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

Student Name: Lucy Sinclair

Student Number: 212207019

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:

Lucy Sinclair

DATE: 23/8/15

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. yes

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

Created 3 quality problem picture photos. yes

The photos MUST be original photos taken by yourself. yes

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

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

yes

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.

yes

Open-ended questions are creative and engaging. yes

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

yes

Each question is accompanied by three possible correct responses. yes

Cross-curriculum links are made to each photo. yes

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

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

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

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

File size of the word document is under 4mb. yes

Assignment is uploaded to the Cloud Deakin dropbox. yes

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

Wu, (1994) states that the introduction of open-ended problem pictures in the classroom allows mathematical education to get one step closer to real world mathematics. Wu, also suggests that open ended problem pictures actively engage students and prompt them to actively explore various possibilities. This research is backed up by Bragg and Nicol (2008), when they state that open-ended problem pictures provide a context for learning of mathematical concepts and promote mathematical inquiry. Bragg and Nicol also go on to suggest that open-ended tasks allow students to construct ideas on what it means to do mathematics, and that students develop beliefs about the idea of mathematics from their experience in the mathematics classroom through activities. Cifarelli and Cai, (2005) also state the benefit of open-ended problems or as they state ‘mathematical exploration’ is that some of the task unspecified and requires the students to sometimes re-formulate the problem in order to develop a solution, allowing them to develop mathematical problem solving skills. Bragg and Nicol (2011) also discuss the importance of bringing photographs of familiar objects or places into the mathematics classroom, allows students to relate classroom mathematics to real world leaning in a fun and interesting way. Open-ended picture problems will support my teaching in the future by understanding that picture problems can be used as the basis of an entire lesson that stands alone or to create a unit of work, depending on the depth of the questions (Liburn and Sulivan, 2014). I also now know that open-ended picture problems are a perfect way to make maths relevant to students lives and therefore they will have purpose and be engaged in the problem, as Bragg and Nicol (2008) state that problem pictures are grounded in real-life activities. However Liburn and Salivan (2014), state that the questions must be good and beneficial, teachers must spend time to construct questions that facilitate meaningful learning, it is not enough just to create an open question with no educative value. I will take this into my future teaching to make sure I am constructing meaningful mathematical open-ended problems derived from pictures. As Bragg and Nicol (2008) state that knowledge is inseparable from the activity and context in which it is used. There for using problem pictures with open-ended questions allows students to connect their classroom context to the real world context and enhance student learning.

References for the rationale:

Bragg, Leicha and 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, issue. 3, pp. 3-9

Cifarelli, V. and Cai, J. (2005). The evolution of mathematical explorations in open-ended problem-solving situations. The Journal of Mathematical Behavior, 24(3-4), pp.302-324.

Lilburn, P. and Sullivan, P. (2014). Extracts from Openended maths activities. Open-ended maths activities: using "good" questions to enhance learning in mathematics, [online] 2(1), pp.5-6, 23, 42, 55, 63, 84, 93. Available at: http://equella.deakin.edu.au/deakin/file/469aab9a-c1a0-2541-edd4-4dbf02481462/1/scan-openended-sullivan-2004.pdf [Accessed 18 Aug. 2015].

Wu, H. (1994). The role of open-ended problems in mathematics education. The Journal of Mathematical Behavior, 13(1), pp.115-128.

Problem Picture 1 Location: My kitchen

Problem Picture 1 - Questions

Grade level: 4

Question 1How many other ways can you make 1 cup of rice, using the measuring cups 1/2, 1/3 and 1/4.

Answers to Question 1

AusVELS - Number and Algebra

Content strand/s, year, definition and code Mathematics, Number and algebra, fractions and decimals, Year 4, Investigate equivalent fractions used in contexts(ACMNA077)

Enabling PromptSee if you can fill up the 1cup to the top using scoops ½ cup, ¼ cup and 1/3 cup. You can use the cups as many times as you like and do not have to use all cups. How many ways can you do this?

