Dealing with difference

Peter Sullivan

NB September 12 difference

Overview

• The Australian context• Articulating the challenge of diversity of

readiness• All groups are mixed ability• Nine strategies for teaching in mixed ability

classrooms

NB September 12 difference

The Australian school context• Decline in specialist mathematics study in senior years• Widespread dissatisfaction (from teachers, students

and parents) with the way mathematics is taught– Emphasis on “telling” followed by practice– Overuse of texts designed for practice rather than learning

(and certainly not for fostering creativity and student decision making)

– Disconnected from meaning and relevance• Belief by teachers (and parents and students) that

some (many?) cannot learn mathematics

There is also

• Difficulty in recruiting secondary mathematics teachers …

• … meaning that some junior mathematics teachers may lack confidence in their own mathematical knowledge

Of course, it is difficult to teach when there are …

• … fast learners in the class who shout answers and criticise others who are thinking through problems that the fast learners have already solved, and who complain to their parents about being under-extended

• … some other learners who have more or less given up believing that they cannot learn, and who prefer to interrupt others

• … extensive and exhaustive lists of content to cover that pressure teachers to skim from topic to topic

• … routines in schools that leave teachers with limited time for collaboration, sharing ideas, innovating, …

Nature of challenge• Teachers stressed by kids who cannot do it• Attendance• Wilful indifference to work• Lots of feeder schools (78?)• How to cater for the dfferences (inc in the top class)• Teacher headsets• VET• Pitching the lessons• How to “hold back”

What makes classroom learning difficult for low achieving students?

• Classrooms are complex settings and there is usually too much going on for those students who are falling behind to know what to focus on

Cognitive load

Sensory register

Working Memory

Long term memory

SelectionAttention

• Classrooms are also social settings ... and no one wants to feel that they cannot cope or contribute

• Performance avoidance, identity and other issues

Social rationale

NB September 12 difference

What classroom organisational factors might restrict opportunities

for high achieving students?

What is the issue with difference?

• ACARA says there is a 5 year gap: Cockcroft reported a 7 year gap

• While we know that there are factors contributing to differences in readiness (Indigenous, SES, rural, gender), even within these subgroups there is the same degree of diversity

• Differences in readiness refer not only to achievement, but aspirations, expectations, resilience, mindsets, confidence, satisfaction, etc

PISA and socio economic background

% at the highest level(s)

% not achieving level 2

High SES literacy 21 5

Low SES literacy 4 23

High SES numeracy 29 5

Low SES numeracy 6 22

But these groups are

also diverse

But my school has very weak students …

Between-school differences account for approximately 22 per cent of the variation in students’ tertiary entrance scores. About half of this between school variation can be accounted for by differences between schools in individual student characteristics.About half of this variation can be accounted for by differences in the academic and socioeconomic mix of students and school sector.

On NAPLAN

• Approximately one-sixth of the variation in achievement scores on both the reading comprehension tests and the mathematics tests could be attributed to differences between schools.

• This finding is similar to findings for Australian students who participated in TIMSS and PISA, two recent international studies of student achievement.

• A little more than one half of this between-schools variance could be explained by differences in the student composition—school socioeconomic status (SES) and the proportion of students from language backgrounds other than English in the school—and the school climate.

What is the Australian mathematics curriculum hoping for?

• development of expert mathematicians• expert users of mathematics in the professions• a workforce capable of meeting all numeracy

requirements• citizens able to use the mathematics they

need

This is even described as an entitlement …•of each student to knowledge, skills and understandings that provide a foundation for successful and lifelong learning and participation in the Australian community. (p.10) The document also makes the explicit assumption …•that each student can learn and the needs of every student are important. It enables high expectations to be set for each student as teachers account for the current levels of learning of individual students and the different rates at which students develop. (p.10)

What are you hoping for?

• Creativity, imagination, adaptability, willingness to think, make decisions, persist, …

• … and life long learners (note concern about following CFC models) …

• … or correct answers, compliance, acceptance of place in life, …

Some fundamental principles

• All can learn• Effort increases ability as well as achievement• We do not learn by listening and teachers do

not foster creativity, insight, etc by telling• Much learning is social, so experiences in

which the whole class participates contribute to building a class community

Some connections with “curriculum”• In what ways should the learning experiences of 8 year

olds who create mathematics easily differ from those of a 13 year old who experiences difficulty in learning?

• Is mathematics a hierarchy of micro skills that need to be taught sequentially or a network of mainly connected ideas?

• How might the ways that teachers address diversity have an impact on the ways students experience the curriculum?

• Which is easier to learn: – comparison of fractions or co-ordinate geometry? – division of fractions or index laws?– reading analogue clocks or vectors?

