Logic and the Development of Scientific Competencies in First-Year General Education

Mandana Sobhanzadeh^{1},
Karim Dharamsi^{2},
Nicholas Strzalkowski^{1},
Peter Zizler^{3},
Eric Roettger^{1}

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1. Introduction

Logic is the formal study of the principles of correct reasoning. Year of research supports a long-held intuition that logic and critical thinking can improve student understanding of the concepts presented across a variety of disciplinary areas (Durand-Guerrier, 2003; Fitch, 2012; Hoyles & Küchemann, 2002; Jenicek & Hitchcock, 2004; Morou & Kalospyros, 2011). Jenicek & Hitchcock (2004) have suggested that:

our entire professional life is a wild world of arguments-meant in the sense of exchanges between people sharing information and giving reasons which form the bases, grounds, and warrants for their claims. Since logic and critical thinking is about rational uses of evidence, a valuable preparation for professional practice would naturally include learning the proper uses of evidence in daily practice and research.

In 2017, MRU’s Department of General Education formed an institution-wide committee to revise GNED 1101, “Scientific and Mathematical Literacy for the Modern World”—a first year course offered to all degree students. The course is offered in multiple sections (>30 of ~40 students each) taught by numerous math and/or science instructors. All instructors cover the same curricular content and a common final exam is delivered across all sections to provide a summative assessment of student learning. An introduction to *logic *is the first unit covered since the guiding presupposition is that subsequent units will exploit the intellectual competencies necessary for the kind of problem solving demanded by science. For instance, after teaching logic, students are introduced to *scientific method*—a procedural framework with which, at minimum, they assume some familiarity. Not without their own controversies, a methodological account is provided for the principle of falsifiability and forms of reasoning, including key distinctions between deductive and inductive judgments. These concepts are seeded as students develop scientific literacy and competencies to evaluate scientific claims.

In subsequent topics such as “Nuclear Energy and Radioactivity”, students learn about seminal experiments leading to significant discoveries. They apply logic to critically assess the experiments and the justificatory inferences required to anchor such discoveries and demarcate between science and pseudoscience.

Here is an example of a class activity in which students use logic to evaluate the arguments about an experiment in physics:

Wolfgang Pauli and Enrico Fermi hypothesized the existence of a third particle in the products of beta decay in 1933. Since the energy of the electron in beta decay has a range of possible values, it means that a third very light particle must also be produced so that it carries the remainder of the available energy. Enrico Fermi coined the word neutrino for the ‘little neutral one’ (Tsokos, 2010). A group of MRU students performed an experiment in which a neutron at rest decayed and released a proton and an electron. They expected both the energy and momentum to stay the same before and after the nuclear decay (conservation of momentum and conservation of energy). They observed that energy and momentum before and after the nuclear decay are not the same and therefore suggested the presence of another particle with appropriate momentum and energy to balance the event (Neutrino).

Use truth tables to evaluate whether the following arguments about this experiment are valid or invalid.

Hypothetical Conditional: If the energy before and after the nuclear decay is not the same and the momentum before and after the nuclear decay is not the same, then there is a new particle involved in this event.

Disjunctive Statement: Energy is not the same before and after the nuclear decay or the momentum before and after the nuclear decay is not the same.

Conclusion: Therefore, a new particle is involved in this nuclear decay.

p: Energy before and after the nuclear decay is the same.

q: Momentum before and after the nuclear decay is the same.

r: There is a new particle involved in this event.

(~p ^ ~q) ® r Premise 1

~p ˅ ~q Premise 2

______

r Conclusion

The argument is not a tautology (true in all cases) and is therefore invalid.

This example highlights the importance of using logical inference in assessing the conceptual design of scientific experiments—and also in establishing the *kind of reasoning* one is exploiting to better understand the relevant relationships between phenomena being studied. Like induction, this experiment involves observations. It does not however establish a general rule. Rather, it attempts to determine through a form of abductive inference a relationship between a cause and an effect.

One of the authors (MS) of this paper used to be a physics laboratory instructor. She noticed that it was common for students’ “discussion and conclusion” sections of lab reports to not be logically valid. For example, one student used a conditional statement instead of a biconditional statement and the conclusion contained the inverse fallacy: if the net force acting on the object is zero then acceleration is zero. Net force is not zero therefore the object experienced an accelerated motion.

p: Net force is zero.

q: Acceleration is zero.

p ® q Premise 1

~p Premise 2

______

~q Conclusion

This argument is invalid (fallacy of inverse).

