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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? VELS & Interdisciplinary Learning 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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Interdisciplinary LearningInterdisciplinary Learning(VELS, VCAA)(VELS, VCAA)
Communication Design Thinking Integration Creativity and Technology ICT (Information & Communications Technology)
*Can all be included as requirements of a completed investigation.*
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Thinking SkillsThinking SkillsThe type of skills normally associated with investigations are generally higher order skills or processes.
These processes fall under the broad heading of Working Mathematically in VELS. (Victorian Essential Learning Standards, VCAA)
“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.
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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
Back to Start
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
Back to Investigation List
Poster, Book and Poster, Book and ModelModel
PosterPoster
InvestigationInvestigation
ExamplesExamples
(6 Slides)
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
Back to Thinking SkillsBack to Investigation List
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 ReflectionsSome 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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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.htm
A sample investigation proforma is also available from the Mathematical A sample investigation proforma is also available from the Mathematical
Association of New South Wales websiteAssociation of New South Wales website www.mansw.nsw.edu.au/studentservices/investigating-
mathematics.htm
(18 slides)Back to Investigation List
Five Five
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- 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.
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
Back to Investigation list
Back to Index
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!
ReferencesReferences www.mav.vic.edu.au/studact/mtq.htm “MTQ Is For You.” June Penney & Agatha Anamourlis, MAV
Publications VELS- Victorian Essential Learning Standards, VCAA, Working
Mathematically (Mathematics) and Interdisciplinary Learning www.mansw.nsw.edu.au/studentservices/investigating-
mathematics.htm 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.
AcknowledgementsAcknowledgementsThank-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 & KewBack to Start