ESM410 Assignment 1:
Problem Pictures Task - Creating openended questions
Student Name:
Amy McNeill
Student Number:
800364143
Campus:
Burwood
PLAGIARISM AND COLLUSION Plagiarism occurs when a student passes off as the student’s own work, or
copies without acknowledgement as to its authorship, the work of any other person. Collusion occurs when a student
obtains the agreement of another person for a fraudulent purpose with the intent of obtaining an advantage in
submitting an assignment or other work. Work submitted may be reproduced and/or communicated for the purpose of
detecting plagiarism and collusion.
DECLARATION I certify that the attached work is entirely my own (or where submitted to meet the requirements of
an approved group assignment is the work of the group), except where material quoted or paraphrased is
acknowledged in the text. I also certify that it has not been submitted for assessment in any other unit or course.
SIGNED:
DATE: 26th August 2012
An assignment will not be accepted for assessment if the declaration appearing above has not been signed by the
author.
YOU ARE ADVISED TO RETAIN A COPY OF YOUR WORK UNTIL THE ORIGINAL HAS BEEN
ASSESSED AND RETURNED TO YOU.
Assessor’s Comments: Your comments and grade will be recorded on the essay itself. Please ensure your name
appears at the top right hand side of each page of your essay.
Rationale for the use of problem pictures in the classroom
The implementation of the AusVELS curriculum will occur next year in primary schools across Victoria, with the aim
of encouraging teachers to ‘help students become self-motivated, confident learners through inquiry and active
participation in challenging and engaging experiences’ (Victorian Curriculum and Assessment Authority, 2012, p.2).
The use of effective open-ended problems provides an approach to the teaching and learning of mathematics that
addresses this aim, as such tasks provide opportunities for engaging all students in successful learning experiences,
while also allowing students to approach the task at varying levels and in varying ways (Sullivan, Mousley &
Zevenbergen, 2005). Open-ended problems also have added advantages of giving students ‘opportunities to
investigate the problem context, make decisions, generalise, seek connections, and identify alternatives’ (Sullivan,
Mousley & Zevenbergen, 2005, p.107).
A problem picture is defined as ‘a photograph of an object, scene or activity that is accompanied by one or more
open-ended mathematical word problems based on the context of the photo’ (Bragg & Nicol, 2011, p.4). Using openended problem pictures provides opportunities for teachers to ‘pose new kinds of problems: those that are openended and grounded in images of real-life activities’ (Bragg & Nicol, 2008, p.202). This creates relevance for students
as they develop an understanding of the connection between mathematics in the classroom and in everyday, real
world contexts (Bragg & Nicol, 2011). Through connecting mathematical concepts to student’s worlds and
experiences, the likelihood of students remembering and applying concepts in future situations is increased
(Carpenter, Frank & Levi, 2003). Another benefit stated by Phillips (2012) includes the ability for problem pictures to
challenge students at a range of levels. Furthermore, Bragg & Nicol (2011) assert that problem pictures ‘can
ultimately enhance an educator’s ability to connect with mathematics in ways that open possibilities for seeing
maths differently’ (p.3). Using open-ended problem pictures in the classroom provides teachers with a strategy for
achieving the aforementioned aim of the AusVELS curriculum, while also developing students’ awareness of
mathematics around them (Bragg & Nicol, 2011).
References:
Bragg, L & Nicol, C 2008, ‘Designing open-ended problems to challenge preservice teachers’ views on mathematics
and pedagogy, in PME 32 : Mathematical ideas : history, education and cognition : Proceedings of the 32nd
Conference of the International Group for the Psychology of Mathematics Education, International Group for
the Psychology of Mathematics Education, Morelia, Mexico, pp. 201-208.
Bragg, L & Nicol, C 2011, ‘Seeing mathematics through a new lens: using photos in the mathematics classroom’, The
Australian Mathematics Teacher, vol. 67, no. 3, pp. 3-9.
Carpenter, T, Franke, M, & Levi, L 2003, Thinking mathematically, Heinemann, Portsmouth.
