Judy Anderson
The University of Sydney
Judy.anderson@sydney.edu.au
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.
Understanding Students 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.
Fluency
Students 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.
Reasoning Students 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
Higher order thinking
5. Evaluate
Problem solving
6. Create
Reasoning
Cognitive
process
What learners need to do
Action verbs
Remember
Retrieve relevant information
from long-term memory
Recognise, recall, define, describe,
identify, list, match, reproduce, select,
state
Understand
Construct meaning from
information and concepts
Paraphrase, interpret, give egs,
classify, summarise, infer, compare,
discuss, explain, rewrite
Apply
Carry out a procedure or use a
technique in a given situation.
Change, demonstrate, predict, relate,
show how, solve, determine
Analyse
Separate 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
Evaluate
Make judgements based on
criteria and/or standards.
Comment on, check, criticise, judge,
critique, discriminate, justify,
interpret, support
Create
Put 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 …
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)
The outcome …
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 …
Teaching for problem solving knowledge, skills and understanding
The outcome …
Problems at the
end of the chapter!
Teaching about problem solving Problems used to
heuristics and behaviours (the strategies ‘practise’ strategies
and processes)
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 …
Teaching for problem solving knowledge, skills and understanding
The outcome …
Problems at the
end of the chapter!
Teaching about problem solving Problems used to
heuristics and behaviours (the strategies ‘practise’ strategies
and processes)
and checklists
Teaching through problem solving posing questions and investigations as
Some success but
key to learning new mathematics
limited
(beginning a unit of work with a problem implementation
the students cannot do yet)
Successful problem solving requires
Deep mathematical
knowledge
Personal attributes
eg confidence,
persistence,
organisation
General reasoning
abilities
Skills and Attributes
Heuristic
strategies
Abilities to work
with others
effectively
Communication
skills
Helpful beliefs
eg orientation to ask
questions
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
Number
and
Algebra
Deep mathematical
knowledge
General reasoning
abilities
Communication
skills
Helpful beliefs
eg orientation to ask
questions
Abilities to work
with others
effectively
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
 Explain the difference between
particular pairs of algebraic expressions,
such as x 2 and 2x
 Compare similarities and differences
between
sets

 of linear relationships, eg.
y  3x, y  3x  2, y  3x  2
Number and Algebra:
Fractions
Deep mathematical
knowledge
 Explain why

 Explain
why
1
8
is less than
1
4

2 1 3
 
3 4 7

Informal and Formal Proof
General reasoning
abilities
Communication
skills
Abilities to work
with others
effectively
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 www.curriculum.edu.au)
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 http://www.curriculum.edu.au
 ABS – http://www.abs.gov.au
 NCTM – http://www.nctm.org
 NRICH website –
http://nrich.maths.org.uk/primary
 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