1. A webinar facilitated by Angela Jones and Anne Lawrence Mathematics and the NCEA realignment

2. Introductions Background to the changes The new achievement standards matrix in mathematics and statistics SOLO taxonomy and Achievement standards Next steps Introductions Background to the changes The new achievement standards matrix in mathematics and statistics SOLO taxonomy and Achievement standards Next steps Mathematics and the NCEA realignment

3. Mathematics and NCEA realignment Angela Jones Senior adviser Secondary Outcomes Team Ministry of Education angela.jones@minedu.govt.nz Anne Lawrence Adviser in Numeracy, Mathematics & Statistics Massey University College of Education a.lawrence@massey.ac.nz Angela Jones Senior adviser Secondary Outcomes Team Ministry of Education angela.jones@minedu.govt.nz Anne Lawrence Adviser in Numeracy, Mathematics & Statistics Massey University College of Education a.lawrence@massey.ac.nz Introductions

4. Mathematics and NCEA realignment Angela Jones Senior adviser Secondary Outcomes Team Ministry of Education angela.jones@minedu.govt.nz Anne Lawrence Adviser in Numeracy, Mathematics & Statistics Massey University College of Education a.lawrence@massey.ac.nz Angela Jones Senior adviser Secondary Outcomes Team Ministry of Education angela.jones@minedu.govt.nz Anne Lawrence Adviser in Numeracy, Mathematics & Statistics Massey University College of Education a.lawrence@massey.ac.nz Introductions

5. Aligned curriculum and assessment The New Zealand Curriculum published in 2007 clearly states that students are to be engaged in thinking mathematically and statistically, and to be modelling situations and solving problems at all curriculum levels. The New Zealand Curriculum is asking us to inspire young people to become successful learners, confident individuals and responsible citizens A mathematics curriculum should focus on mathematics content and processes that are important and worth the time and attention of students. Alignment and coherence of the three elements – curriculum, standards, and assessment – are critically important foundations of mathematics education.

6. Aligned curriculum and assessment (cont) Achievement standards are aligned to The New Zealand Curriculum. Unit Standards with curriculum links have been replaced by achievement standards. Assessment tasks are written to reflect good assessment practices and the type and level of thinking required by the curriculum and the achievement standards.

7. The SOLO taxonomy SOLO stands for the Structure of Observed Learning Outcomes. It was developed by Biggs and Collis (1982). Biggs describes SOLO as “a framework for understanding”. (1999, p.37) SOLO identifies five stages of thinking. Each stage embraces the previous level but adds something more in terms of the level of thinking.

8. The stages of SOLO Prestructural the student acquires bits of unconnected information that have no organisation and make no sense. Unistructural and Multistructural students make connections between pieces of information but not the meta-connections between them Relational the students sees the significance of how various pieces of information relate to one another Extended abstract students make connections beyond the scope of the problem or question, to generalise or transfer learning into a new situation

9. Surface and deep thinking Unistructural and multistructural questions test students’ surface thinking (lower-order thinking skills) Relational and extended abstract questions test deep thinking (higher-order thinking skills) Use of SOLO allows us to balance the cognitive demand of the questions we ask and to scaffold students into deeper thinking and metacognition

10. Multistructural questions Students need to know or use more than one piece of given information, fact, or idea, to answer the question, but do not integrate the ideas. This is fundamentally an unsorted, unorganised list.

11. Relational questions These questions require students to integrate more than one piece of given knowledge, information, fact, or idea. At least two separate ideas are required that, working together, will solve the problem.

12. Extended abstract questions These questions involve a higher level of abstraction. The items require the student to go beyond the given information, knowledge, information, or ideas and to deduce a more general rule or proof that applies to all cases.

14. A change of focus in mathematics The SOLO hierarchy shifts us from the situation in maths of “doing more” and “ doing longer tasks” to qualify for a higher grade. This change recognises and rewards deeper and more complex thinking along with more effective communication of mathematical ideas and outcomes. It is these that are fundamental competencies to mathematics..

15. 1.How many sticks are needed for 3 houses? (unistructural) 2.How many sticks are there for 5 houses? (multistructural) 3.If 52 houses require 209 sticks, how many sticks do you need to be able to make 53 houses? (relational) 4.Make up a rule to count how many sticks are needed for any number of houses. (extended abstract) Houses123 Sticks59__ An example of the levels in algebra

16. Multistructural This cylinder is part of a model for a rocket. The radius is 3cm and the height is 5cm. Neil would like to know the volume of the cylinder

17. Relational This solid is a model of a rocket. The cylinder has a radius of 3cm. The height of the cylinder is 5cm and the height of the cone is 4cm. James would like to know the ratio of the volume of the cylinder to the volume of the cone

18. Extended abstract This solid is a model of a rocket and consists of a cone placed on top of a cylinder. The total height of the solid is 8cm. Investigate the possible ratios of volume of cylinder : volume of cone

