For instance, pick a geometry we have studied which you find interesting. Is there some aspect of it which was discussed briefly in class but which we didn't pursue? Is there some way of changing the rules which intrigues you?
If you're not having much luck, browse through the textbooks. Look at the books on reserve at the library. Talk to me.
Once you have tentatively chosen a topic, write a few sentences explaining it. If you are creating your own model, describe exactly what it is. If there's something missing from a proof, or from the coverage of a topic in one of the books, or whatever, describe what's missing.
Turn in your choice of topic, together with the brief explanation.
Now that you have chosen the topic, you should know at least in principle what geometry model(s) you will be working with. The next step is to decide what questions to ask about it. So make up a list of questions about your model. Does it need a distance function? Do you plan to determine what corresponds to circles?
Select several of these questions (1 is too few; 10 is too many) which you hope to answer while writing your essay. Divide them into appropriate categories. Now you're ready for the outline: Start with an introduction, end with a summary/discussion/conclusion, and put the various (categories of) problems in the middle. Briefly describe each part.
Turn in your outline.
Solve the problems. This is the fun part!
Write up what you did. You need to include enough detail so that people can understand it. Most calculations should be given explicitly. Lots of figures (with suitable captions/descriptions) are a big help. But you also need to include enough words so that people can understand it; theorems and proofs may be appropriate, but are certainly not sufficient.
Turn in your rough draft.
Be a perfectionist. Fix your math mistakes. Fix your grammar mistakes. Fix your spelling mistakes. Make sure your logic is sound. Make sure your reader will know at each stage what you're doing. Perhaps some reminders are needed: ``Now we will solve the Dray conjecture'' or ``We therefore see that the Dray conjecture is false''.
A discussion of the history of non-Euclidean geometry is not appropriate. A comparison of different (historical) versions of neutral geometry might be.
This does not necessarily mean that you must do something nobody's ever thought of before, although you'll certainly get brownie points if that is the case. You do need to work through the math yourself, and present the results in your own words.
You may use whatever references you can find which might be appropriate. But you must give appropriate credit. A direct quote, for instance the statement of a postulate or a theorem, should be clearly labeled as such. A figure which appears elsewhere must be so labeled. It is not appropriate to make minor changes in text, or to redraw a figure, without giving a proper reference; this is plagiarism. By all means paraphrase an argument you find elsewhere. But give credit to the author. And don't fill up the entire essay this way; that's a book report.
Your references should appear separately at the end of your essay, with a section heading such as References or Bibliography. Full publication data must be given, including title, author(s), publisher, and year. Page numbers may be given if appropriate.
Your essay should be easy to read. Ask a friend to read it. Tell them not to worry about the details. Is the argument clear? They should be able to read the introduction and conclusion and tell you what your essay is about. Can they?
Your essay should be easy to read in another sense: Type it (or use a word processor)! Get that new ribbon/cartridge you've been thinking about! Use section headings. Indent your paragraphs. Don't run lengthy equations into the text - display them neatly on separate lines. (You may hand-write equations if you can not type special symbols.)
By all means include lots of figures! These can appear in the text or on separate pages at the end, and may be hand-drawn. Each one should have a label such as Figure 1 as well as a caption. You must describe each figure in the text in enough detail so the reader can figure out why it's there.
Your essay should be 5-7 pages long not counting figures and lengthy equations. Somewhat longer is OK; shorter is not.
It's a good idea to get the math right!
As you know, mathematics is a very complex subject. Learning it requires more than just memorizing sets of facts and examples. If students are not able to process their ideas before, during, and after learning has taken place, they will have trouble really improving their conceptual framework.
This processing can take place orally, through discussion; mentally, through thinking back over the work and learning done; or in writing, by explaining in a narrative form the process they just went through. Ideally, all of these modes are used by a teacher, so that students have multiple opportunities to cement new knowledge, and also so that students who learn in different ways can have their unique learning styles supported.
