Mathematics is a method and language of cognition of the surrounding world. Mathematics is an exact science that does not tolerate arbitrariness in interpretation and various speculations. This is the embodiment of order and rigid logic. It helps to understand the world around us, to learn more about its laws, since these laws are subject to the same order that prevails in mathematics. One way to understand universal mathematical laws is to write a study on this subject. In this article, we will explain how to do this and what mistakes to avoid.

Studying mathematics, students:

- Develop a mathematical style of thinking, the ability to generalize, the ability to analyze;
- Learn to find patterns;
- Form the ability to think and reason logically, formulate thoughts correctly and clearly, draw correct logical conclusions;
- Acquire the ability to quickly think and make decisions;
- Form the skills of planning ahead, the ability to keep several consecutive steps in mind;
- Develop skills of conceptual and abstract thinking, the ability to consistently and logically build complex concepts or operations and keep them in mind.

Basically, every math project writing is carried out according to the four following stages:

1.Preparation. Defining the theme and goals of the project.

2.Planning.

- Identification of sources of information;
- Determination of methods for collecting and analyzing information;
- Determining how results are presented;
- Establishment of procedures and criteria for evaluating results and the process;

3.Research. Information gathering, solution of intermediate tasks. Basic tools to use are:

- Interview,
- Polls
- Observations,
- Experiments,
- Work with additional literature.

4.Results and (or) conclusions.

Let’s find out in detail what should you do on each stage of your paper writing in order to make it well-researched, structured and logical.

In order to begin research, you need to find a problem that you can investigate and that you would like to solve. This problem will tell you how to formulate a research topic. The ability to see the problem is sometimes valued higher than the ability to solve it.

Tired of all the guides and never-ending instructions?

The main task of any researcher is to find something unusual in the ordinary, to see the difficulties and contradictions in those things where for others everything seems habitual, clear and simple. The easiest way to develop the ability to see problems is to learn to look at the same objects from different points of view.

**Experimental**– these are topics that involve conducting your own observations and experiments;**Theoretical**– these are topics on the study and synthesis of information, facts, materials contained in various theoretical sources: books, scientific articles, etc.

The crucial stage in the preparation of the study is the stage of justifying the relevance of the topic. To substantiate the relevance means to explain the necessity of this topic in the context of the general process of scientific knowledge. Justifying the relevance of the chosen topic, you should indicate why this particular mathematical problem is currently relevant. The relevance may consist of the need to obtain new data and the need to test new methods, etc.

Justifying the relevance, it is necessary to briefly highlight the practical reasons and difficulties for which the study of this topic has become necessary.

Once the mathematical problem is found and justified, the researcher can begin to develop a hypothesis. Translated from the ancient Greek language, the hypothesis means “the basis, the assumption, the judgment about the natural connection of phenomena”

The hypothesis must satisfy a number of requirements:

- Be verifiable;
- Contain an assumption;
- Be logically consistent;
- Fit the facts.

The hypothesis is formulated as a complex sentence: **‘if …, then …’**. Such a formulation should direct the attention of the student to the disclosure of the essence of the phenomenon, the establishment of cause-effect relationships. In the course of the experiment (or using another scientific method), the hypothesis is refined, supplemented, developed or rejected.

Before finding the list of references online, follow this algorithm of actions:

- Find out the academic standards for your work. Teachers have more serious requirements for a number of works, for example, only articles published in scientific journals during the current year. Your professor can also specifically identify which sources you need to use in your mathematical research.
- Learn about the reputation of the author. In any field of knowledge, there are such names, whose glory is unshakable. References to the works of such titans will not allow doubts to penetrate into the heads of listeners, and they will help you to gain a certain authority. This does not mean that the names of lesser-known figures should be ignored. Even if their views are fundamentally different from the fundamental statements of thinkers, you can safely use them. The main thing here is not the lack of reputation, but the absence of the author’s bad reputation. Today it is not difficult to find scandals on the Internet in which certain authors or scientists were involved. Plagiarism always pops up at the wrong time, leaving unfortunate explorers with not the most pleasant glory.
- Beware of information online. Such sources are in the public domain, regardless of their professionalism and level of skill. More credible is the information from the websites of government agencies, research centers, and international scientific journals. Here you need to pay attention to the reputation of a company that owns the site.
- Skip articles that were published at the expense of the author, not the publisher. If the publisher did not want to spend money on the release of one or another author, it is unlikely that his opinion has any weight in the scientific community.
- Use the sources that can be your main assistants in the work on a scientific paper. You can safely use the lists of sources found in peer-reviewed or scientific articles. They were written by experts with extensive experience and knowledge in a particular area. Such works are created in order to convey certain information to the community. As a rule, the authors of such articles are highly qualified specialists who can be safely trusted. And the reviews are not only written by such professional authors but also repeatedly checked by experienced experts on the subject of the article.
- Check out alternative views. If you have doubts, you can always ask your professor about the credibility of the source of interest. As a rule, teachers are fairly well informed about the degree of confidence in a particular author. In this case, you should always find out an alternative opinion before the end of the work.