Phillips, R 2012, ‘Problem pictures’, retrieved 20 August 2012.
http://www.problempictures.co.uk/
Sullivan, P, Mousley, J & Zevenbergen, R 2005, ‘Increasing access to mathematical thinking’, The Australian
Mathematical Society Gazette, vol. 32, no. 2, pp. 105-109.
Victorian Curriculum and Assessment Authority 2012, ‘The AusVELS curriculum: Mathematics’, VCAA, Melbourne.
Problem Picture 1
Location:
Flinders Street Station, Melbourne
Problem Picture 1 - Questions
Question 1: Prep
Lower Primary
Choose donuts that are alike to create groups of donuts. Compare the groups of donuts - how are these donuts
similar? How are they different? Which groups have more and which groups have less?
Answers to Question 1
In responding to this question, students could cut out the donuts in this photo and order them into groups of similar
objects.
1) Pink, brown, yellow and no icing:
Pink Icing
Brown Icing
Yellow Icing
No Icing
The donuts in these groups are the same because they all have the same colour icing
The donuts in these groups are not the same because they are different sizes and shapes.
There are more donuts without icing than with pink icing
There are less donuts with brown icing than yellow icing
2) Big donuts and small donuts:
Big donuts
Small donuts
The donuts in these groups are the same because they are all the same size
The donuts in these groups are not the same because they have different colours of icing
There are more big donuts than small donuts
There are more small donuts without icing than donuts with icing
3) Donuts with holes and donuts without holes:
Donuts
Donuts with holes in the centre
without holes
in the centre
The donuts in these groups are the same because they are all the same shape
The donuts in these groups are not the same because they are different sizes
There are more donuts without holes than donuts with holes
There are less iced donuts without holes than the iced donuts with holes
Mathematical intent:
Content strand/s, year, definition and code
Number and Algebra: Foundation
- Compare, order and make correspondences between collections, initially to 20, and explain reasoning
(ACMNA289)
- Sort and classify familiar objects and explain the basis for these classifications. Copy, continue and create patterns
with objects and drawings (ACMNA005)
Proficiency strand/s
Problem Solving includes using materials to model authentic problems, sorting objects, using familiar counting
sequences to solve unfamiliar problems, and discussing the reasonableness of the answer
Reasoning includes explaining comparisons of quantities, creating patterns, and explaining processes for indirect
comparison of length
Question 2: Grade 3
Middle Primary
Collect information about the donuts in this photo and organise the donuts into categories. Create a data display
that presents your information.
Answers to Question 2
1) Donuts organised according to icing:
Pink Icing
Brown Icing
Data Display - Table
Type of Donut:
Yellow Icing
No Icing
Number of Donuts:
Pink Icing
6
Brown Icing
6
Yellow Icing
10
No Icing
16
TOTAL
38
2) Donuts organised according to size and shape:
Big Circular Donuts
Big Donuts with holes in the centre
Small Donuts with holes in the centre
Data Display – Table
Type of Donut:
Big Circular Donuts
Big Donuts with holes in the centre
Small Donuts with holes in the
centre
Icing Colour:
Number of Donuts
Pink
2
Brown
2
Yellow
2
Pink
4
Brown
2
Yellow
5
No Icing
10
Brown
2
Yellow
3
No Icing
6
TOTAL DONUTS
3) Donuts organised according to size, icing and filling:
Large
Large
Large
Large
Small
Pink
Pink
Brown
Brown
Brown
JamDonuts
JamDonuts
Donuts
Filled
Filled
Donuts
Donuts
Large
Yellow
JamFilled
Donuts
38
Large
Yellow
Donuts
Small
Yellow
Donuts
Large Plain
Donuts:
Small
Plain
Donuts:
Number of Donuts
12
10
8
6
4
2
Number of Donuts
0
Mathematical intent:
Content strand/s, year, definition and code
Statistics and Probability: Level 3:
Collect data, organise into categories and create displays using lists, tables, picture graphs and simple column graphs,
with and without the use of digital technologies (ACMSP069)
Proficiency strand/s
Problem Solving includes formulating and modelling authentic situations involving planning methods of data
collection and representation, making models of three dimensional objects and using number properties to continue
number patterns
Question 3: Grade 6
Upper Primary
Select your favourite type of donut and work out the probability of choosing one of these donuts at random. How
could you show this information using simplified fractions, decimals and percentages?