Further, she found that most students in physics labs were not familiar with logical relations, sometimes confusing “converse” and “inverse” conditionals in their reports.

For instance, a student in the Classical Physics I lab (PHYS1201) in MRU, argued that when an object is stationary, then its acceleration is zero (p → q, where p: object is stationary, and q: acceleration is zero). Then he concluded that if the object is not stationary then the acceleration is not zero (~p → ~q, which is the inverse and is not equivalent to p → q). The lab instructor (MS) showed him an example of an object moving with constant velocity and zero acceleration and explained that p → q and ~p → ~q are not equivalent. He argued that he did not mean that “if the acceleration is zero then the object is stationary” (q → p, which is the converse). The lab instructor (MS) explained that inverse (~p → ~q) and converse (q → p) are equivalent and provided an explanation of the logic of conditional, inverse, converse and contrapositive with examples related to the lab topic to clarify the mistakes that some students had made in their lab reports.

As others have indicated, examples of incorrect and incomplete explanations from scientists occur both in laboratory experiments and in other authentic learning environments (Feldon et al., 2010). Authentic learning environment is a pedagogical approach that situates learning tasks in the context of future use. It allows for the construction of meaning grounded in real-life situations and the learners own personal experience (Herrington, Reeves, & Oliver, 2014).

2. The Course

GNED1101 is a 13-week multi-section course with around 1400 students, the majority in their first or freshman year. The course meets three lecture hours per week. Its primary aim is to enhance a student’s capacity to understand core mathematical concepts and their intersection with science and technology—*in the lived experience of their daily life*. Hence the curriculum (ideally) helps students develop and apply critical, mathematical, and scientific reasoning skills through the examination of issues drawn from the real world and current events. The emphasis of this course is to encourage an understanding of how mathematics and science are connected and to encourage students to use quantitative and scientific methods to think about the things they encounter in the press, through politics, industry, and discussions in the public square.

The topics covered include:

1) Logic

2) Scientific method

3) Evaluating and Assessment of information,

4) Theory-change

5) Examining natural disasters (earthquakes),

6) Number system and calculations

7) Personal finance with focus on exponential growth

8) Conventional fossil fuel vs. nuclear energy

9) Understanding radioactivity and doing exponential decay calculations

10) Statistics

11) Prevalence and spread of infectious disease.

The textbook used in *GNED *1101 is *Thinking Mathematically* (Blitzer & Miller, 2011). There is also an online book written by a group of MRU faculty members (*Understanding Our Physical World: How Numeracy and Scientific Thinking Build Knowledge*) used in this course. The grade is based on 10% class activities, 60% assessments (quizzes and assignments) and 30% final exam. The course learning outcomes are provided in Appendix A.

3. Logic in Science

Walter Monroe Fitch (2012) in *The Three Failures of Creationism: Logic, Rhetoric, and Science* shows the usefulness of logic in addressing scientific questions. He explains how logic and the scientific method are employed in studying evolution.

There can be confusion about what counts as science since its methods can, at a general level, be consistently applied to astrology and astrophysics, homeopathy, and vaccine research. For instance, both astrology and astronomy depend on “observation” and “observable phenomena”. The astrologist and the astronomer both seek patterns and aim to identify causal relationships and their consequences by virtue of those patterns. Yet, there is an important difference between the astrologer and the astronomer. Sometimes those differences are accepted, but not explained. Whether it is the principle of falsifiability that helps us confidentially differentiate the astrologer from the astronomer or whether it is a misguided attempt to formulate a strict law where one is absent, the debate about what is and what is not a science seems to ride on both the *kind *of question we are asking and the kind of *evidence* we accept to justify our inferences.

