Numeracy profile example - Year 6

Luna's overall performance suggests slower development of maths knowledge and skill. Of the six items that assess Level 4 Number and Algebra skills, Luna answered 50 per cent correctly; the items that assess spatial knowledge skills. Luna displayed delayed development in both place value and the use of counting.  

Luna was able to:  

• identify place value in a four-digit number
• multiply a three-digit number with a single-digit number
• use subtraction and addition to solve a multi-step problem
• identify a rule and continue a pattern for an additive sequence
• identify the third number in the pattern that results from subtracting 0.15 from 1.95
• continue a pattern with equivalent fractions
• use subtraction and addition to solve a multi-step problem.

In summary, Luna's place value knowledge extends to four-digit numbers, and they can apply, in some contexts, the whole number addition, subtraction and multiplication algorithms that are taught directly. Luna continued the equivalent fractions pattern by using whole-number multiplication and the decimal numbers counting pattern by applying the subtraction algorithm used for whole numbers.  

Luna was less able to complete tasks that could not be solved simply by applying explicitly taught whole-number algorithms. Luna was not, for example, able to estimate, identify factors, solve multi-step problems that required reasoning, or solve word problems and equations that required division.  

Luna had difficulty applying and using the basic numeracy concepts in other sub-strands.  

Luna answered incorrectly the Level 4 fractions and decimals tasks, 75 per cent of the Level 5 tasks, and all Level 6 items. They had difficulty understanding the concept of a fraction, adding fractions with the same denominator, identifying decimal equivalent to one-fifth, solving fractions word problems, converting between simple fractions and decimals, and comparing decimals.  

Luna correctly answered the measurement and geometry items that required minimal use of computational skills. This included Level 4 tasks in the 'Shape' sub-strand and Level 5 tasks in the 'Location and transformation' sub-strand.  

Luna could:  

• identify a shape that can be divided into two equal parts
• locate and describe position using an alphanumeric grid
• compare the area of irregular and irregular shapes using informal units
• interpret a timetable to calculate elapsed time.

Luna's performance on Level 5 items was generally lower. Luna did not correctly answer tasks that assessed Level 5 'Shape', 'Geometric reasoning', or 'Using units of measurement' items.  

Luna had difficulty identifying types of angles, the net of a 3D object, the correct unit of measurement, calculating elapsed time in hours and minutes, the perimeter of an irregular shape, and the volume of a cube.  

The Guttman Chart Report showed that 80 per cent of the questions Luna answered correctly were in the easier half of items. This is consistent with the interpretation that Luna is generally learning maths knowledge and skill more slowly than their same-age cohort.

Luna answered both Level 5 'Chance' items correctly. They identified the probability of an outcome as a decimal and as a fraction.  

Luna answered none of the Level 5 'Data representation and interpretation' items correctly. They had difficulty interpreting column graphs and dot plots numerically and estimating data in a bar chart.  

Luna's school records show evidence of maths underachievement in earlier years. Their Year 3 NAPLAN numeracy score was 219.6 (raw score = 3) which located them in Band 1. Luna's performance on the Mathematics Online Interview indicated a comparative difficulty counting quantities, reading and writing numbers, assembling a set of items to match a spoken number up to 20, and a written number to 10, manipulating sequences of numbers up to 100, and performing simple addition and subtraction.  

These difficulties may indicate the presence of a learning disability, such as dyscalculia.  

This includes languages spoken at home, development of language, identified language issues, reading practices at home, and communication skills.

Luna's home language is English and there are no school records of developmental language delay. Luna's Year 3 NAPLAN reading and spelling scores were 368.3 (raw score = 14) and 380.1 (raw score = 7) respectively, placing them in Band 3 for reading and Band 4 for spelling.

Luna has displayed no history of visual-perceptual or auditory issues.

Luna socialises easily with peers. Luna sometimes becomes frustrated and anxious during maths and numeracy activities (such as counting games).

Physical issues: physical development and motor coordination

Luna has no known physical development and motor coordination issues.

Intellectual development and skills

This includes general reasoning ability, developmental milestones, conceptual maturity, ability to remember, retain ideas in short-term memory.

There is no indication that Luna has delayed or impaired intellectual development and or skills. Their short-term and general memory appear adequate and on par with most peers.

Medical issues: early medical issues that impacted readiness for numeracy learning

No medical known medical issues.

Interventions

In earlier schooling, Luna has not received additional numeracy support or intervention beyond the differentiated, evidence-based delivery of the curriculum.  

Periods of successful learning

Earlier teachers reported that Luna was more likely to learn maths concepts and skills in limited, partial ways when they had access to concrete aids and when they were required to recall and apply what they had been taught directly.  

Breaks in attendance

Luna had not had any significant breaks in attendance.

Family history of numeracy difficulty or delayed development

Luna's parents have reported no family history of numeracy difficulties or delayed development. 

