The potential of posing more challenging mathematics tasks and ways of supporting students to engage in such tasks.

Sullivan MAT Nov 2013

The potential of posing more challenging mathematics tasks and ways of supporting students to engage in such tasks.

Peter Sullivan


Abstract• While most students want to work on more

challenging mathematics, there are still some who require substantial support.

• The workshop will explore examples of tasks with low "floors" but high "ceilings" that allow all students to engage with the tasks at some level, but which can be extended productively for those who are ready.

• A particular lesson structure that supports the work of all students on such tasks will be presented and discussed.


What are the challenges that you are experiencing in teaching mathematics?


Some initial assumptions

• Planning happens at 4 levels: the school, the year, the unit, the lesson

• We are starting at the “planning the lesson” end• The goal is to improve the experience of

students when learning mathematics• We will focus on a particular type of lesson

structure (that is broadly applicable to many types of tasks)


Even though such investigations can be made realistic and authentic…

• The maximum gradient of a ramp exceeding 1520mm in length shall be 1:14.

• Ramps shall be provided with landings at the top and bottom of the ramp and at 9m intervals for a ramp 1:14.

• The length of landings shall be not less than 1200mm.• The gradient of ramps between landings will be consistent.• Ramps shall be provided with handrails on both sides

which do not encroach on the 1000mm minimum clear width.

• Angles of approach for ramps, walkways and landings is preferably zero degrees.


Or even extended to

• Design a ramp for some stairs at the school which do not yet have a ramp

• And write a report for the School Council


Nor are we focusing on games such as

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 a board that looks like

÷

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


Even though such games can be extended to …

How could you place 3, 4, 5 and 6 on a board

like this, to make the answer closest to 100

÷


And I am assuming that you already know how to structure lessons

based on texts


There are plenty of resources of great ways to teach mathematics

• The Shell Centre Materials– http://www.mathshell.com/

• Formative Assessment Lessons and Tasks– http://map.mathshell.org/materials

• nrich– http://nrich.maths.org/frontpage

• transum– http://www.transum.org/

• hotmaths– http://www.hotmaths.com.au/

• tarsia - there is not actually a website with this name, but a number that offer software (example below)

http://www.tes.co.uk/article.aspx?storyCode=6107407&s_cid=RESads_MathsTarsia

http://map.mathshell.org/materials

http://map.mathshell.org/materials

http://nrich.maths.org/frontpage

http://nrich.maths.org/frontpage

http://www.transum.org/

http://www.transum.org/

http://www.hotmaths.com.au/

http://www.hotmaths.com.au/





The following are examples of tasks that exemplify the approach on

which we will focus


For year 8Drawing a single straight line, make two quadrilaterals with the same perimeter

12

9

10

11

13

C

B

A

D

E


For Year 1Basketball scores

Parrots 106Galahs 97

How much did the Parrots win by?

(Work out the answer in two different ways)


What might make teaching a lesson based on one of those tasks difficult at your school?


Proposition Set 1• More of the same is not a feasible response• The pathway to improvement is teaching teams

working collaboratively on planning, on teaching and on assessment

• Each lesson sequence should ideally incorporate a variety of types of lessons, tasks and activities

• All students need to make time (away from school) to develop their own fluency at the skills being taught (you might need to explain the rationale for this)


Proposition Set 2• Students benefit from working on tasks that they do not already

know how to do• Students are more likely to connect ideas if they compare and

contrast related ideas and build networks of concepts for themselves

• These connections are the key to remembering and transferring knowledge

• Asking students to solve and/or represent problems in more than one way helps them to build connections

• There are risks if we build connections too slowly• The goal is that students come to know they can learn

mathematics


Should we start easy and wind it up or start at challenging or wind it back?

• Students can benefit when they move from not knowing how to do something to knowing how to do it.

• In other words, what they have learned is explicit to them. – This does not necessarily happen if they are working on the

“known”. • When confronted with a task that they cannot do,

students need to explore their existing mental structures and schemes, explore links, build connections and identify aspects that are unknown.

Where does the idea of “challenge” come from?

