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
Item | 5 | 3 | 6 | 7 | 17 | 14 | 26 | 34 | 19 | 31 | 41 | 11 | 30 | 37 | 10 | 36 | 38 | 1 | 8 | 35 |
Luna's response | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
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.