Open-ended problem pictures

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

Student Name: Thu Thao Christine Ngo

Student Number: 212143725

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: Christine Ngo DATE: 23/8/2015

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

An open-ended problem picture engages students in a number of different ways and presents tremendous benefits in catering to all different learning abilities. Pictures present a “touch of realism” (Sparrow & Swan 2005, p.2) where students are able to make the connection and realise that mathematics is all around us. Gutstein (2006 as cited in Bragg & Nicol 2008, p.201) argues that good tasks include those that are culturally relevant, namely, those that connect to student’s lives. A picture problem provides a perfect model of how mathematics can be a visual connection to students’ lives as the photos we can use in an open-ended problem can be taken from places that students are familiar with. Bragg & Nicol (2011) suggest that it is significant for students to be able to make connections between mathematics they learn in class and outside of the classroom and not to view them as separate entities. Open-ended questions also gives students the freedom to engage with the task as “students are presented with opportunities to explore varied strategic approaches and encouraged to think flexibly about mathematics” (Bragg & Nicol 2011, p.3), students can choose to answer the question in a way they feel most comfortable with or students can choose to be creative with their solutions, maximizing their full potential making comprehensive use of their skills and knowledge in mathematics. Therefore there are high levels of active participation, as students of varying abilities are able to participate because of the flexibility of answers that the question provides, therefore students can feel confident in providing a unique answer. Confidence is an important element of motivation, research tells us that “confident students will be more cognitively engaged in their learning” (Caine & Caine 2001; Pintrich 2003b as cited in Churchill et al. 2012, p.133), therefore providing students with an open-ended problem picture allows that reach of competence for all students.

The use of open-ended problem pictures will help to support the diverse needs of students in my classroom. It is our role as a teacher to help every child “develop their maximum potential” (Reys et al 2012, p.15) therefore, by providing different ways to cater for these needs. An example by Sullivan, Mousley & Zevenbergen (2005) demonstrated how it was possible for teachers to pose appropriate various to the open-ended tasks, therefore allowing the task to “crate opportunities for extension of mathematical thinking” (Sullivan, Mousley & Zevenbergen 2005, p.106) as well as providing opportunity for teachers to enable the question to reduce the complexity for those who are not quite competent in answering the original posed question.

References for the rationale:

Bragg, L. A., & Nicol, C. (2008). Designing open-ended problems to challenge preservice teachers’ views on mathematics and pedagogy. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sepulveda (Eds), Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 201–208). Mexico: Cinvestav-UMSNH: PME.

Bragg, L. A. and Nicol, C. (2011). Seeing mathematics through a new lens: Using photos in the mathematics classroom. The Australian Mathematics Teacher, 67(3), 3-9

Churchill, R., Ferguson, P., Godinho, S., Johnson, N., Keddie, A., Letts, W., Mackay, J., McGill, M., Moss, J., Nagel, M., Nicholson, M. V. (2011). Teaching Making A Difference. John Wiley & Sons

Reys, R.E., Lindquist, M.M., Lambdin, D.V. and Smith, N.L. (2012, 10th Edition). Helping children learn mathematics. Hoboken, NJ: John Wiley and Sons.

Sparrow, L., & Swan, P. (2005). Starting out: Primary mathematics. Victoria: Eleanor Curtain Publishing.

Sullivan, P., Mousley, J. & Zevenbergen, R. (2005). Increasing access to mathematical thinking. Australian Mathematical Society Gazette,32(2), 105-109. The Society, St Lucia, Qld

Problem Picture 1 Location: Burwood Kmart

Problem Picture 1 - Questions

Grade level: 2

Question 1Using the photograph, find out the total value of any two items of your choice. Show 3 different ways you can make up that value using up to 4 notes and up to 5 coins.

