The relationship between different approaches to research in mathematics can be clarified by using some hermeneutics ideas. Traditionally, it has been assumed that a statement, image, writing, etc. has an absolute meaning. Hermeneutics recognizes that the same text has different meanings when presented in different contexts and when presented to different readers.

Procedure in Mathematics Education Research

In educational research, "text" is usually just "data". Sometimes the fact that the data has to be obtained is obvious. This may be the case when we sit down with someone and ask them some questions and record their answers. Other times this process of obtaining is less obvious. Observing and taking notes on a teacher's actions doesn't seem to "get" evidence. It feels much more like the evidence presented.

However, it's important to realize that the things you choose to take notes on. Even the things we observe depend on our personal theories about what is important. In other words, all data is, in a sense, obtained. This is true even in the physical sciences. Physicist Werner von Heisenberg observed that what we learn is not nature itself, but nature exposed to our methods of questioning.

Characteristics

Educational research can be characterized as a never-ending process of gathering evidence that:

1) particular inferences are justified on the basis of the available evidence;

2) such inferences are more justified than plausible rival inferences;

3) the consequences of such inferences are ethically defensible.

Process for Obtaining and Recording Evidence

For some forms of evidence, the process of obtaining is the same as the evidence registration process. If we ask a school for copies of its policy documents in a particular area, all the evidence we obtain comes to us permanently. However, many of the tests you get are ephemeral, and only some of them are recorded.

We could be interviewing someone who feels uncomfortable with the idea of talking on a tape recorder. Here we have to rely on note-taking. Even if we record an interview, there will be no record of the interviewee's changes in posture that may suggest a different interpretation of what is said and what could be done without the visual evidence. The important point here is that it is very rare for all the evidence obtained to be recorded.

During the process of obtaining and recording, and then the evidence is interpreted. Research based on approaches derived from the physical sciences emphasizes text at the expense of context and reader. The same educational experiment is supposed to yield substantially the same results if repeated elsewhere.

Educational Research and Knowledge

In what sense, then, can the results of educational research be regarded as "knowledge"? The traditional definition of knowledge is that it is simply "a true justified belief". In other words, it can be said that we know something if we believe it, if it is true and if we have a justification for our belief. There are at least two difficulties in applying this definition in educational research.

The first is that even within a subject as precisely defined as mathematics or science, it is now recognized that there are serious difficulties in establishing what, exactly, constitutes a justification or an "order" for belief. The second is that these problems are aggravated in the social sciences because the chain of inference might have to be probabilistic. This instead of being deterministic. In this case, our inference may be justified, but it is not true!

An alternative vision of knowledge, for the basis of perceptual knowledge, offers a partial solution to the problem. The central feature is that knowing something is, in essence, the ability to eliminate other rival possibilities. For example, if a person (let's call her Chris) sees a book at a school, then we'll probably say Chris knows it's a book. However, if we know (but Chris doesn't) that the students at this school are experts in making replicas of books that, to all external appearances, look like books but are solid and cannot be opened. So with a justifiable and true vision of knowledge, we'd say Chris doesn't know it's a book, even if it is.

Goldman's solution

Goldman's solution to this dilemma is that Chris knows that the object he is looking at is a book if he can distinguish it from a possible relevant state of things in which it is not a book. In most cases, the possibility that the book-like object in front of Chris may not be a book is not a relevant state of things, so we'd say Chris knows it's a book.

However, in our particular case there is a relevant alternative situation: the book can be a doll or it can be genuine. Since Chris can't tell the difference between these two possibilities, we'd say Chris doesn't know.

Mathematics Education Research Requirements

Therefore, within educational research, we may consider that the task of producing "knowledge" has two requirements. The first is to establish that the inferences made from the tests are justified. This is something most researchers are relatively good at. The second condition, which is more fulfilled in its non-compliance than in its observance, is to establish that the chosen interpretation is more justified than plausible rival interpretations.

That process can never be complete: there are no "out-of-the-box" methods; just an endless process of obtaining evidence that the chosen interpretation is a) supported by available evidence and b) more justified than plausible rival interpretations.

Plausible Rival Hypothesis

This solution to the problem of "knowledge" in education is only partial, because it leaves open what counts as a plausible rival hypothesis. In practice, even in the physical sciences, this is decided by the consensus of a community of researchers. Sometimes what is and is not plausible becomes absolutely explicit, in the form of a theoretical posture. In other words, an investigator might say "because I work from this theoretical basis, I interpret these results as follows, and I do not consider that alternative interpretation to be plausible." Most often, communities of researchers operate within a shared discourse that dismisses some alternative hypotheses, although these tend to be implicit and often unrecognized.

