Chapter+2+--+Background

=Definitions=
 * **Literature Review:** An argument that (1) identifies the important research findings related to the problem or phenomena you are studying (2) situates your study by discussing how your study is related to or builds upon the work or others, and (3) critiques the current research to identify a "hole" (some phenomenon, issue, or problem that still requires additional research) that your study will address.
 * **Theoretical Framework:** A description of the theoretical lens through which you will (1) identify the phenomena you are studying; (2) make predictions about the phenomena, which in turn informs your study design; and (3) interpret the data you collect. The section you write that contains your theoretical framework will also contain an argument for why your particular theoretical lens is appropriate for your study.
 * **Research Question:** The question or questions that your research will answer.

=What Goes In the Background Chapter?=

There are three main components of Chapter 2: a literature review, a theoretical framework, and research questions. The order of the these three things should be purposeful. For example, you could review the literature and then, using that review as a background, focus in on key constructs and how you view them. Alternatively, you could describe your theoretical framework first and then use that framework for organizational or analytical purposes in the literature review. Note that it often isn't possible to decide what order of these two components works best before you start writing each of the sections. In contrast, research questions typically come at the end of the chapter, because asking your particular questions often doesn't make sense until after you have reviewed the literature and described your theoretical framework. However, the research questions can sometimes be meaningfully placed in between the other two components if they help focus the discussion of what follows. If you feel like they would help focus the discussion of both the literature and your theoretical framework, then place them at the end of Chapter 1 instead of Chapter 2.

The Literature Review
You may be wondering why you still have to talk about research in the field when you have already constructed a rationale and purpose for your study. In some sense you have already "situated" your research in the field by talking about the relationship of your research problem to other important issues, problems, and phenomena. Why do it all over again in the literature review?

In your introduction you used broad strokes to situate your research. In the literature review, you will delve much deeper into the literature to show the reader how you are refining and narrowing your research focus, and to convince the reader that this refined focus is important. Consequently, the literature review is not an exercise in summarizing and paraphrasing. Having identified important related studies, your task is to analyze each one through the lens of your own study (i.e., your research problem, purpose, and questions; your theoretical framework) and then to synthesize your findings into a well organized argument. Although you may, at times, devote an entire paragraph to describing a particularly important study, typically the individual sections of the literature review should be organized by idea (by a point in your argument) rather than by study. Literature reviews typically don't accomplish their purpose if they come across as a collection of paragraphs each of which simply summarizes a related study. The key is to analyze and synthesize, not quote and summarize.

A literature review typically has 2 to 4 major sections in it. Each section consists of a review of the literature related to a different issue, topic, or phenomenon. A literature review needs to draw from at least 2 areas of research in order to have a strong enough theoretical base. On the other hand, drawing from more than 4 areas of research becomes too burdensome to coordinate as you construct the argument in your literature review and conceptualize your study. As you think about your literature review, you should attempt to identify 2-4 major areas of research that are related to your research topic. If a lot of research has already been conducted on your research topic, it is likely that the 2-4 areas of research that you review will be closely related to your topic and narrowly defined. If little research has been done on your topic, you may have to draw from research that is not as closely related to your research topic, perhaps even drawing from a research area outside of the field of mathematics education. Note that in all cases, whether your topic has been studied a lot or a little, you must demonstrate how your study will build on previous work, and how your study will add new knowledge to the field. The following are some examples of the sections in the literature reviews of completed theses. (Note that I have tried to describe the sections in a way that makes sense to someone who has not read what has come before each section, and not just copied the headings of the sections.)
 * **Topic** || **Literature Review Sections** ||
 * Role of writing in learning mathematics || * Research on the connection between writing and cognition
 * Research on the effect of writing on learning mathematics
 * Reasons for why research on writing to learn have been ineffective ||
 * Identifying the general format of conceptually-oriented explanations || * A description of Toulmin's framework for argumentation and the ways it has been applied to the field of mathematics
 * Research on mathematical explanations
 * Role of class conventions in constructing explanations ||
 * How eighth grade students estimate with fractions || * Research on the difference between whole number and fraction reasoning
 * Computational estimation strategies
 * Problems with the methods used to study estimation ||
 * How student positioning can lead to failure in an inquiry-based classroom || * Positioning theory and relevant research
 * Gee's Discourse theory and how its related to positioning
 * Research on student failure in inquiry-based classrooms ||
 * Developing a conceptual framework for what it means to understand logarithms || * Research on students' understanding of logarithms as objects
 * Research on students' understanding of logarithms as processes
 * Research on students' understanding of logarithms as functions
 * Research on students' understanding of contextual problems involving logarithms ||
 * The use of professional development to support a first-year secondary mathematics teacher || * Research on beginning teachers
 * Research on professional development
 * Research that suggests developing a community of practice within a professional development experience might be helpful to beginning teachers ||

