Introduction to Research K P Mohanan and Tara Mohanan

11 February 2021

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CHAPTER 2

METHODOLOGICAL STRATEGIES

2.1 Looking Back 2.1.1 Research is not a linear process 2.1.2 Question-and-Answer or Problem-and-Solution 2.1.3 Questioning the answer

2.2 Examples of Methodological Strategies 2.2.1 On Curricula in Bachelor's Programs 2.2.2 On Biological Evolution 2.2.3 On Triangles

2.3 Wrapping up

The Research Gym

2.1 Looking Back In Chapter 1, we identified the components of research as in Fig. 2.1 (same as 1.1):

Figure 2.1

You might wonder: “Why doesn't the process of looking for an answer come to an end when we find an answer? What is the difference between an answer and a conclusion?” This will become clearer at a later point in this chapter.

We explored ~ the process of research, which aims to make a contribution to our current academic

knowledge, and ~ the outcome of that process — the nature and structure of academic knowledge.

Before we proceed to ‘methodology’, three important remarks are in order.

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2.1.1 Research is not a linear process Figures like Fig. 2.1 may give the false impression that research is a strictly linear process. This is far from the case. Take the idea of a research question. We need a question to even begin to think about the methodology. But in the course of thinking about the methodology, planning it, and implementing it, the question often keeps evolving. By the time we present our conclusions to the academic community, it may well be that the question we have actually pursued is significantly different from the one we started out with. So, let’s modify Fig. 2.1 as Fig. 2.2 to reflect this change:

Figure 2.2

Fig. 2.2 has just two arrows back to the formulation of the research question. Actual research may involve many cycles of the process before the project is done.

2.1.2 Question-and-Answer or Problem-and-Solution?

In some domains of research, what we have called a research question and its answer are called a research problem and its solution. They are often used interchangeably. For example, eradication of poverty is a problem that requires a solution. We can equally well formulate it as a research question: How do we eradicate poverty?

There is, however, a difference in their conceptualisation:

~ A research question articulates what we do not know but wish to find out, in the context of what we do know. The answer changes a region of ignorance into knowledge.

~ A research problem articulates an undesirable state of affairs. A solution changes that undesirable state, to make it a desirable one.

In Chapter 1, the research gym posed two questions. The first question is based on unfairness in the grading system. Therefore, it represents a problem. The idea of allowing students to retake exams offers a potential solution. Your task involved critically evaluating that solution.

The second question, on the relation between lightning and clouds, is an acknowledgement of our ignorance. It represents a question. An explanation that accounts for the relation between lightning and clouds is a potential answer to the question. Note that it doesn’t have to be the ‘right’ answer, but simply a possible answer.

With these questions, we mentioned that there is a difference between exam questions and research questions. There is a similar difference between textbook/exam problems and research problems as well. The problems in textbooks, say, in Math or Science, are largely designed to develop rapid calculation skills (e.g., arithmetic problems, solving quadratic equations, calculating the trajectory of a cannon ball, balancing chemical equations). They are expected to be solved at high speed, without the need for thinking. Research problems, on the other hand, require considerable time: they call for imagination, insight, intuition, clarity of thought and rigour of reasoning.

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2.1.3 Questioning the Answer When we present the outcome of our research to members of the academic community, they adopt the role of a judge-and-jury, and ask, “Why should we accept your conclusion as part of academic knowledge?” A defense is the response to this question.

A Master's or PhD scholar has to go through a thesis defense. Authors of manuscripts submitted for publication have to provide proof, evidence and argument, or rational justification, depending on the domain in question. (In mathematics it is called proof or argument; in empirical research it is called evidence, which is used as the basis for an argument.) It is important for research students in any academic domain to master the art and craft of argumentation relevant to their specialisation.

When a journal reviewer evaluates the publishability of a manuscript, or a thesis committee evaluates the worth of a thesis or dissertation, they are engaged in critical thinking. So the researcher presents the defense, while the committee evaluates its merit by scrutinising the conclusion and its justification, and in some cases, the merit of the methodology as well. However, to produce high quality research, it is imperative that the researcher also thinks critically about his/her own work at every step, whether of the question(s), methodology, its implementation, the answers, or the final conclusion.

