Reviewing the Standards for Mathematics and Statistics

1. Reviewing the Standards forMathematics and Statistics A progress report June 2009

2. What has happened since last year: • The timeline for the alignment project has been extended • The aligned standards are now all achievement standards • The level 1 matrix has been changed in response to the consultation • All of the mathematics and statistics standards have been drafted in a generic format • Some progress has been made on the fate of the lower level unit standards • Subject association contracts are winding up • Consultation has been organised by the MoE

3. What hasn’t changed: • Still only three externals allowed • Duplication may still be a problem • Calculator issue still unresolved

5. The timeline for the alignment project has been extended • The original timeline had all new standards at all three levels being implemented in 2010, along with the curriculum • The new timeline: • All standards drafted in 2009 • Tasks, trialing, and exemplars of student work in 2009-2010 • Level 1 standards implemented in 2011 • Level 2 standards in 2012 • Level 3 standards in 2013

6. The aligned standards are now all achievement standards • We were originally required to cover the curriculum with 24 credits in AS and the balance in US. • This distinction has now been removed. • So for level 1 we have 13 new achievement standards with a total credit value of 44 credits • These will replace all AS and US linked to L6 of the NZC • Unit Standards will exist – but not related to NZC at L6 or above • Both AS and US may have A, M, or E grades

7. The principles, values and key competencies in the new curriculum are mandated from Years 1-13. The achievement objectives (AOs) in English, the Arts, Health and Physical Education, Mathematics and Statistics, Science, Social Sciences and Technology are mandated until the end of Year 10. The intention of the new curriculum for Years 11, 12 and 13, is to be ‘enabling’ and allow schools maximum flexibility in learning programmes within the mandated framework of the principles, values and competencies. The expectation is that there will be more specialisation in Years 11, 12 and 13 than at previous year levels. Quote from Ministry Guidelines 28th May

8. Example 1 Ext 1.3Investigate relationships using tables, equations or graphs4 Ext 1.6 Apply geometric reasoning4 Ext 1.10 Demonstrate understanding of chance and data4 CAT 1.2 Apply algebraic methods4 Int 1.1 Apply numeric reasoning when solving problems4 Int 1.10Investigate a given multivariate data set using the statistical enquiry cycle4 Total credits 24

9. Example 2 Int 1.1Apply numeric reasoning when solving problems4 Int 1.4Apply linear algebra3 Int 1.5Solve measurement problems3 Int 1.7Solve measurement problems involving right-angled triangles3 Int 1.9Apply transformation geometry3 Int 1.10Investigate a given multivariate data set using the statistical enquiry cycle4 Int 1.13Investigate a situation involving elements of chance2 Total credits 22

10. Some progress has been made on the fate of the lower level unit standards • All standards should reflect the appropriate curriculum level, and there should be no standards based on the NZC in the matrix below the appropriate level 6 of the curriculum. • Significant work is under way, led by NZQA and MoE, to develop mechanisms to address the needs of the students in question which could involve: • Alternative literacy and numeracy standards. These might be developed in the context of work already underway in the primary, intermediate and adult sectors; • Advice to schools to encourage a greater use of the range of qualifications on the NQF; and • Advice to schools to encourage some students to achieve the required level for a qualification over a period of time (for example taking two years to achieve NCEA Level 1). • Quote from Ministry guidelines May 2009

11. The level 1 matrix has been changed in response to the consultation • Clear messages from you: • Did not like the apparent downgrading of • number and geometry standards at level 1 • Did not like the limit to three externals • Wanted calculator restrictions on externals • Mixed messages: • Choice of externals • Credit values

12. All of the standards have been drafted in a generic format Design issues have been addressed • Standards or achievement criteria with multiple requirements can lead to unfair results • Replacement evidence not available because Merit questions or Excellence questions address different outcomes from Achieved questions. • Specific content attached to grade levels encourages some students to only attempt to learn what is required for Achieved • Specific requirements constrain examiners or make assessments too predictable

13. Our guidelines: Only one criterion per grade level (without “ands”) and “Students need to demonstrate a better (higher quality) performance on the same outcome in order to achieve higher grades”

14. Our challenge: To describe the expected levels of thinking when solving problems at Achieved, Merit or Excellence in terms of quality of thinking, not in terms of more content or different content • Our aim: • to describe current practice • NOT to alter the expected level of performance

15. Looking at current practice Trig 2008 Achieved Question 1 Merit Question 5 or 7 Excellence Question 8b Number Standard Algebra Standard?

