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Kerryn Agnew – Student #11533562 EMM209 Assessment 2: Developing a problem-based lesson

Draft Lesson Plan – Submitted for review on 16/1/2014Unit/Lesson Title: Cameron the caterpillar grows up (Adapted from Black Douglas Professional Education Services, n.d.b) Stage 3 Year 6RationaleThis lesson with develop students’ abilities to apply mathematical reasoning to real world problems, by identifying the relevant information and rules required to solve the problem. The lesson will develop systematic problem solving strategies and is also designed to create positive attitudes to problem solving and improve lesson engagement.

Syllabus Outcomes (NSW Board of Studies, 2012, p198)MA3-1WM describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventionsMA3-2WM selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigationsMA3-3WM gives a valid reason for supporting one possible solution over anotherMA3-5NA selects and applies appropriate strategies for addition and subtraction with counting numbers of any sizeMA3-8NA analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane

Prior KnowledgeStudents have prior knowledge of addition and subtraction using one, two and three digit numbers, multiplication, reading and solving mathematical word problems, and working in small groups. Students have prior knowledge of recognising, describing and constructing simple number patterns.

Resources Interactive whiteboard and presentation resource presenting problem to be solved (presentation available

at https://eportfolio.csu.edu.au/pebblepad/viewasset.aspx?oid=915326&type=file ) Worksheet outlining the entire problem (one per student) (Appendix A) and highlighters Worksheet with table for students to complete (one per student) (Appendix B) A number of large coloured circles in two different colours plus one large caterpillar head (Appendix C

demonstrates how these would be used) Counters of two different colours plus one snake head per group (photo examples Appendix D) Camera for photographing student work (photo examples Appendix D) Poster of problem solving steps and strategies (Appendix E) Student self-evaluation sheets (Appendix F) Suggested teaching scripts (Appendix G)

Learning IndicatorsStudents can use efficient mental and written strategies to solveproblems (ACMNA291)

Assessments Content / Learning Experiences / Teaching StrategiesIntroduction (Engagement)Using the interactive white board and presentation, introduce Cameron the caterpillar (Slide 1 & 2 https://eportfolio.csu.edu.au/pebblepad/viewasset.aspx?oid=915326&type=file). Tell the students he needs our help and ask can they help him?Problem details: Cameron the caterpillar can’t wait to grow up. When Cameron was born, his body was just one colour – Colour A. Each year, he sheds his skin completely and grows longer. When he sheds, every Colour A section in his old skin is completely replaced by 2 Colour A sections and a Colour B section. The Colour B section is always between the Colour

Class Organisation

Teacher at the front of the class with students seated together on the floor.

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https://eportfolio.csu.edu.au/pebblepad/viewasset.aspx?oid=915326&type=file


This includes:• adding three or more numbers with different numbers of digits without the use of digital technologies’

• selecting and applying efficient mental and written strategies to solve addition and subtraction word problems

• checking solutions to problems’

• provides a rationale to why solution is correct

Assessment 1:Teacher observes whether students highlight appropriate words on smart board, displaying awareness of data required to solve problem.

Assessment 2:

A section (ABA). As each year passes, every Colour B section stays as Colour B and every Colour A section is completely replaced with the Colour ABA combination. What does Cameron look like in his 10th year of life?

The relatable problem of Cameron the caterpillar and his need to grow up helps to engage the students and give the problem relevance (Newman & Wehlage, 1993, p10).Highlight to students that they are going to work through the problem using a systematic strategy. Ask the students to brainstorm their understanding of “working systematically” to clarify students’ prior knowledge as well as putting the language into children’s language. Refer to the poster of problem solving steps and strategies (Appendix E). Have this poster in a prominent position in the classroom for students to utilise. Clarify any areas necessary based on student brainstorming responses. Ask the students what information they need to solve the problem. Students review the problem, brainstorming the key mathematical words required to solve the problem. Using the smart board, have students highlight the relevant language and remove unnecessary language from the problem (Example on Slide 4 of presentation). This is Step 1 in Polya’s Problem Solving Steps, understanding the problem (Jonassen, 2010, p4).Cultural diversity teaching strategy: Analysing the language supports students with English as a second language. Australian Indigenous students benefit from modelling before doing (“8 Aboriginal ways of learning”, 2013).Body (Exploration/Transformation/Presentation)Ask the students how they might start to solve the problem. Refer to the problem solving steps and strategies poster (Appendix E). Allocate one student to be Cameron’s head and give a number of other students Colour A circles and Colour B circles (see Appendix C for demonstration). Ask the class want Cameron looks like in his first year of life. Have a student with colour A come up in front of the class to demonstrate how Cameron looked in his first year of life. Ask the students what happens during Cameron’s second year. Prompt students with the problem criteria and ensure Colour A gets completely replaced by sending the first student away and having 3 new students come to the front. Have students show what happens when Cameron is in his 3rd year, with additional students coming up and representing the different segments of Cameron’s body. Continue to prompt students to reflect on problem criteria. Questions might include: Which colours should stay and which should go? How many new colours do we need? Where should they go? Show how Colour B remains and Colour A is

Teacher at the front of the class with students seated together on the floor. Students actively participating by becoming part of Cameron’s body and through brainstorming solutions.

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Teachers observe student participation in brainstorming problem solutions and students rationale when providing solutions.

replaced with the ABA combination. See Appendix G Script A for suggested teaching script.Kinaesthetic learners teaching strategy: Catering for diversity in the way students learn. In this case, students learn through movement (McDevitt, Ormrod, Cupit, Chandler & Aloa, 2013, p300) Indigenous Australian students teaching strategy: Indigenous Australian students benefit from modelling before doing (“8 Aboriginal ways of learning”, 2013).