In short, the tests are obtained, recorded in some way and interpreted (not necessarily in that order!). The interpretations are validated by the elimination of plausible rival interpretations of the evidence. The definition of what counts as "plausible" is determined by the discourse within which validation is carried out.

Consultation systems

Different research methods were investigated by Churchman, who considered that all kinds of research can be classified into 5 broad categories. Each of which was labeled with the name of a philosopher (Leibniz, Locke, Kant, Hegel, Singer) who he considered to better exemplify the stance involved in the adoption of the system, and in particular what should be considered as evidence.

More detailed accounts of the systems can be found in the work of Churchman and his colleagues. However, it may be easier to understand the framework when applied to a "real" research question in math education: should students be allowed unrestricted access to calculators when learning math?

Rhetorical Tools

One approach to this problem is to use only rhetorical tools to try to establish the truth of the proposition. In a Leibnizian system, reason and rationality are considered the most important sources of evidence. Although there are times when such methods are appropriate in educational research, it is often much more appropriate to use some form of evidence of the situation under study (usually called empirical data) in research.

Common Use of Data in Mathematics Education Research

The most common use of data in research, both in the physical and social sciences, is through what Churchman calls Locke's research system. In such research, the evidence is mainly derived from observations of the physical world. Empirical data is collected, and then an attempt is made to build a theory that will account for the data. This corresponds to what is sometimes called a "naive inductivist" paradigm in the physical sciences. This is more appropriate for well-structured problems.

Difficulties in Approaches

The main difficulty of Lockean's approach is that, as the observations are considered evidence, it is necessary for all observers to agree on what they have observed. Because what we observe is based on the theories we have, different people will observe different things, even in the same classroom.

For less structured problems, or when different people probably disagree on what exactly the problem is, a Kantian research system is more appropriate.

Differences between Lockean, Kantian and Hegelian systems of inquiry

Lockean's researcher displays the "fundamental" data that all experts agree are accurate and relevant, and then builds a consistent story from them. But the Hegelian blinder, using the same data, tells two stories. One story supports the most prominent politics on the one hand, and the other supports the most promising story on the other side.

However, the most important feature of Churchman's typology is that we can research research systems. We can question the values and ethical assumptions that these research systems embody. This inquiry into research systems is itself a research system. It is named Singerian by Churchman after the philosopher E. A. Singer. This approach involves a constant questioning of the assumptions of research systems.

Singerian Research

In a Singerian investigation, there is no solid foundation. Instead of asking what it is, we wonder what the implications and consequences of the different assumptions are about what is considered to be:

"Take it as" is a self-imposed imperative of the community. Taken in the context of the entire Singerian theory of research and progress, the imperative has the status of an ethical judgment. That is, the community judges that accepting their instruction is to provoke an appropriate tactic or strategy. Acceptance can lead to social actions outside of research, or to new types of research, or whatever.

Part of the community's judgment concerns the adequacy of these actions from an ethical point of view. Hence the linguistic puzzle that worried some empiricalrs is not a puzzle at all. In the Singer-Based Indagador, the investigating system speaks exclusively in the "should", being the "is" only a convenient parler function when you want to block the uncertainty of the discourse.

The important thing about adopting a Singerian perspective is that with such a research system, one can never refrain from the consequences of his research. Educational research is a process of modeling educational processes. Models are never correct or incorrect, simply more or less appropriate for a particular purpose.

Meanings and consequences

The above description has dealt with the production of "educational knowledge" which, while recognizing the role of context in the interpretation of the text, continues to emphasize the production of shared meanings within a community of researchers.

During the 1980s, this concern about the exchange of meanings among readers was called into question in what is sometimes called action research. In action research, what is important is the potential of research to transform practice in individual school or even for the individual teacher. Even if research has different meanings for those in different contexts, this is not a problem, as long as it has a meaning for the teacher.

There is no doubt that action research has enormous transformative potential for people involved in the research. However, many have argued that it cannot be classified as research per se, because research makes no effort to produce meanings that are shared beyond context and immediate readers.

Conclusions

In every investigation there is a tension between the meanings and consequences of research. The question, then, should the researcher communicate this to the teacher? In the traditional research paradigm, the answer would be a resounding "no".

However, in traditional research focused on text and context, unfortunate consequences are often justified and legitimized by the need to share meanings. In action research, weakness to the extent that the meanings of the research results are shared is justified and legitimized by the consequences of the investigation.

Therefore, a complete consideration of the nature of educational research should take into account both the consequences and meanings of research.

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