ESM410 Assignment 1

ESM410 Assignment 1: Problem Pictures Task - Creating open-ended questions

Student Name: Naomi Mathew

Student Number: 212198847

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: 28/8/2015

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.

Checklist

All points must be ticked that they are completed before submission.

Requirements checklist: Tick completed

The rationale addressed the rationale prompts in the assignment description.

The rationale included relevant citations/references – which are stated.

Created 3 quality problem picture photos.

The photos MUST be original photos taken by yourself.

Location of photos are stated, e.g. Taken at Deakin foreshore.

Developed an original question for each photo with an accompanying enabling and extending prompt.

If your photo has numbers that you are referring to in the problem, the numbers MUST be clearly visible to be able to read in the photo.

Open-ended questions are creative and engaging.

Matched each problem with the appropriate mathematical content, year, definition and code from the Australian Curriculum: Mathematics

Each question is accompanied by three possible correct responses.

Cross-curriculum links are made to each photo.

Reflecting on the trialling of the questions with an appropriately aged child or children.

The trialling reflection included relevant citations/references – which are stated.

There is evidence of reference to problem-picture unit materials.

Problem pictures were collated into a word document using the assignment template.

File size of the word document is under 4mb.

Assignment is uploaded to the Cloud Deakin dropbox.

In order to pass this assignment you must have fulfilled all aspects of the checklist.

Rationale for the use of problem pictures in the classroom

Using open-ended problem pictures in mathematics can prove to be advantageous and engaging for teachers and students in the modern, primary classroom. Kilpatrick and Swafford (2002, p. 9) describe the five strands of mathematical proficiency as understanding, computing, applying, reasoning and engaging. Through the use of open-ended problem pictures, students would be able to comprehend mathematical problems in a different manner, use the problem picture to accurately create multiple mathematical solutions, practically relate their problem picture to other similar circumstances and provide justifications for their solutions. This would serve to support my future teaching practices in delivering more holistic and challenging lessons. Hacking (2015, p. 2) states that ‘children must be able to see real and relevant contexts where maths is used as part of everyday experiences’ in order to be effective. As such, open-ended problem pictures will allow students to make meaningful connections with their learning. Chen and Weiland (2007, p. 47) suggest ‘[demonstrating] the same mathematical concept using multiple modes’ as an effective learning strategy. Open-ended problem pictures will allow students to view mathematic problems in a more engaging manner as well as support those visual learners who will benefit from the visual delivery of mathematic problems. Using open-ended picture problems would enable me to differentiate my future teaching and support the different types of learners in the classroom. Furthermore, the open-ended nature of the problem pictures will serve to ‘foster some of the more important aspects of learning [mathematics], including investigating, creating, problematising, communicating, and thinking as-distinct from merely recalling procedures’ (Sullivan, Mousley & Zevenbergen 2005, p. 106). Open-ended picture problems call for a more abstract and deeper way of thinking and as such, develops other skills that are necessary for students to become critical problem solvers. Another ‘important aspect of open-ended problem photos is that they create a curiosity in the students and a desire to explore possible solutions’ (Bragg & Nicol 2011, p. 4). This is crucial in developing future engagement with mathematics and dispelling the negative perceptions that students often have about mathematics. Open-ended picture problems will support the way in which I engage and challenge students of all abilities in the learning of mathematics. Open-ended problem pictures will also allow me to deliver mathematical content in a different and more meaningful way than previous traditional approaches which in turn, will promote a more conducive learning environment.

References for the rationale:

Bragg, LA & 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, retrieved 26 August 2015, https://d2l.deakin.edu.au/content/enforced/340075-ESM410_TRI-2_2015/AMT%202011%20Vol%2067%20Issue%203%20-%20Seeing%20maths%20through%20a%20new%20lens.pdf?_&d2lSessionVal=gCEgljFce99SijQ3xaEbT1Xch&ou=340075&_&d2lSessionVal=CjlVQBaevZk79NftrrrTMqZbo&ou=340075

Chen, J & Weiland, L 2007, 'Helping Young Children Learn Mathematics: Strategies for Meeting the Needs of Diverse Learners', Exchange (01648527), no. 174, pp. 46-51, retrieved 26 August 2015, Education Source, EBSCOhost.

Hacking, C 2015, 'The power of the picture book for teaching mathematics in the early years', English 4--11, 53, pp. 2-4, retrieved 26 August 2015, Education Source, EBSCOhost.