Students return to the classroom and work in pairs, using coloured counters and a worksheet containing a table (Appendix B & D). Students are also provided with the problem (Appendix A). Tell the students they are going to use the coloured counters to work out what Cameron looks like each year in a similar way to how they used their bodies to work out what Cameron looks like. They will then record each year on the worksheet table provided. This is Step 2 in Polya’s Problem Solving Steps, making a plan or strategy (Jonassen, 2010, p4). Provide students with a starting point for solving the problem by asking students to solve the problem starting with when Cameron in year 1, year 2 and year 3 as per activity just conducted. This is Step 3 in Polya’s Problem Solving Steps, carry out the plan (Jonassen, 2010, p4). Refer back to the solutions obtained in the previous physical activity. Use presentation slide 5 and 6 to help students visualise the replacing of the sections as each year goes by.

Spatial learners teaching strategy: Visual organisation of ideas and concepts caters for spatial learners (Murray and Moore, 2012, p44).Circulate the classroom and engage in discussions with students about the problem, and prompt students to reflect on solution and problem criteria. Take photos of student’s solutions for class discussion and assessment. Allow students to move to the floor to allow more space for laying out the counters.Ability levels teaching strategy: Students are working in pairs allowing different ability levels to support each other, operating within and advancing the zone of proximal development (McDevitt et al., 2013, p223). Students are given a starting point for working out the problem by prompting them on the solutions discovered in the physical activity using the presentation slides. Once students are starting to run out of counters and space to conduct investigation (at Cameron’s 4th or 5th year), ask the students if they are having trouble getting to Cameron’s 10th year using the counters. Why? Where were they able to get to? This is Step 4 in Polya’s Problem Solving Steps, evaluate strategy effectiveness (Jonassen, 2010, p4).

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Assessment 3:Photos are taken of student’s solutions and displayed on the interactive white board. Students reflect as a group on the solutions provided. Teachers question and observe students rationale.

Assessment 4:Teachers observe which students can solve the mathematical sentences in relation to the problem.

What are their answers so far? As a group, students review photos of student solutions. Teachers prompt students to reflect on whether the photos meet the problem criteria and ask students to provide rationales as to why or why not. Record the answers on slide 9’s table in the presentation. (https://eportfolio.csu.edu.au/pebblepad/viewasset.aspx?oid=915326&type=file). Ask the class what other strategies other than using the counters could they use to get to Cameron’s 10th year? Using slide 10, ask students can they see any patterns in the colour A column? Is looking for patterns a strategy on the problem solving poster (Appendix E)? Prompt students to compare the numbers in the colour A column. What happens to the no. of colour A sections as each year passes? Write the students response to this question using their words. Follow a similar script as used for colour A including: test out the student’s theories about the pattern once tested, have the students explain in words the pattern and write it on slide 10 e.g.

colour B is equal to colour A minus 1 and / or colour B is equal to the total number of sections from the previous year

have the students write a mathematical equation for the pattern e.g. B=A-1 or B Year 1=A year 1 – 1= 1-1=0

have the students use the mathematical equation to work out year 5 and / or 6Ask students how they would work out the total no. of sections and create a mathematical equation to represent this i.e. Total sections s=A+B.Students now return to Step 3 in Polya’s Problem Solving Steps, carry out the plan (Jonassen, 2010, p4) as they are now using a new strategy to solve the problem. Highlight this to students on the problem solving strategy poster (Appendix E), noting that it is appropriate to return to different steps if the first strategy doesn’t work. Students return to working in pairs to complete the rest of the worksheet up to year 10. Circulate the classroom and engage in discussions with students about the patterns discovered and the mathematical representation of those patterns, assisting where necessary.

Students working in pairs at desks or on the floor with teachers circulating.

Assessment 5:Teacher collects completed worksheets from pairs.

Conclusion (Presentation/Reflection)As a group, students review the solution starting from where the class left off (i.e. either year 5 or 6). Pairs of students take turns coming to the board and completing the table on slide 10 of the presentation. Problem Answer: Cameron will have a total of 1023 sections in his 10th year with 512 being the colour A and 511 being the colour B.

Teaching at the front of the class with students seated at desks. Students reflect as a group on learning.

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Assessment 6:Teacher collects individual student reflection sheets about problem solving.

Using Slide 11, take the students through the process of checking the final solution. Ask the students how they might do this. What do all the totals have in common? The total sections are equal to A plus B. They are all odd numbers. Is this enough to check the answer? Ask the students “if Cameron has 1023 total sections, how could you work out how many were A, and how many were B?” Is there connection between colour A and the total number of sections? Is there a connection between colour B and the total number of sections or between colour A and colour B? Prompt the students to work through the problem solving methodology as beforeThis is Step 4 in Polya’s Problem Solving Steps, evaluate strategy effectiveness (Jonassen, 2010, p4).To take the problem further, ask the students could they solve the following problem using the checking method they just used (slide 12):If Cameron has a total of 32767 sections, how many colour A and colour B does Cameron have?Congratulate the students for coming up with the answer and encourage them to be proud of their achievement. Discuss some of the areas students found easy and those students found more difficult. What did they enjoy about the problem and / or the process? Why? What did they find difficult? How did they overcome this? Have the students complete the individual reflection sheet (Appendix F).

Next Lesson or extension: Extension question using same problem: How can we calculate what Cameron will look like in his 25th year? Or 100th? This will involve students using algebra including the introduction of indices. Other problems for extension:Mushroom Hunt (Black Douglas Professional Education Services, n.d.c)

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Peer ReviewsPeer review 1: Name of lesson's author: Kerryn Agnew Name of student reviewer: Laree AshfordCriteria: Ability to plan a high quality lesson? CommentsAlignment and appropriateness of rationale, syllabus outcome(s), stage, prior knowledge, learning indicators, experiences and assessment.

Alignment and appropriateness of syllabus outcome(s), rationale, year-level and prior knowledge? Alignment and appropriateness of learning experiences with intended aim?

Syllabus outcomes are appropriate for chosen stage and year. Rationale explains what students will gain from the lesson and prior knowledge is clear and detailed. Learning experiences align with intended aim.

