# Examples of interpreting student data

The following sections guide interpreting assessment data for students in different age ranges.

• Year 2 student Jordan has completed the Mathematics Online Interview and a KeyMaths assessment.
• Year 6 student Toby has completed PAT Maths 4th Edition and a KeyMaths assessment.
• Year 8 student Rahul has completed PAT Maths 4th Edition, On-Demand Mathematics and a KeyMaths assessment.

The outcomes for each, along with observations of classroom learning, have been examined using the information and questions in Deciding if a student has a learning difficulty in numeracy.

## How do a student's current numeracy knowledge and skills compare to a typical student at this level?

The Victorian Curriculum F–10: Mathematics describes how this is acquired developmentally and specifies what is expected at each year level. It doesn't specify the student's assumed level of learning based on age or years of schooling.

The goal of this step is to identify what level of maths knowledge and skills a student with a numeracy difficulty has acquired. This will also help to demonstrate the extent of the difficulty.

### Jordan – Year 2

#### Maths Online Interview

To examine this question for Jordan, combine the class report with Jordan's outcomes. Table 1 illustrates Jordan's outcomes, as measured by the Maths Online Interview's (MOI) growth points, and the proportion of their class who achieved that growth point in each section of the interview.

The letter 'A' indicates Jordan achieved the growth point. The letter 'U' indicates where Jordan did not achieve the growth point (unidentified).

Table 1. MOI outcomes comparison

 Jordan's outcome for each growth point Percentage of the class who achieved the same growth point 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Early counting A A A U U U U 100 100 100 96 91 83 77 Place value A A U U U U 100 100 91 77 77 65 Addition and subtraction A A A U U U U 100 100 100 91 91 83 77 Multiplication and division A A U U U U U U 100 100 91 87 87 77 73 69 Time A A A U U U 100 100 96 91 91 83 Length A A U U U U 100 100 96 91 91 83 Mass A A U U U U 100 100 96 91 91 83 Shape A A A U U 100 100 100 96 83 Visualisation and orientation A A U U U 100 100 96 91 83

Data indicate that the lowest developmental growth points not achieved by Jordan were achieved by greater than 90 per cent of their cohort. Therefore, Jordan's numeracy ability across all sections was in the tenth percentile range for their year level.

Jordan's performance on each section of the MOI can be mapped onto the Victorian F–10 Curriculum: Mathematics using its achievement standards. The relevant growth points achieved by Jordan pair with the following topics and their content descriptions.

• Early counting – VCMNA069
• Place value – VCMNA070
• Addition and subtraction – VCMNA089
• Multiplication and division – VCMNA074
• Time – VCMMG080
• Length – VCMMG078
• Mass – VCMMG078
• Shape – VCMMG098
• Visualisation and orientation – VCMMG082.

The outcomes of the MOI locate Jordan's overall numeracy ability between Foundation and Level 1.

For the growth points that Jordan did not achieve, there were some they could partially engage with.

For example, although Jordan did not achieve growth point two for 'Place value', they could count to and from 100. While they could not read or write two-digit numbers or locate these numbers on a number line, they did show an emerging ability to partition numbers using place values.

#### KeyMaths

Jordan's outcomes in the KeyMaths general areas of Basic Concepts, Operations, and Applications were 75, 79 and 77 respectively, placing Jordan well below the standard score average (100) and in the lower percentile rank for their age.

Jordan's scores in each sub-test provide a more detailed description of their ability. The average for each scale score is 10, with a standard deviation of three points. Jordan's scaled scores and percentile ranks for each of the areas were:

#### Basic Concepts(scaled score and percentile)

• Numeration – five, fifth
• Algebra – six, ninth
• Geometry – seven, fifteenth
• Measurement – six, ninth
• Data Analysis – six, ninth.

#### Operations (computational skills)

• Mental Computation – six, ninth
• Multiplication/Division – –, –.

#### Applications

• Foundations of Problem Solving – six, ninth
• Applied Problem Solving – six, ninth.

The two assessments and these data provide matching, consistent descriptions of Jordan's numeracy ability when compared to typical students at this level.

### Toby – Year 6

#### PAT Maths 4th Edition

Toby completed Test 5 of PAT Maths 4th Edition. It contains 35 multiple-choice tasks and is intended for the end of Year 5 and the start of Year 6. Toby's raw score was eight, which matched a standard patm score of 109.2 and a seventh percentile rank for Year 6.