Kilpatrick, J & Swafford, J 2002, Helping Children Learn Mathematics, n.p.: Washington, DC : National Academy Press, 2002., retrieved 26 August 2015, DEAKIN UNIV LIBRARY's Catalog, EBSCOhost.

Sullivan, P, Mousley, J & Zevenbergen, R 2005, ‘Increasing access to mathematical thinking’, Australian Mathematical Society Gazette, vol. 32, no. 2, pp. 105-109, retrieved 26 August 2015, http://www.austms.org.au/Publ/Gazette/2005/May05/sullivanMZ.pdf

Problem Picture 1 Location: Taken at bus stop in Westfield Doncaster

Problem Picture 1 - Questions

Grade level: 2

Question 1Locate either of the buses’ route numbers and use the three digits to create different number patterns.

Answers to Question 11. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (blank spaces have been filled. Number pattern is ascending by ones)2. 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 (blank spaces have been filled. Number pattern is descending by ones)3. 0, 14, 28, 42, 56, 70, 84 (Number pattern is ascending by 14 [2+9+3])

AusVELS - Number and Algebra

Content strand/s, year, definition and code Patterns and Algebra, Level 2, Describe patterns with numbers and identify missing elements (ACMNA035)

Enabling PromptWith a partner, locate the bus route numbers and use the three digits to create different number patterns.

Answers to Enabling Prompt1. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (blank spaces have been filled. Number pattern is ascending by ones)2. 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 (blank spaces have been filled. Number pattern is descending by ones)3. 0, 14, 28, 42, 56, 70, 84 (Number pattern is ascending by 14 [2+9+3])

AusVELS Content strand/s, year, definition and code Patterns and Algebra, Level 2, Describe patterns with numbers and identify missing elements (ACMNA035)

Justification for change to the original questionThe original question was modified for the enabling prompt to include working with a partner. This modification was selected because some students have trouble identifying potential number patterns and as such will benefit from the ideas and explanations of a second person.

Extending PromptCreate as many number patterns as possible and explain the rule for each number pattern. Experiment with different starting numbers, using the operations etc.

Answers to Extending Prompt1. 60, 58, 49, 46, 44, 35, 32, 30… (Explanation of rule: Starting at 60, the pattern continues as -2, -9, -

3, -2, -9, -3…)2. 1, 3, 3, 10, 12, 12, 19, 21, 21, 28… (Explanation of rule: Starting at 1, the pattern continues as +2,

+0, +7, +2, +0, +7…) 3. 100, 102, 111, 114, 116, 125, 128… (Explanation of rule: Starting at 100, the pattern continues as

+2, +9, +3, +2, +9, +3…)

AusVELSContent strand/s, year, definition and code Patterns and Algebra, Level 3, Describe, continue, and create number patterns resulting from performing addition or subtraction (ACMNA060)

Justification for change to the original questionThe original question was modified to include more challenging number sentences through the use of different starting numbers, operations and so on as well as the inclusion of an explanation of the rule for each number pattern. These modifications were selected because they encourage more complex mathematical equations and also consolidate students’ understanding of number patterns through the explanations for the rules.

Cross-Curriculum LinksThe photo of the buses at Westfield Doncaster depicts a local, built environment. Students can use that photo to identify and describe specific aspects of the environment that are built or natural. Students should think about what the term ‘man-made’ means and the role it plays in our lives. Students can use this photo as a prompt to think about other places in their local area that depict built environments. In contrast, students should also think about what constitutes a natural environment as well as places local to them which would be considered a natural environment. As the picture of the buses at Westfield Doncaster mainly depicts a built environment, students could also discuss and write about what it would be like if the world was made of only built environments. Following on from that, students can explain why it is important to preserve natural environments and the role we play in doing so. This activity requires students to draw upon their individual knowledge and experiences and as such is relatable to all the students. It is also an important discussion in developing socially conscious individuals.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code Humanities, Level 2, Through observation, they investigate and describe elements of the natural and built environments in their local area.

Report of Trialling Problem Picture 1 Child’s pseudonym, age and grade level: Millie, 7, Level 2

Original Question: Locate either of the buses’ route numbers and use the three digits to create different number patterns.

Child’s response to the question:

Reflection on child’s response:

The original question given to the student was “Locate either of the buses’ route numbers and use the three digits to create different number patterns”. At first, the student was puzzled at which numbers she had to use as there were other numbers depicted on the buses but settled on the correct set of numbers as there were four digits in the other groups. She was also slightly confused by the idea of number patterns. However, after I repeated the question verbally, she remembered what they were.