Quality of learning experiences

Does the introduction set the stage? Is it likely to engage children and draw upon their experiences? Are students guided by teacher questions and prompts? How appropriate are the learning experiences? To what extent do they reflect the models of learning? To what extent do they cater for different learning styles? To what extent does the lesson conclude sensibly and provide children with an opportunity to reflect?

The introduction provides students with the knowledge and understanding to fully engage in the problem. The brainstorming activity will encourage students the draw upon prior experiences. Students are guided throughout the lesson by teacher questioning and prompts. The learning experiences are fun, engaging and appropriate for year 6, however some of the equations may be too hard. Modelling the problem, hands on activities and group work will support different leaning styles and backgrounds. The students are given an opportunity to reflect through the self reflection worksheet, and the lesson concludes with a number of possibilities for further learning in this area.

Quality of the problem Is the problem open? Has a solution been provided? Will it engage the children? Is the difficulty appropriate? Does it cover much mathematics?

The problem is engaging and the group activity with the students forming sections of the caterpillar will make it easier for students to understand. The difficulty of the problem is appropriate for year 6. It covers problem solving and various strategies, addition, subtraction and number patterns.

Communication Is there enough information provided for you to answer these questions? Does the writing contain spelling errors?

It is a well detailed and thought out lesson plan, however the references used regarding learning styles were repetitive. Resources are of high quality and will enhance the lesson. Sufficient information provided and no spelling errors.

Creativity How different is this to a lesson you have seen before? How creative are the learning experiences?

The problem used is one I haven't seen before however the style of learning experiences are similar to others I have viewed in EMM209. The lesson contains creative and fun experiences.

Ability to deliver the 5 marks: 4 marks: 3 marks: <3 marks:6


beginning of a problem-based lesson

( 5/ 5)

The presentation is highly engaging and clearly reflects the lesson plan. It is delivered in very coherent and professional manner.

The presentation is engaging and reflects the lesson plan. It is delivered in a coherent manner.

The presentation is satisfactory and in the main reflects the lesson plan. It is delivered in a satisfactory manner.

The presentation does not reflect the lesson or the presenter does not attempt to teach the lesson. The delivery is poor. Or the presentation does not occur.

Additional comments:Kerryn, Your lesson plan, including resources, has been well thought out. The problem is interesting and will engage students. You have put a lot of thought into how the lesson will be implemented, including possible scripts and questioning. When referencing in the content of your plan, ensure it doesn't become repetitive. I noticed photos are taken of students work, are these to be used in this lesson? They could be used in the reflection at the end of the lesson. Overall it is a great lesson, I really enjoyed reading it. Larree Ashford

Peer review 2: Name of lesson's author: Kerryn Agnew Name of student reviewer: Alysha AllisonCriteria: Ability to plan a high quality lesson? CommentsAlignment and appropriateness of rationale, syllabus outcome(s), stage, prior knowledge, learning indicators, experiences and assessment.


Syllabus outcomes are in alignment with both rationale and stage. Prior knowledge and learning experiences would indicate that this is a stage 3 lesson.

Assessments are timely throughout the lesson. Assessment 6 states that they hand in a reflection sheet – time for this to be completed has not been stated in the plan.



Lesson jumps straight into the problem question – suggestion that there is a statement of learning intentions to open the lesson. Lesson is engaging, especially the graphics used in the powerpoint slides – I can see students finding him amusing. You have thought of the different capability levels and learning styles and have catered for them appropriately.


Reading through the problem I found it difficult to understand the completely replacing colour A each time. Even with your graphics I still had to represent it myself as a tree diagram (below). It could be useful to use arrows to indicate the replacement each time. Solution was provided and was able to

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understand the mathematical pattern (in numbers). In your last slide you provide an equation for solving using ‘s’ to represent a number – you might like to explicitly say in prior learning that you have touched on the concept of equations.

Communication Is there enough information provided for you to answer these questions? Does the writing contain spelling errors?

Lesson plan is full and complete and could be taught off as is. No spelling errors identified.


I have seen this lesson online before and avoided it as I couldn’t work out how to teach it – well done for being able to figure it out! Lesson teaches number patterns in a creative and engaging way.

Ability to deliver the beginning of a problem-based lesson

( ___4.5__/ 5)

5 marks: The presentation is highly engaging and clearly reflects the lesson plan. It is delivered in very coherent and professional manner.

4 marks:The presentation is engaging and reflects the lesson plan. It is delivered in a coherent manner.

3 marks:The presentation is satisfactory and in the main reflects the lesson plan. It is delivered in a satisfactory manner.

<3 marks:The presentation does not reflect the lesson or the presenter does not attempt to teach the lesson. The delivery is poor. Or the presentation does not occur.

Additional comments: Kerryn, you have presented a very well thought out and highly professional lesson plan. I am sure the “replace colour A each time” works within the classroom setting using the students, but for me reading the lesson plan I came to the solution set out below. For this reason I had to take 0.5 marks off. Otherwise excellent and shows the standard that we all must reach.

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Review of Peer’s Lesson Plan: Alysha AllisonCriteria: Ability to plan a high quality lesson? CommentsAlignment and appropriateness of rationale, syllabus outcome(s), stage, prior knowledge, learning indicators, experiences and assessment.


Syllabus outcomes align with stage and year. There is a significant amount of subtraction and addition throughout the solving of the problem so this should be noted in the prior knowledge. The rationale could be extended to include practicing multiplication skills as well as developing the mathematical language of multiplication. This would allow a more direct correlation between the learning experiences and the intended aim.



The introduction sets the stage with the students pretending to be witches, which should be very engaging for the students. Having students put on the witches’ hat is a good way to get engagement in group discussion. The physical use of a cauldron is great, too. The use of the video at the start sets the stage and is a fun element. The children have experience with spiders, bats and lizards as well as witches’ spells so the problem should be relatable and interesting. There are lots of opportunities for the students to think for themselves with the teacher using open-ended questions to prompt student reflection. There are a variety of activities including the use of manipulatives, visual aids, modelling and group work, which will cater to a variety of learning styles. A number of different level options have also been provided. The lesson provides activities which progress students through the language model from children’s language to symbolic language. The problem solving strategy is modelled for the students through a group activity then there is social support in the form of student pairs in carrying out the strategy. The worksheet helps support the structure.