In each section of your literature review, you should work to identify what research has shown and what still remains to be understood. In particular, you need to emphasize research results that have shaped your decisions about where you will conduct your study, what types of instruments you will use to collect data, who your participants will be, and what you'll be trying to find out about them. While you identify the findings of particular research studies and the particular contributions they make to the field, you will also need to identify what these studies do **not** reveal. This does not mean you trash the studies because they didn't provide insight into a particular issue, problem, or phenomenon; no study could ever address every possible issue, problem, or phenomenon related to a particular research topic. Instead, you praise the work that has been done but also discuss what remains to be learned.

A strong literature review does not only contain an argument about what is known and unknown in each of its main sections, but also tries to connect these arguments in a way that constructs a larger, overall argument that points to the need for the research question you are asking. Sometimes it is easy to connect the arguments, because the research in one section will point to the need for research in the next section, which will in turn point to the need for research in the next section, and so on. If such a line of reasoning does not exist, then you will probably want to treat each section as if it addresses a particular part of your research focus. For example, in the study above about logarithms, the graduate student organized the literature review by stating at the very beginning that there are four components to understanding logarithms, and then described the research related to each component. Whichever approach you take, the section-implies-section-implies-section argument or the parts-making-up-the-whole argument, you will probably want to outline your overall argument in an introduction paragraph, and then include a summary or conclusion paragraph at the end that summarizes the big points in your argument and describes what conclusions you are making.

After your readers have read your literature review, they **should**...
 * 1) Know how your study fits in with past research.
 * 2) Understand how your study fills an important hole in the research.
 * 3) Be able to anticipate what your research questions are.
 * 4) Think your research questions are important.

After your readers have read your literature review, they **should** **not**...
 * 1) Wonder why you included a particular result in your review. (In other words, they should not feel like the review is a listing of research results, but rather a carefully crafted argument.)
 * 2) Wonder why you failed to mention a particularly important research paper or study.
 * 3) Wonder why you picked these particular research questions. (In fact, if your literature review was particularly strong, your readers should be able to correctly guess what your research questions are when they finish your literature review.)

The Theoretical Framework
The theoretical framework has many purposes. First and foremost, your theoretical framework makes public the lens through which you are viewing the issue, problem, or phenomenon you are studying, as well as the data you will collect. While conducting research, it is not possible to take a completely objective view that requires no explanation. In fact, in order to actually separate a particular issue, problem, or phenomenon from the surround requires some set of values, beliefs, and theories. For example, studying how children are thinking as they engage in solving a mathematical task requires that you first believe that children think, and that their thinking is at least partially revealed in what they say or do. If you did not believe these things, you would probably not be able to "see" children's thinking in their actions as they complete a task. The research community recognizes the impossibility of an objective stance, and instead insists that you make your particular values, beliefs, and theories explicit. Then they judge your work not on whether or not you uncover objective truth, but rather in terms of how faithful you are in applying the particular stance to studying the phenomenon, and whether or not your application of that stance provides valuable insight into what is being studied.

A big part of making your theoretical lens public is defining terms and describing theories. For example, if you are planning on studying students' understanding of the univalence property of functions, you had best define what you mean by function and the univalence condition. You will also most likely need to define what you mean by "understanding." Does it mean that students can solve the problems at the end of a textbook section, or something more? You will also need to describe the theories you are using. By theories, I mean your beliefs about what processes and factors are involved in a particular issue, problem, or phenomenon, and how those processes and factors are related. Your theories might be formal and well-known by a particular name, such as social constructivism, or they may be more informal and less developed, such as the common assumption that teachers' beliefs influence their instruction. Either way, you will need to identify and describe these theories.

A second purpose of the theoretical framework is to explain to your reader why you chose the particular definitions and theories you did. When there are multiple definitions and/or theories available to draw from, you will need to describe all that are well known in the community and could have been adopted for your study. Consequently, your theoretical framework will also take on a persuasive and/or critical tone. To justify their theoretical choices, graduate students typically appeal to how well a definition or theory fits with the purpose of their study and is accepted by the field.

A third purpose of the theoretical framework is to enable you to identify and make sense of the issue, problem, or phenomenon you are studying. Your theoretical framework can suggest the characteristics of different phenomena so that you can tell them apart. (For example, if you were studying students' beliefs about mathematics, you would need to define beliefs in such a way so that you could tell it apart from knowledge, values, and attitudes.) Your theoretical framework can help you anticipate what will happen under certain circumstances, enabling you to plan ahead so that you can gather data that document what you hope to see. Your theoretical framework can suggest codes you might use as you analyze data. Your committee will be carefully reading your theoretical framework to look for how it will support the design of your study, its implementation, and data analysis.