The student answered my problem picture well in terms of how the number pattern was displayed and exceeded my expectations by drawing the arrows above the number pattern and highlighting what had occurred from the previous number to the next one. Whilst the student did have the right idea when creating the number patterns, in terms of adding and subtracting the individual bus numbers, I was expecting the student to experiment more with the bus numbers instead of completing an addition and subtraction number pattern for each bus number. Apart from one computing error in her response, the student demonstrated the capacity to respond to the original question. As the student had essentially included all the elements of the extending prompt in her answer to the original question, I did not issue an extending prompt to her. She demonstrated the ability to experiment with different starting numbers and provide a form of explanation in her response to the original question and in doing so, exceeded my expectations.

The student demonstrated a number of strengths in her mathematical understanding through her response to the prompt. She showed a good understanding of mathematical pattern in the way she repeatedly used the bus numbers in a certain order and the same operation for each number pattern. Mulligan et al. (2008, p. 11) emphasises that recognition of mathematical pattern and structure can positively influence overall learning in mathematics. Perhaps it was the student’s aptitude for patterns that enabled her to gain a positive result that exceeded the requirements of the original question. The student’s weaknesses were not easily identified through the use of this question as it did not appear to extend her past what she already knew. However, the student’s responses demonstrated a lack of creativity. It is unclear whether this is due to a lack of engagement with the task or a lack of understanding about more complex number patterns. Kruteskii (cited in Bharath 2008, p. 3) talks about mathematical creativity as the ability to ‘abstract and generali[s]e mathematical content’. The student showed some understanding of making generalisations through the number patterns however, that was not extended further.

The original question did address certain aspects of the AusVELS link as it allowed the student to ‘[d]escribe patterns with numbers’ (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013). The question encouraged the identification and description of a pattern which involved adding and subtracting numbers greater than one. However, the question was limited in its exploration of missing numbers in the number sequences.

In light of the student’s responses and my reflective practices, I would ensure that my revised question encourages students to engage creatively with their number patterns. To do so, I would allow them to pick any three numbers from the problem picture. I would also encourage students to test each other so that they would be challenged in identifying the number pattern without knowing the original numbers that were used.

Rephrased Question: Identify three numbers on the bus and create as many challenging and creative number patterns as possible. Make sure you have fun with trying out different starting numbers and operations. Once you have finished, erase a few numbers in your sequence and see if your friends can solve it.

References for reflection on the trial of question 1:

Australian Curriculum Assessment and Reporting Authority 2013, The Australian Curriculum, retrieved 31 August 2015, http://www.australiancurriculum.edu.au

Mulligan, J, Mitchelmore, M, Kemp, C, Marston, J, & Highfield, K 2008, 'Through Pattern & Structure', Australian Primary Mathematics Classroom, 13, 3, pp. 10-15, Education Source, EBSCOhost, viewed 31 August 2015.

Sriraman, B 2008, Creativity, Giftedness, And Talent Development In Mathematics, Charlotte, NC: Information Age Publishing, eBook Academic Collection (EBSCOhost), EBSCOhost, viewed 31 August 2015.

Problem Picture 2 Location: Taken in my bathroom

Problem Picture 2 - Questions

Grade level: Foundation

Question 2List as many shapes as you can from this photo and draw pictures of other things that have the same shape as those you have listed.

Answers to Question 2

AusVELS - Measurement and Geometry

Content strand/s, year, definition and code Shape, Foundation, Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment (ACMMG009)

Enabling PromptList or draw as many shapes as you can find in this photo.

Answers to Enabling Prompt

AusVELS Content strand/s, year, definition and code Shape, Level D, Use direct comparison to sort three dimensional objects and two dimensional shapes (ACMMG009d)

Justification for change to the original questionI modified the original question so that students would list or draw the shapes they found in the picture without having to list other objects that contain the same shape. I selected this modification as it enables students to identify shapes based on a single feature and recognise how they fit into everyday life.

Extending PromptPick one of the 2D or 3D shapes that you have found in this picture and create a table to record the number of faces, corners and edges the shape has.

Answers to Extending Prompt

AusVELSContent strand/s, year, definition and code Shape, Level 1, Recognise and classify familiar two-dimensional shapes and three-dimensional objects using obvious features (ACMMG022)

Justification for change to the original questionThe original question was modified so that students had to identify features of a shape such as the number of faces, corners and edges. This modification was selected as it challenged students to actually state the identifying features of shapes.