So we need to add two further pieces to Fig. 2.2, namely, critical thinking and justification:

Figure 2.3

With that picture in mind, we now turn to methodological strategies in research, which we have defined as ways of looking for answers to research questions in different disciplines. (You may find that this conception of methodological strategies is different from what is conventionally found in textbooks or research courses.) We will build on our notion of methodological strategies in this and the next few chapters. Let us look at some examples.

2.2 Examples of Methodological Strategies

2.2.1 On Curricula in Bachelor's Programs Imagine that you are considering the following question to work on for your Master's thesis:

RQ0: Do Curricula in Bachelor's Programs improve the thinking abilities in students?

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You might find an opinion piece on this question in a popular magazine. You may even have your own opinions on this. Now, in everyday conversations, we may answer such a question without worrying about the meanings of words like 'curricula' (plural for ‘curriculum’) and 'thinking abilities'. But to work on the question for a research project, you have to begin by asking yourself the following questions:

What do we mean by curricula? What do we mean by thinking abilities?

A Google search for the term 'curriculum' yields a confusing jumble of answers. A University of Delaware website lists 44 definitions that point to at least ten different concepts associated with that word. (http://www1.udel.edu/educ/whitson/897s05/files/definitions_of_curriculum.htm)

To be able to answer a question, we must first know what the question means. For this, it is not enough to know the general meaning of the words. We need to either find out the intended meaning of crucial words in a question, or carefully state what we mean when we use the word. In this case, you as a researcher need to begin by making a decision on what you want the word ‘curriculum’ to mean. [You may want to take a few minutes to think of your meaning of the word ‘curriculum’.]

Suppose you adopt the following definition: ‘Curriculum’ refers to all the components of an educational program that help students learn what the program aims at. This includes all the courses of the program along with their syllabi (plural for ‘syllabus’), textbooks, classroom activities, and assessments of each course, along with the modes of teaching and learning, as well as the educational policies at the institutional level.

Obviously, one cannot examine all the bachelor's programs in the world to arrive at an answer to RQ0. Also, the question is not answerable within the time available to you. So you might decide to limit the scope of the question in some way, for instance, to the bachelor's programs in India, and perhaps to a specific aspect of the curriculum, for instance, to the assessment component of the program. You would then have to reformulate your question as:

RQ2.1 Do the final examination questions of the Bachelor's Programs in the IITs and the IISERs test the students' thinking abilities?

Now that we have tentatively defined ‘curriculum’ and revised our question, let’s move on to the term 'thinking abilities'. Thinking could refer to several different mental processes. For example, it may refer to considering what to eat for lunch, worrying about one’s loved ones, making calculations, wondering about the meaning of a sentence, imagining a world where one could become invisible, trying to solve a practical problem, looking for excuses for missing a class, thinking about God, figuring out the proof of a theorem, and so on.

To narrow down the meanings that the term ‘thinking’ can take, let’s limit the scope of the question by referring specifically to 'thinking abilities needed for research'. If so, the question becomes:

RQ2.2 Do the final examination questions of the Bachelor's Programs in the IITs and the IISERs test the students' thinking abilities needed for research?

Our question is narrow enough for us to approach its answer now. But to answer it, we need data. The term ‘data’ refers to the pieces of information, qualitative or quantitative, collected through observation.

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What kind of data do we need to answer our question? Examination scores, or student performance apart from exams? What methods must we use to gather such data? Experimental or non-experimental? Quantitative or qualitative? If we use a qualitative methodology, do we use interviews, or participant observation? (https://en.wikipedia.org/wiki/Participant_observation)

To look for an answer to a research question, we need to carefully consider options like those above, and make an informed choice.