A fourth purpose of the theoretical framework is to establish the importance of your research. You should not only describe the definitions and theories from the literature, but you should also be critical of them. Point out weaknesses that your study is positioned to address. After reading your theoretical framework, your reader should be even more convinced that your research questions are important. The theoretical weaknesses that you identify can later be revisited in the implications section of Chapter 5 to show how your study contributes to the field.

The overall structure of a theoretical framework is not nearly as predictable as the literature review. A good place to start is by discussing the definitions and theories that are pertinent to the main phenomenon that is being studied. If there are competing definitions and/or theories, make sure that you explain why you are choosing one over the other, and that it is clear which one you prefer. Of course, you may reject them all and propose your own definitions and/or theories, or present a modification of them. But once again, you will need to persuade your readers that your rejection or modification is reasonable. A discussion of the main phenomenon may point to additional, related phenomena that need to be defined explained. Often these related phenomenon can be defined as they come up in your discussion of the main phenomenon. Once you have explained and justified the terms and theories needed to understand your main phenomenon, your theoretical framework is done.

After your readers have read your theoretical framework, they **should**...
 * 1) Know what theoretical lens you are using to design the study and collect and interpret data.
 * 2) Understand why you picked the particular theoretical lens you did.
 * 3) Understand the intended meaning of your research questions (because now they better understand the terms you use and theories you reference in these questions).
 * 4) Think your research questions are important.

After your readers have read your theoretical framework, they **should not**...
 * 1) Wonder why you have included a particular definition or theory in your discussion of your framework. (In other words, they should not feel like the review is a listing of definitions and theories, but rather a carefully crafted argument that describes and justifies your theoretical lens.)
 * 2) Wonder why you failed to mention a particularly important theory or an alternate, well known interpretation of a term.
 * 3) Wonder what your research questions mean.

To Combine or Not to Combine?
Many researchers in the field of mathematics education do not separate their literature review from their theoretical framework when writing research papers. This is because they often need to cite many of the same papers in both sections. Often papers make both an empirical contribution and a theoretical one. To save space and improve coherence, researchers sometimes try to achieve the purposes of a literature review section and theoretical framework section through a single background section.

Graduate students don't have the same space constraints that researchers do, and so they don't feel the pressure to combine the two sections. However, you may still want to. The big advantage of combining both sections is that it is often easier to discuss papers a single time rather than twice in two different sections--once for the paper's empirical contributions, and once for the paper's theoretical contributions. Also, if you choose to combine the two sections, you can mix literature review subsections with theoretical subsections, which often allows you to create a very powerful argument for your research questions. The big drawback of combining the two, however, is that it is a lot harder to judge if you've met the purposes of both sections when they're combined. It's much easier to tell if you have a complete literature review and theoretical framework when you keep them separate and judge them individually. A second advantage of keeping them separate is that if you make changes to one, you often don't have to make changes to the other, because they are separate arguments. When you combine both sections, a change in one will almost always necessitate a change in the other to maintain the overall flow of the argument.

So what should you do? Some of you will not have a choice, because your advisor will prefer one format over the other. So check with your advisor first. If you have a choice, start writing them separately and see how it works for you. If you start feeling that you are constantly repeating yourself as you go from one section to the other, then maybe consider combining the two sections.

Research Questions
Your research questions are important part of Chapter 2, and often come at the end of the chapter. The reason they come at the end is that often they don't make much sense until after you review the literature and present your theoretical framework.

Perhaps the best way to present your research questions is with following nearly-impossible-to-misunderstand phrase, "The research questions for this study are as follows:" and then write your research questions in the form of a number list. Sure, there are more clever ways to introduce your research questions, and you're welcome to try one of them. But in reality, what's most important to your committee is the ability to find your research questions and understand them, so make it as simple as possible.

=Writing Tips=


 * Structuring your writing.** Writing your literature review and theoretical framework can be a brutal task. If you are struggling, welcome to the club--most graduate students do. Even seasoned researchers often struggle to put together a strong literature review and theoretical framework.

Perhaps the biggest challenge of writing these two sections is that writing necessitates that you create a linear argument, even though the way you've made sense of the literature is to form a complex network of connections between the different ideas and results. Trying to reduce that complex network into a single line of reasoning can make your head feel like it will explode. So the best thing you can do is get outside of your head. Try writing each idea or result on a card and tape them on the wall or use magnets to pin them to a white board. Now you can begin to move ideas around. You'll be surprised at how much easier it is to start thinking linearly. Once you've got an order that makes sense, begin writing at the place that seems easiest. Expand your writing to nearby topics. Don't take the cards down once you start writing--the structure on the wall or board can be very helpful in giving you a big picture of your argument as you write and help you focus on making an argument and not just describing research.

