Quantitative Methods - research paper

I. A Feedback Model of the Research Process

II. Strategies for Statistical Thinking

The purpose of this section is to provide students basic strategies to practice statistical thinking , in addition to fundamental applications.

Teaching statistical thinking and improving performance involves learning how to resolve a number of ambiguities during the statistical inquiry process that are not found in typical homework problems and exams. Inquiry with ill-structured problems requires a number of skills that need to be developed during the course:

a. “Generating a curiosity about the world that identifies “I wonder problems”;

b. Writing a measurable question that provides insight into these problems;

c. Determining relevant valid and accessible data;

d. Planning and carrying out data collection;

e. Checking, cleaning and organizing data;

f. Recognizing the data's limitations;

g. Analyzing and interpreting data;

h. Articulating findings;

i. Seeking explanations; and,

j. Generating further questions (4)”.

This iterative process often requires revision as new understandings develop and unanticipated problems or opportunities arise. The weekly discussion questions provide an opportunity to develop inquiry skills throughout the course. Inquiry is a well-accepted (but not always implemented) process in other subjects, such as science and social studies, but requires development of skills often absent in statistics courses, like the ones listed below:

· “Ability to cope with ambiguity and uncertainty;

· Re-balance between instructor guidance and student independence;

· Recognition of opportunities for learning in unexpected outcomes;

· Flexible and creative thinking;

· Deep understanding of disciplinary content; and

· Tolerance for periods of noise and disorganization (4)”.

This overview discusses the thought processes involved in statistical problem solving in the broad sense from problem formulation to conclusions. It draws on the literature and on the article published by C. J. Wild and M. Pfannkuch, Statistical Thinking in Empirical Enquiry , aimed at uncovering the statistical reasoning processes. The content for this overview has been excerpted from this article and has been modified and adapted to help students develop a framework for statistical thinking throughout the course.

The process from problem specification to outcome is complex and iterative. Not only is the process iterative, but at each stage one often looks back to the previous step and re-evaluates the validity of the decisions made. The process is described in terms of a sequence of steps labeled PPDAC: Problem; Plan; Data; Analysis; and Conclusions, that is useful for statistical thinking (1). The PPDAC approach is shown in the figure below.

Figure I. PPDAC Model: Problem; Plan; Data; Analysis; and Conclusions (1)

This diagram(1) shows that, although the clockwise sequence (1→5) applies as the principal flow, each stage may, and often will, feed back to the previous stage. In addition, it may well be beneficial to examine the process in the reverse direction, starting with Problem definition and then examining expectations as to the format and structure of the Conclusions. This procedure then continues, step-by-step, in a counterclockwise manner (e→a) determining the implications of these expectations for each stage of the process ( The Pennsylvania State University, 2010 ).

The PPDAC model develops information that is gathered by the analysis data, such as detecting and describing patterns, trends, and relations in data. As something relevant is detected in data, new questions arise, causing specific parts to be viewed in more detail.

Applied statistics is part of the information gathering and learning process which is undertaken to inform decisions and actions. Multiple sectors of society increasingly rely on data for decision making, therefore, statistics has become an integral part of the emerging information era that is used to expand the body of knowledge in many fields. As shown in Figure III (3), learning is much more than collecting information, it involves synthesizing the new ideas and information with existing ideas and information into an improved understanding.

Figure II. Triggers for stimulating descriptive, inferential, and contextual thoughts (Pfannkuch, M., 2010)

Wild and Pfannkuch (1999) paper on statistical thinking describes a four dimensional framework for statistical thinking and inquiry, which is shown in Figure IV (2). It includes an investigative cycle , an interrogative cycle , types of thinking and dispositions . The authors characterize these processes through models that can be used as a basis for thinking tools for the enhancement of problem-solving. A brief description for each of these models is presented in the subsequent paragraphs.

Figure IV. A 4-Dimensional Framework for Statistical Thinking

(C. J. Wild and M. Pfannkuch, 1999)

1. Dimension One: The Investigative Cycle

The first dimension is illustrated in Figure (a) Dimension 1(2). It concerns the way one acts and what one thinks about during the course of a statistical investigation. Certain learning goals must be met to arrive at the desired level of understanding. A PPDAC investigative cycle is set off to achieve each learning goal. Knowledge gained and needs identified within these cycles may initiate further investigative cycles. The conclusions from the investigations feed into an expanded context-knowledge base which can then inform any actions (C. J. Wild, 1999).