The mean patm score for a matched cohort on Test 5 is 122.7, with a standard deviation of 7.4. Toby's score is approximately two standard deviations below the mean for the cohort.

The subsections of the PAT Maths Test 5 are organised into five strands: Number, Geometry, Measurement, Statistics, and Probability. The cohort average patm score of each task is 130.2, 125.9, 122.0, 118.6, and 116.8 respectively.

Toby was able to correctly answer the first two tasks for each strand, except Measurement. These represent the least sophisticated items for each topic. Toby was unable to correctly answer any items in the Measurement strand.

Toby's average patm score for Number was 119.5 (fourth percentile), Geometry 119.3 (sixth percentile), Measurement 82.7 (first percentile), Statistics 114.3 (fourteenth percentile), and Probability 110.2 (tenth percentile). Toby's scores are reasonably consistent across the five strands and indicate that they are well below their peers in maths.

#### KeyMaths

Toby's outcomes in the KeyMaths general areas Basic Concepts, Operations, and Applications, were 61, 60, and 59 respectively, placing them significantly below the standard score average (100) and in the first percentile rank for their age.

Toby's scores in each sub-test provide a more detailed description of their ability. The average for each scale score is 10, with a standard deviation of three points. Toby's scaled scores and percentile ranks for each of the areas were:

#### Basic Concepts (scaled score and percentile)

• Numeration – seven, fifteenth
• Algebra – four, second
• Geometry – nine, thirty-seventh
• Measurement – seven, fifteenth
• Data Analysis – nine, thirty-seventh.

#### Operations (computational skills)

• Mental Computation – five, fifth
• Multiplication/Division – five, fifth.

#### Applications

• Foundations of Problem Solving – five, fifth
• Applied Problem Solving – five, fifth.

The two assessments and these data provide matching, consistent descriptions of Toby's numeracy ability when compared to typical students at this level.

### Rahul – Year 8

#### PAT Maths 4th Edition

Rahul completed Test 7 of PAT Maths 4th Edition. It contains 40 multiple-choice tasks and is intended for the end of Year 7 and the start of Year 8. Rahul's raw score was 10, which matched a standard patm score of 117 and an eighth percentile rank for Year 8.

The mean patm score for a matched cohort on Test 7 is 129.5, with a standard deviation of 8.2. Rahul's score is approximately 1.5 standard deviations below the mean for the cohort.

The subsections of the PAT Maths Test 5 are organised into five strands: Number, Geometry, Measurement, Statistics, and Probability. PAT Maths provides the patm value of each task. Rahul was able to correctly answer:

• Geometry – items 14, 15 and 16
• Measurement – items 21 and 22
• Statistics – item 27
• Probability – items 32, 33, 34
• Number – item 39.

Except for item 39 in Number, these represent the least sophisticated items for each topic, requiring foundational mathematical knowledge, understanding and skills to answer. Rahul's performance was extremely consistent with the items answered correctly, placing their patm score between 112 and 124 for each strand. Rahul incorrectly answered item 37, which assesses a student's ability to add three decimal numbers.

#### KeyMaths

Rahul's outcomes on each general area of the KeyMaths – Basic Concepts, Operations, and Applications – were 56, 60 and 56 respectively, placing them significantly below the standard score average (100), in the first percentile rank for their age.

Rahul's scores in each sub-test provide a more detailed description of their ability. The average for each scale score is 10, with a standard deviation of three points.

Rahul's scaled scores and percentile ranks for each of the areas were:

#### Basic Concepts (scaled score and percentile)

• Numeration – six, ninth
• Algebra – three, first
• Geometry – eight, twenty-fifth
• Measurement – six, ninth
• Data Analysis – seven, sixteenth.

#### Operations (computational skills)

• Mental Computation – six, ninth
• Multiplication/Division – five, fifth.

#### Applications

• Foundations of Problem Solving – five, fifth
• Applied Problem Solving – five, fifth.

The two assessments and these data provide matching, consistent descriptions of Rahul's numeracy ability when compared to typical students at this level.

## Explore if their outcomes are indicative of a learning difficulty or disability

The presence and extent of a student's numeracy difficulty are demonstrated by:

• the mathematical knowledge a student has, the skills they can use independently and how these compare to those of a typical student
• their attitudes toward numeracy and maths
• their self-efficacy as a learner and user of numeracy.