The problem is open with a number of solutions. The solution has not been provided though it is referred to as part of a “solution sheet”. The difficulty is appropriate with a number of different level options available. It covers a significant portion of mathematics with the students being taken from children’s language through to the mathematical language of multiplication.

Communication Is there enough information provided for you to answer these questions? Does the writing contain

There is enough information to answer the questions. It is well laid out. A list of resources has been provided. There are a number of spelling errors which need correcting. Some of the sentences could be reworded to make the intention clearer

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spelling errors? e.g. “Ascertain that all children understand the problem that we are looking at is using multiplication to find the total number of legs to reach the solution” could be changed to “Ascertain that all children understand the question. Do the students understand they need to focus on the groups of legs and hence, multiplication is involved?”.


This lesson is similar to problems that were presented in EMM209 tutorial 2 regarding the legs and tails of a number of horses as well as a problem regarding table and stool legs. These could be used as extension tasks or follow on lessons. The presentation is original and creative as well as interactive. Children will enjoy using the cauldron, bats, spiders and lizards to solve the problem.

Ability to deliver the beginning of a problem-based lesson

( 3.5/ 5)

5 marks: The presentation is highly engaging and clearly reflects the lesson plan. It is delivered in very coherent and professional manner.

4 marks:The presentation is engaging and reflects the lesson plan. It is delivered in a coherent manner.

3 marks:The presentation is satisfactory and in the main reflects the lesson plan. It is delivered in a satisfactory manner.

<3 marks:The presentation does not reflect the lesson or the presenter does not attempt to teach the lesson. The delivery is poor. Or the presentation does not occur.

Additional comments:Dear Alysha, This problem is presented in an engaging way with creativity which I believe will engage year 3 students. The activities and assessments are appropriate and will assist in the student learning. The draft lesson plan could benefit from clarification of the rationale, additional editing and spelling corrections, and inclusion of the problem solution, which I’m sure you plan to do in your final plan. This would raise the standard of the lesson to 4 or more out of 5. Best Regards, Kerryn Agnew

Final Lesson PlanUnit/Lesson Title: Cameron the caterpillar grows up (Adapted from Black Douglas Professional Education Services, n.d.b) Stage 3 Year 6Rationale Syllabus Outcomes (NSW Board of Studies, 2012, p198)

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This lesson with develop students’ abilities to apply mathematical reasoning to real world problems, by identifying the relevant information and rules required to solve the problem. The lesson will develop systematic problem solving strategies and is also designed to create positive attitudes to problem solving and improve lesson engagement. The lesson will look at extending students abilities to use symbolic language and algebra to describe number patterns.

MA3-1WM describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventionsMA3-2WM selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigationsMA3-3WM gives a valid reason for supporting one possible solution over anotherMA3-5NA selects and applies appropriate strategies for addition and subtraction with counting numbers of any sizeMA3-6NA selects and applies appropriate strategies for multiplication and division, and applies theorder of operations to calculations involving more than one operation MA3-8NA analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane

Prior KnowledgeStudents have prior knowledge of addition and subtraction using one, two and three digit numbers, multiplication, reading and solving mathematical word problems, and working in small groups. Students have prior knowledge of recognising, describing and constructing simple number patterns. The students have been introduced to using simple algebra to describe number patterns.

Resources Interactive whiteboard and presentation resource presenting problem to be solved (presentation available

at https://eportfolio.csu.edu.au/pebblepad/viewasset.aspx?oid=915326&type=file) Worksheet outlining the entire problem (one per student) (Appendix A) and highlighters Worksheet with table for students to complete (one per student) (Appendix B) A number of large coloured circles in two different colours plus one large caterpillar head (Appendix C

demonstrates how these would be used) Counters of two different colours plus one snake head per group (photo examples Appendix D) Camera for photographing student work (photo examples Appendix D) Poster of problem solving steps and strategies (Appendix E) Student self-evaluation sheets (Appendix F) Suggested teaching scripts (Appendix G) Optional additions: A caterpillar puppet to introduce the problem. Play dough instead of counters as

manipulatives for constructing Cameron’s body.Learning IndicatorsStudents can use efficient mental and written strategies to solveproblems (ACMNA291)

Assessments

Assessment 1:Teacher observes

Content / Learning Experiences / Teaching StrategiesIntroduction (Engagement)Using the interactive white board and presentation, introduce Cameron the caterpillar (Slide 1 & 2 of https://eportfolio.csu.edu.au/pebblepad/viewasset.aspx?oid=915326&type=file). Tell the students that Cameron needs our help and ask can they help him?Problem details: Cameron the caterpillar can’t wait to grow up. When Cameron was born, his body was just one colour – Colour A. Each year, he sheds his skin completely and grows longer. When he sheds, every Colour A section in his old skin is completely replaced by 2

Class Organisation

Teacher at the front of the class with students seated together on the floor.

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This includes:• adding three or more numbers with different numbers of digits without the use of digital technologies’

• selecting and applying efficient mental and written strategies to solve word problems that involve addition, subtraction, multiplication and division

• checking solutions to problems’ including using inverse operations

• provides a rationale to why solution is correct

. • using a table or similar organiser

whether students highlight appropriate words on smart board, displaying awareness of data required to solve problem.

Assessment 2:Teachers observe student participation in brainstorming problem solutions and students rationale when providing solutions.

