MATHEMATICS
Note: Before reading the following guidance, read the “General guidance for the extended essay” section in this guide.
Note: Before reading the following guidance, read the “General guidance for the extended essay” section in this guide.
An extended essay in mathematics should generally take one of the following approaches.
Explore an issue in mathematics of interest to you that is outside of the syllabus. Sometimes in the course of your studies or in your wider reading, your curiosity might be aroused by a problem in mathematics or a different approach to mathematics that you might be interested in exploring: see examples 1, 2, and 3 under “Example topics, research questions and approaches”. These could be called theoretical essays.
You might learn of how mathematics can be used to investigate an issue you are interested in. This issue could come from your career aspirations, a creativity, activity, service (CAS) project, an interest or hobby of yours, or from another subject. If the issue arises from another subject, you could consider the interdisciplinary pathway for your extended essay: see examples 4 and 5 under “Example topics, research questions and approaches”. These could be called practical essays.
You may have a passion for a sport or an interest that can provide some data. Choose a mathematics extended essay and analyse the data, probably using your knowledge of statistics, to answer your research question. If the topic is something you love, you will especially enjoy writing your extended essay: see example 6 under “Example topics, research questions and approaches”.
Note that your essay does not have to be entirely practical or theoretical. Many essays will be a mixture of both. In any case, note that it is mandatory to include secondary research, i.e., a review of the literature.
Remember that mathematics uses a lot of algebra, so the actual final word count for your extended essay is likely to fall short of 4,000 words.
Explore an issue in mathematics of interest to you that is outside of the syllabus. Sometimes in the course of your studies or in your wider reading, your curiosity might be aroused by a problem in mathematics or a different approach to mathematics that you might be interested in exploring: see examples 1, 2, and 3 under “Example topics, research questions and approaches”. These could be called theoretical essays.
You might learn of how mathematics can be used to investigate an issue you are interested in. This issue could come from your career aspirations, a creativity, activity, service (CAS) project, an interest or hobby of yours, or from another subject. If the issue arises from another subject, you could consider the interdisciplinary pathway for your extended essay: see examples 4 and 5 under “Example topics, research questions and approaches”. These could be called practical essays.
You may have a passion for a sport or an interest that can provide some data. Choose a mathematics extended essay and analyse the data, probably using your knowledge of statistics, to answer your research question. If the topic is something you love, you will especially enjoy writing your extended essay: see example 6 under “Example topics, research questions and approaches”.
Note that your essay does not have to be entirely practical or theoretical. Many essays will be a mixture of both. In any case, note that it is mandatory to include secondary research, i.e., a review of the literature.
Remember that mathematics uses a lot of algebra, so the actual final word count for your extended essay is likely to fall short of 4,000 words.
The extended essay is your opportunity to explore an area of personal interest. If this interest is a branch of mathematics or if you plan to use mathematics to explore an interest, then your next task is to narrow this down to a topic. In more theoretical essays, the topic could clearly demonstrate the area of mathematics you will be exploring. In more practical essays, the topic could indicate how mathematics can be used to explore your area of interest. Once you choose your topic of interest, provided it is narrow and sufficiently focused, you should be able to come up with a provisional research question. As your research proceeds, you may decide that the exact wording of the research question needs to change a little to adapt to the direction your research is taking.
Be cautious when formulating research questions that compare two different approaches to a topic. It can sometimes be challenging to demonstrate a comprehensive understanding of the mathematics behind both. For instance, example 1 under “Examples of topics, research questions and approaches” would be better than “Compare RSA with elliptic curves as a method of encryption”. You would need to explain the mathematics behind both RSA and elliptic curves carefully; however, this will not be fully achievable within 4,000 words. Avoid taking on too much and sacrificing showing a clear understanding of the mathematics in trying to fit it all in.
Be cautious when formulating research questions that compare two different approaches to a topic. It can sometimes be challenging to demonstrate a comprehensive understanding of the mathematics behind both. For instance, example 1 under “Examples of topics, research questions and approaches” would be better than “Compare RSA with elliptic curves as a method of encryption”. You would need to explain the mathematics behind both RSA and elliptic curves carefully; however, this will not be fully achievable within 4,000 words. Avoid taking on too much and sacrificing showing a clear understanding of the mathematics in trying to fit it all in.
Before you begin writing your essay, you will need to do some research. In more theoretical essays, this research will mainly involve reading around the subject. Select a few resources that are most relevant; too many sources can be overwhelming and confusing. Depending on the mathematics you want to explore, the use of subheadings in your extended essay may be helpful.
In your introduction, explain your research question clearly and what you plan to do, in your own words. It is important that you show understanding of the mathematics you will be using and that you start from what is familiar to you in the syllabus. If you do not fully understand it, leave it out and revise your plan accordingly. When you write about mathematics, all essential steps need to be shown and explained in your own words. Do not rely too heavily on your sources and do not copy information directly from them. It must be clear to the reader that you fully understand what you are doing. If there is a complicated theorem you want to use, and you feel the proof is too difficult, it is acceptable to quote the theorem without proving it. However, it is essential that you explain what it is for and provide an illustrative example to demonstrate your understanding.
In more practical essays, the emphasis may be on collecting appropriate data to begin with. When writing your essay, after explaining the background, show the data first (in an appendix if there is a large amount). For examples 4 and 6, you would need to start by graphing the data before deciding what to do with it. For this reason, it is better not to specify the mathematics or statistical methods you use in the research question because this may change once you examine your data. Your research may then involve learning (and explaining) some new mathematics to help you analyze the data. However, note that using mathematics from outside the syllabus is not essential. You may be able to answer your research question using the mathematics you have covered in your syllabus.
Please ensure that you use raw data. It is difficult to do much with data that has already been processed, for example, batting averages in baseball.
EE Guide pg 68-71
In your introduction, explain your research question clearly and what you plan to do, in your own words. It is important that you show understanding of the mathematics you will be using and that you start from what is familiar to you in the syllabus. If you do not fully understand it, leave it out and revise your plan accordingly. When you write about mathematics, all essential steps need to be shown and explained in your own words. Do not rely too heavily on your sources and do not copy information directly from them. It must be clear to the reader that you fully understand what you are doing. If there is a complicated theorem you want to use, and you feel the proof is too difficult, it is acceptable to quote the theorem without proving it. However, it is essential that you explain what it is for and provide an illustrative example to demonstrate your understanding.
In more practical essays, the emphasis may be on collecting appropriate data to begin with. When writing your essay, after explaining the background, show the data first (in an appendix if there is a large amount). For examples 4 and 6, you would need to start by graphing the data before deciding what to do with it. For this reason, it is better not to specify the mathematics or statistical methods you use in the research question because this may change once you examine your data. Your research may then involve learning (and explaining) some new mathematics to help you analyze the data. However, note that using mathematics from outside the syllabus is not essential. You may be able to answer your research question using the mathematics you have covered in your syllabus.
Please ensure that you use raw data. It is difficult to do much with data that has already been processed, for example, batting averages in baseball.
EE Guide pg 68-71
Writing in Mathematics
Websites
Websites
- Probability: Theory and Application
- Math Guide
- zbMath online database
- Math on the Web links to online Math journals
- Math.com "The World of Math Online"
- Search on Math
- WolframAlpha computational search
- Math Journals with free online access
- MIT Math Courses Good for adding theory to your research
- Math Videos
- Physics Forum (includes math) As our name suggests, our main focus is on physics, but we also have forums for most other academic areas, including engineering, chemistry, MATH, etc