# 10 The Thought Process of Constructing a Researchable Question: Think Like a Mathematician

Hany Fahmy

###### Hany Fahmy is an Associate Professor in the School of Business at Royal Roads University, Victoria BC Canada (hany.fahmy@royalroads.ca).

# Rationale

Quantitative analysis, business analytics, quantitative research methods, and applied business methods, among others, are examples of required business statistics courses for non-science majors. In my experience, business and other social science domains present a typical quantitative course sequence covering the following subjects, more or less in this order: descriptive statistics, probability, sampling distributions, estimation, hypothesis testing, regression analysis, experimental design, survey sampling, quality control, time series analysis, (analysis of variance) ANOVA and other topics (Rose et al., 1988, p. 277). The first five topics in the list comprise the necessary *foundations *for the remaining *applied *subjects on the list. Therefore, these five subjects are usually covered in the first course of the business statistics sequence and a combination of the remaining subjects is covered in the second course.

I have long been interested in how I can demonstrate the significance of quantitative analysis in the critical thought process pertaining to formulating the research question and defining the research hypotheses in the early stage of carrying out any applied study. This process is important to graduate students, especially for non-science majors who were not exposed to this type of thinking in their previous degrees.

# Overview

To carry out a traditional applied research study, the researcher decides on a topic, develops a researchable question, carries out a literature review, develops a research plan to test the validity of this hypothesis, documents the research findings, and disseminates knowledge. The importance of statistics and quantitative analysis can be easily demonstrated in each of the previously mentioned stages. However, I’m interested here in demonstrating the importance of quantitative analysis in the early stage where the development of research questions takes place. This stage, in my own opinion and in my graduate students’ view, is the most challenging and most intimidating stage of carrying out an applied research study. The struggle in carrying out a good quantitative analysis lies in transforming a verbal research statement into testable and measurable hypotheses; a job that requires the thought process of a mathematician and the skill of a statistician. It is the mathematician’s thought process I want to present.

The process of developing a quantitative researchable question is a sequence of two steps: a thought process and a parameterization exercise. The thought process is founded on mathematical reasoning. The outcome of this initial step is usually a set of key study variables and a set of study hypotheses to be tested empirically. The second step is about the execution of the outcome of the thought process. Within the quantitative domain, this step becomes a parameterization exercise where statistical and mathematical techniques are used to parameterize the behavioral relations between the study variables according to the hypotheses of the study.

## How mathematicians think

When I teach the thought process of parameterizing an applied research question, I begin with a simple general example that illustrates how mathematicians view the world from a lens which insists on a form of precision not commonly applied by other professions.

## Example 1

Consider the following motivational statement: “To be successful, you should dig deep and rise high.” If you give this advice to a mathematician, s/he will probably ask you to define the words “deep” and “high.” You will then be asked to describe the causality between success as an outcome and digging and rising as actions that lead to this outcome.

To a mathematician, the motivational statement is incomprehensible for it is simply imprecise. An alternative, more rigorous, way of stating the previous statement is: let [latex]d[/latex] and [latex]h[/latex] denote depth and height, respectively, as variables. Let [latex]x=h-d[/latex] be the effort exercised, which is defined, in this example, as the distance between [latex]h[/latex] and [latex]d[/latex]. In other words, effort is thought of as a real number bounded from below by the depth value, i.e., the value taken by [latex]d[/latex], and from above by the value taken by [latex]h[/latex]. Let [latex]d_0[/latex] be the initial depth and [latex]h_0[/latex] be the initial height. Then, an equivalent way of stating the above statement is:

To be successful, exercise a magnitude of effort [latex]x=h-d[/latex] such that [latex]d[/latex] is less than (<) the current depth level [latex]d_0[/latex] (dig deep) and is greater than (>) [latex]h_0[/latex]; the current point from which one can rise from (rise high).

A mathematician is likely to comprehend this statement for it is based on concrete measurable variables that describe a defined relation between effort and depth and height.

Example 1 does not only illustrate to our students how mathematicians think, but it also reveals to them the structure of the thought process used when performing quantitative analysis. This structure is founded on defining variables and relations based on elementary mathematical concepts. Example 1 also teaches that defining a measurable variable, i.e., the exercised effort [latex]x[/latex], as a function of other variables, [latex]h[/latex] and [latex]d[/latex], is the first and most important step in conducting any thought process. This definition, i.e., the functional form [latex]x=h-d[/latex], gives a relation between the variables.