Answers to Enabling Prompt

AusVELS Content strand/s, year, definition and code Mathematics, Number and algebra, fractions and decimals, Year 3, Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole(ACMNA058)

Justification for change to the original questionI chose this modification to the problem, to allow students be more hands on with the investigation. Students are asked to experiment and physically fill up the cups with rice and poor it into a 1-cup container and write down their combinations as they go. This allows students to experiment and rely on trial and error to find the correct solution and have a physical manipulative in which to work through the problem. It also connects to real world learning as the students have most likely cooked and measured in the kitchen at home before with their parents.

Extending PromptThe recipe I am using to cook my rice said I needed 2/3 of a cup, however more people are coming over for dinner then I thought so I want to double to recipe. How much rice do I need and what are some ways could I measure it using 1 cup, 1/3, ¼ and ½ cups?

Answers to Extending Prompt1

= 1 1/3

2

___________ 1 cup_________

3

__________ 1 cup _______

AusVELSContent strand/s, year, definition and code Number and algebra, fractions and decimals, Year 5, Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)

Justification for change to the original questionI wanted to extend the students by asking them to do some addition of fractions with the same detonator. This matches with the year 5 ausVELS and allows the students to think of multiple answers to simple addition problems of fractoins. The question also asks them do complete a couple of steps before they can begin to solve the problem adding to the complexity.

1/3

___________1 cup_____________

½ ½ 1/3 = 1 1/3

1 cup1/3

Cross-Curriculum LinksThe photo of the measuring cups could be used in the cross-curricular domain of English. The image of the measuring cups could be used to prompt procedural writing such as instructions or a recipe. Through procedural writing students are able to explore incorporate mathematic language into their writing pieces. Using this image they could be using fractions in their writing and develop and extend their vocabulary.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code English, Writing, Language, year 4, Incorporate new vocabulary from a range of sources into students’ own texts including vocabulary encountered in research (ACELA1498)

Report of Trialling Problem Picture 1 Child’s pseudonym, age and grade level: Jarra, 10, grade 4

Original Question: How many other ways can you make 1 cup of rice, using the measuring cups 1/2, 1/3 and 1/4.

Child’s response to the question:Insert photo of work sample and/or transcript of audio response and/or anecdotal notes.

See reflection for worked in anecdotal notes

Reflection on child’s response:My original question I asked Jarra was ‘how many other ways can you make 1 cup of rice, using the measuring cups 1/2, 1/3 and 1/4.’ Jarra looked at me confused for a moment and then asked if he could use a cup more then once. I told him that he could. I originally asked Jarra to write down on paper what he found but he wanted to use his ipad as he said he did allot of writing at school today. As I did my interviews in afterschool care, I let Jarra write it on his I-pad in notes and collected the working out he did on a scrap piece of paper. Jarra typed ½ + ½ = 1cup immediately without using the rice. However he used the rice to measure out 1/3 cup and found that he needed 3 x 1/3 cups to fill the 1 cup. He made a verbal prediction that he would only need 3 of the ¼ cups to fill the 1 cup. I told him to measure it out and check, he then found that he needed 4 x ¼ cup to fill the 1cup. Then he stated to me “ahhhhh you it tells you how many you need on the bottom number.” When pointing at the denominators of all 3 cups. I asked him if he could think of any other ways to make 1cup and he told me that there

wasn’t any other ways.

Jarra answered the problem much as I expected. I thought maybe he might experiment more and be able to discover other combinations such as ¼ + ¼ + ½ = 1 cup, but he was eager to play with his friends. Jarra did not need to be asked the enabling or extending prompts as I feel the original question was aimed at his current understanding, and as Sullivan, P (2005) states that one of the benefits of open ended questions is that they can be easily accessed by a variety of ability levels.