Cross-Curriculum Links

The photo of the bathroom above would be used as a prompt for students to record events such as getting ready for school. The events that are recorded have to be directly related to the bathroom and have to be events familiar to them. Students may choose to list the events in sequential order starting from when they wake up. Students should endeavour to use the appropriate vocabulary and descriptive words to record down their morning routine. This activity will be particularly relevant and useful for Foundation students as they are in their first year of school and might still be adjusting to their routine in the morning. This activity will also allow students to showcase their abilities and achievements in their morning routine. Links from this activity can also be made to discussions about hygiene and caring for themselves. Finally, this activity would relate to most, if not all, students and therefore be something that they would all be able to relate to.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code English, Literacy, Foundation, Create short texts to explore, record and report ideas and events using familiar words and beginning writing knowledge (ACELY1651)

Report of Trialling Problem Picture 2 Child’s pseudonym, age and grade level: Mylo, 6, Foundation

Original Question: List as many shapes as you can from this photo and draw pictures of other things that have the same shape as those you have listed.

Child’s response to the question:

Reflection on child’s response:

My initial question was “List as many shapes as you can from this photo and draw pictures of other things that have the same shape as those you have listed”. Similar to the first problem picture question, this question was also met with slight hesitation and confusion which did not last long.

The student answered my problem picture better than I expected him to. He identified two common two-dimensional shapes, being a square and a rectangle. Interestingly, the student’s drawings of other objects that contained the identified shapes were all objects pertaining to his classroom. Sharan (2015, p. 83) describes meaningful learning as learning that ‘promotes the construction of knowledge out of learners’ experience, feelings and exchanges’. I believe that the student made connections between what the question was asking him and his learning environment. He was therefore able to use his environment as a resource and correctly associated the shapes he identified with objects familiar to him and relevant to him at the time. When presented with the extending prompt, the student was unable to answer the question

which was what I had expected as he was unfamiliar with faces, corners and edges in the context of a shape.

The student displayed some strengths through answering the question as he was able to identify some shapes and relate those shapes to other objects. He was not given a context and as such had to formulate his own ideas of objects that contained the shapes he had identified. This demonstrates the connections he has formed and the understanding of shapes in his world. However, the student was only able to identify a square and a rectangle and was not able to identify the other shapes present in the picture. Reys et al. (2009) states that students should be able to identify shapes from examples in their world around them. Perhaps the student’s inability to identify other shapes highlights a lack of familiarity with other less common shapes and in effect, this could be seen as an area of improvement in his mathematical understanding. Furthermore, the extending question demonstrated his inability to transfer his knowledge of corners and edges when placed in the context of shapes.

I believe the question addressed the mathematical content that was related to the AusVELS link as the student identified, sorted and described two-dimensional shapes in his environment that were familiar to him (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013). I believe that the question was adequate in drawing out a response from the student that directly related to the curriculum content. However, in light of the child’s response and the reflective process, I would change the wording of the question so that it is more concise and clear. It is my hope that rewriting the question will make the task easier to understand.

Rephrased Question: Find as many shapes as you can in this photo. Where else can you find those shapes?

References for reflection on the trial of question 2:

Australian Curriculum Assessment and Reporting Authority 2013, The Australian Curriculum, retrieved 30 August 2015, http://www.australiancurriculum.edu.au

Reys, RE 2009, Helping children learn mathematics, 9th ed, Hoboken, New Jersey, John Wiley & Sons

Sharan, Y 2015, 'Meaningful learning in the cooperative classroom', Education 3-13, 43, 1, pp. 83-94, Education Source, EBSCOhost, retrieved 30 August 2015.

Problem Picture 3Location: Taken in my kitchen

Problem Picture 3 - Questions

Grade level: 2

Question 3Create an open-ended question about the fruits. Then, survey your class using the question and record their information.

Answers to Question 3

AusVELS - Statistics and ProbabilityContent strand/s, year, definition and code Statistics and Probability, Data representation and interpretation, Level 2, Identify a question of interest based on one categorical variable. Gather data relevant to the question (ACMSP048)

Enabling PromptUse the image to create as many questions as possible that could be used to survey your class and collect data.

Answers to Enabling Prompt

1. Which fruit is your favourite?2. How many of these fruits have you had before?3. Which of these fruits is your least favourite?