Answers to Question 3
1) Large Pink Jam-Filled Donut:
2 of this type of donut out of the total 38 donuts  Probability of picking one of these donuts at random:
Probability = 2/38 = 1/19, 0.053, 5.3%
2) Small Plain Donut:
6 of this type of donut out of the total 38 donuts  Probability of picking one of these donuts at random:
Probability = 6/38 = 3/19, 0.158, 15.8%
3) Large Yellow Donut:
5 of this type of donut out of the total 38 donuts  Probability of picking one of these donuts at random:
Probability = 5/38, 0.132, 13.2%
Mathematical intent:
Content strand/s, year, definition and code
Make connections between equivalent fractions, decimals and percentages (ACMNA131)
Describe probabilities using fractions, decimals and percentages (ACMSP144)
Proficiency strand/s
Understanding includes describing properties of different sets of numbers, using fractions and decimals to describe
probabilities, representing fractions and decimals in various ways and describing connections between them, and
making reasonable estimations
Problem Solving includes formulating and solving authentic problems using fractions, decimals, percentages and
measurements, interpreting secondary data displays, and finding the size of unknown angles
Report of Trialling Problem Picture 1 Question 2
Child’s pseudonym, age and grade level:
Child A, 9 years of age, Grade 3
Original Question:
When trialling this problem picture, the initial wording for this Level 3 question was:
‘Create a display that shows the information in this photo by organising the donuts into categories’.
Child’s response to the question:
Initial attempt at putting objects into categories
(Sp = sprinkles)
Organising donuts into categories
Data Display - table
Reflection on child’s response:
As indicated by this student’s responses, some initial confusion was evident in both how to organise the objects into
categories, and how to best present the data located within this photo. However, once I began discussing the
question with this student, and explicitly addressed the fact that the purpose of organising the donuts into
categories was to assist in making sense of the data, the process became clearer for the student and she was then
able to develop a way of efficiently recording and presenting that data both in the form of diagrams, and as a display
in the form of a table. Given the context where this question was trialled in an informal setting without any prior
discussion or information regarding the mathematical concepts addressed in this question, it is hard to determine
whether this initial difficulty is indicative of a weakness in this student’s ability to interpret the photo and collect the
relevant data. Sullivan, Mousley and Zevenbergen (2005) state the importance of teachers explicitly explaining the
processes in the classroom to their student, including ‘the mathematical focus and purpose of the task’ (p.107). Once
this was made explicit to the student, and the wording of my question was adjusted, the student was able to
complete the task independently.
In referring to the student’s mathematical understanding, this question highlighted significant strengths in organising
objects into categories, and an ability to extend this organising by developing the sub-categories of small and large.
She was also competent at finding an effective way to record the data from the photo, and at displaying that data
she had collected in the form of a table. As such, the mathematical intent of the question was addressed with
reference to the content strand stating that students at this level ‘collect data, organise into categories and create
displays’ (Victorian Curriculum and Assessment Authority, 2012, p.23), all of which are demonstrated in this
student’s responses. The proficiency strand at Level 3 states that ‘problem solving includes formulating and
modelling authentic situations involving planning methods of data collection and representation’ (VCAA, 2012, p.21),
which is again demonstrated in this student’s responses.
Bragg & Nicol (2011) suggest that the process of developing open-ended problem photos is cyclical, with pre-service
teachers rethinking the problem posed as they are devising a question. In this situation, the wording of this problem
was rethought as a result of reflecting on this trialling process, and the final version of this question has been
modified to:
‘Collect information about the donuts in this photo and organise the donuts into categories. Create a data display
that presents your information’.
