................................................................................10

Sample 1...................................................................................................................................11

Sample 2...................................................................................................................................13

Sample 3...................................................................................................................................15

Sample 4...................................................................................................................................17

Sample 5...................................................................................................................................19

Using evidence to plan for future teaching and learning..................................................................21

Teacher reflections...........................................................................................................................21

Appendix 1.......................................................................................................................................22


What is formative assessment?Formative assessment is any assessment that is used to improve teaching and learning. Best-practice formative assessment uses a rigorous approach in which each step of the assessment process is carefully thought through.

Assessment is a three-step process by which evidence is collected, interpreted and used. By definition, the final step of formative assessment requires a use that improves teaching and learning.

For the best results, teachers can work together to interrogate the curriculum and use their professional expertise and knowledge of their students to outline a learning continuum including a rubric of measurable, user-friendly descriptions of skills and knowledge. Teachers can draw on this learning continuum and rubric to collect evidence of each student’s current learning in order to provide formative feedback and understand what they are ready to learn next.

The VCAA’s Guide to Formative Assessment Rubrics outlines how to develop a formative assessment rubric to collect, interpret and use evidence of student learning to plan teaching and learning. For more information about formative assessment and to access a copy of the guide, please go to the Formative Assessment section of the VCAA website.

Using formative assessment rubrics in schoolsThis document is based on the material developed by one group of teachers in the 2019 Formative Assessment Rubrics project. The Victorian Curriculum and Assessment Authority partnered with the Assessment Research Centre, University of Melbourne, to provide professional learning for teachers interested in strengthening their understanding and use or formative assessment rubrics.

This resource includes a sample formative assessment rubric and task and describes how the teachers implemented the task in their schools and also includes representative student work samples.

Schools have flexibility in how they choose to use this resource, including as:

a model that they adapt to suit their own teaching and learning plans a resource to support them as they develop their own formative assessment rubrics and

tasks.

This resource is not an exemplar. Schools are advised to consider whether the sample formative assessment rubric and task meet the needs of their student cohort and is appropriate in the context of their whole-school teaching and learning plan. Additional support and advice on high-quality curriculum planning is available from the Curriculum Planning Resource.

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The formative assessment rubricThe rubric in this document was developed by three Victorian teachers to help inform teaching and learning in Mathematics. It supports the explicit teaching of multiplication and the following skills:

skip counting repeated addition grouping/equal groups arrays automatic recall problem-solving using a range of strategies.

Links to the Victorian Curriculum F–10

Curriculum area: Mathematics

Levels: Level 1 to Level 3

Achievement standards extracts:

Level 1 – Students describe number sequences resulting from skip counting by 2s, 5s and 10s.

Level 2 – Students represent multiplication … by grouping into sets. They recognise increasing and decreasing number sequences involving 2s, 3s, 5s, 10s … to produce sequences by constant addition.

Level 3 – They solve problems using efficient strategies for multiplication with and without digital technologies.

Content descriptions:

Level 1 –

Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (VCMNA086)

Level 2 –

Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences (VCMNA103)

Recognise and represent multiplication as repeated addition, groups and arrays (VCMNA108)

Level 3 –

Recall multiplication facts of two, three, five and ten and related division facts (VCMNA134)

Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (VCMNA135)

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Learning continuum Mathematics, Levels 1 to 3Strand: Number and AlgebraSub-strand: Number and place value

Phase 1 Phase 2 Phase 3 Phase 4 Phase 5 Phase 6

Students skip count to count a collection.

Students organise objects to complete repeated addition.

Students model grouping (3 groups of 4).

Students show working out using arrays.

Students automatically recall multiplication facts of 2s, 3s, 5s and 10s.

Students represent and solve multiplication problems using a range of strategies.

Organising element Action Insufficient evidence

Strategies for multiplication

1. Uses concrete materials to represent multiplication

1.0 Insufficient evidence

1.1 Uses concrete materials including MAB, hundreds charts or counters to skip count by amounts greater than 1 (e.g. 2s, 3s, 5s and 10s)

1.2 Represents repeated addition number sentences using concrete materials

1.3 Represents problems as equal groups using concrete materials

1.4 Uses concrete materials such as counters to represent problems of 2s, 3s, 5s and 10s times tables as an array (e.g. 3 × 4, or ‘3 people have 4 apples each. How many do they have altogether?’)

2. Uses written strategies and/or visual representations


2.1 Records repeated addition as a number sentence (4 + 4 + 4), matching it to groups shown in visual representation

2.2 Draws groups to solve problems

2.3 Draws arrays to solve problems

2.4 Chooses strategies to solve problems demonstrated through visual and written representations (e.g. writing horizontal or vertical equations, or worded problems)

3. Explains their working out


3.1 Skip counts aloud by amounts greater than one (e.g. skip counting 2s, 3s, 5s, 10s by pointing or dragging to each group as they count the objects)

3.2 Applies the language of ‘groups of’ (e.g. ‘3 × 4 is 3 groups of 4’) to solve problems

3.3 Verbalises strategies for automatic recall of 2s, 3s, 5s and 10s (e.g. ‘Multiplying by 4 is the same as doubling and doubling again’; ‘I knew the multiplication fact.’)