Fitch (2012) believes that one of the challenges towards a student’s understanding of the nature of science is the unfamiliarity with the logic of scientific reasoning. He therefore offers a study of basic deductive and inductive logic, including an explanation of the fallacies and rhetorical devices that creationists frequently employ. For example, he explains the fallacy of equivocation using a syllogism based on creationists’ persistent misunderstanding of the term theory. Equivocation results from multiple uses and shifts in meaning of a particular word or expression in a single argument. Ambiguity results when a phrase or term with two or more distinct meanings is used to justify a conclusion. Since equivocation is not a grammatical error, it is often thought to be intentionally misleading, having an interlocutor come to an erroneous conclusion. The syllogism has a long and esteemed history in its categorical form. It is a type of reasoning in which a conclusion is drawn (whether validly or not) from two given or assumed premises, each of which shares a term with the conclusion and shares a common or middle term not present in the conclusion (e.g., *all people who arrive late cannot perform. All people who cannot perform are ineligible for scholarship. Therefore, all people who arrive late are ineligible for scholarships*) (Blitzer & Miller, 2011). Through studying examples of correct and incorrect logical reasoning students can more fully understand the nature of science. Most standard logic textbooks typically illustrate this fallacy and syllogism using trivial examples. Fitch (2012) demonstrates the importance of this fallacy by showing the two very different arguments—one invalid and one valid—that result, respectively, from defining theory as “only a guess,” as creationists do, and defining it scientifically as “a well-supported explanation of many observations” (pp. 10-12):

Premise 1: Evolution is a theory (a = b).

Premise 2: A theory is only a guess (b = c).

Conclusion: Therefore, the theory of evolution is only a guess (c = a).

This argument presents students with an interesting challenge. The syllogism is valid. Validity is only a formal criterion asking whether the form of the argument is syllogistic; it seems to retain transitivity. However, this argument is not sound. While it follows the formal structure of a syllogism, premises 1 and 2 equivocate on the meaning of ‘theory’ and come, therefore, to a misleading conclusion.

Premise 1: Evolution is a theory (a = b).

Premise 2: A theory is only a guess (b* = c).

Premise 3: Therefore, the theory of evolution is only a guess (? = a).

Validity and soundness present a subtle difference in the kind of categorical syllogism used by Fitch. And students may challenge the truth of any premise but fail to grasp that validity is necessary but not a sufficient condition for truth. Note that in the second example *b* and *b** are not synonymous even though they both refer to “theory”.

A sound argument that does not equivocate on “evolution”—and does establish a transitive relation between its terms is:

Premise 1: Evolution is a theory (a = b).

Premise 2: A theory is a well-supported explanation of many observations (b = c).

Conclusion: Therefore, evolution is a well-supported explanation of many observations (c = a).

Here *b *in premise 1 retains its meaning in premise 2. This form of the argument is both valid and sound since it follows the form of the syllogism and its premises are true. Of course, *truth* here still *belongs* to the syllogism. Even if we establish that the conclusion is true, we might be challenged to answer, *true of what?* How do we know, even if we avoid equivocation on ‘theory’ that the truth of the argument is not merely analytic—*formal*?

In GNED1101, the “Scientific Method” is discussed after the unit on “Logic”. We use similar examples to explain the definition of theory and point out the importance of logic in scientific reasoning. Our aim is to draw a line between the deductive form of reasoning used in Fitch’s example and the force of inductive reasoning, core to the justificatory power of scientific method.

Students tend to memorize and mimic language, because they believe that linguistic competency is sufficient to demonstrate knowledge acquisition and comprehension (Arons & Miner, 1990; Eger, 1993; Packer, 2010; Sobhanzadeh, 2015). Of course, students have the right intuitions even if their execution is misguided. They might be assuming as language users and students attempting to grasp complex ideas that their statements are true under certain conditions and that mimicking the form of true statements is tantamount to mimicking the comprehension of when conditions apply to render such statements true. In other words, one reason students memorize the scientific terms and definitions without thinking about their meanings and connections are that they are not familiar with the language of science and the application of logic in science (Arons & Miner, 1990; Eger, 1992; Christiansen & Kirby, 2003). However, they are familiar with how statements about the world are true under certain conditions—given their familiarity with their own language and its meanings. Scientific statements mimic natural language and so also mimic an assumed ready-made relationship between statement and fact, meaning and world. While this is outside the scope of this current study, we think it is important to signal the ingrained realist assumption the guides much of scientific inquiry, namely, that the world is ‘real’, and our true statements align *accurately* with the facts of the world. Since ‘truth’ is to logic what beauty is to art, it is not surprising that scientific method and reasoning covets truth over mere possibility (Fitch, 2012).