Putting steps into the question by ordering the words differently seemed to assist the student in working out a
method for responding to this question. As Sullivan, Mousley and Zevenberg (2005) point out, it would be necessary
for the mathematic terminology to also be explicitly addressed, with reference in this question to the meanings of
terms such as ‘data display’.
References:
Bragg, L & Nicol, C 2011, ‘Seeing mathematics through a new lens: using photos in the mathematics classroom’, The
Australian Mathematics Teacher, vol. 67, no. 3, pp. 3-9.
Sullivan, P, Mousley, J & Zevenbergen, R 2005, ‘Increasing access to mathematical thinking’, The Australian
Mathematical Society Gazette, vol. 32, no. 2, pp. 105-109.
Victorian Curriculum and Assessment Authority 2012, ‘The AusVELS curriculum: Mathematics’, VCAA, Melbourne.
Problem Picture 2
Location:
Federation Square, Melbourne
Problem Picture 2 - Questions
Question 1: Grade 2
Lower Primary
What 2-dimensional shapes can you see in this picture? Identify the names and important parts of these shapes and
outline the shapes you can see.
Answers to Question 1
1) Rectangle: 4 straight lines and 2 pairs of equal sides, 4 corners and 4 edges.
2) Kite: 4 straight lines and 2 pairs of equal sides, 4 corners and 4 edges.
3) Triangle: 3 straight lines and 2 equal sides, 3 corners and 3 edges.
1)
2)
3)
Mathematical intent:
Content strand/s, year, definition and code
Measurement & Geometry: Level 2:
Describe and draw two dimensional shapes, with and without digital technologies (ACMMG042)
Proficiency strand/s
Problem Solving includes formulating problems from authentic situations, making models and using number
sentences that represent problem situations, and matching transformations with their original shape
Question 2: Grade 4
Middle Primary
Outline the 2-dimensional shapes you can see in this photo. Can you combine 2 or more of these shapes to create
new shapes? Can you split shapes that are made up of more than one shape to create new shapes? Outline these
new shapes in the photo in different colours and describe the shapes you found.
Answers to Question 2
1) Combining a quadrilateral with a second quadrilateral to create a parallelogram; splitting a triangle to create a
kite and two quadrilaterals.
Quadrilateral: 4 sided shape, Parallelogram: 4 sided shape with opposite sides that are equal and parallel,
Triangle: 3 sided shape with two equal sides; Kite: 4 sided shape with 2 pairs of equal sides.
2) Combining a triangle and a quadrilateral to create a right-angle triangle; splitting a triangle to create three smaller
triangles and two quadrilaterals.
Quadrilateral: 4 sided shape; Triangle: 3 sided shape; Right-angled Triangle: two perpendicular lines creating a 90°
angle.
3) Combining a quadrilateral with a second quadrilateral to create a rectangle; splitting a square into a triangle and
a quadrilateral.
Quadrilateral: 4 sided shape; Rectangle: 4 sided shape with opposite and equal parallel sides; Square: 4 equal
sided shapes; Triangle: three-sided shape.
Mathematical intent:
Content strand/s, year, definition and code
Measurement & Geometry Level 4:
Compare and describe two dimensional shapes that result from combining and splitting common shapes, with and
without the use of digital technologies (ACMMG088)
Proficiency strand/s
Understanding includes making connections between representations of numbers, partitioning and combining
numbers flexibly, extending place value to decimals, using appropriate language to communicate times, and
describing properties of symmetrical shapes
Question 3: Grade 6
Upper Primary
Outline and investigate angles that you can see in this photo – classify and explain your reasoning when naming the
angle by estimating and measuring the size of the angle in degrees.