Through this open-ended question I was able to see that Jarra had a good understanding of basic fractions of a whole. I believe he only predicted that it would only take 3 ¼ cups to fill the 1 cup because the size of the cups looked similar. Once he thought about the fraction written on the cup and not the cup itself he was able to understand that the cup only contained ¼ of the whole and to get to 4/4 or 1 cup he would have to fill the 1/4 cup 4 times over, this was shown to me when he stated “ahhhhh you it tells you how many you need to fill it on the bottom number.” Jarras weakness however was in inability to look for an answer using a combination of cups. I feel however in a different environment where he couldn’t see his friends playing he would be more willing to experiment more and find other solutions.

I feel that the question did address some of the mathematical intent of “ Number and algebra, fractions and decimals, Year 4, Investigate equivalent fractions used in context” (The Australian Curriculum: Mathematics (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013). I feel like I met the context as it was using ingredients and tools often used In cooking, that students have most likely come across and used in their homes before. Although this question only addresses fractions equivalent to 1 whole, it is a good starting point to develop a basic understanding of equivalent fractions before going on to explore more complex fractions. I feel like this question would build I the basics and fill in the gaps from pervious years and allow students to have a solid base before moving on to harder equivalent fractions. However that being said, If I might modify my original question I would make sure the student understood that they could used the same cup more then once and possibly change the question to prompt answers such as 2 ¼ is the same as ½, to better accommodate my chosen ausVELS standard. I would also bring more contexts into the original question making it more engaging for the student.

Rephrased Question: N/A

References for reflection on the trial of question 1: Sullivan, P. (2005). Teaching mathematics to classes of diverse interests and backgrounds. [online electure], Deakin University

Australian Curriculum Assessment and Reporting Authority. (2013). The Australian Curriculum. Retrieved August, 22, 2015, from http://www.australiancurriculum.edu.au

Problem Picture 2 Location: My house

Problem Picture 2 - Questions

Grade level: 1

Question 2What shapes can you find in this picture of a house?

Answers to Question 2

AusVELS - Measurement and Geometry

Content strand/s, year, definition and code Measurement and Geometry, Shape, year1, Recognise and classify familiar two-dimensional shapes and three-dimensional objects using obvious features (ACMMG022)

Enabling PromptHow many circles and triangles in this picture? Highlight what you find

Answers to Enabling Prompt

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

Justification for change to the original questionI simplified the problem by giving the names of shapes I asked the students to find. Making the problem more specific and reducing the number of steps. I also chose very obvious shapes for students to be able to pick out of the image that would be familiar to them.

Extending PromptDraw the shapes you see in this image and tell me about their features.

Answers to Extending Prompt

Triangles have 3 corners and 3 sides. Squares have even sides, Rectangles have 4 corners & 4 sides 4 corners & 4 sides. However can have uneven sides.

AusVELSContent strand/s, year, definition and code Measurement and Geometry, Shape, Year 2, Describe and draw two-dimensional shapes, with and without digital technologies (ACMMG042)

Justification for change to the original questionI changed the question to match the year 2 ausVELS by asking the student to draw the shapes they could see in the image and then discuss with me their features and how they drew them. This connects with the year 2 AusVELS for shape and challenges the students to start thinking about and verbalising features of common shapes.

Cross-Curriculum LinksThis image could be used to prompt understanding of different types of plants. Students could compare the living things seen in the image and describe the differences and similarities between the plants. Note some are evergreen some are not; this would be a good talking point. Questions could be raised such as ‘Why does the tree on the right have no leaves?’ or ‘How many types of plants are shown, how do you know this? This would help students to understand that plants have different external features.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code Science, Biological sciences, year 1, Living things have a variety of external features (ACSSU017)

Report of Trialling Problem Picture 2

Child’s pseudonym, age and grade level: Emily, 6, year 1

Original Question: What shapes can you find in this picture of a house? Highlight what you find.