• Guidelines for school and system improvement (see, e.g., City, Elmore, Fiarman, & Teitel, 2009)

• The motivation literature (Middleton, 1995; 1999).



This connects to “mindsets”

• Dweck (2000) categorized students’ approaches in terms of whether they hold either growth mindset or fixed mindset


Students with growth mindset:

• Believe they can get smarter by trying hard• Such students

– tend to have a resilient response to failure;– remain focused on mastering skills and knowledge

even when challenged; – do not see failure as an indictment on themselves;

and – believe that effort leads to success.


Students with fixed mindset:

• Believe they are as smart as they will even get• Such students

– seek success but mainly on tasks with which they are familiar;

– avoid or give up quickly on challenging tasks; – derive their perception of ability from their

capacity to attract recognition.


Teachers can change mindsets

• the things they affirm (effort, persistence, co-operation, learning from others, flexible thinking)

• the way they affirm • You did not give up even though you were stuck• You tried something different• You tried to find more than one answer

• the types of tasks posed

In the video to follow

• The first child says something like “when you are confused it means you are learning”

• The second child says “the best part is being confused because you can think about what you can do”

• The third child says you “learn from being confused”

• The fourth child says “you can learn by yourself”


Proposition Set 3

• Posing challenging tasks requires a different lesson structure

• The lesson should foster the sense of a classroom community to which all students contribute with the intention that students learn from each other

• The experience of engaging with the task happens before instruction

• Few rather than many tasks• All students are given time to engage sufficiently to

participate in the review


• This is relevant whether or not the students are grouped by their achievement

• And is applicable with crowded (and even badly behaved) classrooms


achievement

ability


The conventional mathematics lesson

• Review homework• Explain the concept and model the techniques• Students practice the techniques• Solutions are corrected (by the teacher)• Homework is set


Japanese Lesson Study and Lesson Structure

How many squares?


There are Japanese words for parts of lessons• 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• Neriage

– Carefully managed whole class discussion seeking the students’ insights• Matome

– Teacher summary of the key ideas



A five-component cyclic Chinese lesson structure

• Reviewing • Bridging • Variation • Summarising, and• Reflection/Planning

A revised lesson structureLappan et al. 2006

Launch Explore Summarise



The summarise phaseSmith and Stein (2011)

• anticipating potential responses • monitoring student responses interactively• selecting representative responses for later

presentation• sequencing student responses• connecting the students’ strategies with the

formal processes that were the intention of the task in the first place.

A further revised lesson structure

• In this view, the sequence– Launch (without telling)– Explore (for themselves)– Summarise (drawing on the learning of the students)

• … is cyclical and might happen more than once in a lesson (or learning sequence)


Launch

ExploreSummarise

The notion of classroom culture

• Rollard (2012) concluded from the meta analysis that classrooms in which teachers actively support the learning of the students promote high achievement and effort.


Some elements of this active support :• the identification of tasks that are appropriately

challenging for most students; • the provision of preliminary experiences that are

pre-requisite for students to engage with the tasks but which do not detract from the challenge of the task;

• the structuring of lessons including differentiating the experience through the use of enabling and extending prompts for those students who cannot proceed with the task or those who complete the task quickly;


• the potential of consolidating tasks, which are similar in structure and complexity to the original task, with which all students can engage even if they have not been successful on the original task;

• the effective conduct of class reviews which draw on students’ solutions to promote discussions of similarities and differences;

• holistic and descriptive forms of assessment that are to some extent self referential for the student and which minimise the competitive aspects; and

• finding a balance between individual thinking time and collaborative group work on tasks.


Getting started “zone of confusion”

“four before me” •representing what the task is asking in a different way such as drawing a cartoon or a diagram, rewriting the question …•choosing a different approach to the task, which includes rereading the question, making a guess at the answer, working backwards … •asking a peer for a hint on how to get started•looking at the recent pages in the workbook or textbook for examples.



The lessons consist of• One or more challenging task(s)• One or more consolidating task(s) (see Dooley, 2012)• preliminary experiences that are pre-requisite but which do

not detract from the challenge of the tasks• supplementary tasks that offer the potential for

differentiating the experience through the use of – enabling prompts (see Sullivan, et al., 2009) which can reduce the

number of steps, simplify the complexity of the numbers, and vary the forms of representation for those students who cannot proceed with the task;

– extending prompts for students who complete the original task quickly which often prompt abstraction and generalisation of the solutions.