Suppose you have gathered data relevant to RQ2.2. What you have is your research finding. The findings may indicate one of two possible answers to RQ2.2: yes or no. Can your finding now be extrapolated to arrive at a conclusion on the research abilities tested in examinations in the Bachelor's programs in India in general? That depends on the finding.

Suppose you have found that final examination questions of Bachelor’s programs in IITs and IISERs do not test the thinking abilities of students. One way to generalise it to the Bachelor's programs in India is through the following argument:

Our findings show that the exams in IITs/IISERs do not test thinking abilities needed for research.

IITs/IISERs offer some of the best Bachelor's programs in India. It is unlikely that the exams in other programs test thinking abilities needed for

research. Hence, it is reasonable to generalise our findings to Bachelor's Programs in India. Conclusion: Examination questions of the Bachelor's Programs in India do not test

thinking abilities needed for research.

Now we need to decide if this argument is sound, or legitimate. Examining the legitimacy of such an argument is called interpretation in research. For instance, how legitimate is the assumption that IITs and IISERs offer the best Bachelor's programs in India? What does 'best' mean? Does the fact that these are some of the most coveted or most 'prestigious' programs in India mean that they are among the best? IITs and IISERs do not offer Bachelor's degrees in, say, medicine, law, or management. Could it be that programs in these areas do indeed test thinking abilities needed for research? What criteria did the research use to distinguish questions that test thinking abilities and those that don’t? How legitimate are these criteria? Unless such questions have been carefully considered, there would be a gap between our research finding and the legitimacy of the argument we present.

Suppose your finding is ‘yes”. Can we generalise the answer to arrive at a conclusion on Bachelor's programs in India? No, because the finding does not mean that lesser programs ask questions that test thinking abilities. In other words, the programs in the IITs and IISERs are not representative of the programs in India.

For now, we hope this discussion has given you a sense of the importance of methodological strategies to answer research questions. Let’s move on to an example in a different domain.

2.2.2 On Biological Evolution Let us turn to another question:

RQ2.3 Is the theory of biological evolution true?

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As in our previous example, this question requires the clarification of the concepts of ‘theory’, ‘evolution’, and ‘biological evolution’. We would also need to clarify what we mean by 'the theory' of biological evolution.

What is a theory?

We will engage with the concept of theory in greater detail in later chapters, but until then, here are some starting points for an initial understanding:

A. A theory is made up of a set of statements. Here are some examples of theoretical statements:

“Similar poles/charges repel each other; opposite poles/charges attract each other.” “The earth revolves around the sun.” “No straight line, even when extended indefinitely, will meet itself.” “A species is a population of organisms that can interbreed to produce fertile offspring.”. B. The logical consequences of a set of theoretical statements are called theorems in

mathematics, and predictions in science. For example, take a theory of geometry that has the statement, “No straight line can meet itself.” When combined with other theoretical statements, this statement yields the theorem: “The sum of angles in a triangle is equal to two right angles.”

Similarly, the heliocentric theory of the solar system, which assumes that the earth revolves around the sun, predicts that it will be summer in Australia when it is winter in Finland, and vice versa. We may think of the theoretical statements as premises, and the theorems/predictons as conclusions.

C. When the predictions of a scientific theory agree with (are consistent with) what we observe, we judge the theory to be credible. When they are inconsistent with the observations, that is, if our observations contradict the predictions of the theory, we say that the theory is flawed. Flawed theories must be either modified or abandoned.

D. A theory that makes no testable predictions is not a scientific theory.

We may summarise A-C as in Fig. 2.4:

Premises ! Derivation ! Conclusion Figure 2.4

We will call this the Premises-Derivation-Conclusion (PDC) structure of theories. By 'derivation', we mean the steps of reasoning (including calculation, which is also a form of reasoning) that yield the conclusion from the premises. It you have done derivations in, say, algebra or physics problems, you know what the word 'derivation' means.

Given that the theory of biological evolution is a scientific theory, and given that scientific theories must make testable predictions, it follows that a theory of biological evolution must make testable predictions.

What is biological evolution?