### Jordan – Year 2

Jordan's MOI and KeyMaths data indicate that they are well behind peers at their year level. Outcomes on both these assessments place Jordan in the tenth or lower percentage for their age. These results, when demonstrated consistently over a significant period, indicate a high likelihood of a learning difficulty.

Similarly, Jordan's engagement with and attitudes toward numeracy learning are not positive. For example, they do not enjoy maths and will often 'zone out' or talk to friends during class. Jordan is reluctant to begin numeracy activities and needs to be continually encouraged to engage and complete tasks. Jordan experiences frustration and is reluctant to work independently.

Jordan does not hold a high opinion of their numeracy abilities and potential. While they do not explicitly use negative language, they will often say things like 'I don't get it' or 'I don't want to' for example, when the class is playing a numeracy-based game.

When taken together these data indicate the presence of a numeracy learning difficulty, and potentially a learning disability, such as dyscalculia.

While Jordan demonstrates some characteristics that might be indicative of a learning disability, their MOI data suggests that the cause may be due to a general developmental delay or difficulty. This interpretation is substantiated by the fact that Jordan's difficulty was not limited purely to numeracy tasks, they also showed similar developmental levels during spatial and measurement tasks.

### Toby – Year 6

Toby's PAT Maths and KeyMaths data indicate that they are well behind peers at their year level. Outcomes on both these assessments place Toby consistently in the lowest percentiles when compared to their cohort.

Toby's patm scores put them more than two standard deviations below their peers. Results in their KeyMaths assessment place Toby in the first percentile for their age. The KeyMaths tests show that Toby's geometry and data analysis (statistics) comprehension is higher than their number comprehension. The KeyMaths has more tasks in each area than the PAT Maths and enables a more differentiated description of Toby's maths and numeracy outcomes. These results, when taken over an extended period, indicate a learning difficulty and, possibly, the presence of a learning disability such as dyscalculia.

Toby's performance in the PAT Maths assessment indicated a difficulty using place value concepts, enumerating fractional and decimal quantities, counting in decimals and using counting in measurement tasks. Consequently, Toby has difficulty applying addition, subtraction, multiplication and division algorithms with two and three-digit numbers.

Toby completed most tasks in the KeyMaths assessment slowly, suggesting they were not able to recall relevant numeracy knowledge fluently.

Toby's outcomes on the KeyMaths sub-tests were in the lowest 10 percentile range for tasks that required the application or use of maths comprehension and skill.

Toby's teacher has observed that Toby uses less sophisticated strategies for counting and other activities (for example, counting on their fingers) when their peers have moved on to more advanced strategies. They have also observed that Toby has difficulty managing time, whether allowing enough time to complete certain activities in class or having a sense of the passage of time (for example, Toby often asks the teacher on yard duty how long it is until lunchtime ends). These signs, along with the data collected, indicate the likely presence of a learning disability.

For more information about identifying, assessing, planning and implementing evidence-based teaching strategies to support students with numeracy learning disabilities, enrol in the departments online professional learning course Supporting students with learning difficulties, including dyscalculia.

#### Error analysis

Further information about the cause(s) of a student's numeracy difficulty can be gained by analysing the types of errors a student makes when completing tasks or learning new concepts. For example, two errors that Toby made are

356 –

147 =

211.

In this example, Toby selected an inappropriate algorithm for subtraction. When asked to describe their thinking, Toby replied 'Six from seven is one, five from four is one and three from one is two.

74 x

23 =

1,412.

In this example, Toby has also selected an inappropriate algorithm for multiplication. Additionally, they do not recognise the place value features of the task.

Both error patterns are associated with difficulty using place values to partition, rearrange and regroup numbers to at least 10,000 to assist calculations and solve problems (VCMNA131)

Toby can do the subtraction and multiplication components of each task. They have difficulty understanding the relationship between a written number, for example, 356, and grouped quantities such as three bundles of 100, five bundles of 10 and six loose sticks.

Analysing errors can help you to identify the cause of a student's numeracy learning difficulty. Toby has difficulty applying place value understanding to numbers. In the examples, Toby has treated each digit in each number as a separate entity. Error analysis can also help you see which ideas a student has in place and what you need to teach next. In this case, Toby needs to be taught to read each number by noting its place value properties.

For more information, visit Deciding if a student has a learning difficulty in numeracy.