Heterogeneous grouping can have negative impact if …

• … teachers set expectations and starting points based on low achieving students

• … teachers over direct the learning (assuming low achieving students cannot cope) which has the effect of encouraging a fixed mind set in the students

• … there is negative peer pressure on hard working students• … teachers ignore the diversity of readiness and instead treat

everyone as the same (possibly by giving routine tasks that everyone can and is willing to do)

• … teachers teach different content to different groups• … low achieving students “performance avoid” either by

– misbehaving or – being a group work passenger or – pretending to work (they are good at it)

Homogeneous grouping can restrict student opportunities if …

• … teachers teach different content to different groups, thereby narrowing options of students in some groups

• … there is limited or no movement between groups (if there is no chance of “promotion” why would students try hard?)

• … teachers are not conscious of the impact of “self-fulfilling prophesy” effects

• … steps are not taken to avoid development of poor self concept of some members of the upper sets (Big fish little pond effect)

• … the top set are taught (in a routine way) the content from the following year (for example) rather than fostering mathematical creativity (for example) using the current content

• … the grouping fosters a sense of “entitlement” in top stream students

Self-fulfilling prophesyStep 1: Teachers form early differential expectations for

studentsStep 2: As a result the teachers behave differently to different

students and this differential behaviour communicates the teachers’ expectations to the students. If such treatment of the students is consistent, and if the students do not resist, it will have an effect on their self- concept, achievement, motivation, aspirations and classroom conduct.

Step 3: The student responses will actually reinforce the teacher’s original expectations. Ultimately there will be a difference in student achievement and outcomes.

The ways that self fulfilling prophesy works

• Teachers “need for control”• Attribution• Challenge

Brophy argued that it is common for teachers, when interacting with “low expectation”

students, …• - wait less time for them to answer questions;• - give them the answer or call on someone else rather

than waiting;• - use inappropriate reinforcements;• - criticise them more for failure and praise them less

frequently;• - do not give them public feedback on their responses;• - call on them less to respond;• - demand less; and• - have less friendly verbal and non-verbal contact.

But it is hard …

• … if the curriculum is based on stratification to ensure that all students have the same opportunities to learn

• … if you see the need of low stream students is development of skills in isolation to communicate connections, meaning and relevance

• … to communicate to students in low streams that you think they can learn well (especially if they have a restricted curriculum)

You have heard about the importance of “evidence”

• Stratification, streaming, tracking, setting has “… minimal effect on learning outcomes and profound negative equity effects. (Hattie, 2009, p. 90)

But every group is mixed so lesson planning needs to anticipate

differences in readiness

achievement

ability

Nine task based strategies for dealing with diversity while offering

experiences covering common content

Common to all 9 approaches• Building classroom community• Task based, considering the trajectory of tasks

(what comes next!)• Explicit pedagogies• Different ways of solving the tasks, and the

different approaches are themselves educative• Representing solutions in different ways is

both engaging and important mathematically

A possible way of structuring…• Lappan used the structure – Launch, – Explore, – Summarise

• Better is (from Walqui)– Preparing learners– Interacting with the concept– Extending understanding

• But that process is cyclical and might happen multiple times in a lesson (or learning sequence)

• A visualiser helps

What these approaches are not!

• Asking questions that are so easy that everyone can do them

• Setting up groups that allow some students to hide

• Excessive repetition (of course, some is needed)

• …

Strategy 1

• Asking questions with multiple entry points and multiple exit points

• Such questions will nearly always be open-ended

Write a sentence with 5 words, with the mean of the number of letters in the words being 4. Do not use any words of 4 letters.

Draw some rectangles with a perimeter of 20 cm. Work out the area of each of your rectangles.

A set of 36 cubes is arranged to form a rectangular prism.

What might the rectangular prism look like?

What is the surface area of your prisms?

Strategy 2

• Using enabling and extending prompts• These apply to any type of challenging task

Suppose we posed this task:

Seven people went fishing. The mean number of fish they caught was 5, the median was 4 and the mode was 3. How many fish might each of the people have caught? (Give at least 3 answers)

Some enabling prompts

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

• Seven people went fishing. The mode number of fish they caught was 3. How many fish might each of the people have caught?

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.

Extending prompt

• Seven people went fishing. The mean number of fish they caught was 5, the median was 4, the mode was 3, and the range is 6. How many fish might each of the people have caught? (Give all possible answers)

What might be enabling and extending prompts for these tasks?

A brick weighs the same as 3 kg plus half a brick. What does the brick weigh?

Represent your answer in two different ways, one of which involves drawings

A brick weighs the same as 3 kg plus half a brick. What does the brick weigh?

3 kg

SA coaches day 5 September 12

A brick weighs the same as 3 kg plus half a brick. What does the brick weigh?

3 kg

SA coaches day 5 September 12

3 kg

• I used 1 metre of string to tie up this box. The bow takes 300 mm. What might be the dimensions of the box?

• Race to 5x + 5y

Strategy 3

• Realistic investigations that are multi faceted, take time and are meaningful for collaborative group work

Which fits better: a round peg in a square hole or a square peg in a

round hole?

1mm of rain on 1 sq m of roof is 1 L ofwater.Design a tank for this building that captures all of the rain that usually falls this month.