There are at least two different ways to answer this question. We could define ‘biological evolution’ based on what we have learnt in our textbooks and proceed to answer RQ2.3. But if we want to be able to make connections across academic disciplines, there is a more fruitful way to approach the question.

The concept of ‘evolution’ appears across various disciplines. Physicists are interested in the evolution of the universe, historians and sociologists are interested in the evolution

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of society and culture, and biologists are interested in the evolution of forms of life. Therefore, evolution is a transdisciplinary concept.

So let’s begin with the broad notion of evolution and then narrow it down to our specific question.

~ What do we mean by evolution, such that evolution of the physical universe we live in, evolution of the solar system, evolution of the earth, evolution of physics, evolution of Carnatic Music, and so on are all examples of evolution?

~ Within this transdisciplinary concept of evolution, what is biological evolution?

The purpose of raising these questions is to highlight the importance of thinking about them, before proceeding to the next step. If you would like to contribute to academic knowledge such that people across academic domains will benefit from your contribution, then this is an important part of your research process.

(If you are interested in pursuing these questions further, we suggest that you take a look at the article, "Conceptual Foundations of Evolutionary Thought," at https://www.ias.ac.in/article/fulltext/jgen/096/03/0401-0412)

It is not practical to explore the transdisciplinary questions further in this chapter, so we will take a shortcut and ask a simpler question. Charles Darwin wrote the book, The Origin of the Species, and is considered the founder of the theory of biological evolution.

What did Charles Darwin mean by biological evolution?

Let us state what we think is Darwin’s central claim (Claim D) in his theory of biological evolution: Claim D: All the existing and extinct species on the earth evolved from a single

ancestor species.

We must note that Darwin himself left open the issue of whether there was a single ancestor species or a very small number of ancestor species. The ‘single ancestor species’ idea, however, seems to be prevalent, especially in textbooks.

When posing RQ2.3 earlier in this chapter, we had hinted at the need to clarify what we mean by 'the' theory of biological evolution. This was because, in principle, it is possible to have multiple theories of something in a subject. But no matter how these theories vary, they all share Claim D. So we might interpret 'the' theory of biological evolution as the central claim that all variants of evolutionary theory share, namely, Claim D.

For example, monkeys and apes are said to have evolved from a common ancestor that lived around 25 million years ago. (https://pandasthumb.org/archives/2014/10/the-family-tree.html)

Darwin’s claim is expressed in a diagrammatic form as the so-called ‘tree of life’, familiar to everyone who has gone through the topic of biological evolution in school. Notice that we have called this part of Darwin’s theory a claim. A claim needs to be supported by a legitimate argument to be a part of academic knowledge.

For now, suppose we assume the idea of a single ancestor species. A tree representation of the idea would be as in Fig. 2.5, where every node, labelled A, B, C and so on, stands for a species, with A being the ultimate ancestor.

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Figure 2.5

In Fig. 2.5, no species (A, B, C, and so on) can have two distinct ancestors. In other words, you cannot have a node connected to two higher level nodes. Suppose we find that there are, indeed, many instances of a species having two distinct parent species. If so, the textbook tree of life will have to be revised to allow for the kinds of possibilities illustrated in Fig. 2.6.

Figure 2.6

In the context of such a revision, should we judge Claim D to be false?

No. Claim D does not say anything about the choice between Fig.2.5 and Fig.2.6. That textbooks present Fig.2.5 without considering the possibility of Fig.2.6 is not a shortcoming of Darwin’s theory. In fact, the concept of hybrid species, well accepted in the current biology community and accepted by Darwin himself, supports the conceptualisation in Fig. 2.6.

To decide whether to accept Claim D, and to look for evidence both for and against any claim of this nature, one strategy is to adopt the steps suggested in questions (1)-(3) below:

1. What are the kinds of phenomena (observations) that would come under the scope of the theory? What does the theory seek to explain or predict?

For example, Darwin’s theory seeks to explain the wide range of organisms we see on our planet — it offers an explanation for the ‘biodiversity’ on Earth. To do this, he begins by observing attributes that different species have in common, and the ways in which they are different.