3.4 Applies the commutative law to solve problems (e.g. 3 × 4 is the same as 4 × 3)

The formative assessment taskThe following formative assessment task was developed to elicit evidence of each student’s current learning and what they are ready to learn next.

Description of the task (administration guidelines)

Assessment purpose

Students are given the following task to demonstrate their knowledge of multiplication. This can be used as an assessment of prior knowledge (pre-test) and/or their understanding at the end of a teaching cycle (post-test). It assesses the following skills:

efficient counting strategies, e.g. skip counting by 2s, 3s, 5s and 10s repeated addition number sentences and visual representations modelling equal groups organising arrays automatic recall (2s, 3s, 5s and 10s) commutative property solving multiplication problems using a range of strategies

Task instructions

Familiarise yourself with the rubric and the following teacher guidelines. Before beginning this lesson, consider the way you will collect and record student

responses to map against the rubric. For example, you could collect evidence via:

teacher observation checklist based on rubric students recording themselves on devices (saving them in SeeSaw, Debut, Explain

Everything or another platform) students writing a reflection to record their thinking (e.g. ‘I skip counted by 2s’, ‘I

know that 3 × 4 is the same as 4 × 3’) students complete a Think Board worksheet (see Appendix 1) to show their ideas

and understanding.

Set up materials (such as counters, Think Board worksheet, computers/devices, MAB, mini whiteboards, working out paper).

Provide each student with 12 counters (or use MAB, teddies or alternate counting materials).

Introduction Write the learning intention on the board: I am learning to solve multiplication problems

using a range of strategies. Write the following question on the board: Here are a number of counters. How can you

count them? Enabling prompts include the following: Could you use addition? Can you show them in

groups? Can you make an array?


Extending prompts include the following: What can you do with them? Is there a faster way to count them? How could you organise them? What do you know about this number?

Share student responses and discuss the different strategies that they applied. Tally the types of strategies that students used from the following: skip counting, repeated addition, grouping/equal groups, arrays, automatic recall, horizontal/vertical multiplication.

Task Write the following question on the board: How many ways can you show, make, say

and write 3 × 4? Provide students with a think board, working out paper and concrete materials if

needed. A grid paper resource could be used. Remind students to show and explain their working out on the Think Board as they are

doing this. Model an example such as ‘I placed these in 4 groups because there are 3 each.’

Encourage the use of the devices to record their responses.

Extension tasks Consider extending students to use a more complex multiplication problem if they

completed the initial problem competently. Give students two dice and ask them to roll and create their own multiplication problems

to apply their skills. They may use higher numbers. The following worksheet is also an optional extension task.

Evidence collected from this task student work samples teacher observation checklist based on rubric student self-assessment, reflections or explanations (video or audio) visual representations such as poster or anchor chart

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Interpreting evidence of student learning Evidence collected from each student was mapped against the rubric:

The quality criteria that were achieved was shaded in blue. The phase that the student is ready to learn next was shaded in green.

Please note, the following annotated student work samples are representative examples only.

Setting the sceneThis formative assessment task was implemented in a Grade 3 class in two government primary schools in Melbourne.

Both teachers followed the script and prompts on the task instructions.

Both classes completed think boards prior to the lesson.

One teacher conducted the activity as a pre-test, to assess students’ prior knowledge and use this to inform her decisions how to commence the learning sequence. The other teacher completed the activity after a week of teaching multiplication.

In both classes the activity was completed in 50–60 minutes.

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Sample 1

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2.2 Draws groups to solve problems




Sample 1: Evidence of student learning

Annotations 1.3 – Showed groups of 2.2 – Drew groups of 3.2 – Used the language ‘groups of’

She showed that she has automatic recall of times tables facts. She demonstrated different factors including 6 × 6 and 3 × 12, and modelled as groups.

What is the student ready to learn next?The student was assessed as ready to learn Phase 4 next.

The student did not draw arrays, even though teacher judgement indicates that she has some familiarity. She chose the option that she felt more comfortable with. Because she did not show her knowledge of arrays, she needs to consolidate this phase of her learning. This will be a short-term goal in her learning.

Any feedback given

The student was given feedback about drawing ‘groups of’, how she showed this in a clear way and how she labelled these with the correct multiplication sentence. It was highlighted to her that she can show her knowledge in different ways and she was encouraged to show arrays next time as she knows that format as well, to extend her thinking and to provide multiple strategies.

