esm assignment: problempicturestask( creating)open ... · jessica’mcdonald’ ’ 212161352’...

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures

ESM$%& Assignment *: Problem Pictures Task -‐ Creating open-‐

ended questions

Student Name: Jessica McDonald

Student Number: 212161352

Campus: Burwood

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures Rationale for the use of problem pictures in the classroom According to Sparrow (2008) the use of problem pictures within the primary classroom, can have positive effects on both the students as well as the teacher. It is through their real world relevance, in which students tend to benefit the most from problem pictures, as they require students to apply mathematical concepts (Sparrow 2008) contextually. It is through student’s ability to engage with mathematical concepts (Sparrow 2008) in which they can develop deeper levels of thinking. In fact, Sparrow (2008) highlights the importance in having students connect classroom happenings with contexts experience by people outside the classroom, allowing them to understand how what they are learning is relevant to them. Whilst society is changing and education is transforming, Sulivan (2005) highlights problem pictures ability to encourage the shift towards active learning and away from the rote-‐learning environment. Through the use of problem pictures in the classroom, teachers can increase student curiosity about mathematics (Bragg and Nicol 2011) in the environment external to the school, and develop students’ desire to explore the countless solutions (Bragg and Nicol 2011). To further reiterate the benefits behind having students engage with their learning, Sullivan (2005) identifies the ability of open-‐ended problem pictures to foster individual differences as they allow for multiple entry points (Bragg and Nicol 2011).

Throughout this assignment, I have developed a huge appreciation for the benefits in which open-‐ended picture problems can provide students. Throughout my future career as a teacher I will endeavour to incorporate as open-‐ended problem pictures within my numeracy classroom in order to create authentic experiences. In fact, Bragg and Nicol (2011) reiterate the importance of involving the real world within the numeracy classroom as they highlight the need for teachers to see external environments as a sequence for potential rich, mathematical experiences. Within my future teaching, I will aspire to relate as much of the mathematical curriculum, to familiar contexts (Bragg and Nicol (2011) in hope to engage my students further. By incorporating a diverse range of authentic experiences such as that of an open-‐ended problem picture, I hope that I can maximise the learning opportunities (Sullivan, Mousley ad Zevenbergen 2005) for my students and create engaging learning sequences in hope to increase my students willingness to explore mathematical concepts (Sullivan, Mousley ad Zevenbergen 2005). Through my engagement in this assignment I believe that I am more equipped to develop and implement problem picture lessons to assist my students in developing deeper levels of thinking through the construction of knowledge as appose to being passive learners (Inoue and Buczynski 2011).

References for the rationale: 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.

Inoue, N and Buczynski, S 2011, ‘You Asked Open-‐Ended Questions, Now What? Understanding the Nature of Stumbling Blocks in Teachign Inquiry Lessons’, The Mathematics Educator, vol.20, no.2, pp. 10-‐23.

Sparrow, L 2008, ‘Real and relevant mathematics: is it realistic in the classroom? Len Sparrow reminds us what real and relevant mathematics means to children and outlines how teachers can plan purposeful activities and provide relevant contexts in the classroom’, Australian Primary Mathematics Classroom, vol.13, no.2, p.4.

Sullivan, P, Mousley, J & Zevenbergen, R 2005, ‘Increasing access to mathematical thinking’ Australian Mathematical Society Gazette, vol. 32, no. 2, pp.105-‐109, The Society, St Lucia, Qld.

Sullivan, P 2005, ‘Teaching mathematics to classes of diverse interests and backgrounds’ [online video], Deakin University.


Problem Picture 1 Location: My house, East Bentleigh, a photo of the cupcakes for my mums birthday journeys

Problem Picture 1 -‐ Questions Grade level: Two Question 1 If I have 24 Cupcakes, and I wanted to transport them in a box, what are some of the ways I could arrange the cupcakes in the box.

Answers to Question 1


AusVELS -‐ Number and Algebra Content strand/s, year, definition and code Number and Algebra, Number and Place Value, Year 2:

• Recognise and represent multiplication as repeated addition, groups and arrays ACMNA031

Enabling Prompt I have 12 cupcakes and need to transport them in a box. How could I place the cupcakes in the box so that none are on top of each other?

Answers to Enabling Prompt

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures AusVELS Content strand/s, year, definition and code Number and Algebra, Number and Place Value, Year 2:


Justification for change to the original question State the modification you made to the original question: I modified the question by using a smaller number. This in turn also assisted in making the task more attainable as it required less steps.

Why did you select this modification to make to the problem? I selected a smaller number, as I knew that the number 12 had fewer factors than 24, therefore involving less box possibilities. 12 is also a number which would be more familiar to students who require further simplification.

Extending Prompt I have 24 cupcakes and need to transport them to a party. If each cupcake is 5cm long and 5 cm wide, what are some of the box sizes I could use?