This might sound funny, but mathematicians don’t just scribble mathematical notations on chalkboards until arriving at a profound new theorem, or answer to a problem. In addition to doing math, mathematicians are also required to write clearly and effectively. How else could they explain their ideas to people who don’t have their specific skill sets—or, in some cases, even to those who do?
In addition to these considerations, mathematicians also need to keep a complete record of their ideas and work, and be in a position where they can communicate their findings to the world. The list below takes a look at some of the strategies that can be used to engage math students in essay writing.
This is a writing strategy that a teacher uses throughout or at the end of a lesson to engage students to develop big and better concepts and ideas. In mathematics, you can use these strategies to foster this kind of engagement.
1. CALLA Strategy.
The Cognitive Academic Language Learning Approach is a strategy designed to provide students with support when it comes to learning content as well as learning how to learn. What the strategy demands is that the students read and record whatever it is that the problem is requesting.
Students are guided throughout the process to a particular solution by asking them to solve, check and explain every step of their work. This process demands that students write about the things that made the problem difficult, or any other strategies that helped them to solve the problem. This strategy will provide support for ESL learners in content and learning strategies, and go ahead to help them get organized. It also gives students a powerful tool for any kind of future learning endeavor, which is the ability to self-reflect and improve their own learning process.
2. Column Notes Strategy
The Column Notes strategy involves the double and at times the triple entry journal as a column graphic organizer. Students are record important and factual information from a lecture and/or text on the left hand side of the column, and the right hand side is normally used by students to record and process their personal responses and information.
The students can use the third column for the triple journal to summarize their understanding of concepts. This strategy is basically used to help students recall information, gives them an opportunity to clarify information, helps them make personal connections with the gathered new information, and encourages them to analyse and question information that is presented.
3. Compare and contrast strategy.
This strategy has students collect information relating to two or more mathematical concepts. The student will then be required to record the key attributes on a two column graphic organizer that will clarify both the similarities and the differences. This strategy ultimately encourages students to examine the systems being compared analytically, and helps them to clarify any information that they provide in a personal manner that will make the most sense to their own unique learning process.
4. Content definition map strategy.
The Concept Definition Map Strategy is basically a visual representation whereby examples, vocabulary terms, and sub-concepts are related to a main topic. This goes to help students make connections that are in existence between the ideas, gives the students amazing opportunities for review, provides them with tools that allows them to reflect on the changes in their understanding, and gives them the opportunity of accessing prior knowledge.
5. The Frayer model of concept development.
In this strategy, students are required to use a variety of modes and methods including: oral, written, and visual content to develop not only a personal but an in-depth understanding of some of the key mathematical concepts and terms as a whole. The strategy was generally developed to help students gain a better understanding of concepts.
The most intriguing aspect about the model is that it relies on a graphic organizer help with the understanding of the concepts. The concepts are explained to you through definitions, characteristics, and examples.
6. The Gist strategy.
The General Interactions between Schemata and Text is a strategy that involves a step by step process of summarizing text materials. This strategy in particular is very useful for students who have difficulties putting what they have read into their own words.
What students are required to do for the GIST Strategy is to restate their main ideas from mathematics texts while at the same time omitting specific examples that have been used to illustrate concepts. This strategy basically helps students gain a better comprehension of mathematics text, allows them to process new information, and establishes connections with their own ideas while providing students with a structure for identifying and remembering some of the key ideas covered.
Any of these strategies can help to engage a math student into writing. You should explore them all, and adopt the one—or ones!—that prove most effective in supporting your students.
You can still combine two or even more of these approaches to enhance your chances of supporting every individual student. But maybe the best way to engage students into writing is rewards—for instance, by offering some kind of prize or reward to those students who participate, or who do well on a given assignment. You can also show students how easily they can earn some extra money by participating in different writing contests and competitions (this has been working with my students!).
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About the Author
Jessica Millis is an educator and freelance blogger. She works as an essay editor and writer at EssayMama.com – a service that connects students and writers all over the world. Follow EssayMama on Facebook, G+ and Twitter!