INVESTIGATION - MATHSINVESTIGATION - MATHS Maths Talent Quest Maths Talent Quest

June PenneyJune PenneyMAV – Student Activities CommitteeMAV – Student Activities Committee

INDEXINDEX What is a Maths Investigation? Why do a Maths Investigation? The Australian Curriculum Thinking Skills Reflection of Investigation Learning Developing Your Maths Investigation References & Acknowledgements

What is a Maths Investigation?What is a Maths Investigation?

An investigation may be defined as“a situation originating in mathematics or the

real world which lends itself to inquiry.”

Inquiry ---making observations, asking questions and pursuing

investigations has always been a fundamental approach to

understanding the world.

A mathematics investigation allows students to examine situations using various techniques and in the process of their exploration develop skills that can be applied to other problems.

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Why do a Mathematics Why do a Mathematics Investigation?Investigation?

It caters for student diversity and investigative work is viewed as a key way to engage and motivate learners.

Students need to formulate their own questions from a given situation.

By formulating their own questions, students give teachers a clear indication of their level of knowledge and/or understanding of the topic.

It requires students to use mathematical processes to understand the problem or situation.

First hand data – generated by the student is much better for learning than second hand data!

Students develop a systematic record of their work not only an end product.

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Aims for students to ‘recognise connections between the areas of mathematics and other disciplines and appreciate mathematics as an accessible and enjoyable discipline to study.’

Students need to recognise that mathematics is constantly used outside the mathematics classroom and that numerate people apply mathematical skills in a wide range of familiar and unfamiliar situations.

Using mathematical skills across the curriculum both enriches the study of other learning areas and contributes to the development of a broader and deeper understanding of numeracy.

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Mathematics is organised around the interaction of three content strands and four proficiency strands.

The proficiencies reinforce the significance of working mathematically within the content and describe how the content is explored or developed.

‘The curriculum anticipates that schools will ensure all students benefit from access to the power of mathematical reasoning and learn to apply their mathematical understanding creatively and efficiently…..’ ‘It encourages teachers to help students become self-motivated, confident learners through inquiry and active participation in challenging and engaging experiences.’

The Australian Curriculum & AusVELSThe Australian Curriculum & AusVELS

www.australiancurriculum.edu.au www.ausvels.vcaa.vic.edu.au

Mathematics, Thinking Processes, Communication, Design, Creativity and Technology, Information and Communications Technology, Interpersonal Development, English, Languages, Personal Learning, Science, The Arts, The Humanities, The Humanities Economics, The Humanities Geography, The Humanities History, Civics and Citizenship, Health and Physical Education. MTQ -Maths Investigations- using and developing mathematical skills and connections across the curriculum.

General Capabilities

Content and Proficiency

The AusVELS Domains

Thinking ProcessesThinking Processes

The type of skills normally associated with investigations are generally higher order skills or processes.

• These processes fall under the broad heading of Working Mathematically-

The Australian Curriculum proficiencies reinforce the significance of working

mathematically within the content and describe how the content is explored or

developed.• Thinking Processes- AusVELS Domain – students will be developing

creative and critical thinking abilities, and applying them to the expansion of their knowledge and skills.

• “We are preparing students for an uncertain future, there will be problems to solve that probably don’t yet exist.”

• The aim of developing thinking skills through the investigation process is - to provide students with the ability to apply and transfer knowledge to

new and different circumstances as they arise throughout life - to understand and act effectively in the ever changing world in which they

live.Back to Index

Grade 4 Reflection of Investigation Learning

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Developing Your Maths Developing Your Maths InvestigationInvestigation

First Decide Getting Started Final Presentation Format Examples Investigation Steps Being A Mathematician Procedure Ideas Investigation Model Strategies/ Toolbox of strategies Reflections Self-Assessment Log/Journal What to include? Assessment A Learning Journey

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FIRST DECIDE FIRST DECIDE

Class- Organisation and Roles (Maximum 30 students)Class- Organisation and Roles (Maximum 30 students)

Group- Organisation and Roles (2-6 students) An example Group- Organisation and Roles (2-6 students) An example next slide.next slide.