Answers to Question 3
1) Acute angle because it is less than a 90° right angle
Estimate = 50°
Size of Angle = 53°
2) Right angle because the lines are perpendicular and a right angle is equal to 90°
Estimate = 90°
Size of Angle = 90°
3) Obtuse angle because the angle is greater than a 90° right angle
Estimate = 115°
Size of Angle = 112°
Mathematical intent:
Content strand/s, year, definition and code
Measurement & Geometry: Level 6:
Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite
angles. Use results to find unknown angles (ACMMG141)
Proficiency strand/s
Problem Solving includes formulating and solving authentic problems using fractions, decimals, percentages and
measurements, interpreting secondary data displays, and finding the size of unknown angles
Report of Trialling Problem Picture 2 Question 2
Child’s pseudonym, age and grade level:
Child B, 10 years of age, Grade 4
Original Question:
The original wording trialled for this problem picture for Level 4 question was:
‘Outline the 2-dimensional shapes you can see in this photo. Can you combine 2 or more of these shapes to create
new shapes? Can you split these shapes made up of more than one shape to create new shapes?’
Child’s response to the question:
2D shapes outlined in black texta and blue pen
Summary of shapes found, and examples of combining and splitting shapes
Reflection on child’s response:
This grade 4 student’s final responses to this problem picture were as expected, however during the process of
answering this question, the student required scaffolding to complete components of the question. This student
chose to work through each sentence of this question sequentially, beginning with the identification of 2dimensional shapes. The process from there involved the supporting the student and working with her through each
of these identified shapes, assessing if each one was an example of an individual shape or a composite shape.
Sullivan, Mousley and Zevenbergen (2005) state that a key aspect of developing effective open-ended tasks is
providing ‘prompts to support students who are experiencing difficulty’ (p.105). With this additional support in
processing the problem, the student was then able to combine individual adjoining shapes to create new shapes.
This student was also able to split the composite shapes by identifying the familiar shapes she saw within the original
shape.
This trialling process demonstrated strengths in the student’s understanding and ability to identify and describe twodimensional shapes, and an apparent difficulty in understanding the way that shapes could combine or split to
create new shapes. It was an interesting concept for this student to consider the occurrence of shapes in everyday
settings, and to use this photo as a tool for seeing ways that two or more shapes could join or separate to create
new shapes. This may be a result of a lack of familiarity with questions framed in this way, as she was able to name
and describe each of the two dimensional shapes she found during this task. She required assistance with changing
the lens that she was looking through when honing in on part of a composite shape, and when broadening the focus
to include surrounding shapes that could combine together. The student’s initial response to this problem picture
may be key, as her comment of ‘I hate doing shapes in maths’ is telling. The problem with this question may be due
to the issue raised by Sullivan (2010), whereby students may respond negatively to a task due to the context being
unfamiliar and/or uninteresting for the student.
The process described above indicates that the question addressed some of the mathematical intent of the question,
however the wording of the problem was changed to include directions to describe the shapes found. The content
strand states that students at this level ‘compare and describe two dimensional shapes that result from combining
and splitting common shapes’ (VCAA, p.27).
At the conclusion of this trialling process, I decided to reword the question to also include the direction for students
to use different colours for each aspect of the question, to assist in the process of highlighting the different shapes,
and the ways that they combine and split, in a more effective way than the response given during this trial. The
wording of the question was modified to instead read:
‘Outline the 2-dimensional shapes you can see in this photo. Can you combine 2 or more of these shapes to create
new shapes? Can you split shapes that are made up of more than one shape to create new shapes? Name and
describe these new shapes, and outline them in different colours.’
References:
Sullivan, P 2010, ‘Teaching mathematics to classes of diverse interests and backgrounds’, Primary Mathematics
Education CD-ROM, Deakin University, Geelong.
Sullivan, P, Mousley, J & Zevenbergen, R 2005, ‘Increasing access to mathematical thinking’, The Australian
Mathematical Society Gazette, vol. 32, no. 2, pp. 105-109.
Victorian Curriculum and Assessment Authority 2012, ‘The AusVELS curriculum: Mathematics’, VCAA, Melbourne.
