Chapter+1+--+Introduction

=Definitions=
 * **Research Problem:** "A statement about an area of concern, a condition to be improved, a difficulty to be eliminated, or a troubling question that exists in scholarly literature, in theory, or in practice that points to the need for meaningful understanding and deliberate investigation" (from USC Libraries).
 * **Rationale:** An argument for why your research problem is important and worth studying, both for the field in general and for you in particular.

=What Goes In Your Introduction= There are three main purposes for your introduction chapter: These purposes and how to achieve them are described in more detail below. Note that these three purposes should NOT become three separate sections in your Chapter 1. In particular, the introduction of your research problem should actually form a part of your rationale, and your research purpose may be stated somewhere in the middle or towards the end of your rationale. The sections of your Chapter 1, instead, should be organized in a way that supports the readers' sense making of your ideas and arguments, and not around the three purposes of an introduction.
 * 1) To introduce your research problem.
 * 2) To present a rationale for your research.
 * 3) To state the purpose of your research.

Introducing your research problem.
The first part of the introduction section is usually devoted to introducing your research problem. This is often done in a way that also establishes its importance. Writers often accomplish both purposes (introducing the problem and establishing its importance) by establishing a connection between the research problem and a larger issue or problem that is familiar to and valued by the vast majority of mathematics education researchers. The introduction chapter typically begins with this larger issue, because it is something to which the reader can easily relate, and thus makes a good starting point. Some examples of the larger problems or issues that past students have started with include the following:


 * The importance of students being able to engage in mathematical discourse.
 * The need to support new mathematics teachers as they begin their teaching careers.
 * The difficulty of helping students develop an understanding of functions.
 * The difficulty mathematics teachers have in changing their teaching practice.
 * The failure of mathematics education reform movements.

Once the larger issue has been described and its importance established, the writer then forms an argument that links the research problem to the larger problem. The argument is typically constructed by putting together a series of two different types of connections: a //vertical// connection or a //horizontal// connection. The //vertical// connection consists of identifying a subproblem of the larger problem and establishing its importance through its relationship to that larger problem. In contrast, a //horizontal// connection consists of linking the larger problem to a related, smaller problem, and once again establishing the importance of the smaller problem by appealing to its relationship with the larger problem. The writer creates a chain of these vertical and horizontal connections that ends with the research problem.

To illustrate the creation of this chain of connections, consider the following line of reasoning. A graduate student wants to establish the importance of her research problem, which is the lack of knowledge in the field about the types of mathematical questions that a teacher asks during inquiry-based instruction. She starts with a large issue, namely the need for students to learn how to engage in mathematical discourse. She then makes a vertical move by talking about the importance of helping students learn to ask mathematical questions during inquiry-based instruction. She follows with the horizontal move by discussing the key role that teachers' play in providing students with examples of mathematical questions. Lastly, she makes a horizontal move to the research problem, noting that there isn't much information on the types of mathematical questions that a teacher asks while engaged in inquiry-based instruction. By creating this chain of vertical and horizontal connections, the graduate student was able to both introduce the research problem //and// establish its importance.

When constructing your chain of connections, it is sometimes necessary to define terms that are either not generally known, or that have multiple meanings that are in use in the mathematics education community. For example, if you were writing about understanding, you would need to explain what you mean by the term "understanding," since there are many different meanings for understanding in use among members of the mathematics education community. However, when you define a term in the introduction chapter, you need to make those definitions as short as possible. Save extended discussions about why you chose that particular meaning for the term or where the meaning comes from for Chapter 2. An extended discussion of terms and theories at this point will bog down and obscure your argument for why your research problem should be studied.

After you complete your chain of connections, you may want to share why you personally think this research problem is important. Often times researchers select research problems for personal reasons. Sharing those reasons is appropriate in a thesis or project. In fact, if you are engaging in action research, your committee will be looking for your personal reasons for conducting the research, so you need to make sure you include them.

Presenting a rationale for your research.
If you took the above approach to introducing your research problem, then you've already completed part of your rationale. Your introduction has situated your research problem with respect to other issues in the field, and thus established its importance. Another important part of your rationale is to establish that there is at least some part of this problem that has not been adequately addressed in the research literature. Usually the writer does this by first briefly summarizing what is known about the research problem, citing some of the findings from the literature. Then the writer discusses what part of the research problem needs further study, including the particular topic or issue that is the aim of the study. This is followed by a statement of the purpose of your research, described below. Finally, you complete your rationale by explaining the contribution that your research will make to the field. In particular, you focus on how the findings of your study will further the field's attempt to address the research problem. This may include a theoretical contribution, which helps researchers better understand the problem; a methodological contribution, which helps research conduct more fruitful research; or a practical contribution, such as suggestions for addressing or solving the problem.

Stating the purpose of your research.
An introduction chapter is incomplete without a statement of the purpose for your research. The purpose statement should focus on what the research will reveal about the research problem. The purpose of the research should be to //understand// a certain aspect of the research problem, not to //solve// the research problem. For example, if the research problem is students' difficulty in understanding logarithms, the purpose of the research study should not be to help students develop an understanding of logarithms--an attempt to //solve// the problem. Instead, an appropriate purpose would be to better understand why students struggle to understand logarithms or what intermediate understandings students form as they move toward a deep understanding of logarithms.

