Mathematics Assessment

Resources to help you make a consistent and balanced assessment of achievement.

Mathematics Online Interview

Use this tool in a one-on-one interview situation to determine students’ existing mathematical knowledge.

The testing is in relation to growth points, which can be described as key "stepping stones" along paths to mathematical understanding. Refer to the growth points while conducting the interview.

Access the Mathematics Online Interview
Via the Insight Assessment Platform

User guide, tip sheet and support

Schools should keep CASES21 teacher, class and student data up to date for Foundation level students, to ensure the data is accurate on the insight platform, allowing for efficient delivery of MOI assessments.

For further support, please either:

  1. Log a request via the service gateway
  2. Send an email to servicedesk@edumail.vic.gov.au
  3. Phone the service desk on 1800 641 943

Mapping the interview to the Victorian Curriculum

This document links tasks from the interview to the Early Numeracy Research Project (ENRP) growth points, the achievement standard, content description and levels foundation to 5 of the Victorian Curriculum F-10: mathematics. Please note the following document is currently under review and will be updated soon.

 Mapping the mathematics online interview to the Victorian Curriculum F - 10: mathematics (docx - 998.01kb)

 Mapping the mathematics online interview to the Victorian Curriculum F - 10: mathematics (pdf - 1.23mb)

2018 MOI updates

Since launching the interview on the insight platform in 2017, a number of enhancements were identified to improve the usability and functionality of the online interview. The updates include:

  • Edits to improve clarity of instructions.
  • Reduction of visual clutter including green highlighting, sad and smiley faces, and unnecessary free text boxes.
  • Fixed branching issues based on advice from educational experts, for example, the loop in section B back to question 9 has been addressed.
  • Growth point 0 will now be awarded to a student for a section when they answer at least one question (correct or incorrect) in that section, but not enough to achieve a growth point greater than 0.
  • A foundation detour report is now available.
  • Strategies are now required for some questions, a message alerting users to 'please select a strategy' will appear for questions requiring a strategy to calculate a growth point if the required field is left blank.

Fractions and Decimals Online Interview

Use this tool to assess the mathematical understandings and strategies of fractions, decimals, ratio and percentage.

Access the Fractions and Decimals Online Interview
Via the Insight Assessment Platform

It's intended for students in years 5 to 8 but is also valuable for assessing high achieving students in year 4, or at-risk students in year 10.

The interview provides collated data that can be compared within schools to understand student achievement and monitor progress.

User guides

A user guide, script and equipment checklist is being developed and will be available here when it's complete.

Mapping the Interview to the Victorian Curriculum

This document links each task from the interview to the overarching big idea to which it relates.

The task and the big idea are then related to the content description from the Victorian Curriculum F-10: Mathematics which provides the best match. Key aspects of the proficiencies to the interview task and related overarching big idea are also included.

Mapping the fractions and decimals online interview to the Victorian Curriculum F-10 (docx - 887.58kb)

Big ideas linked to the interview

This is a list of knowledge, skills and behaviours which students who have connected understandings of fraction ideas may possess. These capabilities have been divided these into overall ideas and those that relate to particular constructs of rational number.

As fractions is a major focus with the Interview, there is some occasions where only fractions is referred, while in other cases, the broader term rational number is used (to include fractions, decimals, percentages, etc.).

The numbers in the parentheses refers to the corresponding questions in the interview.