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures Answers to Extending Prompt AusVELS Content strand/s, year, definition and code Number and Algebra, Number and Place Value, Year 2:


Number and Algebra, Number and Place Value, Year 3:

• Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies ACMNA057

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures Justification for change to the original question State the modification you made to the original question: To extend the original question, I specified the size of the cupcakes so that they had to consider the array of the cupcakes as well as the actual dimensions of the box.

Why did you select this modification to make to the problem? I chose this modification as I thought that those who are more competent in mathematics will know their factors of 24 a lot more quickly than others. Therefore, in order to complicate the task further, I created a more problem-‐based question where they also had to take the size of the cupcakes into consideration.

Cross-‐Curriculum Links This image could be used to help students understand the process of a chemical reaction. The students could look at the photo and write a hypothesis about the type of chemical reaction, which occurs when cooking. They could then perform a science experiment where they investigate the effects that mixing the cake mix ingredients together.

AusVELS -‐ Cross-‐curriculum Cross-‐curriculum area, Content strand/s, year, definition and code Science, Chemical Science, Year 2:

• Different materials can be combined, including by mixing, for a particular purpose (ACSSU031)


Report of Trialling Problem Picture 1 Child’s pseudonym, age and grade level: Rosie, 8 years old, Grade 2

Original Question: If I have 24 Cupcakes, and I wanted to transport them in a box, what are some of the ways I could arrange the cupcakes in the box.

Child’s response to the question:

Reflection on child’s response: The original question asked the student, ‘If I have 24 Cupcakes, and I wanted to transport them in a box, what are some of the ways I could arrange the cupcakes in the box’. After reading the question aloud to ensure that the student understood the vocabulary within the question, it was clear that by looking at the photo and comprehending the question being asked, the student understood the requirements and began drawing possible arrangements.

After watching the student complete the task, the task was completed as expected through the use of a realistic representation (Hussey and Smith 2003). However, when completing the tasks myself, I used my knowledge of factors to complete the problem. It was evident that as Rosie worked through the problem she was using her knowledge of one number and building on that information. For example, she knew that 3 went into 24 and simply drew three rows of cupcakes and simply added one to each row until she had reached 24. Due to the lack of understanding behind her multiplication, this caused Rosie to miscount the cupcakes in one group.

Rosie appeared to demonstrate a strong understanding behind the concept of creating an array. She was able to recognise that although each box required 24 cupcakes, there were 3 different perimeter possibilities (Reys et al. 2012) when arranging the cupcakes, exhibiting a solid readiness for multiplication (Reys et al. 2012). Through Rosie’s responses to the task, she demonstrated a thorough understanding of the multiplicative structure (Reys et al. 2012), as she was able to create an arrangement of countable objects (Reys et al. 2012) within her representation. Through Rosies diagram, it was evident however, that she didn’t count the cupcakes individually (Reys et al. 2012) as she completed the task as one of her responses lacked 3 cupcakes. This question incorporated the mathematical component of the question, as the main focus was to create various arrays to represent possible layouts for transporting cupcakes. In grade 2, The Australian Curriculum Assessment and Reporting Authority [ACARA] (2013) states that students should be proficient in acknowledging and expressing multiplication as an array. This is apparent within the open-‐ended problem picture

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures task, as it required the students to identify the different ways in which the cupcakes could be arranged within a box when transporting them.

Based on my reflections, I wouldn’t make any direct modifications to the question, as the student was able to identify possible arrays for creating the number 24. However, due to some initial clarification of how many boxes they are aloud to transport the cupcakes in, I believe there needs to be further guidance within the question relating to this. By rephrasing the question to specify that the cupcakes must be transported in one box, it will allow me to avoid particular misunderstandings (Hussey and Smith 2003) when constructing arrays.

Rephrased Question: If I have 24 Cupcakes, and I wanted to transport them in one box, what are some of the ways I could arrange the cupcakes in the box’

References for reflection on the trial of question 1: Australian Curriculum Assessment and Reporting Authority 2013, The Australian Curriculum, Retrieved August 23, 2015, <http://www.australiancurriculum.edu.au>.

Hussey, T & Smith, P 2003, ‘The Uses of Learning Outcomes’, Teaching in Higher Education, vol. 8, no.3, pp.357-‐368.

Reys, R, Lindquist, M, Lambdin, D, Smith, L, Rogers, A, Falle, J, Frid, S & Bennett, S 2012, Helping children learn mathematics (1st Australian ed’n). Milton, Qld: John Wiley and Sons.