AusVELS Content strand/s, year, definition and code Statistics and Probability, Data representation and interpretation, Level 1, Choose simple questions and gather responses (ACMSP262)

Justification for change to the original questionI reduced the number of steps that students would need to complete by removing the requirement to collect and record data. I did this because students will still need to think critically about the types of questions that are relevant to the image and appropriate without spending time on recording the data, which might slow down some students.

Extending PromptWhat statements or facts can be made about the data?

Answers to Extending Prompt

1. Five more people prefer apples over strawberries.2. One quarter of the class have brought apples to school today3. The number of people who brought bananas to school today is equal to the total number of apples

and strawberries that were brought in.

AusVELSContent strand/s, year, definition and code Statistics and Probability, Data representation and interpretation, Level 2, Create displays of data using lists, table and picture graphs and interpret them (ACMSP050)

Justification for change to the original questionThe modification made to the original question asked students to interpret the data that they had collected. This modification was selected as it encourages students to reflect on their data and create links between the data which in turn will be more meaningful to them than if they were to record their data and not reflect on it.

Cross-Curriculum Links

The above photo of the fruits in the fruit basket can also be utilised in The Arts. Students may use the photo as a starting point instead of a real basket of fruits as a more convenient alternative. As the photo highlights various textures and colours, the photo will be useful in allowing students to demonstrate their ability to use the elements of art in order to create a unique and expressive form. As the students would be fairly familiar with the fruits, it provides all students with an unbiased starting point and an understanding of what they should look like. The fruits are also relatively simple to create and as such would be appropriate for Level 2. Students who need extension can focus on more complex aspects of the photo such as the viewpoint or angle of the photo as well as the textures that are apparent. This photo of the fruits provides students with an opportunity to engage and experiment with their growing understanding of the art elements, conventions and principles.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code The Arts, Creating and Making, Level 2, They demonstrate an emerging ability to select, arrange and make choices about expressive ways of using arts elements, principles and/or conventions.

Report of Trialling Problem Picture 3 Child’s pseudonym, age and grade level: Orlando, 8, Grade 2

Original Question: Create an open-ended question about the fruits. Then, survey your class using the question and record their information.

Child’s response to the question:

Reflection on child’s response:

My original question was “Create an open-ended question about the fruits. Then, survey your class using the question and record their information”. The first few questions the student created were all closed-ended questions. The student recognised them as closed-ended questions as soon as he thought of them and finally settled on the question at the top of the page, depicted in the photo above.

The student answered my problem picture as I had expected and intended him to, once he had thought of an appropriate open-ended question. He surveyed the class by asking them to raise their hands and recorded the information in the table he had prepared. He demonstrated some creativity in the way he set out his table which was visually appealing. The student also ensured that the table was labelled in order for me to read it accurately.

The student also answered my extending prompt as I had expected as he was able to give me a few simple facts about the information in the table. He was also able to identify that the information he recorded would be easier to read if his ticks lined up with the ones in the other rows. Bruner and Haste (cited in Monk & Silman 2011, p. 16) state that ‘[b]eing active means that the young child engages with experience, actively (as opposed to passively) bringing his or her existing knowledge and understanding to bear on what is currently under investigation’. The student was actively learning as he was reflecting on his own practices and understanding in order to improve the process that he had created. This demonstrated his active involvement, both physically and cognitively, with the problem picture question.

As mentioned previously, the student demonstrated an understanding of appropriate methods to record data and identified some features of tables such as the labelling, columns and rows. His ability to conduct a survey, record the data, reflect on his solution and interpret the data was also evident in his responses. The student demonstrated his knowledge of open-ended questions as he persisted in finding a question that was not closed. As the student was easily able to provide responses to the original question and extending prompt, it was difficult to identify misconceptions or weaknesses in his mathematical understanding.

Whilst the question appeared to be easy for the student, I believe that it addressed the mathematical intent of the question. The question was formulated to identify the student’s capability in creating a question based on a categorical variable and collecting data related to the question (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013). The student essentially addressed those concepts through the creation of an open-ended task, gathering of data and the ability to record the data in an appropriate format.

I believe that I would retain the same wording and content in the question as it directly related to AusVELS and the student was able to successfully interpret and answer the question.

References for reflection on the trial of question 3:

Australian Curriculum Assessment and Reporting Authority 2013, The Australian Curriculum, retrieved 30 August 2015, http://www.australiancurriculum.edu.au.

Monk, Jenny; Silman, Catherine 2014, Active Learning in Primary Classrooms : A Case Study Approach, e-book, retrieved 01 September 2015, http://deakin.eblib.com.au/patron/FullRecord.aspx?p=1596440.