While covering “logic” in GNED1101, one might ask students to discuss the following question with an aim to understand the application of contrapositive in science and the difference between conditional and biconditional statements.

Newton’s first law tells us that an object in motion subject to no net force will continue to move in a straight line forever. Similarly, an object at rest subject to no net force will remain at rest forever.

This means that if *F*_{net} = 0, then *a* = 0 (uniform motion).

a) Write the following statement out using logic notation:

*If the motion is non-uniform then there must be a net force acting on the object. *

b) Explain in words what the following statement means:

$\sum F}=0\leftrightarrow \text{d}v/\text{d}t=0$

Part a. shows the importance of contrapositive arguments in science.

p: There is an object in motion or at rest subject to no forces.

q: acceleration is zero (uniform motion).

p ® q If *F*_{net} = 0, then *a* = 0 (uniform motion)

~q Motion is non-uniform (acceleration is not zero).

____

~p Net force is not zero.

Part b. shows that Newton’s first law is a biconditional statement: *motion is non-uniform if and only if there is a net force acting on the object. *

4. Student Interviews

We conducted semi-structured student interviews to study the importance of group size in completing class activities in GNED 1101. A total of six first year students were interviewed from the fall 2019, winter 2020 and fall 2021 semesters. Two class activities were conducted from the *logic unit*. Here we share the parts of the interviews about what students thought about the importance of logic to their study of scientific concepts. The interview questions relevant to this paper are found in Appendix B. The interviews helped us address a number of questions about the addition of logic to the GNED1101 curriculum. They helped us to gather information about how the *logic* unit is viewed by this sample of students. Most importantly, they also allow us to see if students understand the importance of logic in science. Any hermeneutic analysis attempts to *interpret* meanings and their relations. Interviewing is an important source of the kind of data we require to establish an assessment of student understanding. Interviews allow us access what students are thinking—“what is in and on someone else’s mind” (Patton, 1982). Interviews permit a lot of detail to be collected that would not normally be easily obtained by other research designs. In this research work, we followed the coding strategies presented by grounded theory approach to analyze the interview transcripts. Coding in grounded theory involves the twin practices of abstraction and generalization. Abstraction practice involves separating a whole into elements that are distinct from one another. These distinct elements shape their original context. Generalizing practice involves finding what is common or repeated among these elements (Charmaz & Belgrave, 2007; Corbin & Strauss, 2008; Packer, 2010). After coding the entire text, we made a list of all code words and then grouped similar codes and found the redundant codes to reduce a list of codes to a smaller, more manageable number. We used the NVivo software to explore the trustworthiness of the emerged codes and conceptual categories.

The first interviewee was a student who intended to study international business at MRU. We refer to him as, *Ahmed*. We refer to the second and third students, who were chemistry students as *Bill* and *Catherine,* respectively. The fourth student wished to enter the biology major beginning in his second-year and we call him, *Dan*. We refer to the fifth student as Bob (he planned to get a Bachelor of Arts, criminal justice degree), and the sixth student as Zara (she was studying nursing).

The points mentioned by all interviewees explaining their perspectives on *logic* in GNED1101 can be classified into two broad categories: first, the influence of logic on learning outcomes listed below, and second the kind of reflective and analytical capacities to reason logic seems to enhance, even in relation to everyday life events:

1) Students demonstrate better use of scientific concepts on written assignments. Logic helps students in writing scientific texts such as lab reports.

2) Logic helps students make better understand scientific concepts.

3) Logic improves problem solving skills.

All interviewees acknowledged that logic not only has helped them in writing lab reports in science but also in understanding the materials presented in science textbooks. The language of science is a language that scientists use to talk about the natural world (Eger, 1993). Since meanings are normally constitutive of their uses, for the uninitiated student (or member of the public) scientific terms can be unmoored abstractions. Memorizing concepts, detached from their particular uses in the understanding of nature, presents significant pedagogical challenges for students and faculty. Of course, lab and field experiences can ameliorate student discomfort with scientific concepts; their uses would make sense and students would presumably apply concepts under appropriate correctness conditions. However, “in-world” experiences are not enough to establish relations between our practices. Eger (1993) rightly believes that memorizing the terms and definitions without thinking about the meaning and understanding the concepts is the most important problem for science education. This suggests that ‘in-world’ experiences are important—and no doubt they are. Problems of meaning arise because students learn concepts in the vacuum of textbooks and classrooms. However, meanings also have relations. While our practices can establish the conditions under which the concept is correctly used, such practices in themselves cannot establish either the validity or soundness of our reasoning. Learning logic can help address a conceptual gap between concept use and consistency in science.