Answers to Question 1

1. $24 + $59 = $83 I. $50, $20, $10, $2, $1

II. $50, $10, $10, $10, $1, $1, $1 III. $20, $20, $20, $20, $2, 50c, 50c

2. $15 + $39 = $54I. $50, $2, $2

II. $20, $20, $10, $1, $1, $1, $1III. $20, $10, $10, $10, $2, $1, $1

3. $65 + $10 = $75I. $50, $20, $5

II. $20, $20, $20, $10, $2, $2, $1III. $50, $10, $10, $5

AusVELS - Number and AlgebraContent strand/s, year, definition and code Level 2:

Number and place value: Solve simple addition problems using a range of efficient mental and written strategies (ACMNA030)

Money and financial mathematics: Count and order small collections of Australian coins and notes according to their value (ACMNA034)

Enabling PromptUsing the photograph, find out the total value of any two items of your choice. Show 3 different ways you can make up that value using any notes ($100, $50, $20, $10, $5) and any coins ($2, $1, 50c, 20c, 10c, 5c).

Answers to Enabling Prompt

1. $24 + $20 = $44 I. $20, $20, $2, $2

II. $10, $10, $10, $10, $1, $1, $1, $1III. $20, $20, $2, $1, 50c, 50c

2. $15 + $39 = $54I. $50, $2, $2

II. $20, $20, $10, $1, $1, $2 III. $10, $10, $10, $10, $10, $2, $2

3. $10 + $15 = $25I. $20, $5

II. $10, $10, $5III. $10, $5, $5, $5

AusVELS Content strand/s, year, definition and code Level 2:

Number and place value: Solve simple addition problems using a range of efficient mental and written strategies (ACMNA030)


Justification for change to the original questionState the modification you made to the original question:

Allowed students to use as many notes and coins as they want Prompted students with the possible notes and coins they can use

Why did you select this modification to make to the problem?

As the main objective was to acquire students to count and order collections of Australian coins and notes, it was possible to remove the restriction of how many notes and coins they were allowed to use for the freedom and ease of calculation. This modification allows students to use any notes and coins they are comfortable with without the restrictions, giving students the opportunity to truly focus on counting money to get to a particular value. The enabling prompt also provided students with the values of the Australian coins and notes for those who may be still unfamiliar with them.

Extending PromptUsing the photograph, find out the total value of three items. When you received change, you received a $10 note and some coins back. Work out how much you gave the cashier (in notes) and what coins you received back. Show 3 different ways you could have received your change.

Answers to Extending Prompt

** Note that the question says ‘coins’, therefore responses must contain more than one coin

1. $39 + $15 + $20 = $74I. I gave the cashier $50, $20, $20 ($90), therefore I could have received back

a. $2, $2, $2 b. $2, $2, $1, $1 c. $1, $1, $1, $1, $1, $1

2. $59 + $65 + $24 = $148II. I gave the cashier $100, $20, $20, $20 ($160), therefore I could have received back

a. $1, $1b. 50c, 50c, 50c, 50c c. $1, 50c, 50c

3. $39 + $65 + $49 = $153III. I gave the cashier $50, $50, $50, $20 ($170), therefore I could have received back

a. $2, $2, $2, $1 b. $2, $2, $2, 50c, 50c

c. $1, $1, $1, $1, $1, $1, $1

AusVELSContent strand/s, year, definition and code Year 2:

Number and place value: Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)


Year 3:

Money and financial mathematics: Represent money values in multiple ways and count the change required for simple transactions to the nearest five cents (ACMNA059)


Working out specific values to receive a certain amount of change Additional item added – finding out the value of 3 items


Another item was added to the total to increase the difficulty of working with larger numbers. This equation goes beyond just doing simple addition to acquire a provided value and gets students to use quick mental subtraction strategies to think about how much they need to give in order to receive a specific amount back. Only an extra item was added to the value, as the main focus is the mental subtraction strategy they apply when calculating the change, adding extra items will lead to the same strategy.

Cross-Curriculum LinksEnglish

Students can create their own multimodal narrative using the image as their starting point. The image may be used in a number of ways to assist students in developing their story such as the character or setting development. For example, the Minions may be used as a main character or their setting could be someone in a shopping centre looking at toys.