(C. J. Wild and M. Pfannkuch, 1999)

2. Dimension Two: Types of Thinking

A number of general and fundamental types of thinking are shown in Figure (b) Dimension 2(2). The four dimensional framework seeks to organize some of the elements of statistical thinking during data-based enquiry. The thinker operates in all four dimensions at once. For example the thinker could be categorized as currently being in the planning stage of the Investigative Cycle (Dimension I), dealing with some aspect of variation in Dimension 2 (Types of Thinking) by criticizing a tentative plan in Dimension 3 (Interrogative Cycle) driven by skepticism in Dimension 4 (Dispositions). Who is doing this thinking? Anyone involved in enquiry, either individually or as a member of a team. While this approach is not peculiar to statisticians, the quality of the thinking can be improved by gaining more statistical knowledge (C. J. Wild, 1999).

3. Dimension Three: The Interrogative Cycle

The Interrogative Cycle illustrated in Figure (c) Dimension 3(2), is a generic thinking process in constant use in statistical problem solving. It appears that the thinker is always in one of the interrogative states while problem solving. The cycle applies at macro levels, but also at very detailed levels of thinking because the interrogative cycle is recursive. Sub-cycles are initiated within major cycles, e.g. the "checking" step of any cycle can initiate a full interrogative sub-cycle. The ordered depiction on a wheel is an idealization of what perhaps should happen. In reality steps are often missed. We discuss the Interrogative Cycle as we observed it, being applied to statistical enquiry and statistical critique. The "thinker" is anyone involved in these activities (C. J. Wild, 1999).

(C. J. Wild and M. Pfannkuch, 1999)

4. Dimension Four: Dispositions

Dispositions are the personal qualities that affect, or even initiate, entry into a thinking mode; they are summarized in Figure (d) Dimension 4 (2). While these elements are generic, they are discussed in the context of statistical problem solving.

(C. J. Wild and M. Pfannkuch, 1999)

· Curiosity and Awareness - Discoveries are triggered by someone noticing something and reacting to internal questions like "Why?', or "How did that happen?", or "Is this something that happens more generally?', or "How can I exploit this?" Being observant (aware) and curious are the well-springs of the question generation process that all innovative learning results from. Wild (1994) formed the slogan "Questions are more important than answers" to emphasize this point (C. J. Wild, 1999).

· Engagement - It occurs when you become intensely interested in a problem or area; a heightened sensitivity and awareness develops towards information on the peripheries of the experience that might be related to the problem. People are most observant in those areas that they find most interesting. Engagement intensifies each of the "dispositional" elements curiosity, awareness, imagination and perseverance (C. J. Wild, 1999).

How do we become engaged? Spontaneous interest is innate; background knowledge helps - it is hard to be interested in something one knows nothing about. Being paid to do a job helps, as does the problem being important to people we care about. This may be our main difficulty in getting statistics students to think. They simply do not find the problems they are asked to think about interesting enough to be really engaged by them. We observed the effects on performance of engagement with some tasks and not others in the statistics students (C. J. Wild, 1999).

· Imagination - It is hard to overemphasize the importance of imagination to statistical thinking. The formation of mental models that grasp the essential dynamics of a problem is a deeply imaginative process, as is viewing a situation from different perspectives, and generating possible explanations or confounding explanations for phenomena and features of data (C. J. Wild, 1999).

· Skepticism: By skepticism, we mean a tendency to be constantly on the lookout for logical and factual flaws when receiving new ideas and information. It is a quality statisticians both possess and value. Some writers refer to this as "adopting a critical attitude" (C. J. Wild, 1999).

· Being logical - The ability to detect when one idea follows from another and when it does not and, to construct a logical argument is clearly important to all thinking. Synthesis of new information with existing knowledge is largely a matter of seeing implications. Logical reasoning is the only sure way to arrive at valid conclusions. To be useful, skepticism must be supported by ability to reason from assumptions or information to implications that can be checked against data (C. J. Wild, 1999).

A propensity to seek deeper meaning means not simply taking things at face value and being prepared to dig a little deeper. Of the other "dispositions", openness helps us to register and consider new ideas and information that conflict with our own assumptions and perseverance is self-evident (C. J. Wild, 1999).

Can "dispositions" be taught? - A person's "dispositions" are typically problem dependent - they change according to the degree to which the person is engaged by the problem. While some people are skeptical and others are credulous, it seems that credulousness in a particular area is a result of ignorance. That is, as you gain experience and see ways in which certain types of information can be unsoundly based and turn out to be false, you become more skeptical. What we want from skepticism is a prompting to raise certain types question concerning the reliability of information, which can be taught (C. J. Wild, 1999).

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