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Sample 2

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1.4 Uses concrete materials such as counters to represent problems of 2s, 3s, 5s and 10s times tables as an array




Annotations 1.4 – Used concrete materials to show arrays and multiplication facts 2.4 – Chose strategies to solve problems through visual and written representations 3.3 – Verbalised strategies for automatic recall

The student achieved all levels of the concrete materials activity part, including skip counting organisation. She explained her strategy of recalling known multiplication facts (10s) and adding extra groups.

What is the student ready to learn next?The student was assessed as ready to learn Phase 6 next. The student demonstrated understanding of written and modelled work to show a multiplication fact. She didn’t explore different factors of 36.

The student has confidently shown her understanding in arrays, groups of and the language of multiplication. She will continue learning different strategies such as commutative law, other factors and then prepare for division.

Any feedback given

The student was given feedback about showing her thinking in different ways and recording the equation as vertical or including other language (e.g. ‘multiply’, ‘by’).

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2.4 Chooses strategies to solve problems demonstrated through visual and written representations

3.3 Verbalises strategies for automatic recall of 2s, 3s, 5s and 10s



Sample 3

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Annotations 1.3 – Showed groups of 2.3 – Showed arrays 3.2 – Used the language ‘groups of’

What is the student ready to learn next?The student was assessed as ready to learn Phase 4 next. She used concrete materials to organise as groups of and arrays. She demonstrated two different multiplication facts for 36 – 6 × 6 and 9 × 4. The student did not draw anything for groups of or repeated addition

The evidence highlighted that the student is able to organise her materials and draw models into arrays. She had a good understanding that the amount of groups multiplied gives the answer. She did not show skip counting because she has developed her automatic recall skills.

Any feedback given

The student was given feedback about her understanding of groups of and how this is a strength in her learning. She was able to write and verbalise her thinking.

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Annotations 1.3 – Showed groups of 2.3 – Drew arrays and circled group (2.2) 3.1 – Skip counted by 2s

The student used concrete materials to organise into groups. He explained that the number was ‘equal’, and we assumed that he means it is ‘even’ (36 is an even number).

The student did not record a multiplication equation and did not demonstrate an understanding of factors.

What is the student ready to learn next?The student was assessed as ready to learn Phase 4 next, even though quality criterion 3.2 (under Phase 4) was not met. The evidence highlighted that the student is able to skip count by 2s, but he did not connect this with repeated addition or groups of. He did not demonstrate different ways of showing his thinking. Therefore his next phase of learning should focus on applying the language of ‘groups of’ to solve problems as well as automatic recall of facts and using different strategies to show his understanding.

Any feedback given

The student was encouraged to demonstrate his thinking in more than one way and to demonstrate how skip counting links to groups of.

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Sample 5

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3.3 Verbalises strategies for automatic recall of 2s, 3s, 5s and 10s

2.4 Chooses strategies to solve problems demonstrated through visual and written representations

1.4 Uses concrete materials such as counters to represent problems of 2s, 3s, 5s and 10s times tables as an array




Annotations 1.4 – Used concrete materials to show arrays and multiplication facts 2.4 – Chose strategies to solve problems through visual and written representations 3.3 – Verbalised strategies for automatic recall of 2s, 3s, 5s and 10s

The student achieved all levels of the concrete materials activity part, including skip counting organisation. She showed her division knowledge as a connection in ‘What else do you know about it?’

The student recorded the multiplication equation in many ways; however, she did not use commutative law or all the factors of the numbers.

What is the student ready to learn next?The student was assessed as ready to learn Phase 6 next. She had a sound understanding of arrays, groups of and the language of multiplication. She will continue learning different strategies such as doubling and commutative law, and explain her thinking.

Any feedback given

The student was given positive feedback about showing her thinking in different ways and marking a connection with division.

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Using evidence to plan for future teaching and learningThe data showed that the majority of students demonstrated understanding of skip counting and use of materials (groups and arrays). A small number of students completed the final two phases of the rubric, which highlighted that the task needed more instruction for the expectations to be met. Most students were working at the expected level.

The data collection showed clear pathways for student learning, including recalling multiplication facts and using multiple strategies.

A rubric covering higher levels of the curriculum could be developed to accurately assess those who completed all phases on this rubric.

Teacher reflectionsAfter trialling the rubric with Year 3s, we found it would also be well suited to Years 1 and 2 to assess student understanding of specific types of modelling (groups and arrays).

We found that the task did not prompt students to show the entire range of their skills and strategies to demonstrate multiplication. The students had sufficient time to complete the task, but the knowledge elicited did not align with the rubric because the task did not provide enough guidance for the students. We believe it would be more beneficial to complete the task in a small group setting, instructed by the teacher. This would enable the teacher to gather more information about the students’ use of concrete materials, and their explanation of strategies and behaviours.

The task sheet was later revised and updated to include a more structured question sequence to show all of their knowledge/strategies (see Appendix 1, which has been updated with an equation). Essentially the criteria from the rubric was written in student instructions.

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Appendix 1

Using formative assessment rubrics in Mathematics · Web viewFormative assessment is any assessment...

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