using formative assessment rubrics in mathematics · web viewwith the implementation of a new...

Using formative assessment rubrics in Mathematics

Multiplication and divisionLevels 4–6

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 – Multiplication and division, Levels 4–6

ContentsWhat is formative assessment?.........................................................................................................4

Using formative assessment rubrics in schools..............................................................................4

The formative assessment rubric.......................................................................................................5

Links to the Victorian Curriculum F–10...........................................................................................5

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

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

Evidence collected from this task...................................................................................................8

Interpreting evidence of student learning...........................................................................................9

Setting the scene............................................................................................................................9

Sample 1...................................................................................................................................10

Sample 2...................................................................................................................................12

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

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

Using evidence to plan for future teaching and learning..................................................................20

Teacher reflections...........................................................................................................................20

Appendix 1 – Worksheet (revised)...................................................................................................21


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 Primary teachers to help inform teaching and learning in Mathematics. It supports the explicit teaching of multiplication and division.

The rubric was designed to provide teachers with information about what students are currently demonstrating in relation to carrying out mental, written and technology-based computation involving multiplication and division. It can be used in conjunction with a formative assessment task that will give students the chance to show that they can:

explore number patterns based on multiples represent worded problems as a number sentence use a range of mental and written strategies calculate increasingly complex equations.

Links to the Victorian Curriculum F–10

Curriculum area: Mathematics

Strand: Number and Algebra

Sub-strand: Number and place value

Levels: Levels 4–6

Achievement standards extracts:

Level 4 – They choose appropriate strategies for calculations involving multiplication and division …

Level 5 – Students identify and describe … multiples.

Level 6 – They solve problems that involve multiplication and division.

Content descriptions:

Level 4 –

Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (VCMNA153)

Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (VCMNA154) Recall multiplication facts up to 10 × 10 and related division facts (VCMNA155) Develop efficient mental and written strategies and use appropriate digital technologies

for multiplication and for division where there is no remainder (VCMNA156)

Level 5 –

Identify and describe factors and multiples of whole numbers and use them to solve problems (VCMNA181)

Use estimation and rounding to check the reasonableness of answers to calculations (VCMNA182)

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https://victoriancurriculum.vcaa.vic.edu.au/Curriculum/ContentDescription/VCMNA182







Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (VCMNA183)

Solve problems involving division by a one digit number, including those that result in a remainder (VCMNA184)

Level 6 –

Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers and make estimates for these computations (VCMNA209)

Further information about Multiplicative thinking can be found on the Department of Education website.

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https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/multithink.aspx





Learning continuum Mathematics, Levels 4–6Strand: Number and Algebra Sub-strand: Number and place value

Phase 1 Phase 2 Phase 3 Phase 4 Phase 5

Students can say or record their understanding of sequencing number patterns involving multiples.

Students can match their understanding of multiplication and division facts to a problem.

Students can use efficient mental and written strategies to solve multiplication and division problems involving single-digit division with no remainder.

Students can use efficient mental and written strategies and appropriate digital technology to solve multiplication and division problems involving large numbers.

Students can multiply and divide decimals and make connections between the powers of 10.

Organising element Action Insufficient evidence

Multiplication and division 1. Skip counts by … 1.0 Insufficient evidence

1.1 Skip counts by 3s, 4s, 6s, 7s, 8s and 9s.

1.2 Continues skip counting and creates simple sequences using decimals(e.g. skip counts to solve 7 × $0.25)

2. Represents worded problems as a number sentence

2.0 Insufficient evidence

2.1 Matches an operation to a worded problem by representing the words as a number sentence(e.g. ‘There were 6 dogs and each had 9 fleas on them.’Represents this as 6 × 9 = )

2.2 Represents multi-step word problems that involve multiplication or division (e.g. ‘There were 26 dogs that had 9 fleas on each of them, 453 cats with 6 fleas on them, and 5 kids with 16 headlice on them. How many fleas were there altogether?’)

