Research into practice: What we can learn from research

into good tasks

Peter Sullivan

AISNSW 2010

What are the challenges you are experiencing in

teaching mathematics your school?

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• Timeframe (too much curriculum)• Kids did not problem based teaching• Making it relevant, especially for low achievers• Buildinhg confidence• Diverse ability range• Retention • Busy lives of kids• Extension • External influences incluidng naplan, parent expectations n…• Levels of concentration• Gaps in prior knowledge• Disruptions to school routine

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Overview

• Findings from research• The Australian mathematics curriculum• 5 principles for improving teaching

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TASKS AND TEACHER ACTIONS• We investigated ways that particular types of

mathematics classroom tasks create different opportunities for students and different challenges for teachers.

• the type of task influences the nature of the learning (e.g., Christiansen & Walther, 1986; Hiebert & Wearne, 1997)

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Task processing model from task to lesson (Stein, 1996)

• Mathematical task as presented in instructional materials– which, influenced by the teacher goals, their subject matter

knowledge, and their knowledge of students, informs …• … mathematical task as set up by the teacher in the

classroom– which, influenced by classroom norms, task conditions,

teacher instructional habits and dispositions, and students learning habits and dispositions, influences …

• … mathematical task as implemented by students – which creates the potential for …

• … students learning.

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Teachers transform tasks

• Stein et al. (1996) noted the tendency of teachers to reduce the level of demand of tasks.

• Doyle (1986) and Desforges and Cockburn (1988) attribute this to complicity between teacher and students to reduce risk

• Tzur (2008) … two key ways that teachers modify tasks: – at the planning stage; – if responses are not as intended.

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Our goals were to describe

• how the tasks respectively contribute to mathematics learning

• the features of successful exemplars of each type• constraints which might be experienced by

teachers • teacher actions which can best support students’

learning

Teachers Tasks Students

Teacher knowledge, attitudes

Cluster meetings

Classroom implementation

Student survey

Teacher responses to tasks

Planning units of work

Teaching units of work

Reflection on teaching

Student responses to units

Planning 3 lessons

Teaching 3 lessons AISNSW 2010

Self ratings of perception of mathematics in % (n = 930)

Rating How good are you at maths? (Q1)

How happy are you in maths class? (Q2)

0 0 11 1 42 3 93 11 184 21 225 33 246 24 157 8 8

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While there was not much difference overall between levels, there was a big difference (mean 3.5 to 5.5) between classes

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In summary

• At each of these middle years levels there is a range of satisfaction and confidence, and teachers should be aware of this

• Teachers make a difference and they need support to both find out the students levels of satisfaction and confidence, and to do something about it if they are low

Qu 9In this table there are three maths questions that are pretty much the same type of mathematics content asked in different ways.

Put a 1 next to the type of question you like to do (learn) most, 2 next to the one you like (learn) next best and 3 next to the type of question you like (learn) least.

We don’t want you to work out the answers.

Movies tickets are $13 for adults and $7 for children. How much does it cost for 2 adults and 4 children to go to the movies?

(word problem)

2 adults and 4 children went to the movies. They spent $120 on tickets. How much might the adult and children’s tickets cost?

(open-ended)

(2 ×13) + (4 × 7)=

(number work)

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Preferences for liking, and learning from, the task types as a % (Q9)

Task Like most Learn most

Number work 54 40

Word problem 35 23

Open-ended 12 37

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11. ... Put a 1 next to the type of question you like to do (learn from) most, 2 next to the one you like (learn from) next best, and 3 for the type of question you like (learn from) the least:

Find the area inside the following shape:

A shape has an area of 10 square units.

What might the shape look like?

A running track has straights that are 100 m

long with half circles at the end. The inside

is all grass. What is the area of the grass?