2. How does the theory seek to explain/predict those similarities and differences?

We have not explored Darwin’s theory in sufficient detail here to answer this question. We need to add further theoretical statements to Claim D to form an argument. Once we have done that, we would have a set of theoretical statements that explain the phenomena and make predictions.

3. How successful is the theory in explaining/predicting what it seeks to explain/predict?

As an example, let us take the anatomical, physiological and behavioural attributes that all insects share (what biologists call 'homologies'), and how these attributes vary across different types of insects.

Here are some shared anatomical attributes of insects:

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They have the same body design, with ~ three major regions: head, thorax, and abdomen, and ~ two legs attached to each segment of the thorax. ~ wings found only on the second or third thorax (in winged insects). and ~ antennae and compound eyes.

How about anatomical differences? ~ Not all insects have wings, and among even those, ~ the patterns on the wings vary in interesting ways.

And here are some other interesting differences: ~ Flight behaviour varies across winged insects. ~ Nutrition: Mosquitoes rely on blood, and butterflies on nectar.

Darwin's evolutionary theory explains shared properties by assuming a common ancestor from which all existing insect species evolved, and explains the differences in terms of mutation. Darwin refers to this idea as “descent (inheritence through ancestry) with modification (mutation)”. Molecular biology provides evidence to support descent with modification at the molecular level: the levels of DNA, proteins, and RNA.

As with the previous example, the discussion of the methodological strategies for the research question on biological evolution illustrates two points:

A. Before we start looking for methodological strategies to answer a research question, we must clearly understand what the question is asking. And for this, we need to understand the concepts that the words in the question denote.

B. The strategies of data gathering in scientific research, and the kinds of data we gather (evidence in support of or against a claim) must be grounded in the understanding of the concepts in the question.

Our earlier example (Section 2.2.1) illustrated the methodological strategies of observational science. The example of biological evolution illustrates the methodological strategies of theoretical science. These two sets of strategies complement one another. [At this point, if your preoccupation so far has been with experimental research, you may be hazy about the the terms theoretical science and experimental science. We will return to issue in a later chapter.]

2.2.3 On Triangles Our third question lies in the domain of mathematics:

RQ2.4 Do straight-angled triangles exist?

To look for an answer to this question, we need to understand two concepts:

What is a straight angle? What is a triangle?

Chances are that most of you think that you learnt answers to these questions in your secondary school. You may have learnt, for instance, that

An acute angle is an angle of less than 90º. A right angle is an angle of 90º. An obtuse angle is an angle of more than 90º but less than 180º. A straight angle is an angle of 180º. A reflex angle is an angle of more than 180º.

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How can you tell whether an angle is, say, 90º? Easy, you might say, take a protractor, and measure it. This answer does not work once we realise that in Euclidean geometry, quantities are not subject to measurement. Why so?

You would remember the secondary school characterisation of a line as an entity that has length but no width.

Can you draw a line with a pencil or pen on a piece of paper? Sure, you might say, and draw something like in Fig. 2.7 or Fig. 2.8:

But the line that you are looking at does have width, however small. The width of the lines in these figures, for instance, is about half a millimetre. The narrowest line that you can draw with a word processing program is probably about 0.1 millimetre. If someone were to draw a line that is 0.001 millimetre, you will not be able to see it. Thus, a line with zero width is invisible. Hence, when two lines meet, you can't measure the angle with a protractor or some other instrument.

In the system of geometry the way Euclid conceptualised it, points, lines, and angles cannot be observed or measured. They are entities constructed by human imagination, and categorised in terms of what we can perceive with our 'mind's eye'.

Imagine line AB. Select any point C on AB such that AC and CB are two line segments, and C is a vertex. Angle ACB is a straight angle, as in Fig. 2.9. Now in your mind, rotate CB such that it lies on top of CA, as in Fig. 2.10. Angle ACB is now zero. Continue to rotate BC such that we are back to the original configuration, and CB has undergone a full rotation (Fig. 2.9). Continue to rotate CB, this time one fourth of a full rotation. Now if we extended AC to ACE (as in Fig. 2.11), angle ACB = angle BCE.