2.3 Represents worded problems using the order of operations

3. Uses strategies for multiplication and division


3.1 Skip counts to calculate the answer to a number sentence(e.g. solves 7 × 5 by skip counting: 0, 5, 10, 15, 20, 25, 30, 35 )

3.2 Demonstrates one multiplicative or division strategy used to solve a problem(e.g. 5 × 5 = 25 25 5 = 5)

3.3 Selects one multiplicative or division strategy that is efficient for the student to solve a problem(e.g. doubling and halving, split strategy, landmark numbers, jump and hop)

3.4 Justifies the reasonableness of an answer by using a strategy or strategies (e.g. estimation, inverse operation, use of known facts)

4. Calculates number sentences


4.1 Solves one-digit number sentence with no remainder(e.g. 7 × 6 = 54 8 = )

4.2 Solves a multiplication and division number sentence using multiple digits with no remainder(e.g. 45 × 3 = 135 5 = )

4.3 Solves a multiplication and division number sentence using larger numbers with a remainder(e.g. 3421 × 7 = 32 × 15 = 6743 4 = 1374 12 = )

4.4 Applies knowledge of powers of 10 to multiply and divide decimals.(e.g. 4.25 × 100 = 0.06 10 = )

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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)

Instructions:

Familiarise yourself with the rubric and these guidelines before the lesson. Distribute the worksheet to students. This problem sheet is divided into sections that

match the progression of the phases of the rubric.* Ask students to independently complete as many questions on the worksheet as they

can. Once they reach a section that they find too challenging, ask them to hand their worksheet back to you.

Ask students to write the applicable number sentence, show their working out, on the worksheet and also write their thinking in words if they can. This is so you can assess the strategies that have been used to solve the problems.

After the task, map the students’ work against the rubric.

Resources:

A copy of the revised worksheet.*

Evidence collected from this taskThe completed student ‘problem sheets’ were collected by the teacher, noting that students were asked to stop when they found the problems too challenging.

*The problem sheet for this task was revised after implementation. The annotated student work samples contain the original worksheet, and the revised worksheet can be found in Appendix 1.

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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 an open-plan inner-city Melbourne Primary school with a high socioeconomic cohort.

The task was presented to three whole classes – two Years 3 and 4 composite classes and one Years 5 and 6 composite class – in the middle of units of work on multiplication and division.

For all classes, a number fluency game related to multiplication and division was conducted and then students were presented with the worksheet. As a class, they discussed the instructions and expectations before completing the worksheet independently. It was made explicit that students should tackle only what they were comfortable with.

Students were given 1 × 50-minute session to complete as much as they could.

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Sample 1

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Sample 1: Evidence of student learning

Annotations 1.1 – Demonstrated they can skip count by 3s and 6s. 2.1 – Used their knowledge of 6s to break up the equation in Question 5b. They knew

they then had to add 6 two more times to find the answer. 3.2 – Demonstrated they use known facts to answer all questions. 4.1 – Evident in all sections completed by the student.

What is the student ready to learn next?While the student showed that they feel confident in solving and attempting multiplication problems and has been assessed as meeting all the criteria under Phase 2, they still require scaffolding to complete division so they have not been assessed as ready to learn Phase 3 next.

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4.1 Solves one-digit number sentence with no remainder

3.2 Demonstrates one strategy used to solve a problem

3.2 Demonstrates one strategy used to solve a problem

2.1 Matches an operation to a worded problem by representing the words as a number sentence

1.1 Skip counts by 3s and 6s



Sample 2

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3.3 Selects a strategy that is efficient for the student to solve a problem

1.1 Skip counts by 3s, 4s, 6s, 7s, 8s and 9s.



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4.2 Solves a multiplication and division number sentence using larger numbers with no remainder

2.2 Represents multi-step word problems that involve multiplication or division




Annotations 1.1 – Demonstrated in Question 2a and Question 7a, where the student split the two-

digit number and skip counted (repeated addition) to solve. 2.2 – Demonstrated in Question 7a: student correctly found full price of toy before using

repeated addition to solve. 3.3 Demonstrated in Question 6a – student solved a 4 × 45 problem by doubling twice. 4.2 Demonstrated in Question 7a – student solved a two-digit × two-digit problem using

doubling strategy.

What is the student ready to learn next?I assessed the student as ready to learn Phase 4 next. They have developed efficient strategies but need to move forward to multiplicative thinking to develop more efficient strategies. They can use split strategy to break down larger numbers, but they are using repeated addition to solve. Their strategies with larger numbers are not yet efficient.

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Sample 3

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Annotations 2.1 – Evidence is shown in Questions 1a and 7a. 3.4 – Evidence is shown in Questions 3a and 7b. 4.3 – Evidence is demonstrated in the answer to Question 7. The student shows the

multiple greater than the required amount.