But if I try this, …

• I will not have enough time for the rest of this topic

• I do not have time to prepare lessons like this• My students will not persist enough to engage

with the task• I am not sure I will be able to control the class• My students will not learn the mathematics by

themselves. I need to tell them.• …


Epmc perimeter


A primary example


• There are many ways to find the difference between two numbers


Basketball scores





Basketball scores

Wombats 26Possums 18

How much did the Wombats win by?



Enabling prompt(s)


Basketball scores

Eels 18Carp 13

How much did the Eels win by?



Basketball scores

Cats 8Dogs 3

How much did the Cats win by?



Extending prompt


Darts scores





Consolidating task(s)


Football scores

Seagulls 63Kingfishers 55

How much more did the Seagulls score?


Southern Adelaide Region

A junior secondary lesson


Surface area = 22

• A rectangular prism is made from cubes.• It has a surface area of 22 square units.• Draw what the rectangular prism might look

like?


Enabling prompt:

• Arrange a small number of cubes into a rectangular prism, then calculate the volume and surface area.


Extending prompt:

• The surface area of a closed rectangular prism is 94 cm2.

• What might be the dimensions of the prism?


A consolidating task

• The surface area of a closed rectangular prism is 46 cm2.

• What might be the dimensions of the prism?


What does that task do?


What does the curriculum say?


YEAR 7• Establish the formulas for areas of rectangles, triangles and

parallelograms and use these in problem solving • Calculate volumes of rectangular prisms

YEAR 8• Choose appropriate units of measurement for area and volume and

convert from one unit to another • Find perimeters and areas of parallelograms, trapeziums, rhombuses

and kites • Investigate the relationship between features of circles such as

circumference, area, radius and diameter. Use formulas to solve problems involving circumference and area

• Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume

http://www.australiancurriculum.edu.au/Glossary?a=M&t=volume

http://www.australiancurriculum.edu.au/Glossary?a=M&t=volume


Year 9

• to solve problems involving circumference and area of circles and part circles, and the surface area and volume of prisms and cylinders

Describing the proficiencies• Understanding

– (connecting, representing, identifying, describing, interpreting, sorting, …)

• Fluency – (calculating, recognising, choosing, recalling, manipulating, …)

• Problem solving – (applying, designing, planning, checking, imagining, …)

• Reasoning – (explaining, justifying, comparing and contrasting, inferring,

deducing, proving, …)



Connecting to the proficiencies: • YEAR 7

– Understanding includes making connections between representations …– Fluency includes …calculating areas of shapes and volumes of prisms– Problem Solving includes formulating and solving authentic problems using

Measurements– Reasoning includes investigating strategies to perform calculations

efficiently• YEAR 8

– Understanding includes explaining measurements of perimeter and area– Fluency includes evaluating perimeters, areas of common shapes and their

volumes and three dimensional objects– Problem Solving includes formulating, and modeling practical situations

involving … areas and perimeters of common shapes,– Reasoning justifying the result of a calculation or estimation as reasonable

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Perimeter


The achievement standards:

• By the end of Year 7, students use formulas for the area and perimeter of rectangles and calculate volumes of rectangular prisms.

• By the end of Year 8, students … solve problems relating to the volume of prisms … convert between units of measurement for area and volume, … perform calculations to determine perimeter and area of parallelograms, rhombuses and kites.


http://www.australiancurriculum.edu.au/Glossary?a=M&t=Volume

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Volume



The lessons consist of• One or more challenging task(s)• One or more consolidating task(s) (see Dooley, 2012)• preliminary experiences that are pre-requisite but which do

not detract from the challenge of the tasks• supplementary tasks that offer the potential for

differentiating the experience through the use of – enabling prompts (see Sullivan, et al., 2009) which can reduce the

number of steps, simplify the complexity of the numbers, and vary the forms of representation for those students who cannot proceed with the task;

– extending prompts for students who complete the original task quickly which often prompt abstraction and generalisation of the solutions.