Self-efficacy

Luna has a low opinion of their maths ability and expects not to be successful in completing maths tasks. When they have difficulty with a task or answer it incorrectly, Luna sees it as confirmation of a lack of ability. Luna frequently becomes frustrated. This low self-efficacy impacts set goals.;

Attitude toward mathematics

Luna does not enjoy maths. They are reluctant to begin numeracy activities and need to be encouraged to engage and complete tasks. Luna is not prepared to use their well-developed general reasoning ability to take risks or to act independently when learning maths and instead seeks to be told the steps to take.

Use of appropriate task-organisational strategies

Luna lacks self-agency in maths learning. Luna has difficulty recalling what they know about a maths task, planning how they will work through it, thinking ahead, maintaining on-task attention and monitoring their progress and learning activity.

Conclusion

Luna's numeracy profile provides a base for planning a program of learning and relevant interventions. Across all numeracy knowledge areas, Luna needs to learn explicitly how to transfer and generalise their foundation numeracy knowledge and skills.  

Luna needs to:  

• extend Level 5 number and place value, and patterns and algebra skills for whole numbers. This includes learning to apply them in contexts that require reasoning, for example, estimating and calculating the properties of numbers such as factors
• improve measurement skills by learning to apply numeracy skills to spatial attributes such as the measurement of the perimeter, area, volume and angle
• develop a concept of angle as the amount of 'openness' between two straight lines and categorise simple angles
• improve data representational skills by learning to apply numeracy knowledge and skill to items in pie charts and bar graphs and to identify the questions answered by data.

Luna's engagement with and attitudes toward numeracy learning are not positive. They experience anxiety and frustration and doubt their numeracy abilities and potential. To build Luna's skills and confidence, Luna's teacher should ask reflective questions during their interactions.

For example:  

• What do you know now that you didn't know before? What can you do now that you couldn't before?
• What did you do to help learn the new ideas?
• How do you think you might use what you have learned in the future?

Luna can also use these questions to self-direct a learning activity.

Luna's confidence can also be improved by asking them reflective questions at the beginning of a teaching session:

• What do you know about these ideas already? What similar ideas have you learned?
• How did you learn the earlier ideas?
• What might you do to help you learn these new ideas?

It's important to help students like Luna see that numeracy is something that we use in all areas of daily life. Luna can be encouraged to talk about examples of numeracy and mathematics that might be used at home, for example, when travelling, shopping, playing, or cooking.

Next steps

The Guttman Chart report for the Linear Mathematics test on the DAL shows the test items in order of difficulty and the relative difficulty of the skills that match each item.

Each item is linked with its 'item intent' and content description in the Victorian Curriculum F–10: Mathematics.  

Table 1 has been created from Luna's Guttman Chart report for the Level 5 test. It shows the order of difficulty of the easiest 21 items on the test and Luna's response to each (one for a correct response, zero for an incorrect response).

Table 1. Luna's Guttman Chart

It also shows where Luna's performance begins to alternate between correct and incorrect responses. This is where to begin explicit teaching. For Luna, you would teach the skills that match items 14, 19, 31, 41, 11 and 30.

Start with the skills that Luna can do and what they know and build on those step by step.

For example, to teach the skills that match:  

• items 14 and 19, you could begin with the understanding Luna used to solve item 3
• items 31 and 30 you could begin with skills Luna used to complete item 26
• item 11 you could draw on the skills Luna used for item 10.

Encourage each step in Luna's knowledge or skill and fluency by providing scaffolding and modelling, such as telling them how to think about the ideas at hand, for example, to visualise a fractional quantity changing, and by providing opportunities for practice and questions. As Luna's competence and confidence improve, support can gradually be removed.  

Consider the following questions when planning for interventions or adjustments to support a student's learning:  

• What does the student need to learn next? If you use the VCAA DAL tests, the Guttman Chart and the Item Response Summary Reports allow you to match the test item outcomes for a student with the Victorian Curriculum F–10: Mathematics and locate the student's existing knowledge and skills. You can then see what to teach.
• How will you sequence what they need to learn? Certain knowledge and skills may need to be developed before the student can demonstrate to others or meet numeracy goals.
• How will you develop their attitudes toward numeracy and themselves as learners and users of numeracy and mathematics?
• What organisational skills will you teach students to develop their capacity to teach themselves?
• When working on maths tasks, students can learn to ask themselves:
• What type of maths task is it? What task is it like that I already know?
• What does the first/second part say? How will I do it?
• Does my answer make sense? How can I tell?
• When needing to learn a maths algorithm or to recognise a pattern, the students can learn to ask themselves:
  
• How are these two examples similar/the same? What do they both have?
• What is the pattern/rule here?
• How does the pattern/rule fit with what I already know? How does it change what I know?
• When might I use it in the future?
• How will you adapt your classroom to be a better environment or culture for learning?
• What will continue to require scaffolding? How will you gradually release responsibility for learning back to the student? For more information visit Helping students to become independent learners.