### Rahul – Year 8

PAT Maths and KeyMaths data indicate that Rahul is well behind his peers at their year level. Outcomes on both assessments place Rahul consistently in the lowest percentiles when compared to their cohort.

The items that Rahul was able to answer during the PAT Maths test for Geometry, Measurement, Statistics, and Probability either involved the manipulation of numbers less than 10 or did not require the use of numerosity about understanding the other strands. This suggests that Rahul is confident solving single-digit problems but is not equipped with the knowledge of, or understanding to use, more complex numerosity skills in these strands.

The item that Rahul answered correctly in the Number strand involved the addition of two negative numbers, each less than 10. Each item on the PAT Maths assessment is linked with a standard from the Victorian F–10 Curriculum: Mathematics. The items that Rahul answered correctly, and their matching content descriptors, are listed below.

• Measurement – item 21 (VCMMG141) at Level 3 and 22 (VCMMG259) at Level 7.
• Statistics – item 27 (VCMSP235) at Level 6.
• Probability – items 32 (VCMSP267) and 33 (VCMSP266) at Level 7, item 34 (VCMSP177) at Level 4.
• Number – item 39 (VCMNA241) at Level 7.

Levels 5 to 7 require a knowledge of numerosity that covers whole numbers generally, proportional reasoning, directed numbers and algebra. PAT Maths items only provide a partial assessment of each standard.

Rahul completed most tasks in the KeyMaths assessment slowly, suggesting they were not able to recall relevant numeracy knowledge fluently. Rahul's difficulties with enumeration likely contributed to their mental computation, addition/subtraction and multiplication/division outcomes and their application of maths in problem-solving. Rahul's performance on the KeyMaths sub-tests showed a consistent pattern; their outcomes were in the lowest 10 percentile range for tasks that required the application or use of maths comprehension and skill.

#### Dynamic assessment

Dynamic assessment is a procedure for investigating a narrow maths learning domain by constructing a mini-learning progression.

The steps required to solve a problem are specified to the student, and the teacher and student collaborate to solve the problem. The student may not be able to solve the entire problem independently, but with help from the teacher, they can complete the steps one by one to solve the problem. The teacher might use prompts such as, 'Okay, what do we do now?' or 'What do you think we need to do next?', handing the student responsibility to then solve like problems.

During this exercise, the teacher can also gauge the student's learning needs, specifically, the nature and amount of help needed to solve problems independently. Dynamic assessments can help students to internalise problem-solving concepts and procedures.

The following example is a dynamic assessment that might be used to support Rahul.

Rahul's teacher could use vocabulary for symbols to help Rahul to solve this problem a² + b² = c². The teacher might ask prompting questions such as, 'What do we call these symbols?' (for example, plus, add, minus), and 'What's another word for them?

When prompted, Rahul remembered 'add' and knew 'minus' but not 'takeaway'. They were unfamiliar with 'addition' and 'subtraction'

Further prompting, such as 'When the two is small and above a number or letter, what do we call it?', revealed Rahul knew 'square' and 'squared' and that they meant for something to be multiplied by itself. Rahul was unfamiliar with the term 'to the power of' and still seemed a little unsure after it was explained.

To further support Rahul to solve the problem, the teacher could:

• Ask Rahul to describe the concept in their own words
• Explain the task and what is required
• Make connections between the current topic and prior learning
• Break the task down into smaller steps
• Test numeracy idea(s) with friendlier numbers
• Represent information visually
• Interpret information by manipulating concrete materials (such as MAB blocks).

While it is clear from these data that Rahul requires significant support in their numeracy knowledge, understanding and skills, it is also important that the teacher's observations and the possibility of a literacy learning difficulty are also considered.

Rahul's family emigrated from Sri Lanka in the middle of last year. Rahul attends a supplementary English as an Additional Language class once a week. Rahul's teacher has noticed that they have difficulty with long-form maths questions but is reluctant to seek help. When questioned further, Rahul admitted that this is because at times they do not completely understand the language of the question or what it is asking for. This may indicate literacy learning difficulties.

## Demonstrating this difficulty consistently over time

For a description of either a learning difficulty or learning disability to be accurate, a student must be monitored and demonstrate evidence of significant academic challenge for an extended period. It's also crucial that the cohort they are compared to is typical for that level.

Monitoring a student's progress through regular observation will help keep their numeracy profile current and provide you with information about how they learn and respond to different approaches. Aim to monitor students for a short period (for example, one month) before trialling adjustments.