A probability task


First do this task

• 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.Sullivan MAT Nov 2013

Assume we have 10 people1 2 3 4 5 6 7 8 9 10

BP BP BP BP BP BP

MP3 MP3 MP3 MP3 MP3 MP3 MP3



BP BP BP BP BP BP




BP BP BP BP BP BP




BP BP BP BP BP BP



A consolidating task


• How many passengers might be on the train? • What is the maximum and minimum number of

possibilities for people who have both a backpack and an MP3 player?

• Represent each of your solutions in two different ways.


An enabling prompt

• On a train, there are 10 people. • Six of the people have a backpack, and 7 of the people

have an MP3 player. • How many people might have both a backpack and an

MP3 player? • What is the smallest possible answer for this? • What is the largest possible answer?


An extending prompt

• On a train, the probability that a passenger has a backpack is 2/3, and the probability that a passenger has an MP3 player is 2/7.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.


Student preferences for teaching approaches

A container and 3 eggs weighs 170 grams. The same container and 5 eggs weighs 270 grams.

What is the weight of the container?

PRE TEST

Answer Response %

7 g 21 6%

50 g 98 29%

30 g 55 16%

20 g 167 49%

Total 341 100

POST TEST

Answer Response %

7 g 8 3%

50 g 63 24%

30 g 33 12%

20 g 163 61%

Total 267 100%

I prefer … much harder than the egg.

I prefer …about as

hard as the egg.

I prefer … much easier

than the egg.

I prefer …egg question through working by

myself47 67 5 119

I prefer … egg question through working with

other students 22 125 27 174

I prefer …egg question by listening to

explanations from the teacher before I work

on the question

4 25 18 47

Total 73 217 50 340

PEP Nov 25 symposium

The “Advanced” (generic) statements

• Understanding– I chose, used and showed relevant ideas and

connected them together. I used mathematical words correctly

• Fluency– My working out was complete with no errors, I

used appropriate formulas if they were needed, and I presented calculations efficiently, incorporating relevant shortcuts


• Problem solving – I explained clearly how I planned and solved the

problem, my method was creative and I checked that my solution(s) had no errors.

• Reasoning– The steps I took are shown, and I used examples

to explain and justify my thinking.

You need a MYKI card before you can travel on public transport in

Melbourne. It costs $4 to buy a MYKI card and you need to put extra cash on the card to travel. If each journey costs

$2.50, what is the total cost of 6 journeys?

Answer Response %$15 511 57%

$6.50 64 7%$19 289 32%$10 28 3%Total 892 100%

PRE TEST

Answer Response %

$15 228 47%

$6.50 26 5%

$19 209 43%

$10 19 4%

Total 482 100%

POST TEST

I prefer …much harder

than MYKI

I prefer … about as hard as MYKI

I prefer … be much

easier than MYKI

I prefer …questions like the MYKI question working by myself

195 84 11 290

I prefer …MYKI question working with

other students38 81 14 133

I prefer …the MKYI question by listening to explanations from the teacher before I work

on the question

9 26 12 47

Total 242 191 37 470


• Missing number multiplicaiton• Patterns with remainders• Jigsaw• SA = 22• 2/3 and 201/301


Curriculum documentation should presumably inform …

• School planning• The planning of the program for the year in

mathematics by level• The planning of units of work (lesson

sequences)• The planning of teaching of lessons and

assessment of student learning

The Curriculum includes content descriptions, such as …

• Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292)


http://www.australiancurriculum.edu.au/Curriculum/ContentDescription/ACMSP292

http://www.australiancurriculum.edu.au/Curriculum/ContentDescription/ACMSP292

An example of a task that can be made into a lesson


• 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.



… and proficiencies

• Problem Solving includes … using two-way tables and Venn diagrams to calculate probabilities

• Reasoning includes justifying the result of a calculation or estimation as reasonable, deriving probability from its complement …

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Probability


… and achievement standards

• By the end of Year 8, students … model authentic situations with two-way tables and Venn diagrams. They choose appropriate language to describe events and experiments …

• Students … determine complementary events and calculate the sum of probabilities

http://www.australiancurriculum.edu.au/Glossary?a=&t=Describe

• Some people came for a sports day. • When the people were put into groups of 3

there was 1 person left over. • When they were lined up in rows of 4 there

were two people left over. • How many people might have come to the

sports day?