19. Achievement standard 1.4 Apply linear algebra in solving problems Apply linear algebra, involving relational thinking, in solving problems Apply linear algebra, involving extended abstract thinking, solving problems Achieve Merit Excellence Achievement standard 1.4

20. What does the SOLO taxonomy suggest that this might look like for achieve, merit and excellence? AS1.4: Applying linear algebra in solving problems Excellence MeritAchieve

21. Explanatory notes: 1. The following achievement objectives taken from the Equations and Expressions, and Patterns and Relationships threads of the Mathematics and Statistics learning area, are related to this standard: form and solve linear equations solve linear equations and inequations and simultaneous equations in two unknowns relate graphs, tables, and equations to linear relationships, relate rate of change to the gradient of a graph.

22. Explanatory note 3: Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts. The situation will be set in a real-life context. The phrase ‘ a range of methods ’ indicates that there will be evidence of at least three different methods. Students will be expected to be familiar with methods related to: using formulae forming, graphing or manipulating linear models such as C= 8+0.75 t when solving problems relating rate of change to the gradient of a graph using simultaneous equations, inequations or graphs when solving problems such as those involving simple linear programming.

23. What does the standard say for achieve? Applying linear algebra will involve using a range of methods in solving problems, demonstrating knowledge of algebraic concepts and terms, and communicating solutions which would usually require only one or two steps.

24. What does the standard say for merit? Relational thinking will involve one or more of: selecting and carrying out a logical sequence of steps connecting different concepts and representations demonstrating understanding of concepts forming and using a model, and relating findings to a context, or communicating thinking using appropriate mathematical statements.

25. What does the standard say for excellence? Extended abstract thinking will involve one or more of: demonstrating understanding of abstract concepts developing a chain of logical reasoning, or proof forming a generalisation, and using correct mathematical statements, or communicating mathematical insight.

26. Task 1.4A Taxi charges At Wiaawe airport, there are three different taxi companies available for hire: Fred’s Taxi Company, P and G Taxi Company and Flatrate Taxi Company The Resource page gives the hire charge for each company and the distances to common destinations in Wiaawe from the airport. Represent the three taxi companies' charges using the same representation, for example, three equations or three graphs with the same variables and scale.

27. Fred’s Taxi Company Come with us! Fixed charge: $5.00 Per kilometre: $0.50 P and G Taxi Company Cheap rates! C = 0.65D + 2 where C is the cost in dollars, and D is the distance in kilometres Flatrate Taxi Company Our advantage is clear! $25.00 flat fee to any destination (up to 60 km) Additional charges apply over 60 km. Recommend which taxi company to use for a trip to two of the common destinations. Recommend distances for which would it be cheapest to use P and G Taxi Company. Fred, who owns Fred’s Taxi Company, wants to be the cheapest taxi company people can use to travel to any destination. Write and describe at least two different ways Fred could realistically change his charges to achieve this goal. Include specific examples of the rates he could use. DestinationDistance (km) City Centre17 Port24 Bethlehem45

28. Evidence for achievement A student uses different methods by: using formulae Fred’s cost to city = 0.5 x 17 + 5 = $13.50 forming, graphing or manipulating linear models Fred’s: y = 0.5x + 5 Or Fred’s graph relating rate of change to the gradient of a graph using simultaneous equations, inequations or graphs

29. Evidence for merit A student might demonstrate understanding of concepts by solving Fred and PG equations simultaneously C = 0.5D + 5 and C = 0.65D + 2 to give distance of 20km, where their charges are the same, so PG is cheapest for distances less than 20km Clearly communicated, with evidence of method and use of mode An example of selecting and carrying out a logical sequence of steps: To go to the Port the costs are: Fred = 0.5 x 24 + 5 = $17, PG = 0.65 x 24 + 2 = $17.60, Flat = $25. Use PG because it is the cheapest Clearly communicated, with evidence of method and use of model.

30. Evidence for excellence A student might provide a chain of logical reasoning For all distances over 40km he needs to drop to a flat rate (like Flatrate Taxi Company). The rate needs to be anything less than $25 per journey. e.g. Rate = $24 for any length journey over 40km And for any journey less than 20 km Fred needs to have a charge which is lower than P&G. e.g. His charges could be C = 0.65D + 1.5. His fixed charge is $1.50 and his rate is the same as P&G.

31. Evidence for A, M, E Final grades will be decided using professional judgement based on a holistic examination of the evidence provided against the criteria in the Achievement Standard.

32. Process from here Online forum : How does this work? Homework: Trial the task, Mark student work - keep notes about judgements but leave student work unmarked (raw) Select student work to cover a range of grades Submit raw student scripts

33. Next steps Trial the task Mark Select and submit student scripts Participate in online forum