Judy Anderson The University of Sydney

Key messages … 1. Balance is important 2. Evaluate the types of questions and tasks used during mathematics lessons 3. Assessment, assessment, assessment!!! 4. Alignment between curriculum, teaching and assessment

Mathematics teaching should include opportunities for (Cockcroft, 1982): exposition by the teacher; discussion between teacher and pupils and between pupils themselves; appropriate practical work; consolidation and practice of fundamental skills and routines; problem solving, including the application of mathematics to everyday situations; and investigational work.

UnderstandingStudents build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. FluencyStudents develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily.

Which tasks would support these proficiencies? Examine the types of questions and tasks you use during mathematics lessons.

Gould, 2006 Because three is a larger number than 2 Because four is a larger number than three Because six is a larger number than 3 Because 5 & 6 are larger numbers than 2 & 3 Because 12 & 13 are larger numbers than 9 & 10 ✔ ✖ ✖ ✖ ✔ ✔

Problem solving Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. ReasoningStudents develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying, and generalising.

Which tasks would support these proficiencies? Examine the types of questions and tasks you use during mathematics lessons.

Bloom’s Taxonomy 1. Understand 2. Remember 3. Apply 4. Analyse 5. Evaluate 6. Create Higher order thinking Problem solving Reasoning

Cognitive process What learners need to doAction verbs RememberRetrieve relevant information from long-term memory Recognise, recall, define, describe, identify, list, match, reproduce, select, state UnderstandConstruct meaning from information and concepts Paraphrase, interpret, give egs, classify, summarise, infer, compare, discuss, explain, rewrite ApplyCarry out a procedure or use a technique in a given situation. Change, demonstrate, predict, relate, show how, solve, determine AnalyseSeparate information into parts and determine how the parts relate to one another. Analyse, compare, contrast, organise, distinguish, examine, illustrate, point out, relate, explain, differentiate, organise, attribute EvaluateMake judgements based on criteria and/or standards. Comment on, check, criticise, judge, critique, discriminate, justify, interpret, support CreatePut elements together to form a coherent whole, or recognise elements into a new pattern Combine, design, plan, rearrange, reconstruct, rewrite, generate, produce

Thinkers Bills et al. (2004) Give an example of … (another and another) Open-ended Explain or justify Similarities and differences Always, sometimes or never true Odd-One-Out Generalise Hard and easy

Approaches to teaching problem solving … The approach …The outcome … Teaching for problem solving - knowledge, skills and understanding (the mathematics) Teaching about problem solving - heuristics and behaviours (the strategies and processes) Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet)

Approaches to teaching problem solving … The approach …The outcome … Teaching for problem solving - knowledge, skills and understanding (the mathematics) Problems at the end of the chapter! Teaching about problem solving - heuristics and behaviours (the strategies and processes) Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet)

Approaches to teaching problem solving … The approach …The outcome … Teaching for problem solving - knowledge, skills and understanding Problems at the end of the chapter! Teaching about problem solving - heuristics and behaviours (the strategies and processes) Problems used to ‘practise’ strategies and checklists Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet)

Approaches to teaching problem solving … The approach …The outcome … Teaching for problem solving - knowledge, skills and understanding Problems at the end of the chapter! Teaching about problem solving - heuristics and behaviours (the strategies and processes) Problems used to ‘practise’ strategies and checklists Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet) Some success but limited implementation

Successful problem solving requires Skills and Attributes General reasoning abilities Deep mathematical knowledge Heuristic strategies Personal attributes eg confidence, persistence, organisation Communication skills Helpful beliefs eg orientation to ask questions Abilities to work with others effectively Stacey, 2005

Which tasks or problems?

Types of problems??? Open-ended Rich tasks Real-world problem Challenge Investigation Inquiry Problem-based Reflective inquiry

Which tasks or problems? Content specific questions requiring a range of levels of thinking

Area and Perimeter in Year 5/6 Which shape has the largest perimeter? Please explain your thinking. Design a new shape with 12 squares which has the longest possible perimeter. Deep mathematical knowledge General reasoning abilities Communication skills Heuristic strategies

Which card is better value? Please explain your thinking. Deep mathematical knowledge General reasoning abilities Communication skills Heuristic strategies

Number and Algebra

1.Make up an equation where the answer is x = 2 2.Make up an equation where the answer is x = 3 3.Make up an equation where …. Another idea: Change one number in the equation 4 x – 3 = 9, so that the answer is x = 2. Number and Algebra Deep mathematical knowledge General reasoning abilities Communication skills Helpful beliefs eg orientation to ask questions Abilities to work with others effectively

Number and Algebra Explain the difference between particular pairs of algebraic expressions, such as and Compare similarities and differences between sets of linear relationships, eg.

Number and Algebra: Fractions Explain why is less than Explain why Deep mathematical knowledge General reasoning abilities Communication skills Abilities to work with others effectively Informal and Formal Proof

Constructive alignment ( Biggs, 2004) Curriculum Instruction Assessment

Planning for Implementation (including Problem Solving and Reasoning) Identify the topic (mathematical concepts) Examine curriculum content statements Use data to inform decisions on emphasis Select, then sequence, appropriate tasks/activities Identify the mathematical actions (proficiencies) in which you want students to engage Design assessment for ALL proficiencies

Favourite Sources MCTP (Maths 300 through Bills, C., Bills, L., Watson, A., & Mason, J. (2004). Thinkers. Derby, UK: ATM. Downton, A., Knight, R., Clarke, D., & Lewis, G. (2006). Mathematics assessment for learning: Rich tasks and work samples. Fitzroy, Vic.: ACU National. Lovitt, C., & Lowe, I. (1993). Chance and data. Melbourne: Curriculum Corporation. Sullivan, P., & Lilburn, P. (2000). Open-ended maths activities. Melbourne, Vic: Oxford. Swan, P. (2002). Maths investigations. Sydney: RIC Publications.

Resources: MCTP (Maths 300) – Curriculum Corporation website ABS – NCTM – NRICH website – Others???

Key messages … 1. Balance is important 2. Evaluate the types of questions and tasks used during mathematics lessons 3. Assessment, assessment, assessment!!! 4. Alignment between curriculum, teaching and assessment