SA Sullivan and Aulert 2013

Some “enabling” prompts

• Some people came for a sports day. When they were lined up in rows of 4 there were two people left over. How many people might have come to the sports day?

• • Some people came for a sports day. When the

people were put into groups of 3 there was no-one left over. When they were lined up in rows of 4 there was no-one left over. How many people might have come to the sports day?


An extending prompt

• Some people came for a sports day. When the people were put into groups of 3 there was 1 person left over.

• When they were lined up in rows of 4 there was 1 person left over.

• When they were lined up in columns of 5 there was 1 person left over.

• How many people might have come to the sports day?


The “consolidating” task

• I have some counters. • When I put them into groups of 5 there was 2

left over. • When they were lined up in rows of 6 there

was the same number in each column and none left over.

• How many counters might I have?



How does that lesson connect to algebra?


Multiplication content descriptions• Year 4: Develop efficient mental and written

strategies and use appropriate digital technologies for multiplication and for division where there is no remainder

• Year 5: Solve problems involving division by a one digit number, including those that result in a remainder

• Year 6: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

http://www.australiancurriculum.edu.au/Glossary?a=M&t=multiplication+

http://www.australiancurriculum.edu.au/Glossary?a=M&t=remainder

http://www.australiancurriculum.edu.au/Glossary?a=M&t=number

http://www.australiancurriculum.edu.au/Glossary?a=M&t=remainder


Patterns

• Explore and describe number patterns resulting from performing multiplication (ACMNA081)

• Solve word problems by using number sentences involving multiplication or division where there is no remainder (ACMNA082)

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Number

http://www.australiancurriculum.edu.au/Curriculum/ContentDescription/ACMNA081


http://www.australiancurriculum.edu.au/Glossary?a=M&t=Multiplication%20

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Remainder

http://www.australiancurriculum.edu.au/Curriculum/ContentDescription/ACMNA082


Connecting to the proficiencies

• Year 5: • Understanding includes making connections between

representations of numbers, …• Fluency includes … using estimation to check the

reasonableness of answers to calculations • Problem Solving includes formulating and solving

authentic problems using whole numbers … • Reasoning includes investigating strategies to perform

calculations efficiently, continuing patterns …


Year 6:

• Understanding includes describing properties of different sets of numbers, … and making reasonable estimations

• Fluency includes … using brackets appropriately, • Problem Solving includes formulating and solving

authentic problems …• Reasoning includes explaining mental strategies

for performing calculations, describing results for continuing number sequences



The achievement standards:

• Year 5: By the end of Year 5, students solve simple problems involving the four operations using a range of strategies. They check the reasonableness of answers using estimation and rounding.

• Year 6: By the end of Year 6, students …solve problems involving all four operations with whole numbers. They write correct number sentences using brackets and order of operations.

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Rounding


http://www.australiancurriculum.edu.au/Glossary?a=M&t=Order%20of%20operations


Where is the “ceiling”?The lowest possible (though not necessarily correct) solution is 6 (Largest option out of 4 plus 2 and 3 plus 1) so we know the solution must be 6 or larger.Consider integers from 6 onwards ie 6+n (n is an integer larger than or equal to zero)The lowest number of people at sports day will be the lowest value of n for which

• (6+n-1)/3 AND (6+n-2)/4 are both integers • • (6+n-1)/3 = (5+n)/3 and (5+n)/3 is an integer when n=1, 4, 7…• (6+n-2)/4 = (4+n)/4 and (4+n)/4 is an integer when n=0, 4, 8…• • Therefore n=4 and the lowest number of people at sports day is 10.• • I’ve tested this approach on a couple of other examples and seems OK so could be

expressed in general terms if we really wanted to…

AVAILABLE TO DOWNLOAD FREE FROM

http://research.acer.edu.au/aer/13/ aer


Documents

The potential of posing more challenging mathematics tasks and ways of supporting students to engage in such tasks.