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Preferences for the Task Types as a percentage

Type Like most Learn most

Area by counting 39 33

Practical calculation 20 44

Open-ended 42 22

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From a different survey after a unit of work

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Student Preferences (%)Student Preferences (%)

TaskTask Task Task TypeType

Task favouriteTask favourite 22ndnd favourite favourite Task best for Task best for learninglearning

22ndnd best for learning best for learning

Finding similar graphsFinding similar graphs 11 0.0%0.0% 0.0%0.0% 2.1%2.1% 2.1%2.1%

Clues on cardsClues on cards 11 0.0%0.0% 3.1%3.1% 18.0%18.0% 10.4%10.4%

Using excel to present dataUsing excel to present data 11 14.3%14.3% 6.0%6.0% 12.2%12.2% 9.2%9.2%

Matching graphsMatching graphs 11 2.0%2.0% 0.0%0.0% 2.0%2.0% 10.4%10.4%

Weather projectWeather project 22 8.0%8.0% 6.0%6.0% 2.0%2.0% 2.1%2.1%

Two way tablesTwo way tables 22 2.0%2.0% 8.0%8.0% 6.0%6.0% 6.3%6.3%

Average height of classAverage height of class 22 0.0%0.0% 6.0%6.0% 8.0%8.0% 2.1%2.1%

This goes with thisThis goes with this 22 4.0%4.0% 2.0%2.0% 4.0%4.0% 14.6%14.6%

Most commonly used lettersMost commonly used letters 22 6.0%6.0% 2.0%2.0% 8.0%8.0% 2.1%2.1%

Average height in schoolAverage height in school 22 26.0%26.0% 18.0%18.0% 8.0%8.0% 17.6%17.6%

A sentence with 5 wordsA sentence with 5 words 33 4.0%4.0% 12.0%12.0% 8.0%8.0% 4.2%4.2%

Rock paper scissorsRock paper scissors 33 22.0%22.0% 27.8%27.8% 0.0%0.0% 4.2%4.2%

Seven people went fishingSeven people went fishing 33 10.0%10.0% 4.0%4.0% 18.0%18.0% 12.5%12.5%

Conducting a surveyConducting a survey 33 2.0%2.0% 2.0%2.0% 2.0%2.0% 2.1%2.1%

Total:Total: 5050 5050 5050 4848AISNSW 2010

Student opinions about their mathematics classes: Some free format responses

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• In summary, it seems that the students were extraordinarily articulate about what they wanted in their maths lessons

• In synthesising the responses, students like lessons that – used materials (although these were not structured materials), – were connected to their lives, – involved games, – were practical with some emphasis on measurement, – in which they worked outside,– The see “like” and “learn” as different – with the method of grouping being important, and– over half of the students claim to like to be challenged.

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What does this mean?

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In summary

• The students were extraordinarily articulate about what they wanted in their maths lessons

• The first characteristic of the responses is the diversity of aspects on which the individual students commented, suggesting that there is no commonly agreed ideal lesson, and there are many ways to teach well.

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• The ways of working in class are clearly important for students, and however the teacher intends that the student work, the reason for this needs to be clarified for the students..

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A preliminary task

• In 5 words or less, write down an aspect of teaching mathematics that you would advise beginning teachers to ensure they think about in all of their lessons

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But first ...

• An underlying assumption is that at least some learning should come

• from engagement of individuals • with

– tasks – each other

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Some of the key decisions

• Mathematics success creates opportunities and all should have access to those opportunities

• The curriculum should prioritise teacher decision making

• The curriculum should foster depth and important ideas rather than breadth

• Students can be challenged within basic topics, including the advanced students

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There are 3 content strands

• Number and algebra• Measurement and geometry• Statistics and probability

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… and 4 proficiency strands

• Understanding• Fluency• Problem solving• Reasoning

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Key teaching idea 1:

• Identify big ideas that underpin the concepts you are seeking to teach, and communicate to students that these are the goals of your teaching, including explaining how you hope they will learn

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An example for us to discuss

• Write a sentence that has5 words, with an average of four letters per word (no 4 letter words)

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Some questions

• What is the mathematical point of that task?• What is the pedagogical point of that task?• How do you make these points explicit to

students?

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Some questions

• What mathematical actions can be addressed by working on that task?

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Which card is better value?

Please explain your thinking.

