Identifying and fostering quality teaching in numeracy

Identifying and fostering quality teaching in numeracy

Kaye StaceyUniversity of Melbourne

[email protected]

Improving Numeracy in Australia

Now The Future

Learning standards

Numeracy ~ Mathematics

Increasing age

NumeracyMathematics

Mathematics

Mathematics

PISA “Take the test” www.oecd.org

• The following three items are publically released, and available from the PISA “Take the test” website. This contains items on scientific literacy, reading literacy and mathematical literacy (which is essentially numeracy in the Australian usage).

PISA “Take the test”www.oecd.org

PISA “Take the test”www.oecd.org

Q1. .......

Q2. Explain how the graph shows that on average the growth rate for girls slows down after 12 years of age.

6.3 According to this graph, on average, during which period in their life are females taller than males of the same age?

Master of Numeracy – Numeracy Subject

PISA 2006 international rankings

PISA 2006 tested 14170 Australian 15-year-olds

Statistically, 8 countries better than Australia

Australian score range (5th to 95th percentile) is narrower than the OECD average

Australia does well (here and other studies)

From Thompson & De Bortoli (2007) PISA in Brief from Australia’s Perspective, ACER







mean

5th & 95th percentiles







Achievement of top 25%


Percentage of students reaching TIMSS 2003 benchmarks

About 60 countries participated: Australia, USA, France, England, Indonesia, Singapore, Japan, Brazil…

10

26

12

18

25

25

24

19

36

26

37

30

22

16

21

25

7

7

5

9

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Year 8 Maths: Australia

Year 8 Maths: International mean



below Low benchmark Low Intermediate High Advanced

Yr 4 - INT

Yr 4 - AUS

Yr 8 - INT

Yr 8 - AUS


10

26

12

18

25

25

24

19

36

26

37

30

22

16

21

25

7

7

5

9

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%






Yr 4 - INT

Yr 4 - AUS

Yr 8 - INT

Yr 8 - AUS

Percentage of students exceeding high TIMSS 2003 benchmarks

About 30% of Australian students achieving “high benchmark”


10

26

12

18

25

25

24

19

36

26

37

30

22

16

21

25

7

7

5

9

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%






Yr 4 - INT

Yr 4 - AUS

Yr 8 - INT

Yr 8 - AUS

Percentage of students exceeding high TIMSS 2003 benchmarks

About 70% of Singapore students achieving “high benchmark”

About 30% of Australian students achieving “high benchmark”

Schooling Issues Digest(Stacey & Stephens)

• Performance by mean scores• Performance by content area

– Which square has 2/3 shaded? (Int 57%, Aust62%, Singapore 93%)

• State and Territory comparisons• Performance by student characteristics

– Gender– Home language and migration background– Remote and Rural location– Indigenous students– Socio-economic background

• School, Classroom and Learning Influences– School environment and student attitudes– Students’ learning strategies– Teachers and classes– Instruction

Discussion Time

What does a numerate student need to know and be able to do?

Australian Curriculum: Mathematics (2010)

Proficiency strands: actions in which students can engage when learning and

using the content • 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. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information.

• Fluency– Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately,

efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

• Problem Solving– Students develop the ability to make choices, interpret, formulate, model and investigate problem situations,

and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

• Reasoning– Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing,

proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.

Mathematical Proficiency

• Content strands – knowledge base is essential

• Understanding — comprehension of mathematical concepts, operations, and relations;

• Fluency — skill in carrying out routine procedures flexibly, accurately, efficiently, and appropriately;

• Reasoning — capacity for explanation, justification, logical thought,

• Problem solving — ability to devise methods to solve problems, within or beyond mathematics

• Productive Disposition — habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy

• WHAT• HOW• WHY

• USE

Quality teaching attends to all aspects of mathematical proficiency

• Fluency (fashionable to under-rate this!)• Understanding (Box Plots)• Reasoning • Problem solving (PISA, Barbie Bungee)• Productive Disposition (Engagement article)

Practical activity to illustrate aspects of quality teaching – especially “understanding”

What is a box plot and how is it interpreted?

Building a boxplot

Click here for Box Plot Construction

Discussion point

• What issues of learning maths/numeracy are highlighted by this example?

• What features of good numeracy teaching were demonstrated?

• In what ways was the data set well designed for this ‘lesson’?

• Suggestions for improvement?

• Except for the obvious points – no practice provided, little attention to meaning of the statistical indicators,....

Key points about good numeracy teaching

• Obstacle: establishing relevance to individual– affective: careful choice of context– cognitive: demonstration of value of concept/technique

• Obstacle: abstractness of representation– (student – score – point on number line – disappears into

box plot)– so start with concrete data set, return as needed

• Obstacle – your suggestions?