Firstly, students will be asked to develop a plan for their narrative noting down their characters, the setting, a problem or conflict and a resolution. Next students will be required to write out their story, with a beginning middle and end. Once the story is completed, and students have gone back and reread and edited their piece they will have a chance to draw their scenes on a piece of paper, 4-5 scenes will be drawn. Then once students have completed their drawings they will photograph their images on their iPad/tablet and load it onto a story making application (e.g. Story Creator), where they will be able to add audio and text to their images.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code English, Level 2

WritingLiterature

Create events and characters using different media that develop key events and characters from literary texts (ACELT1593)

Literacy

Create [a] short imaginative text 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)

Reread and edit text for spelling, sentence-boundary punctuation and text structure (ACELY1672) Write legibly and with growing fluency using unjoined upper-case and lower-case letters (ACELY1673) Construct texts featuring print, visual, audio elements using software (ACELY1674)

Report of Trialling Problem Picture 1 Child’s pseudonym, age and grade level: Evelyn, Age 8, Grade 2

Original Question: Find out the total value of any two items of your choice. Show 3 different ways you can make up that value using up to 4 notes and up to 5 coins.

Child’s response to the question:Answer to original Question 1:

The student had no problem doing the original question so I trialled the extending prompt

Reflection on child’s response:

My original question was “Find out the total value of any two items of your choice. Show 3 different ways you can make up that value using up to 4 notes and up to 5 coins.” which required Evelyn to recall the Australian dollar denominations and apply addition strategies to obtain the cost of two items. With the restrictions Evelyn had no difficulty working out a different solution every time. I asked her how she solved the problem and she said “well I picked any note and wrote that number down and then I picked another note and did the sum in my head first before I wrote it down in case the total went over, so first I picked $20 and then I thought I would add another $20 and I calculated that in my head and it was $40 so I wrote down $10 and then I knew to get to $50 I would only need $10 and since there is a $10 note I knew I could write down that too”. The level of mental strategies that were exhibited throughout the calculation of this problem already demonstrates her strong number sense. Shumway (2011) describes someone who has a strong number as someone who understand numbers, ways to represent numbers, relationship among numbers, and number systems. Students who make reasonable estimates, computes fluently and who uses reasoning strategies to figure out a problem.

As it was evident through Evelyn’s thinking strategies on the original question that she posed a strong number sense, it was anticipated that she was capable of answering the extended prompt. The question was answered as expected, where responses varied with their potential to show equivalent values of small change. For example Evelyn was able to demonstrate her knowledge that 50 cents is the same as two 20-cent coins and a 10-cent coin. I asked her what her strategy was for answering this question and she said “I added 10 to the answer because the question said I got back $10 and some coins, then I went to the nearest 10s number so 89 + 10 equals 99 so the nearest number was 100”. This example further establishes the fact that she possesses a strong number sense.

Much of Evelyn’s strengths in her mathematical understandings were highlighted through these questions. She had a strong understanding of Australian currency and was able to “identify equivalent values in collections of coins and notes” (Australian Curriculum Assessment and Reporting Authority [ACARA], 2013) and was able to recognise when subtraction or addition was required to solve the problem. Through each of her answers she demonstrated that she is able to work from any given number. She challenged herself numerous times as she worked out different combinations as well as working out a new total for her enabling prompt to get a different value. It is apparent that Evelyn is comfortable working with larger numbers, and when children learn the “verbal count list and understand cardinal values for numbers, they learn to represent larger numbers exactly and see that each number has a unique successor” (LeCorre & Carey, 2007; Sarnecka & Carey, 2008 as cited in Jordan, Glutting & Ramineni 2009, p.82). The dots she used under her third answer of the enabling prompt revealed that she is capable of skip counting. Evelyn’s use of mental addition and subtraction strategies that she used before writing the answer down showed that she thought about the problem before rushing into writing numbers down, meaning she’s an “effective problem solver” (Reys et al. 2012, p.113) as she planned ahead what she would do in order to solve a problem. She also used a “guess-and-check strategy” (Reys et al. 2012, p.126) where she made repeated educated guesses, using what has been learned from earlier guesses to make subsequent guesses well. Throughout her problem solving, she realized the value of the money she was adding on so she added a smaller value on.

The mathematical intent of the question was addressed, as Evelyn was required to solve simple addition to obtain the value of two items as well as “count collections of coins or notes” (ACARA, 2013) to make up that particular value. She understood that there were many combinations she should create that make up the same value.