If you get stuck in your writing, share what you've got with others. Your advisor is a natural choice. Most advisors can find time to read 2-3 pages with a very short turn around time. Longer sections may take much longer for them to get to, so don't send them the whole chapter. Pick the 2-3 pages where you are most stuck. Also, schedule a time when you can come by for 5 minutes of feedback. Once again, most advisors can fit that into their busy schedules, and you'll be amazed at how useful 5 minutes can be. Besides you advisor, there are many other people who can help you. Exchange writing with your office mate. Talk about your writing to a family member. Go on a walk a friend and take advantage of his or her curiosity in your thesis to test out some ideas. Just getting the ideas out can often help you gain insight into how to structure them into an argument.


 * Tracing your argument through topic sentences.** When you feel like your argument in your literature review or theoretical framework is getting lost in all the information you are providing, try examining the topic sentences in your paragraphs. Topic sentences are typically found at the beginning of a paragraph, but may also be the second sentence in the paragraph if the first sentence is a transition sentence, one that connects the previous idea to the next idea. A topic sentence introduces the main idea of the paragraph, and all the content of a paragraph should be clearly related to the topic sentence. If you can't find a topic sentence for a paragraph, or if the content of a paragraph isn't directly related to the topic sentence, you may have found your source of confusion. Add a topic sentence and/or fix the content in the paragraph. If you still feel like you haven't clarified your argument, look at the sequence of topic sentences. It should present a coherent flow of the main ideas in your argument. If the flow isn't there, rearrange the topic sentences, or modify, add, or delete topic sentences until you've created a clear, persuasive argument that flows well. Then go back to the paragraph level and fix each paragraph so that the content fits with and supports the topic sentence.


 * Writing research questions.** There are many good sources online and in research methods manuals on writing good research questions. One source I particularly like is Maxwell's (2013) treatment of research questions. Below is the bibliographic information for Maxwell:

Maxwell, J. A. (2013). //Qualitative research design: An interactive approach// (3rd ed.). Thousand Oaks, CA: Sage.

=Common Graduate Student Mistakes=
 * Padding the literature review and theoretical framework.** The tendency when writing literature reviews and theoretical frameworks is to throw in everything and the kitchen sink. You read it; you want credit for having read it, especially if it nearly killed you. Avoid this temptation. Theories and research findings that aren't central to understanding your research focus, justifying your research design, or establishing the importance of your study should be jettisoned. They just clutter up your argument and make it difficult for your reader to understand what you are trying to do. Ask yourself as you place each card in your linear argument whether that card is essential or not. You may think that in order to please your committee, everything you've read needs to end up in your literature review or theoretical framework. They will want to know that you're well read and know your stuff. In reality, your committee doesn't want to spend time reading about research that is not pertinent to your study, and including results or theories in your background chapter that are not essential to understanding your research can mislead your committee about what you are attempting to do in your research study. So remove the extra baggage.


 * Waiting until you've "read enough" to start writing.** Almost all graduate students feel that they don't know enough on their topic to begin writing a literature review or theoretical framework, regardless of how much they've read. There always seems to be more to read. That's why you can't wait until you've finished reading the research to write. Begin writing from the very beginning. At the very least summarize (using sentences and paragraphs) the findings or theoretical contribution of the papers you read. Then make a card for each one of the important ideas and put it on your wall or board. As you add cards, arrange them into the linear form of an argument. You'll soon have enough cards in some segments to write part of your argument. You can cut and paste parts of your summaries into your argument. Soon you will have made enough progress that you will be able to make better decisions about what to read next based on what parts of your argument you have yet to flesh out.


 * Producing a literature review or theoretical framework that lacks a clear argument.** This is a common error for graduate students. To create a persuasive argument, use the ideas in "Writing Tips" above. Once you've got your outline, start fixing what you've written. Keep in mind that it may be easier to start with a series of topic sentences that follow the outline of your argument and then cannibalizing what you've written before. Oft times starting with what you've already written keeps you in the same rut you were in, and despite attempts at revision, you still end up with a text that lacks a clear argument.


 * Quoting too much.** The temptation when reviewing the literature is to quote extensively from the works that you are drawing from. Seldom do your readers want to read a quote. They would rather read your paraphrased version. You should only quote when you absolutely need a particular wording to prove your point, or on the rare occasion that you believe someone's prose is simply too insightful and beautifully written to paraphrase. When you do quote, don't expect the reader to be able to make the same sense of the quote as you did. Explain what the quote says and what it does for your argument. A good rule of thumb is that you should devote at least the same amount of space to your explanation of a quote as the space taken up by the quote itself.