The following points mentioned by interviewees are related to the reasoning influences of logic on everyday life events:

1) Logic integrates awareness and encourages one to be more critical about the world events around.

2) It helps students make better arguments in different situations of everyday life.

3) It improves critical thinking, comparison, reasoning, and explaining about various world events around.

Notice that these three capacities assume a relationship between logical inference and scientific content. Logic is silent on what kind of content a student reasons about. Rather, logic provides a formal guide to assess our reasoning. All interviewees explained that learning logic in GNED1101 helped them to be more critical about what they see and hear in everyday life. By this we assume that the capacities to better evaluate one’s inferences helps render consistency to one’s judgements about the trials of everyday life. Given this, the last two points are highly related to each other. Logic improves critical thinking and reasoning and as a result helps students make better arguments in different situations from friendly conversations to formal debates and writing articles and essays.

Some statements made by Interviewees about the influence of logic on learning skills (problem solving, writing and understanding scientific texts) and everyday life events are presented in Table 1. Eysink et al., (2011) state that reasoning occurs in all sciences and in all possible contexts. This is an unsurprising claim since as language users living out our lives in linguistic communities, we are familiar with having to justify our beliefs and provide reasons for why we think something is the case. Logic can render some consistency to our reasoning in many of these everyday situations. Of course, formalized rules about reasoning are silent on specific content; the same rules apply in every situation under which appropriate conditions apply for the relevant rule. This makes logic abstract and general and, in some cases, formalizable. Introducing students to formal logic, the first order predicate calculus for example, can help demonstrate the structural clarity logic can provide to natural language. In everyday life, people can develop naïve notions about logical reasoning. Of course, knowing the rules does not in itself make you a better reasoner and not all good reasoning involves explicit understanding of logic. We have been in the academy long enough to not assume this. In the context of scientific reasoning, when our ideas about the world are vetted by way of experiment and explicit methodological decisions about research design, logic can aid in consistency, in helping us understand the limitations of our concepts and their meanings. It can also help us identify cases of equivocation and other fallacious forms of reasoning. Perhaps most importantly, if the learners develop certain ideas, it is difficult to change their minds and to convince them they should replace their prejudices or preferences with the new ideas that contradict their pre-existing ideas. The neutrality of sound arguments on our desires and particular interests may be the greatest virtue of logic’s essential place in scientific thinking.

Teachers of logic are often confronted with the problem of how to teach students to solve problems and to translate natural language statements into formal

Table 1. Some statements made by Interviewees about the influence of logic on learning skills and everyday life.

statements. By incrementally increasing the complexity of problems and relating them to both real-world phenomena and our theories, teachers can make learning logic interesting and show students the application of logic in various contexts. As can be seen in Table 1, most interviewees found logic confusing at the beginning of the semester and when they were exposed to the class discussions and the class activities about the application of logic in everyday life, they started realizing the importance of logic in various contexts.

5. Conclusion

In this paper, we discussed the importance of connecting the study of logic to the study of science in the first year of a general education program. The interviews with students show that one of the most important parts of teaching logic is to make connections among logic, its application in everyday life and the application of logic in various disciplines. The interviewees acknowledged the positive impact of studying logic on their learning skills such as problem solving, writing and understanding scientific texts along with everyday life events. It is very important to provide examples related to everyday life while teaching logic unit. Creating worksheets and class activities to discuss the fallacies and validating the arguments related to everyday life, science labs and various disciplines are very helpful in teaching logic. The results of this study show the importance of logic in science laboratories, where students need to prepare a lab report as well as reading science textbooks, where students need to make sense of the scientific arguments. Logic would help students make arguments that are logically valid in the lab reports and make sense of the materials presented in the science textbooks such as understanding the differences between conditional and biconditional scientific statements.

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