References for reflection on the trial of question 1:

Australian Curriculum Assessment and Reporting Authority. (2013). The Australian Curriculum. Retrieved July 30, 2015, from http://www.australiancurriculum.edu.au

Jordan, N.C., Glutting, J., Ramineni, C. (2009). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences. 20, 82-88. http://dx.doi.org/10.1016/j.lindif.2009.07.004


Shumway, J.F. (2011). Number sense routines: building numerical literacy everyday in grades K-3. Portland: Stenhouse Publishers

Problem Picture 2Location: Bogong Park, Glen Waverley

Grade level: 2

Question 2Identify and describe all the 2D and 3D shapes you can see in the photo and draw all the 2D shapes. Choose a shape and show two different types of transformations.

Answers to Question 2

AusVELS - Measurement and GeometryContent strand/s, year, definition and code Level 2:

Shape:

Describe and draw two-dimensional shapes, without digital technologies (ACMMG042) Describe the features of three-dimensional objects (ACMMG043)

Location and transformation:

Investigate the effect of one-step slides and flips without digital technologies (ACMMG045) Identify and describe half and quarter turn (ACMM6046)

Enabling PromptWhat 2D and 3D shapes you can see? How many faces, edges and corners does each shape have? Draw all the 2D shapes and choose one and show two different types of transformations (reflection, slide or rotation).

Answers to Enabling Prompt

Shapes are the same as question 1 – Here are other possible transformations.

AusVELS Content strand/s, year, definition and code Level 2:

Shape:

Describe and draw two-dimensional shapes, without digital technologies (ACMMG042) Identify the features of three-dimensional objects (ACMMG043)


Investigate the effect of one-step slides and flips without digital technologies (ACMMG045)

Identify and describe half and quarter turn (ACMM6046)


Simplified language and direct instruction for describing the shape Prompted students with the 3 types of possible transformations


This modification allows for the students who are not yet familiar with the terms of describing and transforming to be guided with direct instruction. The prompts allow students to focus on remembering and applying the technique that they have previously learned.

Extending PromptIdentify, draw and describe all the 2D and 3D shapes you can see in the photo. What other 3D shape (one you have not mentioned) can be made using one of the 2D shapes as a base? Describe it and how two different types of transformations using this 3D shape.

Answers to Extending PromptAll the 2D listed in the original answer. 3D Shape drawings are shown below.

AusVELSContent strand/s, year, definition and code Level 2:

Shape:

Describe and draw two-dimensional shapes, without digital technologies (ACMMG042) Describe the features of three-dimensional objects (ACMMG043)


Investigate the effect of one-step slides and flips without digital technologies (ACMMG045) Identify and describe half and quarter turn (ACMM6046)


Extended students thinking by getting them to think of other 3D shapes using the 2D shapes they found as base


The modification moves students beyond what 3D shapes they can see and apply their knowledge of other 3D shapes, using the 2D shapes they found in their picture as a catalyst to their thinking. The prompt also extends student’s thinking on how the 3D objects would look when they have been transformed.

Cross-Curriculum LinksScience

The focus of the science lesson will be to investigate the physical sciences of the push and pull affects. Using the image as a starting point students will describe what happens to the shape of the left swing chair when someone sits on it, and again describe what happens when someone pushes the person sitting on the swing. Students will then be asked to see if there is anything else in the image of the playground, where a push and full affect can be applied (e.g. swing, ropes etc). They will then investigate what happens when they pull a rubber band and let it go (doing this against a wall, away from students). While completeling a POE (predict, observe and explain) sheet, students will firstly predict what happened and write their observations down then test their theory by pulling on the rubber band and finally writing their observation and explanation down. Once the whole class is finished, they will communicate through a class discussion, sharing their hypothesis and explanations.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code Science, Level 2

Science Understanding

Physical sciences

A push or pull affects how an object moves or changes shape (ACSSU033)

Science as a Human Endeavour

Nature and development of science

Science involves asking questions about, and describing changes in, objects and events (ACSHE034)

Science Inquiry Skill

Questioning and predicting

Respond to and pose questions, and make predictions about familiar objects and events (ACSIS037)

Planning and conducting

Participate in different types of guided investigations to explore and answer questions, such as manipulating materials, testing ideas, and accessing information sources. (ACSIS038)

Evaluating

Compare observations with those of others (ACSIS041)

Communicating

Represent and communicate observations and ideas in a variety of ways such as oral and written language, drawing and role play (ACSIS042)


Original Question: Identify and describe all the 2D and 3D shapes you can see in the photo. Draw all the 2D shapes. Choose a shape and show two different types of transformations.