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Some questions

• What is the mathematical point of that task?• What is the pedagogical point of that task?• How do you make these points explicit to

students?

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year 7

• Determine mean, median, and range and use these measures to compare data sets explaining reasoning including use of ICT

AIZ Zone 2 & 3 Day 1

year 7

• to understand and become fluent with written, mental and calculator strategies for all four operations with fractions, decimals and percentages

AIZ Zone 2 & 3 Day 1

year 8

• Generalise from the formulas for perimeter and area of triangles and rectangles to investigate relationships between the perimeter and area of special quadrilaterals and volumes of triangular prisms and use these to solve problems

AIZ Zone 2 & 3 Day 1

year 9

• Work fluently with index laws in both numeric and algebraic expressions and use scientific notation, significant figures and approximations in practical situations

AIZ Zone 2 & 3 Day 1

year 9

• Solve problems involving linear simultaneous equations, using algebraic and graphical techniques including using ICT

AIZ Zone 2 & 3 Day 1

year 10

• Understand and use graphical and analytical methods of finding distance, midpoint and gradient of an interval on a number plane

AIZ Zone 2 & 3 Day 1

Key teaching idea 4:

• Interact with students while they engage in the experiences, and specifically planning to support students who need it, and challenge those who are ready

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Establish classroom ways of working

• Examples of “norms”– errors are part of learning – all students must persist – all students must be willing to justify their thinking– working as a community of learners benefits

everyone

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An idea we can discuss

• 5 people went fishing. The mean number of fish caught was 4, and the median was 3. How many fish might each person have caught?

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Some questions

• What mathematical actions can be addressed by working on that task?

• What might be the challenges in turning this into a lesson?

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What are enabling prompts?• Enabling prompts can involve slightly varying an aspect of the

task demand, such as – the form of representation, – the size of the numbers, or – the number of steps,

so that a student experiencing difficulty, if successful, can proceed with the original task.

• This approach can be contrasted with the more common requirement that such students – listen to additional explanations; or – pursue goals substantially different from the rest of the class.

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Factors contributing to difficulty• It may not be clear which aspects may be contributing to a

particular student’s difficulty, but by anticipating some of the factors, and preparing prompts that, for example, – reduce the required number of steps, – simplify the modes of representing results, – make the task more concrete, or – reduce the size of the numbers involved,

• the teacher can explore ways to give the student access to the task without the students being directed towards a particular solution strategy for the original task.

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How might you adapt that task for students experiencing difficulty?

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How might you adapt that task for students who finish quickly?

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Sample Task 2

• Seven people went fishing. The mean number of fish caught was 4, and the median was 3. How many fish might each person have caught?

Considering the quality of response

• Basic– “The number of fish could be 1, 2, 2, 3, 4, 5, 11”

• Developed– “The following are some possibilities:

• 0, 0, 0, 3, 4, 5, 16• 0, 0, 1, 3, 4, 5, 15• 0, 0, 0, 3, 4, 5, 16• etc

• Advanced– “The total fish caught is 28. The middle number is

3, so the first 3 numbers of fish caught must have a numbers that are 3 or less. For each set, the latter three numbers make up the total. So if the first 3 numbers total 5, then the latter 3 numbers must total 20.”

Key teaching idea 5:

• Adopt pedagogies that foster communication, mutual responsibilities, and encourage students to work in small groups, and using reporting to the class by students as a learning opportunity

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How do we turn this into lessons?

• Launch• Explore• Summarise • review

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The Japanese have better words

• Hatsumon– The initial problem– Kizuki - what you want them to learn

• Kikanjyuski– Individual or group work on the problem– Kikan shido – thoughtful walking around the desks

• Nerige– Carefully managed whole class discussion seeking the

students’ insights• Matome

– Teacher summary of the key ideas

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My suggested words

• Task introduction• Facilitation of student engagement• Whole class discussion• Teacher summary

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Key teaching idea 3

• Engage students by utilising a variety of rich and challenging tasks, that allow students opportunities to make decisions, and which use a variety of forms of representation

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A story from Mr T (sent last week)• In year 9 revision, the students were working

on this problem– You earn $12 per hour for 22.5 hours. You pay

26% of your earnings in tax. How much tax will you pay?