Overall ideas
  • Articulates rational number thinking using appropriate language [1-20]
  • Forms and manipulates a variety of physical and mental models (areas and regions, sets, number lines, ratio tables, etc), in continuous and discrete situations [1-20]
  • Understands the subconstructs of rational number (part/whole [1-4, 6], division [6], measure [8-11], ratio, and operator [4]) as well as their interrelationships
  • Understands that rational numbers are largely about relationships [1-3, 6, 7, 9, 12-15, 20]
  • Thinking multiplicatively rather than additively when appropriate (relative Vs absolute thinking) [17]
Part-whole
  • Understands that fractions are equal shares that are not necessarily congruent and that the subdivision of the whole must be exhaustive [1]
  • Recognises that a given fraction (continuous/discrete) of a may not be the same size as that fraction ofb [19]
  • Moves from the whole to a given part, from the part to the whole and from the part to the part flexibly [2, 3, 6, 7]
  • Understands that if a is a certain fraction of b, we can determine what fraction b is of a, through the reciprocal relationship
Connecting concepts with symbols/equivalences 
  • Understands the meaning attached to each part of a fraction (e.g., the denominator shows what ‘denomination’ is being counted, the numerator ‘enumerates’ how many of these parts) [9, 12, 14, 18]
  • Understands that fractions (including whole numbers, mixed numbers and improper fractions) are entities that can be counted (e.g. 4/5 represents four things called “fifths”) and can recognise and use counting patterns and equivalences [3, 5, 9, 12, 14]
  • Uses appropriate symbols to represent rational numbers (e.g. fractions, decimals and percentages) and can flexibly move between these as appropriate
Fractions as a number
  • Understands and operates with the ‘density’ of rational numbers (meaning that between any two rational numbers there is an infinite number of rational numbers), relating them appropriately to whole numbers [11]
  • Can identify a rational number on a number line with consideration to the calibrations and the intervals specified [10]
Fractions as a division
  • Recognises a/b as a divided by b [6, 16]
  • Can solve whole number division problems understanding the significance of the size of the quotient (e.g. 4 ÷ 5 will result in an answer less than one) and/or treating remainders appropriately [6, 17]
  • Has appropriate strategies in sharing-type problems [6]
Relative size/benchmarking
  • Readily compares and orders rational numbers, using efficient and understood strategies [5, 9-11, 13, 15]
  • Relates a given rational number to key benchmarks (e.g. 0, ½, 1), using place value as appropriate [5, 8-11, 13, 15]
  • Understands the inverse relationship between the denominator and the size of the parts [7, 9]
Operators and operations
  • Combines and partitions rational numbers using appropriate physical or mental tools, renaming as appropriate [4, 6, 10-12, 14, 16, 18]
  • Estimates the answer appropriately in a rational number calculation [8, 16, 17, 19, 20]
  • Can nominate a problem situation to which a particular rational number operation might apply, and conversely can represent a relevant rational number operation given the problem situation [17, 19, 20]

Fraction pairs: Definitions of strategies

This information will assist teachers in developing their understanding in relation to the types of strategies that students use to demonstrate their fraction understanding, for example residual thinking.

Benchmarking

Correct benchmarking is evidence that a student understands the relative size of fractions. It is also useful for comparing decimals.

When benchmarking, a student will compare a fraction to another well known fraction, usually a half, or to a whole number such as zero or one.

For example, when comparing 5/8 and 3/7; 5/8 is greater than a half, and 3/7 is less than a half, therefore 5/8 is bigger.

Residual thinking

The term residual refers to the amount which is required to build up to the whole. For example 5/6 has a residual of 1/6.

This thinking is useful for comparing the size of fractions such as 5/6 and 7/8. 5/6 has a residual of 1/6 and 7/8 has a residual of 1/8. Therefore 7/8 is a larger fraction because it has the smaller residual – the smaller amount to make the whole.

Sometimes, however, residual thinking alone is not an efficient strategy. When comparing 3/7 & 5/8, measuring up the residuals of 4/7 & 3/8 is not a helpful strategy as you are left with two residuals that are no easier to compare than the original pair. In this case, the residuals then need to be benchmarked to 1/2 and 1 to prove which is larger. If students use residual thinking alone with this pair, it should be classified as an unsatisfactory explanation.

Residual thinking with equivalence

In order to use residual thinking effectively, creating an equivalent residual sometimes makes the justification clearer. For example, when comparing 3/4 and 7/9 a student may state that 3/4 has a residual of 1/4 or 2/8. Therefore the residual for 7/9 (2/9) is smaller than the residual for 3/4 (2/8). The fraction with the smaller residual is the larger fraction.

Residual thinking with some other proof

Sometimes residual thinking alone is not the most appropriate strategy. For example if a student uses residual thinking alone to compare 3/4 and 7/9, they must then convince the interviewer that they can justify which of the residuals is bigger (1/4 or 2/9).

An example of residual with proof might be, “I know one quarter of nine is more than 2 because 2 is a quarter of eight, so 2/9 must be less than 1/4 therefore 7/9 is the bigger fraction”.

Please note: an explanation of residual thinking without proof, should be recorded as “other (unsatisfactory explanation for either a correct or incorrect solution)”.

Gap thinking

This strategy is a form of whole number thinking, where the student compares the whole number difference between the numerator and denominator.