Individual-OrganisationIndividual-Organisation Will the investigation be-Will the investigation be- A home investigation.A home investigation.

A school investigation.A school investigation.

A home/school partnership investigation.A home/school partnership investigation.

Who is doing the investigation?Who is doing the investigation?

Which investigations will be entered in the MTQ?Which investigations will be entered in the MTQ? Every school investigation is entered.Every school investigation is entered.

Own School judging of investigations to select those to Own School judging of investigations to select those to be be

entered.entered.Back to Investigation List

Getting Started.Getting Started.

Be Creative and Individual.Choose something that interests you.

Use: Mind mapping Brainstorming Lateral thinking

Begin: Brainstorm Ideas and Inspiration for Investigation. Begin a Log or Journal Clearly Explaining the Investigation, Thinking

and Learning Processes and Understandings. Generate Ideas for Topic and Related Mathematical Content. Select Topic and Outline Investigation Content. Ensure Acquisition and Management of Resources. Develop a Timeline. Consider Presentation Format. ( Posters, Booklets, Display Folders, ICT/CD and Support Materials such

as Models are all acceptable. Models must relate to the mathematics not just be an artistic display “add on”.)

Topic

Idea Idea

Idea

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Poster, Book and Poster, Book and ModelModel

PosterPoster

InvestigationInvestigation

ExamplesExamples

(6 Slides)

ModelModel and Books

Small Small Display Display FolderFolder

Big BookBig Book

Visual Arts Visual Arts BookBook

Large Display Large Display FolderFolder

Presentation Presentation PackagePackage

Maths of Arts & Maths of Arts & CraftsCrafts

Maths of CulturesMaths of Cultures

Maths of Maths of Hobbies or Hobbies or InterestsInterests

Investigation StepsInvestigation StepsInvestigation Aim:

A question is a problem if the procedure or method of solution is not immediately known to you but requires you to apply creativity and previous knowledge in new and unfamiliar situations.

If the procedure or solution is obvious then it is not a problem but an exercise.

Conclude. Draw Together. Summarise Findings. Repeat Process. Extend the Situation by Formulating Further

Questions. Reflections. Explain or Justify Results. Test Conjectures. (Use strategies) Make Conjectures. (Formation of opinion on incomplete grounds.) Explore Systematically. (Use strategies) Get to Know the Situation and Formulate Questions. Define your AIM. Choose your TOPIC.Begin Log or Journal.

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Title:

Aim:                

Investigation skills/resources used:

Conclusion:

Plan:

Doing the investigation: Document the mathematics of your investigation.

*The development of your investigation must be clearly demonstrated in your log/journal.

*A final publication/presentation of your investigation is to be completed & submitted along with your log/journal.

A Concise, Problem to SolveOutline and Guide

Mathematics Processes/StrategiesWorking document.Raw data, methods , findings &personal thinking.

Clearly answers aim.

Interesting & Worthwhile

Remember to use the judging rubric.

http://www.blackdouglas.com.au/taskcentre Back to Investigation List

Question

Task

Evaluate Problem

Issue

Some Procedure IdeasSome Procedure Ideas

*George Polya (Hungarian Mathematician) “How To Solve It”, 1945.

* See, Plan, Do, Check

An issue to explore A question to ask A problem to solve A task to complete

5E’s of Investigation

Engage Explore Explain Elaborate Evaluate

Toolbox/Graphic Organiser

“Toolbox” Reference - “Work It Out,” Tom Hill, Oxford University Press.

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Garafalo & Lester (1985)

Orientation Organisation Execution Verification

NSW BOS (2002)

Questioning Applying Strategies Reasoning & Communication Reflecting

Investigation ProcedureInvestigation Procedure(One Model*)(One Model*)

See - Understanding the Problem

Plan - Deciding on a Strategy or Plan

Do - Solving the Problem

Check - Checking Your Results

(8 slides)(8 slides)

* George Polya- “How To Solve It”,1945

SeeSeeUnderstanding the ProblemUnderstanding the Problem

Identify the problem you want to answer.Identify the problem you want to answer. Read the problem carefully.Read the problem carefully. Pick out the various parts of the problem.Pick out the various parts of the problem.