Child’s response to the question:

Reflection on child’s response:I started by asking the original question “What shapes can you find in this picture of a house? Highlight what you find.” Emily studied the house for a minute or so and turned to me and said ‘I’m not sure, is it a square?’ To me that showed me that Emily was trying to tell me the shape of the entire house not the features. This is discussed by Reys et al (2009) when he suggest that young children often only focus on visual cues that are prominent to them, for Emily this would be the house as a whole. She seemed confused and unsure of herself so I decided to try the enabling prompt instead. I asked her to “How many circles and triangles are in this picture? Highlight what you

find.” Once Emily started looking for particular shapes she found the 3 circles on the fence straight away. She then started searching for a triangle, I prompted her after a few minutes by suggesting ‘If you were to draw a house where might you use a triangle’ She had a think then exclaimed ‘the roof!’ She then excitedly highlighted the triangle on the roof of the house on the image.

Emily did not answer my original question, as she seemed unsure and overwhelmed. Knowing Emily quite well I almost expected this reaction; she is quite low levelled and is suspected to have undiagnosed learning difficulty. However I created the original question before I chose Emily as my subject, and decided to ask her the original question anyway to see what she came up with. However once I gave her more specific instructions by using the enabling prompt she was able to complete the question.

I found that Emily had good knowledge of her shapes and their features, but was originally unable to locate them in a real world image as it was allot of information and many possible shapes at once. She was able to recognise and identify circles and triangles within the image once she was able to focus on fewer things. My enabling question was developed for the foundation standard ‘Measurement and Geometry, Shape, Foundation, Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment’ (The Australian Curriculum: Mathematics (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013. I feel that my enabling prompt addressed the mathematical intent by asking Emily to name and locate familiar 2D shapes in the environment. I feel the enabling prompt worked quite well and Emily was able to focus on particular shapes and identify them correctly and confidently. If I was to modify the prompt in any way, I may of asked Emily to also find a 3D shape in the image, to extend her slightly and connect more with the ausVELS level.

Rephrased Question: -Used enabling prompt: Can you find any circles or triangles in this picture? Highlight what you find

-If you were to draw a house where might you use a triangle?

References for reflection on the trial of question 2: Australian Curriculum Assessment and Reporting Authority. (2013). The Australian Curriculum. Retrieved August 22, 2015, from http://www.australiancurriculum.edu.au

Reys, R Lindquist, M Lambdin, D & Smith, N 2009, ‘Helping Children Learn Mathematics’, John Wiley & Sons, United States of America

Problem Picture 3 Location: McDonald’s Burwood

Problem Picture 3 - Questions

Grade level: 4

Question 3Conduct a survey of all the students in their class on their favourite soft drink; design what you think is the best way to display your results.

Answers to Question 3

AusVELS - Statistics and ProbabilityMathematics, Statistics and probability, Data representation and interpretation, year 4, Evaluate the effectiveness of different displays in illustrating data features including variability(ACMSP097)

Enabling PromptConduct a survey of 10 students on their favourite soft drink from the image is; how can you show your results.

Answers to Enabling Prompt

Number of students Total

Coke III 3

OJ II 2

Sprite I 1

Coke zero I 1

Fanta II 2

Diet coke I 1

AusVELS Content strand/s, year, definition and code Mathematics, Statistics and probability, Data representation and interpretation, year 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)

Justification for change to the original questionI changed the original question by making it more specific, by asking the student to only ask 10 students about their drink preferences from those shown in the image. This allows the student to work with base 10 in their calculations and will make it easier to interpret data and develop a way to effectively present it.

Extending PromptConduct a survey of all the students the class on their favourite soft drink; design what you think is the best way to collect your data and display your results. Why did you choose this way?

Answers to Extending Prompt

I think that a bar graph is the best way to show the results as it makes the data easy to read and see how many people voted for each drink. It also is easy to see what drinks were more and less popular by quickly looking at the height of the bars.

I think that a pie chart is the best way to display the results as it shows you the results as a part of a whole. It lets you quickly see what is more or less popular. It also makes it easy to make estimations for examples that coke it preferred by almost half of the students.