For example, 5/6 and 7/8 both have a difference of “one” between the numerator and denominator. A student using “gap thinking” might claim therefore that these fractions are the same size. When comparing 3/4 and 7/9, a student using gap thinking would choose 3/4 as larger because it has a smaller “gap”, thereby choosing incorrectly.

There are some instances where “gap thinking” will lead students to a correct choice. For example, comparing 3/8 & 7/8. This is an inappropriate strategy for comparing the size of fractions.

“Higher” or “larger” numbers

With this strategy, fractions are deemed to be bigger if they contain larger digits. For example, when comparing 4/7 and 4/5 students may incorrectly claim that 4/7 is larger because it has a “larger number”. Also in comparing 2/4 and 4/8, a student would choose 4/8 as it has “higher numbers”.

Sometimes students will directly compare the numerators or denominators and conclude a larger digit at the top or bottom of a fraction means that it is a larger fraction.

This is an inappropriate strategy for comparing the size of fractions.

Other (satisfactory explanation with a correct solution)

There are very few correct solutions with appropriate strategies that do not already fall into the provided categories, but it is possible for this to occur. For example: a student may be able to mentally convert a fraction to a decimal and then compare or use some other mathematically correct strategy.

This option is only for correct solution and appropriate explanation.

Other (unsatisfactory explanation with either a correct or incorrect explanation)

There are many explanations (too numerous to mention) that may fall into this category. It is a “catch-all” for any strategy that cannot be placed in the other categories. Typically it will include any explanation that is mathematically incorrect, partially correct or vague.

Sometimes students relate fractions to an image of an area model. Their justification might be “7/9 is larger than 3/4 because if I imagined a picture of them, 7/9 would look more”. This reasoning is not evidence of understanding the size of the fractions.

In a situation where the student provides a partially correct or vague explanation, it is appropriate to ask for further information in a non-leading way. For example, “can you tell me more about how you know? So you think it seems larger, but how can you be sure?”

Scaffolding Numeracy in the Middle Years

Scaffolding Numeracy in the Middle Years (SNMY) : assessment materials, the Learning and Assessment Framework, learning plans, authentic tasks and the research findings from the project that investigated a new assessment-guided approach to improving student numeracy outcomes in Years 4 to 8. The project identified and refined a Learning and Assessment Framework for the development of multiplicative thinking in the middle years using rich assessment tasks.

Assessment for common misunderstandings

Assessment for common misunderstandings : assessment tools based on a series of highly focussed, research-based Probe Tasks. The Probe Task Manual includes a number of additional tasks and resources which have been organised to address common misunderstandings.

Early Numeracy Research Project

The Early Numeracy Research Project (ENRP) was a three-year (1999-2001) prep to year 2 research project. It involved:

  • 35 trial schools
  • 28 schools representative of government primary schools across the state (including one special school)
  • four CEO schools
  • three AISV schools.

They were involved in developing a comprehensive approach to mathematics that brings together a set of design elements identified by Hill and Crévola in the Early Literacy Research Project.

The aim of the project was to improve mathematics learning and it was necessary to quantify such improvement. A framework of key growth points in numeracy learning were created. Students' movement through these growth points in trial schools could then be compared to that of students in the reference schools. It was intended that the framework would:

  • reflect findings of relevant research in mathematics education
  • emphasise important ideas in early mathematics understanding in a form and language readily understood and, in time, retained by teachers
  • reflect, where possible, the structure of mathematics
  • allow mathematical knowledge and understanding to be described
  • form the basis of planning and teaching
  • provide a basis for task construction for assessment via interview
  • allow the identification and description of improvement in learning
  • enable a consideration of those students who may benefit from additional assistance
  • have sufficient "ceiling" to describe the knowledge and understanding of all children in the first three years of school.

Not all possible mathematical domains are included in this framework. The project focused on these strands:

  • Number (incorporating the domains of Counting, Place Value, Addition and Subtraction, and Multiplication and Division Strategies).
  • Measurement (incorporating the domains of Length, Mass and Time).
  • Space (incorporating the domains of Properties of Shape, and Visualisation and Orientation). 

We describe growth points​ as key "stepping stones" along paths to mathematical understanding. They provide a kind of conceptual landscape. However, we do not claim that all growth points are passed by every student along the way. "The order is more or less the order in which strategies are likely to emerge and be used by children. ... intuitive and incidental learning can influence these strategies in unexpected ways." (ENRP Final Report, p. 39)

Download the Early Numeracy Research Project Summary (pdf - 646.56kb)