Colour A sections and a Colour B section. The Colour B section is always between the Colour A section (ABA). As each year passes, every Colour B section stays as Colour B and every Colour A section is completely replaced with the Colour ABA combination. What does Cameron look like in his 10th year of life?The relatable problem of Cameron the caterpillar and his need to grow up helps to engage the students and give the problem relevance (Newman & Wehlage, 1993, p10). A Cameron puppet, who introduces the problem, could be added to add an element of fun to the introduction. Highlight to students that they are going to work through the problem using a systematic strategy. Ask the students to brainstorm their understanding of “working systematically” to clarify students’ prior knowledge as well as putting the language into children’s language. Refer to the poster of problem solving steps and strategies (Appendix E). Have this poster in a prominent position in the classroom for students to utilise. Clarify any areas necessary based on student brainstorming responses. Ask the students what information they need to solve the problem. Students review the problem, brainstorming the key mathematical words required to solve the problem. Using the smart board, have students highlight the relevant language and remove unnecessary language from the problem (Example on Slide 4 of presentation). This is Step 1 in Polya’s Problem Solving Steps, understanding the problem (Jonassen, 2010, p4).Cultural diversity teaching strategy: Analysing the language supports students with English as a second language. Australian Indigenous students benefit from modelling before doing (“8 Aboriginal ways of learning”, 2013).Body (Exploration/Transformation/Presentation)Ask the students how they might start to solve the problem. Refer to the problem solving steps and strategies poster (Appendix E). Allocate one student to be Cameron’s head and give a number of other students Colour A circles and Colour B circles (see Appendix C for demonstration). Ask the class want Cameron looks like in his first year of life. Have a student with colour A come up in front of the class to demonstrate how Cameron looked in his first year of life. Ask the students what happens during Cameron’s second year. Prompt students with the problem criteria and ensure Colour A gets completely replaced by sending the first student away and having 3 new students come to the front. Have the students demonstrate what happens when Cameron is in his 3rd year, with additional students coming up and representing the different segments of Cameron’s body. Continue to prompt students to

Teacher at the front of the class with students seated together on the floor. Students actively participating by becoming part of Cameron’s body and through brainstorming solutions.

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to record methods used to solve problems

• create number sentences to describe patterns

Use equivalent number sentences involving multiplication and division to find unknownquantities (ACMNA121)

Additional indicators include:• using appropriate language to describe patterns

• using brackets to write number sentences, perform calculations involving grouping symbols and mixed operations without the use of

Assessment 3:Photos are taken of student’s solutions and displayed on the interactive white board. Students reflect as a group on the solutions provided. Teachers question and observe students rationale.

reflect on problem criteria. Questions might include: Which colours should stay and which should go? How many new colours do we need? Where should they go? Show how Colour B remains and Colour A is replaced with the ABA combination. See Appendix G Script A for suggested teaching script. Kinaesthetic learners teaching strategy: Catering for students that learn through movement (McDevitt, Ormrod, Cupit, Chandler & Aloa, 2013, p300)

Students return to the classroom and work in pairs, using coloured counters and a worksheet containing a table (Appendix B & D). Students are also provided with the problem (Appendix A). Tell the students they are going to use the coloured counters to work out what Cameron looks like each year in a similar way to how they used their bodies to work out what Cameron looks like. They will then record each year on the worksheet table provided. This is Step 2 in Polya’s Problem Solving Steps, making a plan or strategy (Jonassen, 2010, p4). Provide students with a starting point for solving the problem by asking students to solve the problem starting with when Cameron in year 1, year 2 and year 3 as per activity just conducted. This is Step 3 in Polya’s Problem Solving Steps, carry out the plan (Jonassen, 2010, p4). Refer back to the solutions obtained in the previous physical activity. Use presentation slide 5, 6, 7 & the start of 8 to help students visualise the replacing of the sections as each year goes by.Spatial learners teaching strategy: Visual organisation of ideas and concepts caters for spatial learners (Murray and Moore, 2012, p44).

Circulate the classroom and engage in discussions with students about the problem, and prompt students to reflect on solution and problem criteria. Take photos of student’s solutions for class discussion and assessment. Allow students to move to the floor to allow more space for laying out the counters.Ability levels teaching strategy: Students are working in pairs allowing different ability levels to support each other, operating within and advancing the zone of proximal development (McDevitt et al., 2013, p223). Students are given a starting point for working out the problem by prompting them on the solutions discovered in the physical activity using the presentation slides.Once students are starting to run out of counters and space to conduct investigation (at Cameron’s 4th or 5th year), ask the students if they are having trouble getting to Cameron’s 10th

year using the counters. Why? Where were they able to get to? This is Step 4 in Polya’s Problem Solving Steps, evaluate strategy effectiveness (Jonassen, 2010, p4). What are their

Students working in pairs at desks or on the floor with teachers circulating.

Students seated together in pairs, either on floor or at desks, reflecting on answers. Teacher at the front guiding the discussion.

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digital technologies

• ability to work in small groups, listen to instructions and reflect on own learning

Assessment 4:Teachers observe which students can solve the mathematical sentences in relation to the problem.

answers so far? As a group, students review photos of student solutions. Prompt students to reflect on whether the photos meet the problem criteria and ask students to provide rationales as to why or why not (examples of photos in Appendix D). Use slide 8 if need be to emphasise the pattern. Record the answers on slide 9’s table in the presentation.

Ask the class what other strategies other than using the counters could they use to get to Cameron’s 10th year? Using slide 10, ask students can they see any patterns in the colour A column? Is looking for patterns a strategy on the problem solving poster (Appendix E)? Prompt students to compare the numbers in the colour A column. What happens to the no. of colour A sections as each year passes? Write the students response to this question using their words. See Appendix G Script B for suggested teaching script.

Once the rule is confirmed, ask the students to create a mathematic statement to represent this e.g. A Year 1= 1, A Year 2= 2xA Year 1 = 2x1= 2 etc. Add this to the slide 10 table. Can the students now use this mathematic equation to work out year 5 or year 6? Have a student come up and write the equation on the slide and work out year 5 and / or 6.Do the same for the colour B column. Prompt students to compare the numbers in the colour B column. What happens to the no. of colour B sections as each year passes? Write the students response to this question using their words. Follow a similar script as used for colour A including: test out the student’s theories about the pattern once tested, have the students explain in words the pattern and write it on slide 10 e.g.

colour B is equal to colour A minus 1 and / or colour B is equal to the total number of sections from the previous year

have the students write a mathematical equation for the pattern e.g. B=A-1 or B Year 1=A year 1 – 1= 1-1=0

have the students use the mathematical equation to work out year 5 and / or 6Ask students how they would work out the total no. of sections and create a mathematical equation to represent this i.e. Total sections s=A+B.Students now return to Step 3 in Polya’s Problem Solving Steps, carry out the plan (Jonassen, 2010, p4) as they are now using a new strategy to solve the problem. Highlight this to students on the problem solving strategy poster (Appendix E), noting that it is appropriate to return to different steps if the first strategy doesn’t work. Students return to working in pairs to complete the rest of the worksheet up to year 10.