Design a cola can that has the same volume as the current cans but requires less aluminium

Calculate the volume of the cylinder that is made by bending an A4 sheet of paper vertically.

Now calculate the volume if the sheet was horizontal.

How many people can we fit into this room?

• A chameleon has a tongue that is half as long as its body ...

• … how long would your tongue be if you were a chameleon?

In what ways are the arch at St Louis and the Sydney Harbour Bridge similar to or different from a parabola, circle, ellipse, hyperbola, sine curve, catenary?

The school is considering building an arch over the front gate. What curve would you recommend? Write the equation (using actual measurements) for your curve.

Strategy 4:

• Using a text book in different ways

NB September 12 difference

Some examples

• Read the last question first. What do you need to learn to be able to do that question? Which of the earlier questions look like they might help?

• Work in pairs. One of you does the odd questions. The other does the even ones. Then each of you can explain your working to the other.

• In what ways are the questions different from each other?

Strategy 5

• Asking questions that emphasise connections and are challenging (but at the right level for the curriculum)

• At the end of the season, the coach noticed that the mean and median of the number of goals kicked by the 20 players was 10. He also noticed that ¼ of the players kicked less than 5 goals, ¼ of the players kicked 5 or more but less than 10, ¼ of the players kicked 10 or more but less than 15, and ¼ of the players kicked 16 or more. How many goals might each of the players have kicked?

CEOM 2012

To give us something to discuss

• On a train, the probability that a passenger has a backpack is 0.6, and the probability that a passenger as an MP3 player is 0.7.

• How many passengers might be on the train?• How many passengers might have both a

backpack and an MP3 player?• What is the range of possible answers for this?• Represent each of your solutions in two different

ways.

Some sample questions

• Find two objects with the same mass but different volumes

• Draw some closed shapes with 6 right angles• Draw a line 1 m long on this page• The perimeter of a rectangle is 20 cm. What might be

the area?• Draw (on squared paper) as many different triangles as

you can with an area of six square units• A number has been rounded off to 5.3. What might be

the number?

Strategy 6

• Creating task sequences where there is no expectation that all students can do the first one(s), but for which all can do the last one(s)

A “constructing” task

• In a tank there are 200 fish, 99% of which are guppies.

• How many guppies do I need to take out to end up with 98% guppies?

A consolidating task

• At the football there are 50 000 spectators, 55% of whom are Collingwood supporters.

• How many Collingwood supporters do I need to expel from the stadium to end up with 50% Collingwood supporters?

A possible sequence of tasks..

Write a sentence with 5 words, with the mean of the number of letters in the words being 4. Do not use any words of 4 letters.

Seven people went fishing. The mean number of fish they caught was 5, the median was 4 and the mode was 3. How many fish might each of the people have caught? (Give at least 3 answers)

• At the end of the season, the coach noticed that the mean and median of the number of goals kicked by the 20 players was 10. He also noticed that ¼ of the players kicked less than 5 goals, ¼ of the players kicked 5 or more but less than 10, ¼ of the players kicked 10 or more but less than 15, and ¼ of the players kicked 16 or more. How many goals might each of the players have kicked?

After the brick task

A box weighs the same as 12 kg plus a quarter of a box. What does the box weigh?(do this in two different ways)

Strategy 7

• Creating task sequences that proceed from simple (to engage students in the task) and which progressively become more difficult

Getting to schoolJohn Hindmarsh

20 km

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

Assume that it costs $2 per km for the full costs of the

journey

Getting to schoolJohn Hindmarsh

73 km

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

Assume that it costs $1.37 per km for

the full costs of the journey

Getting to school

John Mark

5 kmHindmarsh

20 km

How much should Mark give John if he picks him up and takes him home?

Assume that it costs $2 per km

for the full costs

Getting to school

John Mark

17 kmHindmarsh

57 km

How much should Mark give John if he picks him up and takes him home?

Assume that it costs $1.37 per km for the full

costs

Getting to school

John Mark

z kmHindmarsh

x km

How much should Mark give John if he picks him up and takes him home?

Assume that it costs $y per km

for the full costs

Getting to school

John Mark

5 kmHindmarsh

20 km

How much should Mark and Susan give John if he picks them up and takes them home?

Assume that it costs $2 per km

for the full costs

10 km

Susan

Getting to school

John Mark

15 kmHindmarsh

72 km

How much should Mark and Susan each give John if he picks them up and takes them home?

Assume that it costs $1.35 per km for the full

costs

23 km

Susan

Strategy 8:

• Games that are a mix of skill and luck

In turn, players roll a 10 sided die (numbered 0 to 9)

and, after each roll, write the number rolled in one of

the rectangles on their own board that looks like

. ÷ 0 .

The winner has the answer closest to 100 (for example).

Strategy 9

• “Create your own experience” activities on which subsequent questions might build

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Which numbers are missing?

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Which ones are wrong?

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2100 2200 2300 2400 2500 2500 2700 2800 2900 3000

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CEO cohort 6 2012

Which numbers are wrong?

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What is the hardest question you could use this to answer?