Problem Picture 3
Location:
Boathouse Drive, Melbourne
Problem Picture 3 - Questions
Question 1: Grade 2
Lower Primary
I have paid $2.20 for my ticket. Using only the coins listed in this photo, what coins could I have used to buy my
ticket? Write and draw your responses and place your coins in order.
Answers to Question 1
1) $1.00 + $1.00 + 20¢ = $2.20¢
20¢
$1
$1
2) $1.00 + 20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢ = $2.20¢
10¢
10¢
20¢
20¢
20¢
20¢
20¢
$1
3) $2.00 + 10¢ + 10¢ = $2.20¢
10¢
10¢
$2
Mathematical intent:
Content strand/s, year, definition and code
Number & Algebra: Level 2:
Count and order small collections of Australian coins and notes according to their value (ACMNA034)
Proficiency strand/s
Problem Solving includes formulating problems from authentic situations, making models and using number
sentences that represent problem situations, and matching transformations with their original shape
Question 2: Grade 3
Middle Primary
If I was wanting to park here for three hours, how much would I have to pay? What combinations of coins could I use
if I had to use at least three different types of coins? Write and draw your combinations.
Answers to Question 2
1) $1.50 x 3 = $4.50
Combination 1: $2.00 + $1.00 + $1.00 + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ = $4.50
$2
$1
$1
10¢
10¢
10¢
10¢
10¢
2) $1.50 x 3 = $4.50
Combination 2: $2.00 + $2.00 + 20¢ + 20¢ + 10¢ = $4.50¢
$2
$2
20¢
10¢
20¢
3) $1.50 x 3 = $4.50
Combination 3: $1.00 + $1.00 + $1.00 + $1.00 + 20¢ + 10¢ + 10¢ + 10¢ = $4.50¢
$1
$1
$1
$1
20¢
10¢
10¢
10¢
Mathematical intent:
Content strand/s, year, definition and code
Number & Algebra: Level 3:
Represent money values in multiple ways and count the change required for simple transactions to the nearest five
cents (ACMNA059)
Proficiency strand/s
Problem Solving includes formulating and modelling authentic situations involving planning methods of data
collection and representation, making models of three dimensional objects and using number properties to continue
number patterns
Question 3: Grade 5
Upper Primary
Imagine you are wanting to park your car in this car park and you have an unlimited number of coins. Choose an
amount of time that you will stay – work out how much it will cost and create a table that shows the number of
combinations of coins you could use.
Answers to Question 3
1) 20 minutes = 50¢
Coins Used:
Number of Coins Needed:
50¢
1 x 50¢
20¢ + 20¢ + 10¢
2 x 20¢, 1 x 10¢
20¢ + 10¢ + 10¢ + 10¢
1 x 20¢, 3 x 10¢
10¢ + 10¢ + 10¢ + 10¢ + 10¢
5 x 10¢
2) 40 minutes = $1.00¢
Coins Used:
$1.00
50¢ + 20¢ + 20¢ + 10¢
50¢ + 20¢ + 10¢ + 10¢ + 10¢
50¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
20¢ + 20¢ + 20¢ + 20¢ + 20¢
20¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢
20¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢
20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
3) 1 hour = $1.50¢
Coins Used
Number of Coins Needed:
1 x $1
1 x 50¢, 2 x 20¢, 1 x 10¢
1 x 50¢, 1 x 20¢, 3 x 10¢
1 x 50¢, 5 x 10¢
5 x 20¢
4 x 20¢, 2 x 10¢
3 x 20¢, 4 x 10¢
2 x 20¢, 6 x 10¢
1 x 20¢, 8 x 10¢
10 x 10¢
Number of Coins
Needed:
$1.00 + 50¢
1 x $1, 1 x 50¢
$1.00 + 20¢ + 20¢ + 10¢
1 x $1, 2 x 20¢, 1 x 10¢
$1.00 + 20¢ + 10¢ + 10¢ + 10¢
1 x $1, 1 x 20¢, 3 x 10¢
$1.00 + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
1 x $1, 5 x 10¢
50¢ + 50¢ + 50¢
3 x 50¢
50¢ + 50¢ + 20¢ + 20¢ + 10¢
2 x 50¢, 2 x 20¢, 1 x 10¢
50¢ + 50¢ + 20¢ + 10¢ + 10¢ + 10¢
2 x 50¢, 1 x 20¢, 3 x 10¢
50¢ + 50¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
2 x 50¢, 5 x 10¢
50¢ + 20¢ + 20¢ + 20¢ + 20¢ + 20¢
1 x 50¢, 5 x 20¢
50¢ + 20¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢
1 x 50¢, 4 x 20¢, 2 x 10¢
50¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢
1 x 50¢, 3 x 20¢, 4 x 10¢
50¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
1 x 50¢, 2 x 20¢, 6 x 10¢
50¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
1 x 50¢, 1 x 20¢, 8 x 10¢
50¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
1 x 50¢, 10 x 10¢
20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 10¢
7 x 20¢, 1 x 10¢
20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢
6 x 20¢, 3 x 10¢
20¢ + 20¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
5 x 20¢, 5 x 10¢
20¢ + 20¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
4 x 20¢, 7 x 10¢
20¢ + 20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
3 x 20¢, 9 x 10¢
20¢ + 20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
2 x 20¢, 11 x 10¢
20¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
1 x 20¢, 13 x 10¢
10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢ + 10¢
15 x 10¢
Mathematical intent:
Content strand/s, year, definition and code
Number & Algebra: Level 5:
Pose questions and collect categorical or numerical data by observation or survey (ACMSP118)
Proficiency strand/s
Reasoning includes investigating strategies to perform calculations efficiently, continuing patterns involving fractions
and decimals, interpreting results of chance experiments, posing appropriate questions for data investigations and
interpreting data sets
Report of Trialling Problem Picture 3 Question 2
Child’s pseudonym, age and grade level:
Child C, 9 years of age, Grade 3
Original Question:
The initial wording for the trial of this Grade 3 level problem picture was:
‘If I was wanting to park here for three hours, how much would I have to pay? What are some of the combinations of
coins I could use if I had to use at least three different coins?’
Child’s response to the question:
Reflection on child’s response:
In responding to this question, this student demonstrated strengths in both developing multiple ways of
representing money in the forms of drawing the coins required to respond to the question, and in writing a list of the
coins that she chose to use. The student also demonstrated strengths in performing calculations using repeated
addition to ensure that the total of the coins matched the amount that the question required, and in coming up with
a number of solutions, working from straightforward combinations to more complicated arrangements as her
confidence with the task grew. As such, the student responded to the problem picture in the way that I had initially
intended.
Sullivan (2005) suggests using prompts to support students in responding to open-ended tasks, where ‘the choice of
appropriate enabling prompts is based on factors that may contribute to the complexity of a task’ (p.12). One such
factor discussed in this article is ‘the degree of abstraction or visualisation required’ (Sullivan, 2005, p.12). I was
concerned with this problem pictures that students may have difficulties with the visualisation of coins that the task
required. In anticipating this as a factor, I provided the student with a variety of coins to act as concrete materials as
a means of giving the student access to exploring and completing the question (Sullivan, 2005). This student was able
to work independently without requiring the coins as concrete materials to support her in answering this question.
This demonstrated a further strength in her ability to process the given information and in problem solving through
exploring a different approach to the task that enabled her to engage with the question.
While the Level 3 content strand stipulates that students at this level ‘represent money values in multiple ways and
count the change required for simple transactions to the nearest five cents’ (VCAA, 2012, p.22), the information
embedded in the photo specified different requirements, where 10 cents was the smallest coin that could be used to
purchase a ticket. As a result, this question did address the mathematical intent of the question despite this task not
requiring the inclusion of five cent coins. However, this element added to the interactive nature of this photo, where
‘the photo was essential to complete the problem’ (Bragg & Nicol, 2011, p.7).