16. Achieved level performance: • using appropriate methods when solving problems • demonstrating knowledge of number concepts and terms • communicating solutions which would usually require only one or two steps

17. Merit level performance: • carrying out a logical sequence of steps • connecting different concepts and representations • demonstrating understanding of concepts • forming and using a model • relating findings to a context • communicating thinking using appropriate mathematical statements

18. Excellence level performance: • devising a strategy to investigate or solve a problem • demonstrating understanding of abstract concepts • developing a chain of logical reasoning, or proof • forming a generalisation • using correct mathematical statements • communicating mathematical insight

19. Next problem: Find a word or short phrase to include in the criterion for Achieved that summarizes the Merit and Excellence step-up in thinking. Suggestions: Merit demonstrate strategic competence Excellence demonstrate adaptive reasoning Merit relational thinking Excellence extended abstract thinking

20. SOLO Taxonomy provides criteria for assessing the cognitive complexity of students’ understanding when mastering new learning. SOLO is content independent and thus is useful as a generic measure of understanding across different disciplines. Teachers using SOLO criteria can validly and reliably identify ascending cognitive complexity in individual and collective student learning outcomes. For example, the asTTLE assessment tool for teaching and learning used in New Zealand schools is based upon SOLO taxonomy. We use SOLO with teachers and students in both secondary and primary schools to identify the cognitive complexity of a student’s understanding and from this determine their future learning needs. We believe that SOLO can play a pivotal role in teacher and student design of cognitively differentiated learning environments and experiences. From hooked-on-thinking.com

21. SOLO describes five levels of student understanding when encountering new learning. At the prestructural level of understanding, the task is inappropriately attacked, and the student has missed the point. At the unistructural level, one aspect of the task is picked up, and student understanding is disconnected and limited. At the multistructural level, several aspects of the task are known but their relationships to each other and the whole are missed. At the relational level, the aspects are integrated, and contribute to a coherent understanding of the whole. At the extended abstract level, the new understanding at the relational level is re-thought at another level, and used as the basis for prediction, generalisation, reflection, or creation of new understanding.

22. How does all this look in a new standard? Link to measurement standard

23. What about the statistics and probability standards? Merit: with justification With justification involves linking aspects of the statistical enquiry cycle to the context. Justification also involves making supporting statements which refer to evidence such as statistics, data values, trends or features of visual displays. Excellence: with statistical insight With statistical insight means integrating the statistical and contextual information and knowledge to show a deeper understanding. This may involve relevant contextual knowledge, or may involve considering the effects of related variables, or reflecting about the process.

24. Link to Stats standard: With all the statistics standards there is ongoing work lead by Pip Arnold which should result in an excellent Teaching and Learning Guide for the statistics and probability strand of the new curriculum. They are working on producing assessment rubrics which will clarify the qualities we are looking for when awarding Merit or Excellence grades when assessing work related to the PPDAC cycle.

26. Calculator issue still unresolved What is the issue? At the beginning of this project we were asked to write standards suitable for the use of all hand-held technologies, including CAS This was because the official view was that the CAS “pilot” had shown that CAS was good for teaching and learning mathematics, and so CAS would therefore be “implemented” in NZ schools. Also because the NZQA did not want to continue running parallel assessments for CAS users.

27. BIG PROBLEM Initial consultation by NZAMT and the further consultation on the draft matrix in 2008 clearly showed that teachers did not want CAS allowed in exams • Ministry claims: • The consultation questions were not properly explained or put in context • Teachers are conservative, or ignorant about CAS

29. Our response: We have defended the consultation result: We have questioned the claims made for teaching and learning. We have also raised what we believe to be the valid concerns of teachers: cost, value for money, usefulness or otherwise outside school, potential for cheating, equity, difficulty of setting a fair exam (CAS neutral), what happens in other countries, lack of funding for further teacher support or PD in NZ, and workload and pedagogic issues if CAS is forced on us at the same time as the new curriculum and aligned standards!

30. Meanwhile, the REDBACK arrived: Is this a scientific or a graphic calculator?

31. Our current position on calculators: • We are recommending that external exams should be limited to graphic calculators. • As there are only 3 externals, this is similar to the practice in those countries which do allow CAS in exams: the usual situation is 50% CAS and 50% CAS free. • We would be allowing CAS in internals. • The only standards where the use of CAS would be considered an equity issue are those involving algebraic manipulation.

32. Claims made are often factually incorrect: “Victoria allows CAS in exams” Link to VCE Rules 2009

33. The standards review has been initiated to: • improve programme and pathway coherence for learners • improve the design of all standards • align standards with the NZC • enhance support for assessors’ professional practice • improve the credibility of the NCEA and the National • Qualifications Framework (NQF)

34. Your school has been notified about the consultation Consultation Process Research New Zealand have set up a web site for online consultation. The link is: https://surveys.researchnz.com/NCEA/Registration Deadline 3rd July Register to view all materials with your e-mail address. You will receive a “password”. You can re-enter the site as many times as you need to familiarize yourself with the materials, then fill in the response form on-line.

35. The consultation: • Your feedback is valued • Your ideas are useful • Have we achieved the objectives?