Students return to desks in pairs and work together on worksheet completion. Teacher circulating and engaging with students.

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Circulate the classroom and engage in discussions with students about the patterns discovered and the mathematical representation of those patterns, assisting where necessary.

Assessment 5:Teacher collects completed worksheets from pairs.

Assessment 6:Teacher collects individual student reflection sheets about problem solving.

Conclusion (Presentation/Reflection)As a group, students review the solution starting from where the class left off (i.e. either year 5 or 6). Pairs of students take turns coming to the board and completing the table on slide 11 of the presentation. Problem Answer: Cameron will have a total of 1023 sections in his 10th year with 512 being the colour A and 511 being the colour B.Using Slide 12, take the students through the process of checking the final solution. Ask the students how they might do this. What do all the totals have in common? The total sections are equal to A plus B. They are all odd numbers. Is this enough to check the answer? Ask the students if they were told Cameron has 1023 total sections, how could they work out how many were A, and how many were B? Is there connection between the colour A sections and the total number of sections? Is there a connection between colour B and the total number of sections or between colour A and colour B? Prompt the students to work through the problem solving methodology as before including: test out the student’s theories about the pattern once tested, have the students explain in words the pattern and write it on slide 12 i.e.

colour A is equal to half of the total number of sections plus 1, Colour B is equal to colour A minus 1 and the total number of sections equals the sum of A and B.

have the students write a mathematical equation for the patterns i.e. A=(s+1)/2 =(1023+1)/2 =512, B=A-1=512-1=511 and A+B=512+511=1023

This is Step 4 in Polya’s Problem Solving Steps, evaluate strategy effectiveness (Jonassen, 2010, p4).To take the problem further, ask the students could they solve the following problem using the checking method they just used (slide 13):If Cameron has a total of 32767 sections, how many colour A and colour B does Cameron have? Colour A is equal to half of the total number of sections plus 1. What does this look like mathematically? A=(s+1)/2 =(32767+1)/2 =32768/2 =16384Colour B is equal to colour A minus 1. B=A-1 =16384-1=16383Congratulate the students for coming up with the answer encouraging them to be proud of their achievement. Discuss the areas students found easy and those students found difficult. What did they enjoy about the problem and / or the process? Why? What did they find difficult? How did they overcome this? Allow students time to complete the reflection sheet

Teaching at the front of the class with students seated at desks. Students reflect as a group on learning. Teacher leads discussion, prompting reflection. Students come to the front as called.

Students return to desks and complete self-reflection individually.

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(Appendix F).Next Lesson or extension: Extension question using same problem: How can we calculate what Cameron will look like in his 25th year? Or 100th? This will involve students using algebra including the introduction of indices. Students will need to return to the original problem and create an algebraic equation to represent the total number of sections Cameron has for a given year i.e. where s= total sections & y= no. of years, s=2y-1. Therefore, in year 25, s=225-1= 33554431. Students could then use the equation developed previously to calculate the number of colour A and colour B segments i.e. colour A= (s+1)/2=33554432/2= 16777216, colour B=colour A-1=16777215Other problems for extension:Mushroom Hunt (Black Douglas Professional Education Services, n.d.c) (Appendix J)

Reflection on Lesson-Planning ProcessThe lesson-planning process for this assessment involved three key elements. These elements included the selection of the teaching strategy, in this case problem-based learning, the selection of the problem and the peer review process. Each element is important in the lesson planning processes and is underpinned by sound theoretical frameworks, while adhering to the set assessment and curriculum criteria.

The first part of the lesson-planning process involved selection of the teaching pedagogy. In this case, the assessment criteria required a problem-based learning approach as the teaching strategy. Problem-based learning can be defined as “an instructional learner-centred approach that empowers learners to conduct research, integrate theory and practice, and apply knowledge and skills to develop a viable solution to a defined problem” (Savery, 2006, p9). Problem-based learning has been found to have distinct advantages over other learning strategies within the primary education context. Psychological theory and research suggests that this instructional method allows students to learn key concepts while simultaneously acquiring thinking skills (Hmelo-Silver, 2004, p235). Additionally, research suggests that students’ intrinsic motivation can be enhanced when teachers provide moderately challenging and personally meaningful activities, offer variety in the format and nature of tasks as well as offer choice (Anderman & Corno, 2013, p102). All of these elements are components of a problem-based learning approach. Problem-based learning is also geared around substantive conversations which are defined as unscripted, not teacher directed, and open for feedback, questioning, and a variety of answers (Berlark, 1992, p77). These conversations build on collective ideas with the view to promoting knowledge and understanding not just fact recall (Newman & Wehlage, 1993. p10). In considering this assessment, it was important that the lesson plan contain these aforementioned key elements of a problem-based learning experience. These elements guided the selection of a problem which would allow for challenging, meaningful learning, open-ended discovery and substantive conversations while adhering to the curriculum requirements and the assessment criteria.

As outlined, the problem needed offer to opportunity for self-directed investigation while being challenging and engaging for students. In addition, the assessment required the problem to introduce or reinforce concepts from the NSW mathematical syllabus, including working mathematically and content from the Number and Algebra strand. Initially, Cars in a garage (Appendix H) was selected as a problem for this assessment. This is an engaging open-ended problem which would have worked well for this assessment. However, on reflection, this problem would have greater gender appeal to boys over girls and hence, the more gender neutral problem involving a caterpillar was selected to ensure engagement of all children. Cameron the caterpillar grows up was adapted from Make a snake (Appendix I). The outcomes of the problem were retained however, the presentation and wording of the problem was changed to ensure complete clarity around the pattern. The resulting problem was one that offered scope for open-ended discovery, opportunity to work mathematically as well as challenge and engage the students. It also contained clear links to

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the NSW mathematics curriculum, including the use of problem solving skills, the use of the four operations, and pattern identification and description, which was a key factor in the assessment criteria.