Mathematical tasks and student cognition: Classroom-based factors that support and

inhibit high-level mathematical thinking and reasoning

Henningsen & Stein 1997 JRME

• Engagement in high-level mathematical reasoning and thinking important as– To become better problem solvers – To learn maths better

• The tasks that students do in class are the primary vehicle for learning

• High-level tasks often lose their high-level character when used in classrooms

H &S method

• Selected tasks which had desirable task features to promote mathematical thinking

• Observed classroom implementation• Asked

– What classroom factors supported high-level cognitive processes by students?

– What happened when the cognitive demand declined?

• Looked for factors from the literature and from the data

7 mixed

reasons

10

no math’l focus

11 - unsystematic exploration

36 declined

8 - procedures without meaning

144 tasks

58 set up to encourage high level

thinking

22 maintained high level

Promoting engagement in high level thinking


• Built on prior knowledge and extended it• Appropriate amount of time for engaging with task• Active teaching

– High-level performance modelled – Scaffolding that doesn’t remove challenge– Consistently pressing students to explain and justify

• The article goes on to document what caused decline to – procedures without connections, – no mathematical activity, and – unsystematic exploration. (See article for details).

• With the wider use of tasks with potential to engage students in high level thinking, these ‘declines’ are also now very eivdent in our schools.

What caused decline to procedures without connections?

• Removal of challenging aspects, usually by subtle alteration

• Too much or too little time allocated• Focus on right answers only (“what do you

write down here”)

Decline to unsystematic explorationMake a square metre, centimetre etc

• Students allowed too much time to build sq metre – too good

• In building, they focussed on linear not area measures

• Missed the maths – relationships between linear and area dimensions

Not to scale

What caused decline to “no mathematical activity?

• Inappropriate task for those students• Class management problems• Too much or too little time• Sometimes lack of accountability (“didn’t

count” for marks)


• Built on prior knowledge and extended it• Appropriate time for engaging with task• Active teaching

– High-level performance modelled – Scaffolding that doesn’t remove challenge– Consistently pressing students to explain and justify

• Avoiding the pitfalls that lead to– Procedures without connections and meanings– Unsystematic exploration– No mathematical focus

Barbie Bungee

• Application of linear functions and graphing for Year 9• Each student’s own Barbie is going to bungee jump

from a high part of the school (e.g. 7m walkway above asphalt playground) on a rope made from elastic bands. Students collect in the classroom from trial (smaller) jumps and predict number of elastic bands for real jump, then test.

• Google this, or see our RITEMATHS website for Year 9/10 algebra applications with technology for this and other resources http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS

http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS/

y = 22 + 9x

What features may make this an excellent learning activity ? How might it decline from higher order thinking?

Features of Barbie Bungee

• A real world problem, put into classroom-practical form– formulating math’l problem from real world problem– Solving math’l problem– Interpreting and checking in real world

• High motivation to do well with your own Barbie• Can lose mathematical focus• Can become “procedures without connections”• Some teachers’ valuing of this task were entirely about the

motivational features, not the mathematical.

http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS

http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS

Spotlight: Research into Practice VLNS

Max Stephens

Engagement in Mathematics: defining the challenge and promoting good

practices

Stephens: Introductory pages• Wide definition of engagement

– ABC Affective Behavioural Cognitive

• Reasons for disengagement– Not ‘relevant’, difficult, confusing, causing criticism

• Apparent engagement through ‘busy work’• Deeper aspects of engagement

– Procedural fluency • individual improvement, personalised through ICT etc

– Conceptual understanding• deeper connections, made explicit

– Reasoning and problem solving • opportunities maintained at high level

– Individual and collective engagement , for different purposes

Stephens: Approaches to engagement

• Cognitive dimensions– Adaptive: Self-efficacy, valuing learning, mastery orientation– Mal-adaptive : Anxiety, failure avoidance, uncertain control

• Behavioural dimensions– Adaptive: Persistence, planning, task management– Mal-adaptive: Self-handicapping, disengagement

• Attend to students’ voice, belonging, choices, responsibility, success– Article lists associated classroom values and practices, and

specific strategies.

Stephens (citing Baxter and Williams) Scaffolding techniques to promote

engagement

• Analytic scaffolding– Teachers supporting better learning– 7 varieties

• Social scaffolding– Teachers supporting good interactions in class– 4 varieties

• Scaffolding autonomy and self-regulation– Teachers supporting ’positive cognitions’ – 7 varieties

• Different students need different scaffolding

Discussion Time

Discussion Points

• Select ONE type of scaffolding for your table• How can each of the varieties of scaffolding be

put into action?• What experiences are there at your school of

these?• How can teachers be helped to use such

scaffolding?

Thank you

[email protected]

mailto:[email protected]

Identifying and fostering quality teaching in numeracy

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