The Mathematical Exploration

1. The Mathematical Exploration Internal assessment SL

2. The Mathematical Exploration Internal Assessment (IA) in mathematics SL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics (20 % to your final grade for the course). Your work in completing the IA component differs in important ways from the written exams for the course. • No strict time constraints as with written examinations. • Freedom to help decide what mathematical topic you wish to explore. • Writing about mathematics and not just doing mathematical procedures. • Discussion with, and feedback from, your teacher. • You should endeavour to explore a topic in which you have a genuine personal interest. • You will be rewarded for evidence of creativity, curiosity and independent thinking.

3. The Mathematical Exploration • You are required to write a report on a mathematical topic that you choose in consultation with your teacher. This report is formally referred to as the Mathematical Exploration. • The Mathematical Exploration is aptly named because your primary objective in writing this report is to explore a topic in which you are genuinely interested and that is at an appropriate level for the course. • Your teacher may provide you with a list of ideas (or ‘stimuli’) from which to choose a topic or which may help you to develop your own ideas for a topic to explore (see the list of 200 ideas printed later in this chapter). • Your report should be approximately 6 to 12 pages long.

4. Internal Assessment Criteria • Your Mathematical Exploration report will be assessed by your teacher and according to the following five criteria. It is very important that you familiarize yourself with the assessment criteria and refer to them while you are writing your report. The scoring levels for each criteria and associated descriptors are as follows.

5. Criterion A: Communication • This criterion assesses the organization and coherence of the exploration. A well-organized exploration has an introduction and a rationale(which includes a brief explanation of why the topic was chosen). It describes theaim of the explorationand has a conclusion. A coherent exploration is logically developed and easy to follow.   Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document. A - Communication • 0 : The exploration does not reach the standard described by the descriptors below. • 1 : The exploration has some coherence. • 2 : The exploration has some coherence and shows some organization. • 3 : The exploration is coherent and well organized. • 4 : The exploration is coherent, well organized, concise and complete.

6. Criterion B: Mathematical presentation This criterion assesses to what extent you are able to: • use appropriate mathematical language (notation, symbols, terminology) • define key terms, where necessary • use multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models. You are expected to use mathematical language when communicating mathematical ideas, reasoning and findings. You are encouraged to choose and use appropriate ICT tools such as graphic display calculators, screenshots, graphing, spreadsheets, databases, drawing and word-processing software, as appropriate, to enhance mathematical communication. B - Mathematical Presentation • 0 : The exploration does not reach the standard described by the descriptors below. • 1 : There is some appropriate mathematical presentation. • 2 : The mathematical presentation is mostly appropriate. • 3 : The mathematical presentation is appropriate throughout.

7. Criterion C: Personal engagement This criterion assesses the extent to which you engage with the exploration, and present it in such a way that clearly shows your own personal approach. Personal engagement may be recognized in different attributes and skills. These include thinking independently and/or creatively, addressing personal interest and presenting mathematical ideas in your own way, using simple language to describe complex ideas. C - Personal Engagement • 0 : The exploration does not reach the standard described by the descriptors below. • 1 : There is evidence of limited or superficial personal engagement. • 2 : There is evidence of some personal engagement. • 3 : There is evidence of significant personal engagement. • 4 : There is abundant evidence of outstanding personal engagement.

8. Criterion D: Reflection This criterion assesses how well you review, analyze and evaluate the exploration. Although reflection may be seen in the conclusion to the exploration, you should also give evidence of reflective thought throughout the exploration. Reflection may be demonstrated by consideration of limitations and /or extensions and by relating mathematical ideas to your own previous knowledge. D Reflection • 0 : The exploration does not reach the standard described by the descriptors below. • 1 : There is evidence of limited or superficial reflection. • 2 : There is evidence of meaningful reflection. • 3 : There is substantial evidence of critical reflection.

9. Criterion E: Use of mathematics This criterion assesses to what extent and how well you use mathematicsin your exploration. The mathematics explored should either be part of the syllabus, or at a similar level or beyond. It should not be completely based on mathematics listed in the prior learning. If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion. The mathematics can be regarded as correct even if there are occasional minor errors as long as they do not detract from the flow of the mathematics or lead to an unreasonable outcome. E - Use of Mathematics • 0 : The exploration does not reach the standard described by the descriptors below. • 1 : Some relevant mathematics is used. • 2 : Some relevant mathematics is used. Limited understanding is demonstrated. • 3 : Relevant mathematics commensurate with the level of the course is used. Limited understanding is demonstrated. • 4 : Relevant mathematics commensurate with the level of the course is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated. • 5 : Relevant mathematics commensurate with the level of the course is used. The mathematics explored is mostly correct. Good knowledge and understanding are demonstrated. • 6 : Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Thorough knowledge and understanding are demonstrated.

10. Guidance 1) Select a topic in which you are genuinely interested. Include a brief explanation in the early part of your report about why you chose your topic – including why you find it interesting. 2) Consult with your teacher that the topic is at the appropriate level of mathematics. • Find as much information about the topic as possible. Although information found on the internet websites can be very helpful, try to also find information from books, journals, textbooks and other print material. 4) Prepare and organize your material into a thorough and interesting report. Although there is no requirement that you present your report to your class, it should be written so that your fellow classmates can follow it without trouble. Your report needs to be logically organizedand use appropriate mathematical terminology and notation.

11. Guidance 5) The most important aspects of your report should be about mathematical communication and using mathematics. Although other aspects of your topic (e.g. historical, personal, cultural etc) can be discussed, be careful not to lose focus on the mathematical features. 6) Two of the assessment criteria – personal engagement and reflection – are about what you think about the topicyou are exploring. Don’t hesitate to pose your own relevant and insightful questions as part of your report, and then to address these questions using mathematics at a suitable level along with sufficient written commentary. 7) Although your teacher will expect and require you to work independently, you are allowed to consult with your teacher – and your teacher is allowed to give you advice and feedback to a certain extent while you are working on your report. It is especially important to check with your teacher that any mathematics in your report is correct. Your teacher will not give mathematical answers or corrections, but can indicate where any errors have been made or where improvement is needed.

12. You may sometimes find it difficult to know where to start with a task as open-ended as this. While it is hoped that you will appreciate the richness of opportunities for mathematical exploration, it may sometimes be useful to be provided by a stimulus as a means of helping you to get started on your explorations. Possible stimuli that could be given include: Stimuli • sport • archaeology • computers • algorithms • cell phones • music • sine • musical harmony • motion • e • electricity • water • space • orbits • food • Volcanoes • diet • Euler • games • symmetry • architecture • codes • the internet • communication • tiling • population • agriculture • viruses • health • dance • play • pi (π) • geography • biology • business • economics • physics • chemistry • information technology in a global society • psychology

13. A possible mind map for the stimulus “water” During introductory discussions about the exploration, the use of brainstorming sessions can be useful to generate ideas. In particular, the use of a mind map has been shown to be useful in helping students to generate thoughts on this. The mind map below illustrates how, starting with the stimulus “water”, some possible foci for a mathematical exploration could be generated.