The reason why the purpose shouldn't be to solve the problem is that such research tends to have little value. Suppose, for example, that your research consisted of a particular instructional treatment given to a particular population of students. The knowledge that this treatment worked in this particular setting has limited value, because we don't understand what aspects of the treatment or what aspects of the setting contributed to its success. Without this knowledge, it is difficult to apply this finding to other contexts. In contrast, by focusing the research on better understanding the research problem (or the particular factors that seem important in addressing the research problem), the researcher contributes findings that have a wider range of application, and thus are much more useful to the field.

The one exception to the rule of describing your research purpose in terms of understanding rather than solving something is when you plan to do action research. Action research involves an attempt to improve a particular situation, practice, or outcome. Thus, your purpose statement will almost always be phrased in terms of accomplishing a particular objective, such as incorporating informal assessment into your instruction or increasing your students' success at coordinating representations when reasoning about linear functions. Note that both of these purposes are personal--one is about improving your instruction (not every mathematics teachers' instruction) and the other about improving your students' learning (not all students' learning). In action research, the purpose is always to improve a practice or outcome in which you have a personal stake. If you are doing action research, your committee will be looking for this personal connection in your research purpose.

Sometimes graduate students use their research question(s) as the statement of purpose. They will write, "The purpose of this research study is to answer the following questions:" and then list their questions. This is particularly appropriate if your research questions can be understood without defining additional terms or providing additional background for why you are asking these questions, and not others, about your research problem. However, many graduate students wait until Chapter 2 to reveal their research questions, because they need to provide a more in-depth discussion of issues, theories, and research results before their research questions will make sense to their readers. When this is the case, a statement of your research purpose should not include your research questions, but instead refer to a more general purpose for your study, one that makes sense after reading your introduction and that is also compatible with the research questions you will state in Chapter 2.

=Writing Tips=

Your introduction is probably the first formal writing that you will do for your thesis or project. Those first few paragraphs may well be the toughest writing you will do for your thesis or project, because you probably won't feel ready to write. For some people, the easiest way to get started is to write down their stream of consciousness as they think about why they are doing their research. They take the pressure off themselves by realizing that they will probably throw away most of what they write at the beginning, but at least after they get something down on paper, they will have some ideas to work with and will no longer be facing a blank page. Others cannot engage productively in free writing, but instead prefer to write only ideas that are at least somewhat organized and polished. For these people, the best way to start is to imagine presenting an argument to a fellow graduate student, and then create an outline of this argument. Points in the argument can be transformed into topic sentences for paragraphs, which later can be filled out to form paragraphs.

Regardless of which way you start writing, be prepared to do a lot of revision. Because you will start on your introduction as early as your first semester in the program, your understanding of and reasoning about your research will change drastically from when you first start writing to the time when you defend your proposal. Each major change in your thinking will necessitate some revision in your introduction. Why, then, not put off drafting your introduction until you know a lot about your research topic? The reason is that writing about your research, even when you know very little, leads to significant insight that thinking and reading alone cannot accomplish. With this in mind, you can see that spending large amounts of time perfecting your writing in the first few drafts of your introduction is not wise. Instead, use the first few drafts to test out new ideas, to try out different ways of organizing your rationale, and to consider and clarify the purpose of your research. Save the careful editing for later drafts.

=Common Graduate Student Mistakes=
 * Unclear or incomplete argument.** By far the most difficult part of writing a rationale is to make your line of reasoning clear to your reader. When your argument doesn't seem to be coming across, it is often because you have either left out important parts of your argument or failed to adequately link the parts of your argument together. A good place to start identifying these problems in your writing is to create a rich outline that maps out the logic of your overall argument. You may want create this outline by first examining closely what you have written to identify its underlying argument. Be brutally honest and outline what you have already written; include only the ideas and the logic in your outline that are in the rationale your wrote, and not the ideas and logic that you had in your head when you wrote it. Then go back and rearrange, add, modify, and delete ideas and logical connections until you have a strong argument. Then use that outline to rewrite your rationale. If this outlining activity does not identify problems in your argument, then the issue is likely clarity in writing. Try using advanced organizers (a brief summary at the beginning of the chapter that lays out the big points of your argument), transition sentences that make clarify the relationship between the last idea and the next one, and topic sentences that clearly identify the main points in your argument.


 * Too much detail in the introduction.** While you will want to cite relevant literature to support the arguments you are making in your introduction, you do not want to review that literature in depth in this first chapter; doing so will distract from the bigger points you are trying to make. So save the detailed review for the background chapter. Similarly, although it is appropriate in your introduction to refer to theoretical or methodological issues that are relevant to your argument, do not describe those issues in detail.


 * Rationale given only in terms of personal interests.** You may have very strong personal reasons for studying a particular problem. Such strong motivation is good, because it will help you get through the long process of completing a project or thesis. But your project or thesis must make a contribution to the field in addition to providing you with personal insight. So don't state your rationale only in terms of your personal interests. Help others understand why they should be interested, too.


 * Purpose statement is unclear.** The purpose of your research may be clear in your mind, but it is often difficult to communicate to others. Try out your purpose statement on other graduate students, and then have them restate your purpose in their own words. You'll be surprised at how easily your purpose statement can be misunderstood!