Once the problem was selected, the elements of a problem solving lesson could be incorporated into the lesson plan. Using the ELPSARA model, the lesson plan was developed to reflect on students past learning, develop their language from children’s language through to symbolic language, and apply problem solving steps and strategies to the learning. Group discussions formed a key part of the reflection process throughout the lesson. During the introduction phase, students reflect on their use of problem solving steps and strategies from past lessons while having visual aids to prompt recollection. This is revisited during the body of the lesson when the initial investigative method reveals limitations. The language of mathematics is developed during the lesson plan. Initially, group discussions and activities allow the children to identify the problem’s key words using their own language. Children are encouraged throughout the lesson plan to describe using words the problem itself and the emerging patterns. Manipulatives enable students to physically interact with the problem before moving to the symbolic representation. The development of a table offers a transition between the counters and the mathematical language. Extra activities were added to the lesson plan catering to different learning styles and ability levels including visual white board presentation, physical activities involving children as body sections, and the use of starting points and extension options. Finally, students are given additional opportunities to apply their new knowledge through reverse computation for checking, additional questions and extension options. A variety of assessments are incorporated in the plan including teacher observation, written work and student self-reflection.

The last part of the lesson-planning process involved reflection on the lesson construction facilitated by a peer review process. The feedback from peers provided valuable insight in how to improve the lesson while confirming the lesson plan was clear and well structured. There was a concern regarding the difficulty of algebra introduced at the end of the lesson. As this is intended primarily as an extension opportunity, this was not removed. A strong suggestion provided by Alysha Allison (Peer Review 2) included the incorporation of an ABA pattern diagram to ensure clarity. This was added to the final whiteboard presentation as an addition slide. The process of reviewing other lesson plans identified areas where this lesson was strong and weak in comparison to others. For example, this lesson plan appeared to contain greater detail in the development of the key ideas than the two lesson plans reviewed however, appeared weaker in creativity and engagement of the students. As a result, an option was added to the final lesson where counters could be replacement with play dough. This adds an additional element of fun for the students and they are not restricted by the number of counters as they can construct their own. Cameron can also be brought to life through the use of a puppet, another optional addition to the final plan. In final self reflection, MA3-6NA was added as an outcome and indicators were further extended. Additional clarity was given to class organisation and time allocated for student reflection.

In conclusion, it is important during the lesson-planning process to consider the elements of pedagogy and curriculum in creating meaningful, successful learning experiences for students. Lessons that contain challenging problems with opportunities for open-ended investigation allow students to gain knowledge and understanding through the discovery process. Lessons should also engage students and where possible, offer an element of fun. The reflective practice of observing and having others observe the learning experiences prior to implementation ensures teachers also benefit from the effects of collaborative learning and gain greater insights into their own practice.

Appendix A – Worksheet for students outlining the problem

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I’m Cameron the caterpillar and I can’t wait to grow up! I want to know what I’m going to look like when I’m turning 10. Can you help me?

Cameron the caterpillar can’t wait to grow up. In Cameron’s first year of life, his body was just one colour – Colour A. Each year, he sheds his skin completely and grows longer. When he sheds, every Colour A section in his old skin is completely replaced by 2 Colour A sections and a Colour B section. The Colour B section is always between the Colour A section (ABA). As each year passes, every Colour B section stays as Colour B and every Colour A section is completely replaced with the Colour ABA combination. What does Cameron look like in his 10th year of life?


Appendix B18


Cameron the caterpillar worksheet with table for students to complete

Years of Cameron’s life No. colour A sections No. colour B sections Total number of sections

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

Year 7

Year 8

Year 9

Year 10

Appendix CExample of how to demonstrate problem strategies using the students as segments of Cameron the caterpillar

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Figure 1: Student holding caterpillar head

Figure 2: Students representing caterpillar in year 1 (red represents colour A)

Figure 3: Students representing caterpillar in year 2 (blue represents colour B)First student sits down and is replaced with three new students

Figure 4: Students representing caterpillar in year 3Students holding colour A sit down and are replaced with new students. Student holding colour B remains.

Figure 4: Students representing caterpillar in year 4 (if needed)Students holding colour A sit down and are replaced with new students. Students holding colour B remain.

Appendix DCounters being used by students to workout problem (Black Douglas Professional Education Services, n.d.b)

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Figure 1 Figure 2 Figure 3

Appendix EProblem solving posters to prompt students to work systematically

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Figure 1: Skills needed for problem solving (Analyzemaths, n.d.) Figure 2: Strategies for problem solving (Glastonbury Public School, 2013)

Appendix FProblem solving reflection sheet for student’s self reflection on learning (Alaska Department of Education and Early Learning, 1996)

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Appendix G: Suggested teaching scripts for use during the lessonScript ATeacher: As we can see, in year 1, Cameron has one section in colour A, being held by John. What happens in year 2 according to the problem?