• A girl, Emma, wanted help.• Mr T: Do you have a job?• Emma : Yes• Mr T: How much an hour do they pay you?• Emma : I don’t know I just started

MAWA National issues

Mr T: Let’s say you ear $12 an hour and you work for 3 hours. How much is that?

Emma : I don’t know. Do you divide?Mr T: No. Think about earning $12 an hour.

You work one hour, and then another and then another. How much have you earned?

Emma : I don’t know

MAWA National issues

• Kylie (sitting nearby): Is it $36?• Mr T: Yes. Good.• Emma (to Kylie): God, you’re smart.

MAWA National issues

One of the key issues with the Australian Curriculum is teacher

decision making

• Some examples of adapting

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Some activities• Relationships

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The average height of 3 people in this room is 1.7 m. You are one of those

people.Who are the three people?

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Making questions open

Method 1:• Write down a question and work out the

answer.• Make up a new question that includes the

answer as part of the question.Method 2.• Write down a complete question including the

answer.• Remove some of the question parts.

Race to 10

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What questions can we answer?

2

11

4

1+

2

11 + 4

3

4

31 - 2

1

Using empty number lines

• 37 + 45

37

3

40

2

40 80 82

37

1010 10 10

3 2

47 57 67 7780

82

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• Use an empty number line to work out

4

122

2

17-

4

122

2

4

120

4

115

4

314

5

2

1

Key teaching idea 2:

• Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning

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The first part of this

One perspective on building on what they know

• The idea of the self- contained “lesson”

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A sample lesson

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Using isometric paper, draw the shapes made by sticking 3 cubes together (whole faces touching)

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2 cubes might look like

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Draw a shape made from 10 cubes

• Go beyond the straight line

The next task• A block of city buildings is 3 cubes wide and 3 cubes

long• It looks like this from the front

On isometric paper, draw what the set of buildings might look like

The task• A block of city buildings looks like this from the side

• And like this from the front

• On isometric paper, draw what the set of buildings might look like

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Front

Side

Frontview

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In what ways was that different from a conventional mathematics lesson?

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“Enabling prompts”

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A different destination

• A block of city buildings looks like this from the side

• Draw what the set of buildings might look like

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For students experiencing difficulty:

• Have some isometric paper with part of the building already drawn.

• Have some cubes. Ask them to model what the building might look like.

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How might you extend this for students who finish quickly?

• How many different designs that fit the directions can you make?

• Draw a set of buildings on the isometric paper, and draw the front and side view as well.

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• Draw what this representation might mean

1 3

3

1

32

1 1 2

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Key teaching idea 2:

• Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning

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the second part

Getting to workJosie Loreto

20 km

How much does it cost Josie to get to work and back home again?

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Assume that it costs $2 per km for the

full costs

Getting to workJosie Loreto

73 km

How much does it cost Josie to get to work and back home again?

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Assume that it costs $1.37 per km for

the full costs

Getting to workJosie Loreto

x km

How much does it cost Josie to get to work and back home again?

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Assume that it costs $z per km for the

full costs

Getting to work

Josie Carmen

5 km

Loreto

20 km

How much should Carmen give Josie if she picks her up and takes her home?

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Assume that it costs $2 per

km for the full costs

Getting to work

Josie Carmen

13 km

Loreto

59 km

How much should Carmen give Josie if she picks her up and takes her home?

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Assume that it costs $1.37 per km for the full

costs

Getting to work

Josie Carmen

x km

Loreto

y km

How much should Carmen give Josie if she picks her up and takes her home?

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Assume that it costs $z per km

for the full costs

Getting to work

Josie Carmen

5 km

Loreto

20 km

How much should Susan and Carmen give Josie if she picks them up and takes them home?

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Assume that it costs $2 per

km for the full costs

5 km

Susan

A preliminary task

• In 5 words or less, write down an aspect of teaching mathematics that you would advise beginning teachers to ensure they think about in all of their lessons

AISNSW 2010