BC is now perpendicular to AE, and angle ACB, which is equal to angle BCE, is a right angle. This doesn’t involve any measurement, or the use of a protractor.

Let us go back to our question now.

Do straight-angled triangles exist?

To answer this question, let us continue to do things with our mind. In Fig. 2.11, we have three straight lines AC, CE and BC. Rotate CB around C such that B lies on top of E. Now extend CE such that B and E coincide, as in Fig. 2.12:

We now have three straight lines AC, CB, and AB. Imagine that point A, point B and point C are all vertices. The question we are asking is:

Given that A, B, and C are three vertices, and AC, CB, and AB are straight lines,

Is ACB a triangle?

Define triangle in such a way that the answer 'yes' is 'correct', and the statement that straight-angled triangles exist is a theorem ( = is true).

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Now define triangle in such a way that the answer 'no' is correct, and the statement that straight-angled triangles do not exist is a theorem (= is true).

The point we are making is that in mathematical inquiry, unlike in scientific inquiry, whether or not a statement is true depends on the definitions we choose, and, as we saw in Chapter 1, also on the axioms we choose. What we have given here is a taste of how the methodology for research in mathematics is different from the methodology for research in science.

2.3 Wrapping up In Chapter 1, we addressed the question, “What is research?” and presented an answer in terms of the components of research (Fig. 2.1). In Section 2.1, we developed this answer in such a way that the concepts of research question and research problem are equivalent, but viewed from slightly different perspectives. We also removed the strict linearity of Fig. 2.1 to shed light on the idea of research involving several cycles (Fig. 2.2), and introduced the role of critical thinking in various stages of the process (Fig. 2.3).

In Section 2.2., we gave three extended examples of research methodology. In Section 2.2.1, we went through an example from observational inquiry in science; the particular methodological strategy there involved choosing a representative sample. We were open to the choice between experimental and non-experimental observation, and between quantitative and qualitative observation. Section 2.2.2 illustrated research in theoretical science, and Section 2.2.3, in mathematics.

There are fundamental differences between the methodological strategies of research in science and in mathematics. One of them has to do with the very nature of knowledge in these two domains. Science seeks to understand the nature of the world we happen to live in; mathematics seeks to understand the nature of logically consistent imagined worlds. The worlds of flat vs. curved surfaces, gradient vs. discrete surfaces, and rigid vs. non-rigid surfaces are conceptualised through the researchers' imagination, and constructed through definitions and axioms. Having constructed such worlds, the mathematician asks questions about what is true in each of these worlds.

If the above characterisation of mathematics intrigues you, and you wish to find out more, watch the following videos.

1) Roger Penrose - Is Mathematics Invented or Discovered? https://www.youtube.com/watch?v=ujvS2K06dg4 2) Steven Weinberg - Is Mathematics Invented or Discovered? https://www.youtube.com/watch?v=NpMk9G-ddiM&t=84s 3) Stephen Wolfram - Is Mathematics Invented or Discovered? https://www.youtube.com/watch?v=RlMMeqO7wOI&t=158s

In Section 2.2.3, we found out that if we assume one definition, then the statement that straight-angled triangles cannot exist is true, but if we assume another definition, then the statement that straight-angled traingles do exist is true. This is not a logical contradiction, because mathematical truths are dependent on the axioms and definitions of the particular mathematical world in question: what is true in one world can be false in another world.

This is not the case in science, because science is concerned with the particular world that we live in. And within any given world, logical contradictions are prohibited. Is the

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space in our world a flat one? Newton thought it was, modelling it on Euclid’s geometry of flat spaces. In contrast, Einstein assumed that gravity causes curvature of space, and modelled our world on spherical geometry. Given the evidence and arguments that Einstein provided, we now believe that Newton was wrong.