I think a line chart is the best way to display the results as you can easily compare the results for each drink by following the line down or up. It also helps to show the differences for example that 3 more people like Oj then sprite as the line goes up 3 on the x axis from sprite to reach OJ.

AusVELSMathematics, Statistics and probability, Data representation and interpretation, year 5, Pose questions and collect categorical or numerical data by observation or survey (ACMSP118)

Justification for change to the original questionI changed the original question to allow students to extend their knowledge by interpreting the data based on their chosen representation. I wanted students to see how different types of displays have different benefits. Students must also collect data by survey from all the students in the class, the final number may not be base 10 depending on the day so it adds a little extra challenge. It encourages students to think deeper about how they will display their results and the reasons behind that choice.

Cross-Curriculum LinksStudents could use this image as a prompt in a Design, Creativity and Technology focus. Students could possibly design their own drink fountain. Students would need to create a design brief and design flavours or features that would fit in with the requirements of the establishment or home.

AusVELS - Cross-curriculum Design, creativity and Technology , year 3, learning focus: “As students work towards the achievement of Level 4 standards in Design, Creativity and Technology, they begin to provide input into the development of design briefs. They generate ideas from a variety of sources, and recognise that their designs have to meet a range of different requirements”

Report of Trialling Problem Picture 3 Child’s pseudonym, age and grade level: Jarra, 10, grade 4

Original Question: Conduct a survey of all the students in the class on their favourite soft drink; design what you think is the best way to display your results.

Child’s response to the question:Original response

.

Final response

Reflection on child’s response:My original question to Jarra was ‘Conduct a survey of all the students in class on their favourite soft drink; design what you think is the best way to display your results’. Jarra seemed excited about this task and showed me an app on his i-pad that allowed him to create all types of graphs. I allowed this as the The Australian Curriculum: Mathematics (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013) for year 4 ‘Data representation and interpretation’ did not state that the graphs had to be done by hand.

Jarra answered the prompt as I expected, he collected the data from all the students and placed the data into an app on his i-pad. First he created a pie graph out of the data that showed the % of students that preferred each drink. After the pie graph was created Jarra and I had a discussion about if we were to show this graph to all the students in afterschool care that possibly the younger students may not to be able to understand percentages. Jarra agreed and suggested that maybe he should make a column graph so the younger students would be able to read it easier. Jarra then created a column chart and decided that he would make the x axis count up by 2. He then suggested that all the students should be able to read the chart as in a column graph you can see exactly how many preferred which drink, unlike the original pie chart. Jarra did not need to answer the enabling or extending question as the original question was at a good level for his abilities.

Jarra showed strength in comparing and contrasting 2 ways of displaying data after a slight prompt from myself and discussed the benefits and weaknesses of each display. It was obvious to myself however through trailing 2 open-ended questions with Jarra, that he is not confident in his

mathematical representations without using technology. He really pushed using his I-pad for all aspects of solving the problem and was reluctant when I prompted him to do it by hand.

The question I feel met the mathematical intent of the question. Jarra displayed his ability to evaluate the effectiveness of different data displays, in discussing the data features (The Australian Curriculum: Mathematics (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013). This standard supported by Reys et al, (2009) when they state that constructing and interpreting different types of graphs and the knowledge associated is an important part of mathematics instruction.

In light of Jarra’s response I would make it clear that I wanted a graph that would be best for the multi-age group of students in afterschool care, so he would be able to choose the best graph for his audience. In Jarra’s situation I would also ask the student to create the graph by hand.

Rephrased Question: N/A

References for reflection on the trial of question 3: Australian Curriculum Assessment and Reporting Authority. (2013). The Australian Curriculum. Retrieved August 23, 2015, from http://www.australiancurriculum.edu.au

Reys, R Lindquist, M Lambdin, D & Smith, N 2009, ‘Helping Children Learn Mathematics’, John Wiley & Sons, United States of America