Child’s response to the question:

Extending prompt answers

Reflection on child’s response:My original question was “Identify and describe all the 2D and 3D shapes you can see in the photo. Draw all the 2D shapes. Choose a shape and show two different types of transformations”. Evelyn had no problem connecting the shapes she knew and identifying them in the playground.

The problem was answered as expected, it was evident that she has had a lot of practice and experience in describing the properties of shapes as she used the correct terminology as she counted the ‘edges’ and ‘corners of the 2D shapes, as well as ‘faces’ for 3D shapes. Her ability to apply previous knowledge of describing shapes using the correct terminology demonstrated that she was “moving toward a more precise description of classes of shapes” (Reys et al. 2012, p.376). Evelyn also exhibited great capability in her knowledge of transformations, where she was able to successfully reflect the shape on the x-axis.

Evelyn’s answer on the extending prompt was what I was expecting. I asked her how she was able to draw the 3D shapes and she said she had used the image to help her and she copied the way it was drawn in the image, and simply connected the lines together to form a closed 3D shape. She was also able to make the connection between what she knew about 2D shapes and 3D shapes and applied her understanding through her answer, that the bottom of a cone is a circle.

Many strengths of her mathematical understanding were highlighted through her answers to these questions. As “children begin forming concepts of shapes long before they enter school” (Clements & Sarama 2000a, p.82), it is apparent that she has had a high capability of making continuous connections with shapes to real life objects as she is learning. Her thinking moved beyond simply distinguishing shape features to actually describing them with detail. Clements & Sarama (2000b) describe this thinking level as the descriptive level, where children recognise and can characterize shapes by their properties. Evelyn was also able to apply the relationship between 2D and 3D shapes in the enabling prompt, and this is important, as “students need concepts from 2-dimensional shapes in order to more

completely describe 3-dimensional objects” (Reys et al. 2012, p.376). Her competent knowledge of 2D shapes also helped her to describe and count the edges, faces and corners of the 3D shapes in the photo.

Overall, the question did address the mathematical intents of describing 2D shapes as she was required to find different shapes in the photo and identify key features by counting the edges and corners (ACARA, 2013).


Australian Curriculum Assessment and Reporting Authority. (2013). The Australian Curriculum. Retrieved July 30, 2015, from http://www.australiancurriculum.edu.au

Clements, D.H., & Sarama, J. (2000). The Earliest Geometry. Teaching Children Mathematics, 7(2), 82-86

Clements, D.H., & Sarama, J. (2000). Young Children’s idea About Geometric Shapes, Teaching Children Mathematics, 6(8), 482-488


Problem Picture 3Location: My house


Grade level: 2

Question 3Using the photograph of the fruits, generate a question to conduct a survey with your peers. With the data gathered, represent your findings in the best possible way and interpret them.

Answers to Question 31. Which one of these fruits is your most favourable?

Fruit Students

Orange |||| ||

Banana ||||

Apple |||| ||||

Pear ||

Mandarin |||

My Class’s Most Favourable Fruit

Apples are the most favourable fruit in the class

= 1 student

2. Which one of these fruits is your least favourable?

Orange

Banana

Apple

Pear

Mandarin

3. On average, how often do you eat bananas?

Everyday 1 to 2 3 to 4 5 to 6 Never0

2

4

6

8

10

12

Times a week

Num

ber o

f Stu

dent

s

AusVELS - Statistics and Pro babilityContent strand/s, year, definition and code Level 2

Data representation and interpretation

Bananas are the least favourable fruit in the

More people in the class never

eat bananas

Fruit Students

Orange ||||

Banana |||| ||||

Apple |

Pear ||||

Mandarin ||||

Frequency Students

Everyday |||

1-2 times a week

||||

3-4 times a week

|||

5-6 times a week

||||

Never |||| ||||

Identify a question of interest based on one categorical variable. Gather data relevant to the question (ACMSP048)

Collect, check and classify data (ACMSP049) Create displays of data using lists, table and picture graphs and interpret them (ACMSP050)

Enabling PromptUsing the photograph of the fruits, think of a question you can ask the class. Gather the data and record the information you collect in a table, and represent the information in a graph. Looking at the graph, explain what you have found out.