QuestionsQuestions What is the problem asking me?What is the problem asking me? Are there any words I don’t understand?Are there any words I don’t understand? What do I already know?What do I already know? What am I trying to do?What am I trying to do?

PlanPlanDeciding on a Method/Plan to Get a Deciding on a Method/Plan to Get a

SolutionSolution Gather together all available informationGather together all available information Make some predictions or guesses.Make some predictions or guesses. Think about the different strategies you may use.Think about the different strategies you may use. (Refer Strategies/Toolbox of Strategies in Index)(Refer Strategies/Toolbox of Strategies in Index) Decide which strategy or strategies will suit your Decide which strategy or strategies will suit your

problem.problem. Write down your plan.Write down your plan.QuestionsQuestions How am I going to solve the problem?.How am I going to solve the problem?. Have I seen the problem or a similar one before?Have I seen the problem or a similar one before? How can the known help me with the unknown?How can the known help me with the unknown? Can I restate the problem?Can I restate the problem?

Trial and error. Guessing, checking, improving. Gathering data. Drawings, diagrams, graphs. Working backwards. Looking for patterns. Writing an equation. Using a formula. Simplifying the problem. Do I know a similar problem? Elimination of possibilities. Using a list or a table. Using materials. Using models. Acting it out.

Test conjecture by- examples and counter-examples.

Strategies for Solving Unfamiliar Strategies for Solving Unfamiliar ProblemsProblems

www.blackdouglas.com.au/taskcentre

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DoDoCarry out the Plan to Solve the ProblemCarry out the Plan to Solve the Problem Work through, one step at a time.Work through, one step at a time. Do each step carefully.Do each step carefully. Explain and show how you reach your answer.Explain and show how you reach your answer. Reflect on where you are at.Reflect on where you are at. Re-think and modify your strategies as needed.Re-think and modify your strategies as needed. Create a new plan if necessary.Create a new plan if necessary.QuestionsQuestions What do I do next?What do I do next? Have I proved I am correct?Have I proved I am correct? Do I continue with my plan?Do I continue with my plan? Is my plan working?Is my plan working? Do I need to change my plan?Do I need to change my plan?

SOME of the Mathematical Processes UsedSOME of the Mathematical Processes Used

Questioning

Predictin

g

Collecting DataGeneralising

Estimating

Reflecting

Interpreting

Analysing

Comparin

g

Classifyi

ng

Experimentin

gProving

Explorin

g

Justifying

Hypothesising

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CheckCheckThink Carefully and Examine Your Think Carefully and Examine Your

AnswerAnswer Write your answer in a complete Write your answer in a complete

sentence.sentence.

QuestionsQuestions How can I check my result? How can I check my result? Have I used all the important information?Have I used all the important information? Does the answer make sense?Does the answer make sense? Does it answer the whole problem or question?Does it answer the whole problem or question?

ReflectionsReflections How could the problem relate to other problems?How could the problem relate to other problems? Is there another strategy I could use to get the answer?Is there another strategy I could use to get the answer? How can I use this method to solve further problems?How can I use this method to solve further problems? Back to Investigation List

Investigation ReflectionsInvestigation Reflections

Some Ideas: I have learnt…. I have found…. I have discovered… I now need to… Today I/Tomorrow I… Something new…. Something challenging… Further thoughts…. Can I check this another way? What happens if? How many solutions? What else can I learn from this?

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Self Assessment

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Log/Journal/Rough WorkingsLog/Journal/Rough Workings ( (COMPULSORY)COMPULSORY)

NameName TitleTitle Key question/s of investigationKey question/s of investigation Investigation Plan- Point FormInvestigation Plan- Point Form - Mind Map- Mind Map Investigation Development- Ongoing demonstration and Investigation Development- Ongoing demonstration and

explanation of the Mathematics used.explanation of the Mathematics used. Carefully read judging criteria Carefully read judging criteria

www.mav.vic.edu.au/studact/mtq.htmwww.mav.vic.edu.au/studact/mtq.htm

A sample investigation proforma is also available from the A sample investigation proforma is also available from the

Mathematical Association of New South Wales websiteMathematical Association of New South Wales website www.mansw.nsw.edu.au/studentservices/investigating-mathematics.htm

(18 slides)Back to Investigation List

Log/JournalLog/Journal

ExamplesExamples From PreviousFrom Previous

MTQ InvestigationsMTQ Investigations

Organise your group/class in a way that suits Organise your group/class in a way that suits

youyou..