In spite of the difference between mathematics and science, research in these two domains have significant shared attributes. We will have occasion to see the similarities as we proceed. We will also have occasion to see that research in the humanities shares these attributes, and that their methodological strategies sometimes resemble the axiom-and-definition-based strategies in mathematics, and at other times, the evidence-based strategies in science.

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THE RESEARCH GYM [GYM 2]

In the Research Gym in Chapter 1, Question 1 was about the feasibility and ethical desirability of a policy. Your task was to give an argument for (reasons in support of) your answer. This was in preparation for justification in Fig. 2.3 in this chapter.

In Question 2, we pointed to what looked like a logical contradiction in the textbook account of lightning in terms of positively and negatively charged clouds. Given the principle that similar charges repel each other, it follows that water molecules with similar charges cannot come together to form a cloud. And when molecules with opposite charges come together through attraction, the charges would be neutralised. So, positively charged clouds and negatively charged clouds cannot exist. Yet textbooks tell us that lightning is caused by electric current from charged clouds. How do we come up with an account of lightning that does not involve this logical contradiction?

Activities involving arguments and logical contradictions are common in research. In Chapter 1, Section 1.2.2, we gave an argument against the existing way of classifying academic knowledge, by showing a series of logical contradictions in the system. Such examples in the chapters, together with the Gym exercises, are designed to develop your research muscles. With that in mind, answer one of the questions given below.

QUESTION 1 Have you heard of mathophobia? This is what Wikipedia says:

“Mathematical anxiety, also known as math phobia, is anxiety about one's ability to do mathematics. It is a phenomenon that is often considered when examining students' problems in mathematics.

Mark H. Ashcraft defines math anxiety as “a feeling of tension, apprehension, or fear that interferes with math performance” ...The academic study of math anxiety originates as early as the 1950s, where Mary Fides Gough introduced the term mathemaphobia to describe the phobia-like feelings of many towards mathematics...

Ashcraft suggests that highly anxious math students will avoid situations in which they have to perform mathematical calculations. Unfortunately, math avoidance results in less competency, exposure and math practice, leaving students more anxious and mathematically unprepared to achieve.”

https://en.wikipedia.org/wiki/Mathematical_anxiety

Given that the phenomenon of math anxiety is widespread, one might entertain at least three hypotheses about it:

Hypothesis I: Learners suffer from math anxiety because of their own and others’ impression that they lack the capacity for mathematical thinking.

[How legitimate is that impression?]

Hypothesis II: Math anxiety is the result of poor quality teaching of mathematics. [If a math teacher humiliates learners when they fail to give the 'correct’ answer, they associate math with boredom, pain, and fear.]

Hypothesis III: The anxiety is not about real mathematics which can be joyful, but about what is taught as math in textbooks and classrooms.

Consider the following possibilities: Of the three hypotheses, a. One (it could be any) is true, and the other two are false. b. Two are true, and one is false. c. All of them are true. d. None of them is true.

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Write a research plan to find out which of these four possibilities is the case. Here is what you need to do for this. Think carefully about the question, then describe in a page or so what kind of data would help you to engage with the question, and how you would gather the data. [Note: It is okay to take two or three days to write that page!]

QUESTION 2

Let us go back to the lightning problem in Gym 1, where we assumed the principle that two similarly charged entities repel each other, and that this principle predicts that charged clouds cannot exist. Now, we may have been wrong in this assumption. The charged molecules in a cloud are not in contact: they are at a distance from one another. The principle does not predict that molecules with similar charges cannot exist at a distance.

But the question remains: How do clouds with similar charges form?

Do a google search for information on the formation of charged clouds, and see if you can either find an answer in one of the websites, or come up with an answer using the information you find. (The Wikipedia entries on lightning and on static electricity may turn out to be useful.)

Once you have (or think you have) a solution, subject it to critical thinking. (Critical thinking is an important component of research (Fig. 2.3). We used it in our argument against the existing classification of knowledge, as well as in Gym 1, Question 2.)