Problem Picture 2

Location: Chamford Gymnastics Club, Murrumbeena

Problem Picture 2 -‐ Questions

Grade level: Two Question 2: What shapes can you see in the photo? Draw two of the shapes in the photo and label their properties, then write down whether they are 2D or 3D



AusVELS -‐ Measurement and Geometry Content strand/s, year, definition and code Measurement and Geometry, Shape, Year 2:

• Describe and draw two-‐dimensional shapes, with and without digital technologies ACMMG042

• Describe the features of three-‐dimensional objects ACMMG043

Enabling Prompt Can you draw and name any 2D or 3D shapes that you can see in the photo?


AusVELS Content strand/s, year, definition and code Measurement and Geometry, Shape, Year 1:

• Sort, describe and name familiar two-‐dimensional shapes and three-‐dimensional objects in the environment ACMMG009

Measurement and Geometry, Shape, Year 2:



Justification for change to the original question State the modification you made to the original question: I simplified the activity to require the students to draw the shape first and then label their names.

Why did you select this modification to make to the problem? I made this modification in order to make the task more attainable. It is more attainable because all they have to do is draw the shapes and then name them. Their difficulty lies in the labelling of the shapes.

Extending Prompt What shapes can you see in the photo? Draw two shapes and identify their numerical properties. You might like to include: number of faces, edges, corners, or vertices.

Answers to Extending Prompt

AusVELS Content strand/s, year, definition and code Measurement and Geometry, Shape, Year 2:


• Describe the features of three-‐dimensional objects ACMMG043

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures Justification for change to the original question State the modification you made to the original question: I built on the original question to increase the number of tasks required. Once the student has identified the shapes they not only have to label the features of the shapes, but have to include the numerical values.

Why did you select this modification to make to the problem? For a student who knows their shapes, identifying them is easy. By having the students think further about the shapes they can see, it required them to think critically to identify the numerical values I.e. a cube has 6 faces.

Cross-‐Curriculum Links This image could also be used for a writing stimulus for English. Using the picture, the students would be expected to use the objects in the picture to create a story. For example, a student may wish to write a story about the little mouse that got stuck in the gymnasium. They could then use the shapes along with their mathematical vocabulary to incorporate aspects of the shapes throughout their story.

AusVELS -‐ Cross-‐curriculum Cross-‐curriculum area, Content strand/s, year, definition and code English, Creating Texts, Year 2:

• Create short imaginative, informative and persuasive texts using growing knowledge of text structures and language features for familiar and some less familiar audiences, selecting print and multimodal elements appropriate to the audience and purpose (ACELY1671)


Report of Trialling Problem Picture 2 Child’s pseudonym, age and grade level: Fiona, 7 years old, Grade 2

Original Question: What shapes can you see in the photo? Draw two of the shapes in the photo and label their properties, then write down whether they are 2D or 3D


Reflection on child’s response: The initial question for this task required the student to first identify the shapes they could see in the photo and then draw two of them and label their properties, then write down whether they are 2D or 3D. After presenting Fiona with the problem picture and informing her of the task, she was able to understand the task requirements and began listing the shapes she could see.

Whilst observing Fiona throughout the task, it was evident that the open-‐ended nature of the task enabled Fiona to complete the task regardless of her knowledge of shapes (Hussey and Smith 2003). The first part of the problem-‐picture task was completed as expected however; Fiona lacked the ability to label particular properties of the shapes.

Fiona wass able to identify the names (Reys et al. 2012) of the shapes within the photo however, Reys et al. (2012) recognises Fiona’s inability to recognise and label the features of shapes being a result of poor exploration behind the properties of geometric shapes. It is evident that Fiona is able to recognise the types of shapes within the problem picture (Reys et al 2012), however due to a lack of direction within the question (Hussey and Smith 2003), Fiona was unable to complete the second part of the task successfully. Although Fiona struggled to complete the task as expected, it is apparent that the task was ‘capable of inclusion’ as Fiona was able to correctly identify the shapes she could see. It is evident that Fiona possesses a strong understanding behind the names of the geometric shapes, however could benefit from further development of mathematical vocabulary and ideas (Reys et al. 2012).

This task integrated the mathematics component of the task, as the main emphasis was identifying the 3D and 2D shapes within the photo. In grade 2, ACARA requires the students to describe the features of 3D shapes whilst drawing and labelling the features of 2D shapes. This was evident within the task, as the students were expected to identify, draw and label the features of two shapes they could identify within the picture.

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures In terms of rephrasing the question, I believe that the question needs to be altered in order to specify that all the student needs to do is label the features. This will help to ensure that the student understands what exactly is required when they are told to label their properties.

Rephrased Question: What shapes can you see in the photo? Draw two of the shapes in the photo and label their properties. You may like to label each shapes faces, edges, vertices, corners and dimension.



Reys, R, Lindquist, M, Lambdin, D, Smith, L, Rogers, A, Falle, J, Frid, S & Bennett, S 2012, Helping children learn mathematics (1st Australian ed’n). Milton, Qld: John Wiley and Sons.


