INVESTIGATION - MATHS Maths Talent Quest June Penney MAV – Student Activities Committee
INDEX • 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? 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. Back to Index
Why do a Mathematics 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. Back to Index
The Australian Curriculum & AusVELS Content and Proficiency • Mathematics is organised around theinteraction 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.’ General Capabilities • 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. The AusVELS Domains • 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. www.australiancurriculum.edu.au www.ausvels.vcaa.vic.edu.au Back to Index
Thinking 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
Back to Index Grade 4 Reflection of Investigation Learning
Developing Your Maths Investigation • 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
Back to Investigation List FIRST DECIDE Who is doing the investigation? • Class- Organisation and Roles (Maximum 30 students) • Group- Organisation and Roles (2-6 students) An example next slide. • Individual-Organisation Will the investigation be- • A home investigation. • A school investigation. • A home/school partnership investigation. Which investigations will be entered in the MTQ? • Every school investigation is entered. • Own School judging of investigations to select those to be entered.
Topic Idea Idea Idea Back to Investigation List 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”.)
Investigation Examples Poster, Book and Model (6 Slides) Poster
Model and Books Small Display Folder
Visual Arts Book Big Book
Large Display Folder Presentation Package
Maths of Hobbies or Interests Maths of Arts & Crafts Maths of Cultures
Investigation Steps Investigation 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. Back to Investigation List
MTQ Investigation Remember to use the judging rubric. Title: Interesting & Worthwhile A Concise, Problem to Solve Aim: Plan: Outline and Guide Investigation skills/resources used: Mathematics Processes/ Strategies Working document. Raw data, methods , findings & personal thinking. 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. Clearly answersaim. Conclusion: MTQ Investigations Title: Aim: Plan: Investigation skills used: Maths used: Document the development of the mathematics of the investigation. This should be demonstrated in your log/journal. Conclusion:
Back to Investigation List http://www.blackdouglas.com.au/taskcentre
Issue Question Evaluate Problem Task Some Procedure Ideas • Garafalo & Lester (1985) • Orientation • Organisation • Execution • Verification *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 • NSW BOS (2002) • Questioning • Applying Strategies • Reasoning & Communication • Reflecting “Toolbox” Reference - “Work It Out,” Tom Hill, Oxford University Press. Back to Investigation List
Investigation Procedure(One Model*) See - Understanding the Problem Plan - Deciding on a Strategy or Plan Do - Solving the Problem Check- Checking Your Results (8 slides) * George Polya- “How To Solve It”,1945
SeeUnderstanding the Problem • Identify the problem you want to answer. • Read the problem carefully. • Pick out the various parts of the problem. Questions • What is the problem asking me? • Are there any words I don’t understand? • What do I already know? • What am I trying to do?
PlanDeciding on a Method/Plan to Get a Solution • Gather together all available information • Make some predictions or guesses. • Think about the different strategies you may use. (Refer Strategies/Toolbox of Strategies in Index) • Decide which strategy or strategies will suit your problem. • Write down your plan. Questions • How am I going to solve the problem?. • Have I seen the problem or a similar one before? • How can the known help me with the unknown? • Can I restate the problem?
Strategies for Solving Unfamiliar Problems • 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.
Back to investigation List www.blackdouglas.com.au/taskcentre
DoCarry out the Plan to Solve the Problem • Work through, one step at a time. • Do each step carefully. • Explain and show how you reach your answer. • Reflect on where you are at. • Re-think and modify your strategies as needed. • Create a new plan if necessary. Questions • What do I do next? • Have I proved I am correct? • Do I continue with my plan? • Is my plan working? • Do I need to change my plan?
SOME of the Mathematical Processes Used Questioning Collecting Data Generalising Exploring Hypothesising Analysing Predicting Reflecting Comparing Interpreting Justifying Classifying Experimenting Estimating Proving Back to Investigation List Back to Thinking Skills
CheckThink Carefully and Examine Your Answer Write your answer in a complete sentence. Questions • How can I check my result? • Have I used all the important information? • Does the answer make sense? • Does it answer the whole problem or question? Reflections • How could the problem relate to other problems? • Is there another strategy I could use to get the answer? • How can I use this method to solve further problems? Back to Investigation List
Investigation 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? Back to Investigation List
Self Assessment Return to Investigation List
Log/Journal/Rough Workings (COMPULSORY) • Name • Title • Key question/s of investigation • Investigation Plan- Point Form - Mind Map • Investigation Development- Ongoing demonstration and explanation of the Mathematics used. • Carefully read judging criteria www.mav.vic.edu.au/studact/mtq.htm A sample investigation proforma is also available from the Mathematical Association of New South Wales websitewww.mansw.nsw.edu.au/studentservices/investigating-mathematics.htm (18 slides) Back to Investigation List
Log/Journal Examples From Previous MTQ Investigations
Example One Organise your group/class in a way that suits you. (One Class Organisation 2007) • Writers Group • Research Group • Calculator Group • Artists Group • Photographers Group
Example Three Sections • What is the project about? • What is the main question? • What are the other questions? • What maths is involved? • What method/s are used to investigate the problem/s? • What materials and equipment will be needed?
Example Four Sections • Date? • What are we doing for our project today? • What maths did we use? • What did we discover or learn today? • What do we need to do next?
Example Five 2007 Home/School Partnership Timeline Example- 1 0f 3 Pages
Which investigations will be entered in the MTQ? The above school had their own school judging of investigations to select those to be entered. Some schools have a school expo to display the quality of their investigations and every school investigation is entered. While other schools submit all investigations and have a display to celebrate their maths later in the year. 2007 Home/School Partnership Timeline Example- 3 0f 3 Pages
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 separatelog, journal or rough workings. • It also needs to include reference to the progress of the mathematical investigation being undertaken, the problems undertaken and the mathematicalconclusions reached. • A bibliography listing all references used. • Acknowledgementof anyassistance given. Back to Investigation List
Assessment 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 points Download judging criteria from MAV Website- http://www.mav.vic.edu.au/studact/mtq.htm Back to Investigation list Back to Index
A 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
REMEMBER Curiosity and interest are the centre of inquiry! Choose your investigation wisely!
Hope you are excited about your Investigations! Back to Investigation List
References • 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. Acknowledgements 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 Back to Start