Consider the following research questions:

## Example 2

*This study investigates the effect of human capital information on stakeholders’ decisions.*^{[1]}

The research statement in Example 2, which I will denote by [latex]W[/latex], is too general for analysis for it contains terms that need to be defined precisely. In particular, the statement is not clear about the type of human capital information that will be used. Also, it is not clear what type of stakeholders and decisions will be considered. I would ask students how they would go about refining this statement into a more precise and quantifiable version. A refinement of the statement [latex]W[/latex], for instance, could read as follows:

## Example 3

*This study investigates the effect of human capital information in management reports on investors’ decisions.*

Let [latex]W'[/latex] denote the statement in Example 3. It is easy to see that [latex]W'[/latex] is an improvement over [latex]W[/latex] for it contains a better, yet still imprecise, definition of the *source *of human capital information and the *type *of stakeholders. The decision to be studied, however, is still not clear. Thus, more refinement is warranted. We might ask the students to try and list the relevant human capital information pertaining to [latex]W[/latex] that could be studied, e.g., workers’ education, training, skills, health, among others.^{[2]} More formally, let the set {[latex]A_1,A_2,…[/latex]} be the chosen set of human capital attributes pertaining to a particular publicly traded company. If an attribute is measurable, i.e., has a quantifiable scale, it is considered a variable. Otherwise, a proxy can be used for quantification. For instance, workers’ years of schooling, [latex]X_1[/latex], could be used as a proxy for the education attribute, [latex]A_1[/latex]. The number of workshops that workers have attended, [latex]X_2[/latex], could be used as a proxy for the workers’ training attribute, [latex]A_2[/latex], and so on. The outcome of this process is a set of measurable variables or proxies {[latex]X_1,X_2,…[/latex]} that capture the human capital attributes {[latex]A_1,A_2,…[/latex]} of the statement [latex]W[/latex].

In the refinement process of [latex]W[/latex], the student might also be directed to decide on the type of investors, e.g., retail traders, institutional investors, or both. If the decision is to study institutional investors, i.e., fund and portfolio managers, then the decision and nature of the focus of the study ought to be related to the way financial portfolios are constructed; the so-called asset allocation and security selection process associated with the portfolio management process (PMP).

With this degree of focus, the students might see a further clarification of the question under study to:

## Example 4

*This study investigates the hypothesis that human capital adds value to the PMP. In particular, the study focuses on studying the value added of including a set of key human capital attributes, in addition to the fundamental and technical analyses, in the decision-making process in the security selection stage of the PMP.*

The statement [latex]W''[/latex] in Example 4, which is considered a refinement of [latex]W'[/latex], is precise enough to warrant empirical investigation. It is a quantifiable statement since it comprises a set of measurable variables; namely, the set [latex]X=[/latex]{[latex]Y,X_1,X_2,…,X_n[/latex]}, where [latex]Y[/latex] is the rate of return or the risk of the constructed portfolio, and [latex]X_1,X_2,…,X_n[/latex] are the measurable human capital attributes. The researcher can now begin to choose a methodology to measure the value added from including human capital information in the analysis. In other words, the researcher can now decide on how to quantify the relation between [latex]Y[/latex] and [latex]X_1,X_2,…,X_n[/latex].

# Reflection

In general, the mapping from [latex]W[/latex] to [latex]X[/latex] is the most important and most challenging part of defining a researchable question when carrying out an applied study. The thought process often begins with an abstract process [latex]W[/latex] and progresses, through a set of refinements, to narrow down the key measuring attribute(s) of this process into a set [latex]X[/latex] of measurable component(s). The next step is to choose a research approach, normally quantitative but could involve mixed methods, which would then include qualitative approaches, to investigate the research question.

Taking this quantitative approach, a parameterization of the hypothesized relation between the key variables ought to be defined and statistical analysis used to make sense of the data.

Learning mathematics could be for its pure beauty or for its incredible power of analysis. Aside from the motive, however, the subject is crucial for the thought process of any researcher. For that reason, in my teaching, I strive to encourage my students to think like mathematicians regardless of the type of research (e.g., qualitative, quantitative, or mixed methods) they intend to implement.

Learning the effective use of mathematics in constructing a researchable question is equivalent to the task of assembling a chain that comprises of several dependent links in a sequence. If one link is missing, it will be impossible to pull on the chain. This missing link is what renders the thought process of thinking like a mathematician challenging for some students. However, with some effort and instructor’s support, overcoming this challenge is possible.

### References

Rose, E. L., Machak, J. A., & Spivey, W. A. (1988). A survey of the teaching of statistics in M.B.A programs. *Journal of Business and Economic Statistics*, 6(2). 273-282.

- I would like to thank Arianna Bhagwansingh, a candidate in the DBA program in the School of Business at Royal Roads University, for providing this research statement from her DBA proposal. At the time of writing this chapter, Arianna and I had a chat about how to translate this statement into a researchable question. ↵
- The researcher here is basically defining the key variables of the study. A thorough literature review is advised at this stage to pinpoint the key variables that were considered by previous studies and the other potential variables (if any) that the researcher is thinking of entertaining ↵