Problem Picture 3 Location: At my house, after opening a packet of The Natural Confectionery company, Party Mix

Problem Picture 3 -‐ Questions Grade level: Two Question 3 Can you sort the Party Mix into groups to work out which lolly I have the most of?



AusVELS -‐ Statistics and Probability Content strand/s, year, definition and code Statistics and Probability, Data Representation and Interpretation, Year 2:

• Collect, check and classify data ACMSP049

• Create displays of data using lists, table and picture graphs and interpret them ACMSP050

Enabling Prompt Can you group the lollies into different groups?



AusVELS Content strand/s, year, definition and code Statistics and Probability, Data Representation and Interpretation, Year 2:



Justification for change to the original question State the modification you made to the original question: To simplify the problem, I took away the complexity of having to compare the number of lolly in each group in order to make an educative decision based on the group with the least amount of lollies.

Why did you select this modification to make to the problem? I chose to take away the aspect of interpreting the data in order to assist the students in the task. Instead of going that one step further and having the students interpret the data, I had them practice their grouping strategy and focus on getting the right number of lollies in each group.

Extending Prompt Can you sort the Party Mix into groups? Which group has the most? And which group has the least?

Answers to Extending Prompt

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures AusVELS Content strand/s, year, definition and code Statistics and Probability, Data Representation and Interpretation, Year 2:



Justification for change to the original question State the modification you made to the original question: To extend the task, I added on an extra component, requiring the student to extend on their knowledge from the information and make a decision based on which lolly had the most and which lolly had the least.

Why did you select this modification to make to the problem? I chose to make this modification as I thought it was a simple extension to the problem. By having the student complete the task they not only had to work out which group had the biggest number of lollies, but they also had to identify the group with the least amount of lollies.

Cross-‐Curriculum Links This photo could be used in the area of Drama. The students could look at the lollies on the plate and act out what it would be like and feel like to be a lolly on the plate.

AusVELS -‐ Cross-‐curriculum Cross-‐curriculum area, Content strand/s, year, definition and code Drama, Year 2:

• Use voice, facial expression, movement and space to imagine and establish role and situation (ACADRM028)


Report of Trialling Problem Picture 3 Child’s pseudonym, age and grade level: Sarah, 7 years old, Year 2

Original Question: Can you sort the Party Mix into groups to work out which lolly I have the most of?


Reflection on child’s response: My original open-‐ended task required the students to ‘sort the Party Mix into groups to work out which lolly I have the most of?’ The student understood the requirement of the task and eagerly began searching the photo for groups to place the different lollies into.

To an extent, the question was completed as expected as the student grouped the lollies according to similar attributes i.e. dinosaurs with the dinosaurs and fruit with the fruit however, upon my completion of the different questions, most responses included a graph-‐style response which is easily read and interpreted. Whilst the task was both responsive and flexible (Hussey and Smith 2003), Sarah was able to identify one of the methods in grouping the pre-‐specified (Hussey and Smith 2003), as she grouped the lollies according to their shape. Sarah exhibited a strong understanding behind the methods involved in grouping the lollies, as she was able to simplify the problem by eliminating the context and any unrequired information (Reys et al. 2012). Through Sarah’s ability to create data, which was easier to work with, she demonstrated the capacity to make decisions, model her interpretations and investigate (Reys et al. 2012) the relationships within the data. Within her response, Sarah demonstrated a competency in using a diagraming strategy (Reys et al. 2012) as she classified the data into groups according to their shape and created a representation using a one-‐to-‐one correlation. This allowed her to easily illustrate the relationships within the data and make an educative response in regards to which lolly I had the most of. If Sarah had of created more of a graph-‐like response as expected, she may have found it easier to identify the biggest group and therefore make more of an argument for why the dinosaurs had the most lollies.

Jessica McDonald 212161352 ESM410 AT1: Problem Pictures This problem-‐picture task focused on the mathematical concept identified within the question, through its main focus on the sorting and interpretation of data collected. In line with ACARA (2013), students in grade 2 are expected to be competent in collecting, checking and classifying data, whilst creating displays and interpretations. This was demonstrated throughout the child’s response, as they were required to collect the data presented to them through the picture, check the one-‐to-‐one correspondence and create a detailed representation. The students were then required to make an educated interpretation based on which lolly the party mix had the most of. At this point in time, I would not make any modifications based on the students response to the task. This is because the student was able to successfully complete the requirements specified by ACARA (2013).

Rephrased Question: N/A



Reys, R, Lindquist, M, Lambdin, D, Smith, L, Rogers, A, Falle, J, Frid, S & Bennett, S 2012, Helping children learn mathematics (1st Australian ed’n). Milton, Qld: John Wiley and Sons

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