Answers to Enabling Prompt1. Do you prefer eating oranges or mandarins?

Oranges Mandarin02468

1012141618

Oranges or Mandarins?

Fruit

Num

ber o

f Stu

dent

s

2. Would you prefer eating an apple, banana or pear?

Apple Banana Pear02468

101214

Which do you prefer?

Fruit

Stud

ents

3. Looking at the photo, which fruit do you think is the heaviest?

People in the class prefer eating mandarins

The students in the class prefer eating apples out of

Fruit Students

Oranges |||| |||

Mandarins |||| |||| |||| ||

Fruit Students

Apple |||| |||| ||

Banana |||| ||

Pear |||| |

Red Orange Yellow Green0123456789

10

Heaviest Fruit

Fruit

Num

ber o

f Stu

dent

s

AusVELS Content strand/s, year, definition and code Level 2





Communication of strategies were more explicit Language of the problem is simplified


The terminology was simplified as students are still learning these terms. The prompts guide students in a more obvious direction where the steps control the students to record and represent their data in a particular way.

Most of the students in the class think that the orange is the heaviest fruit and no one in the

Fruit Students

Orange |||| |||| |||| ||

Banana |

Apple ||||

Pear ||

Mandarin

Extending PromptUsing the photograph of the fruits, generate a question to conduct a survey with your peers. With the data gathered, represent your findings in the best possible way. From your interpretation of the data, explain whether the results represent the whole school.

Answers to Extending Prompt

1. Which fruit is your most favourable?


1

2

3

4

5

6

7

8

9

10

Most Favourable Fruit

Fruit

Num

ber o

f Stu

dent

s

2. Would you prefer the colour red, yellow, orange or green?


2

4

6

8

10

Preferred Colour

Colour

Stud

ents

3. Which fruit do you mainly see at your house the most?

The most favourable fruit in the class are apples and the least are pears. The results cannot determine

school would be apples

students in the school, so surveying 25 students is

The most preferred colour

can’t represent the whole

fraction of the school was

Fruit Students

Orange |||| ||

Banana ||||

Apple |||| ||||

Pear ||

Mandarin |||

Colour Students

Red |||| ||

Orange ||||

Yellow |||| ||||

Green |||||

Orange Banana Apple Pear Mandarin None0

1

2

3

4

5

6

7

8

9

Fruits Mostly Seen at Home

Fruits

Num

ber o

f Stu

dent

s

AusVELSContent strand/s, year, definition and code Level 2





Extending the interpretation of data


The interpretation requires them to think beyond their classroom and to consider the appropriateness of the data collection method in regards to whether the results they have collected can represent the whole school. It focuses students to think about the relationship between their generated question and the data collection method (only asking peers), and whether it is enough to make a generalised statement.

Cross-Curriculum LinksHealth

As students learn about healthy eating and having a healthy life style, the image of fruits can be incorporated into a health lesson where the students learn to make healthy food choices. As the food pyramid gets introduced into the lesson, students can think about where the fruit goes into the pyramid. Using the image as a starting point, students are to list other similar foods that would fit in the same category. Then a discussion can be formed around the benefits of eating fruit daily, examining the vitamins, minerals and nutrients that fruits can provide. The lesson will further explore the number of servings of particular food groups that children their age should be consuming daily, the 4 food groups will be broken into vegetables and fruit, grain products, milk and alternatives and meat and

Most students in the

different households

Fruit Students

Orange ||||

Banana |||| ||

Apple |||| |||

Pear

Mandarin ||||

None

alternatives, and the types of foods in that group students should choose to consume (for example, grain products which are low in fat, sugar or salt). Which will lead to a further discussion of the benefits of eating other food groups.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code Health and Physical Education – Level 2

“Learn to make healthy food choices according to healthy eating models, and to consider the factors that influence their choice of foods. They begin to recognise the importance of variety and frequency of food consumption for an active and healthy life” (VCAA, 2013)


Original Question: Using the photograph of the fruits, generate a question to conduct a survey with your peers. With the data gathered, represent your findings in the best possible way and interpret them.