Indirect evidence

1.1 – The student is skip counting by 12s, and it assumed they can skip count by lower numbers.

What is the student ready to learn next?The student has achieved most but not all of the quality criteria on the rubric for Phase 4, so the student is assessed as ready to learn Phase 4 next.

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4.3 Solves a multiplication and division number sentence using larger numbers with a remainder

2.1 Matches an operation to a worded problem by representing the words as a number sentence

3.4 Justifies the reasonableness of an answer by using a strategy or strategies



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4.4 Applies knowledge of powers of 10 to multiply and divide decimals



2.3 Represents problems using the order of operations

1.2 Continues skip counting and creates simple sequences using decimals




Annotations 1.2 – Evidence is shown through understanding and correctly answering Questions 8a

and 8b. 2.3 – Evidence is shown in the working out of Question 9a. 3.4 – Evidence is shown in answers to Questions 2, 3 and 4, as well as Questions 7a,

7b and 9a. 4.4 – Evidence is demonstrated in answers and arrows in answering Questions 4 and 8.

What is the student ready to learn next?The student has achieved all the quality criteria on the rubric. The next phase of their learning could be identified as beginning to work towards Level 7 in the Mathematics curriculum.

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Using evidence to plan for future teaching and learningThe data gathered when implementing the task illustrated:

while many students showed competence with multiplication, their skill/confidence with division was much less visible

the need to clearly differentiate this topic for groups of students the range of abilities and how that informs what each student is ready to learn next the need for extension tasks/options for the rubric beyond Phase 5.

One teacher gave copies of the rubric to her students so they could co-assess, discuss their next steps and write goals from their data.

The information received showed clearly that most students needed further exposure to the concept of fact families – the relationship between multiplication and division. Teachers needed to really emphasise the inverse nature of the two operations and why they are taught together. This could be achieved through repeated practice and rich tasks.

Teacher reflectionsWith the implementation of a new scope and sequence this year, our school is teaching Maths using a much more spaced practice approach, so I can see myself using it again before the end of the year, especially as this will provide some important ‘post’ data to show growth. Using our data and collaborative feedback from the first use of the task, we have revised both the task and the rubric so that they complement each other better.

While my current Years 5 and 6 cohort showed they were in general quite competent with the task, we have extended it slightly, and clarified a couple of components, so I will definitely use it again with other cohorts in the future. Next time, I would provide an ‘I can’ version of the rubric to my students prior to completing the task so they can use it as self-assessment and see clearly what their next steps are.

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Appendix 1 – Worksheet (revised)Section 1

1a. 6 × 7 =

63 ÷ 9 =

2a. 33 × 6 =

136 ÷ 8 =

3a. 32 × 15=

1374 ÷ 12 =

4a. 3.67 × 100 =

0.4 ÷ 10 =

Section 2

2a. The Richmond Tigers kicked 45 points in each quarter. What was their score for the whole game?

Number Sentence:

Working Out:

2b. The Grade 3/4 students are attending school camp. There are 120 kids going on the camp. If the cabins at camp sleep 6 kids each, how many cabins do they need?

Number Sentence:

Working Out:

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Section 3

3a. Baby Shark toys are on sale for ½ price and are only $26. If Julian bought twelve baby sharks at full price, how much would he pay?

Number Sentence:

Working Out:

3b. Easter eggs are sold in packs of one dozen. In the Scout group, there are 187 kids, if everyone in the group gets one egg, how many packets does the Scout leader need to buy and how many eggs will be left over?

Number Sentence:

Working Out:

Section 4

4a. The cafe at the Tennis sells bowls of strawberries and cream for $4.25 each. If they sell 100 bowls on Tuesday, how much money will they make?

Number Sentence:

Working Out:

4b. Natasha measured the total thickness of 100 books and found that it was 673mm. How thick is each book?

Number Sentence:

Working Out:

Section 5

9a. Robert bought 2 burgers for $3.50 each and 3 medium French fries for $1.20 each. How much did he spend?

Number Sentence:

Working Out:

9b. Sammy is buying decorations for her Halloween party. She buys 5 packets of black bats and each packet has 10 bats. Sammy also buys 6 packets of spiders that have 4 spiders in each packet. However, one of the spiders is missing a leg so she decides not to use it. How many spiders and bats does Sammy use to decorate the Halloween party?

Number Sentence:

Working Out:

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