In reflecting on the trialling process of this photo, the wording of the question was altered as a means of clarifying
the requirements of the question, while still ensuring that the problem was open-ended:
‘If I was wanting to park here for three hours, how much would I have to pay? What combinations of coins could I
use if I had to use at least three different coins?’
References:
Sullivan, P 2010, ‘Teaching mathematics to classes of diverse interests and backgrounds’, Primary Mathematics
Education CD-ROM, Deakin University, Geelong.
Sullivan, P, 2005, ‘Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms’,
International Journal of Science & Math Education (in press), retrieved 10 August 2012.
http://springerlink.metapress.com/content/u010160k22847126/?p=84249d720536452490b346cdf33b394c&pi=0
Victorian Curriculum and Assessment Authority 2012, ‘The AusVELS curriculum: Mathematics’, VCAA, Melbourne.
Overall Conclusion
Through the process of undertaking this Problem Picture task, I have developed a new appreciation for the
occurrence of mathematics in everyday items, locations and situations through changing the perspective in which I
view the world around me as being full of potential for authentic settings to cultivate mathematical understandings
and thinking. The simple task of having my digital camera with me when on an expedition in the city provided me
with many photos in which a variety of mathematical concepts could be developed and explored. After this step, I
followed the second approach outlined by Bragg & Nicol (2011) as starting with the photo, then posing problems
based on relevant curriculum content and proficiency strands for each level. Trialling these problem pictures with
both students and peers allowed me to clarify my wordings for the questions posed to ensure that the mathematical
intent was consistent with the set task.
The effective use of open-ended questions as presented by Sullivan (2010) and the specific approach of open-ended
problem pictures discussed by Bragg & Nicol (2011) have changed my perspectives on a new approach to catering
for diversity in the classroom and in the differentiation of tasks to allow all students to engage in the same task, with
the use prompts as means of scaffolding to support students experiencing difficulties and extend students who
complete the initial task readily. Through the course of this task, open-ended problem pictures have provided rich
opportunities for exploring mathematics for myself, the students involved in the trialling, and my peers who I tested
the questions on. The experience of completing these problems proved to be engaging, enjoyable and provided
challenges, as well as connecting the world around me to maths. This is invariably a fundamental connection that
these problem pictures assisted in developing.
The findings of Bragg & Nicol (2008) state that the process of pre-service teachers designing problem pictures
‘prompted a shift in their understanding of pedagogical approaches and ways in which they viewed mathematics’
(p.206-207). Bragg & Nicol (2011) also state that problem pictures have the potential for ‘broadening preservice
teachers’ pedagogical repertoire to incorporate maths in the environment’ (p.5). This has certainly proved to be true
for myself, where I now feel confident in employing this as a pedagogical approach that provides students with
opportunities to see and understand the connections between mathematics in the classroom context and in the
real-world environment (Bragg & Nicol, 2011). This task has also expanded my understanding of what is possible in
teaching mathematics and developing meaningful learning experiences for all students (Bragg & Nicol, 2009).
References:
Bragg, L & Nicol, C 2008, ‘Designing open-ended problems to challenge preservice teachers’ views on mathematics
and pedagogy, in PME 32 : Mathematical ideas : history, education and cognition : Proceedings of the 32nd
Conference of the International Group for the Psychology of Mathematics Education, International Group for
the Psychology of Mathematics Education, Morelia, Mexico, pp. 201-208.
Bragg, L & Nicol, C 2009, ‘Designing problems : what kinds of open-ended problems do preservice teachers pose?, in
PME 2009 : Proceedings of 33rd Annual Meeting of the International Group for the Psychology of Mathematics
Education, PME, Thessaloniki, Greece, pp. 4-225-4-232.
Bragg, L & Nicol, C 2011, ‘Seeing mathematics through a new lens: using photos in the mathematics classroom’, The
Australian Mathematics Teacher, vol. 67, no. 3, pp. 3-9.
Sullivan, P 2010, ‘Teaching mathematics to classes of diverse interests and backgrounds’, Primary Mathematics
Education CD-ROM, Deakin University, Geelong