Child’s response to the question:

Extending prompt was also asked – “From your interpretation of the data, explain whether the results represent the whole school.”

[Transcript of conversation]

Me: What’s your interpretation of the data?

Evelyn: There are more people in the class who prefer eating bananas

Me: Could these results represent the whole school?

Evelyn: I don’t think we can say the whole school would prefer to eat bananas because there’s so many people and so many classes in the school and I only asked the people in one class and in grade 2, maybe the older or younger kids might like something else

Me: If we didn’t have time to ask everyone, what would be a good way to see what the whole school prefers?

Evelyn: ummm… maybe we could ask some students in each class but then what if we happen to only ask the people who like a certain fruit then it wouldn’t be very fair

Me: So do you think there would be any other way?

Evelyn: Maybe it would be best to just ask most people because then it would be more fair… we can’t just say everyone in the school prefers like say bananas if we only asked some students

Reflection on child’s response:My original question was “Using the photograph of the fruits, generate a question to conduct a survey with your peers. With the data gathered, represent your findings in the best possible way and interpret them.”, Evelyn had no difficulty in interpreting the question, she immediately knew that conducting a survey meant she had to use some form of a table and a tally to collect the data. This understanding meant she answered the question as expected as she was able to represent her findings in some easily understood form, and that using tally marks is extremely useful. I was rather impressed by her efforts in creating the graph; she has obviously had some great experience in creating graphs in the past. I noticed she was looking around at the graphs they had previously created in the class, which means she uses her resources around her very well. When interpreting the data, she was able to “understand the relationship between the data and the context of the graphic display in which they appear” (Ontario 2007, p.24), therefore being able to see that the number ‘6’ next to ‘bananas’ means nothing if you look at it alone, and that it is required of her to look at the whole table to get a holistic understanding that 6 is the highest number, therefore bananas is the most preferred fruit in the class.

Reys et al. (2012) describes how data analysis and statistics provide a meaningful context for promoting problem solving and critically thinking. The extending prompt was answered as expected and allowed Evelyn to go beyond just interpreting what the data says and extends her thinking to how the data can/cannot represent the whole school. Learning should “focus on the process of exploration” (Pratt 2006, p.16), not simply just finding an answer. The extending prompt allows students to critically think and provide reason to a question, students are able to explore possible solutions to what it means to look at data in a different way.

Throughout Evelyn’s answers she has provided a strong mathematical understanding on representing and interpreting data. It is an important part of mathematics learning to have the knowledge related to constructing and interpreting data (Reys et al. 2012, p.436) as students encounter ideas of statistics outside of school every day. It is evident through my conversation with Evelyn about the data that she has a strong understanding of how data is looked at, she reads beyond the data and “makes inferences about the data” (Ontario 2007, p.25) where she applies background knowledge to interpret information that is not explicitly stated in the graph. It is important for teachers to “support learners in coming to understand these ideas in new ways” (Pratt 2006, p.16, therefore someone with the critically skills like Evelyn would need to be extended even further, to promote a deeper level of understanding on how data can be interpreted.

I believe the question has addressed the mathematical intention of the question, as students were required to generate a question and gather relevant data to create a graph. Bohan, Irby & Vogel (1995) discuss how it is beneficial for students to identify their own questions as it gives them that ownership of the analysis, therefore their motivation for the study will be high. The open-endedness of the task is highly engaging and “fosters more important aspects of learning mathematics” (Sullivan, Mousley & Zevenbergen 2005, 106).


Bohan, H., Irby, B. & Vogel, D. (1995). ‘Problem solving: Dealing with data in the elementary school’. Teaching Children Mathematics, 1(5)(January), pp.256-260

Ontario (2007). A Guide To Effective Instruction In Mathematics, Kindergarten To Grade 3 - Data Management and Probability. Ontario Education

Pratt, N. (2006). Interactive Maths Teaching in the Primary School. London: Paul Chapman Publications


Sullivan, P., Mousley, J. & Zevenbergen, R. (2005). Increasing access to mathematical thinking. Australian Mathematical Society Gazette,32(2), 105-109. The Society, St Lucia, Qld

Open-ended problem pictures

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