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Students: Colour A is completely replaced by Colour A, Colour B and Colour A.Teacher: Ok. John, you sit down. Let’s have Jane, who has Colour A, Jill, who has Colour A, and Liam, who has Colour B, come up. Where should they stand?Students: Liam should stand in between Jane and Jill as he has Colour B and it always has to be in the middle. Teacher: Great. Jill, you stand next to the head, then Liam, then Jane. So, in Cameron’s 2nd year, he has 2 Colour A and 1 Colour B in a pattern of ABA. Does this meet the problem criteria?Students: yes. Teacher: Ok. Who can tell me what happens in the 3rd year of Cameron’s life?Students: Colour A is replaced by colour A, Colour B and Colour A again. Teacher: Ok. So who should sit down and who should stay standing?Students: Jill and Jane should sit down and Liam should stay standing as he has the Colour B. Teacher: Right. That’s because the problem states that Colour B always remains while Colour A gets completely replaced. So Jill and Jane, you sit down. Now, who should come up in their place?Students: Bill, who has colour A, Bob who has colour B, and Hamish, who has colour A should go up and stand where Jill was standing. Mia, who has colour A, Ryan, who has colour B, and Isaac who has colour A should go and stand where Jane was standing.Teacher: Ok. Do they need to stand in a particular order?Students: Yes. It should go Bill, Bob, Hamish then Liam then Mia, Ryan and Isaac.Teacher: Ok, let’s do that. So now we have the pattern colour ABA B ABA. Cameron now has 4 colour A sections and 3 colour B sections. Does this meet all the problem criteria?Students: Yes.Teacher: Great. Does everyone understand the problem and what happens each year?

Script BTeacher: Can you see a pattern in the colour A column? What happens to the number of coloured A sections as the years go by?Students: Each year the numbers of colour A doubles from the year before.Teacher: Let’s test that out. In year 1, there is 1 colour A and year 2, there is 2 colour A sections. Does this fit the theory that the colour A doubles each year?Students: yes.Teacher: Great. Let’s try a couple more. In the second year, there is 2 colour A sections and in the third year, there is 4 colour A sections and then, in year 4, there is 8 colour A sections. Does the rule of colour A doubling each year still fit?Students: yes.Teacher: So the rule is, each year colour A doubles. Appendix H: Cars in a garage (Black Douglas Professional Education Services, n.d.a)Summary of the problemCars and garages are familiar to students of all ages. In this task the challenge is to find all the ways to park the cars in the garages.

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How many parking solutions are there? How do we know when we have found them all?

Possible investigations for this task are: What happens if we change the number of cars? What happens if we change the number of spaces? What happens if there are more spaces than cars? What happens if we make conditions on where particular cars can park?

The calculation in Cars in a Garage revolves around increasingly long multiplication strings of consecutive numbers and the card challenges students to work out the answer for 100 cars.Offer the students simple four function calculators and invite them to carry out the multiplications in sequence ... 2x1 ... 3x2x1 ... 4x3x2x1 ... 5x4x3x2x1 ... etc. This work can be shared around the class. To the students' surprise there will come a time when the calculator runs out of screen digit space. At which number of cars does this happen? To this point primary children can be involved in this exploration which highlights the limits of the technology.

Appendix I: Make a snake (Black Douglas Professional Education Services, n.d.b)Summary of the problemStudents recreate the pattern of coloured rings on the body of Mungo the Maths Snake by using coloured beads or blocks in a defined order. Mungo is born with only one colour ring (Colour A), but each season she sheds her skin and replaces Colour A with an ABA pattern. The visual patterns in the problem can be difficult to 'get hold of' because they stretch out a

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long way quite quickly, but once key elements are noticed, they can help to identify and explain the significant number patterns in the problem.

Appendix J: Extension question – Mushroom hunt (Black Douglas Professional Education Services, n.d.c)

Six people collect mushrooms and on return compare the numbers in their baskets. They notice two special things:

The total number of mushrooms collected is 63. By combining the numbers in their baskets in different ways they can make every number up to 63.

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The challenge is to find the numbers that must have been in each basket.

The answer to this problem is similar to the Cameron the Caterpillar problem where the total number of mushrooms (m) is equal to 2 to the power of the number of baskets (b) minus 1 i.e. m=2b-1=26-1=63

References

8 Aboriginal ways of learning. (2013). Retrieved October 2, 2013, from the 8 Ways Wiki: http://8ways.wikispaces.com/

Alaska Department of Education and Early Learning. (1996). A collection of assessment strategies. Retrieved from http://www.eed.state.ak.us/tls/frameworks/mathsci/ms5_2as1.htm

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Analyzemaths. (n.d.) Skills needed for problem solving. Retrieved from http://www.analyzemath.com/

Anderman, E. & Corno, L. (2013). Handbook of educational psychology. Retrieved from Ebooks library.

Black Douglas Professional Education Services. (n.d.a). Cars in a garage. Retrieved from http://www.blackdouglas.com.au/taskcentre/002cars.htm

Black Douglas Professional Education Services. (n.d.b). Make a snake. Retrieved from http://blackdouglas.com.au/taskcentre/005mksnk.htm

Black Douglas Professional Education Services. (n.d.c). The mushroom hunt. Retrieved from http://blackdouglas.com.au/taskcentre/038mushr.htm

Glastonbury Public School. (2013). Problem solving strategies. Retrieved from https://www.glastonburyus.org/curriculum/math/gradesk-5/problemsolving/Pages/ProblemSolvingStrategies.aspx

Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266. doi: 10.1023/B:EDPR.0000034022.16470.f3

Jonassen, David H. (2010). Learning to solve problems: A handbook for designing problem-solving learning environments. Retrieved from http://www.eblib.com

McDevitt, T.M., Ormrod, J.E., Cupit, G., Chandler, M., & Aloa, V. (2013). Child development and education. Frenchs Forest: Pearson Australia

Murray, S. & Moore, K. (2012). Inclusion through multiple intelligences. Journal of Student Engagement: Education matters 2012, 2(1), 42–48. Retrieved from http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1014&context=jseem

Newmann, F.M. & Wehlage G.G. (1993). Five standards of authentic instruction. Educational Leadership, 50(7), 8-12. Retrieved from http://www.ascd.org/publications/educational-leadership.aspx

NSW Board of Studies. (2012). Mathematics K-10 syllabus. Retrieved from http://syllabus.bos.nsw.edu.au/download/

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http://blackdouglas.com.au/taskcentre/005mksnk.htm


Savery, J. R. (2006). Overview of problem-based learning: Definitions and distinctions. Interdisciplinary Journal of Problem-based Learning, 1(1), 9-20. doi: 10.7771/1541-5015.1002

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