The specific content of the main part of your work depends on the type of mathematical research, which can be:

**Empirical**– based on the processing and interpretation of materials of experimental or applied research, conducted by a student to confirm the hypothesis proposed in the introduction;**Theoretical**– based on evaluative analysis of information contained in scientific sources, which serves as the foundation for the presentation of their own, original, and possibly disagreeing with the generally accepted point of view of the author’s views on the problem under consideration;**Combined**– combining the first two approaches to research and clearly structured into two parts: theoretical and practical, in which information obtained from scientific sources is confirmed by the student’s personal empirical research;

This type may include computational-graphic works or case studies, which imply, for example, the development of a physical model or the solution of a specific mathematical problem.

If your task is to do theoretical and analytical research, then the started research is developed in the main part. In this case, the main part of the work is the in-depth continuation of the problem analysis based on the theoretical framework tools (the methods listed in the introduction). This can be a comparison and detailed consideration of the views of different scientists and authors in order to substantiate their own point of view, which will be demonstrated in the conclusions.

You should start the main part of this study by listing the tasks indicated in the introduction. Then propose ways to solve them and describe the results of the application development. As a confirmation of the effectiveness of the methods, data of experiments, observations, as well as illustrations and help on implementation are systematized using tables, graphs, diagrams.

Usually, such works are written by future engineers, technologists, in order to suggest ways to improve the subject of research or to develop an independent project from scratch. In this case, the main share of the main part of the work consists of drawings, formulas, calculations, dependency graphs, and the like.

Of course, mathematical research is impossible without mathematical calculations. However, the main task of the student is not only to make the correct calculations, use the necessary formula or approach, but also to be able to explain his choice with words. Therefore, when you write a study in mathematics, you must explain each of your actions logically, using mathematical terms, and do so in such a way that the reader cannot make a double interpretation of your words. Explain each point from the new line and make sure that your paper is convenient to read and perceive. It is not recommended to use arrows when you move from one step to another, just write about your intentions with the help of the words.

To better understand the basics of the correct approach to writing math research, we consider the most common mistakes of students who lead them to a dead-end and do not allow them to test their hypothesis in the right way and get the correct result.

- First, many students begin research, forgetting generally accepted rules, definitions, or formulas. Study the material on the topic completely before you start writing the work.
- Often, students know the rules and formulas, but simply do not understand them. You should not just study the material, you should penetrate into it and realize the purpose of applying the received information.
- It happens that a student perfectly knows all the material and understands it, but for a specific task, he chooses a completely different formula or a wrong research method. Try a few easier tasks to learn how to choose the right formulas and methods.
- One of the most common mistakes is neglecting a careful study of the conditions of the problem. As many experienced teachers say, half of the solution to the problem is already inherent in it, and this is absolutely true. The correct understanding of the condition gives the correct direction of thoughts when testing a hypothesis.
- A very annoying but common problem for many students is constant computational errors. Professional teachers say that many talented students simply do not pay attention to elementary calculations, as a result of which they come to the wrong answer.
- A common mistake in solving geometric problems is the neglect of the properties of geometric shapes. Often such problems are based on these properties, so this is the first thing you should pay attention to.
- When students perform certain actions using formulas, one must remember the properties of mathematical operations. This material is taught in primary school, but often students, being passionate about the study itself, forget about it.

Of course, math research writing cannot be easy because you face two tasks at the same time – to calculate correctly and to write properly. Be sure to follow the right scientific methods and use up-to-date sources, and this will facilitate achieving your goal several times.

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