(One Class Organisation 2007)(One Class Organisation 2007)

Writers GroupWriters Group Research GroupResearch Group Calculator GroupCalculator Group Artists GroupArtists Group Photographers GroupPhotographers Group

Example Example OneOne

Example ThreeExample Three

What is the project about?What is the project about? What is the main question?What is the main question? What are the other questions?What are the other questions? What maths is involved?What maths is involved? What method/s are used to What method/s are used to

investigate the problem/s?investigate the problem/s? What materials and equipment will be What materials and equipment will be

needed?needed?

SectionsSections

Example FourExample Four

Date?Date? What are we doing for our project What are we doing for our project

today?today? What maths did we use?What maths did we use? What did we discover or learn today?What did we discover or learn today? What do we need to do next?What do we need to do next?

SectionsSections

2007 Home/School Partnership Timeline Example- 1 0f 3 Pages

Example Example FiveFive

2007 Home/School Partnership Timeline Example- 2 0f 3 Pages

2007 Home/School Partnership Timeline Example- 3 0f 3 Pages

Which investigations will be entered in the

Which investigations will be entered in the MTQ?MTQ?

The above school had their own school judging of The above school had their own school judging of investigations to investigations to

select those to be entered.select those to be entered.Some schools have a school expo to display the quality of Some schools have a school expo to display the quality of

their investigations and every school investigation is entered.their investigations and every school investigation is entered.

While other schools submit all investigations and have a While other schools submit all investigations and have a display to celebrate their maths later in the year.display to celebrate their maths later in the year.

ExampleExample SevenSeven

What to Include?What to Include?

All completed investigations must have:

• An investigation title.• A list of all the components of the investigation.• Each component clearly labelled.• Documented evidence of student’s investigation thoughts,

processes and development to be included as separate log, journal or rough workings.

• It also needs to include reference to the progress of the mathematical investigation being undertaken, the problems undertaken and the mathematical conclusions reached.

• A bibliography listing all references used.• Acknowledgement of any assistance given.

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AssessmentAssessment

Primarily based on communication, evidence of mathematical content and the understanding of the

investigation appropriate to the student’s year level.

Assessment Format - Evaluation Rubric

Communication 16 points

Mathematical Content and Understanding 16 points

Ideas and Resources 8 points

Presentation 4 points

TOTAL 44 pointspoints

Download judging criteria from MAV Website-Download judging criteria from MAV Website-http://www.mav.vic.edu.au/studact/mtq.htmhttp://www.mav.vic.edu.au/studact/mtq.htm

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A Learning JourneyA Learning Journey

“Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost

and discover more than they thought possible.”

W.S.Anglin

REMEMBERREMEMBER

Curiosity and interest are the centre of inquiry!

Choose your investigation wisely!

Hope you are excited about

yourInvestigations!

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ReferencesReferences www.mav.vic.edu.au/studact/mtq.htm www.australiancurriculum.edu.au ausvels.vcaa.vic.edu.au www.mansw.nsw.edu.au/studentservices/investigating-

mathematics.htm “MTQ Is For You.” June Penney & Agatha Anamourlis, MAV Publications Donna Ludvigsen, Working Mathematically, MAV Conference, Dec 2006 Doug Williams, Black Douglas Professional Education Services, MAV Conference, Dec 2006 www.blackdouglas.com.au/taskcentre “Work It Out.” Tom Hill, Oxford University Press “How To Solve It.” George Polya, (Princeton 1945) “Identifying problem solving in school mathematics: students’ and teachers’

perspectives.” Judy Anderson, Connected Maths, MAV Annual Conference 2008.

AcknowledgementsAcknowledgements Thank-You To 2007, 2008 & 2009 MTQ Participants For- A variety of 2007, 2008 & 2009 MTQ Investigation Examples. Investigation Timeline & Self-Assessment, Mark Smith & Steve Wilson, Carey Baptist

Donvale & Kew

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