Concepts in Curriculum Design

Problem Solving, Decision Making, , and Research K P Mohanan and Tara Mohanan 12th July 2019

A brief reflection on the attempts to improve the quality of student learning in schools and colleges in India reveals a deep rooted problem in the very system of formal education and of educational reform.

THE PROBLEM The Kothari Commission Report in 1964, two National Education Policies in 1968 and 1986, the National Curriculum Framework in 2005 have pointed to the need for our curricula to aim at Higher Order Learning Outcomes (HOLO) such as critical thinking, inquiry, , , innovation, clear and precise communication, team work, and leadership. Yet, these recommendations have not been translated into student learning: the fact that nearly the same recommendations have been made over and over again implies that the quality of student learning in India has not improved since Independence as far as these educational goals are concerned. There is a chance that the recommendations in NEP 2019 too will have the same fate.

A CAUSAL ANALYSIS OF THE PROBLEM Given our educational culture of Test Oriented Learning, students learn only what is needed for doing well in Board Examinations, Entrance Tests, and Aptitude Tests. That means that unless the desired learning outcomes are probed into by the questions in these examinations and tests, - students will not invest time and energy to achieve these outcomes, - teachers/lecturers/professors will not teach in such a way that these outcomes are achieved, and - textbooks will not incorporate them into their exposition and exercises. It is also the case that the concepts that the terms denoting HOLOs have not been fleshed out with sufficient clarity and precision, such that their implications for syllabus design, textbook design, school/college administration, classroom teaching, and design of examination questions have not been worked out. If we don’t clarify the concepts and derive their logical consequences, it is natural that syllabus designers, textbook writers, class room teachers, school/college administrators, and exam paper setters would continue with their habituated practices. This is exactly what has been happening.

A RECOMMENDED SOLUTION The above problem can be solved to a certain extent if the UGC (a) creates a set of MOOCS to nurture HOLOs like problem solving, critical thinking, inquiry, clear and precise communication, creativity and innovation, team work, and so on, (b) ensures that the final examinations, entrance tests, and aptitude tests probe into the HOLOs, , and (c) persuades employers to use, for candidate selection, their performance specifically in the examinations for advanced level MOOCs for the HOLOs.

To implement the above solution effectively and efficiently, we need to clarify the concepts denoted by the terms for each of the HOLOs. The note that follows, on the concept of problem solving, is an example to illustrate this enterprise.

1 Introduction 2 What is Problem Solving 3 What is a Problem 4 What is Inquiry and What is Research 5 Problem Solving as Mechanical Application 6 Problem Solving as Thoughtful Innovative Application 7 Problem Solving as Inquiry 8 Problem Solving in Knowledge Application 9 A Recommendation

1 Introduction In documents on educational policies and education reform, the terms cognitive skills, metacognitive skills, non-cognitive skills, higher order thinking skills, and so on often come up as what educational programs ought to aim at. And cognitive skills generally include skills of research, inquiry, decision- making, and problem-solving. Article 46 of UNESCO’s Education 2030 Framework, for instance, says: “A narrow focus on work-specific skills reduces graduates’ abilities to adapt to the fast-changing demands of the labour market. Therefore, beyond mastering work-specific skills, emphasis must be placed on developing high-level cognitive and non-cognitive/transferable skills, such as problem-solving, critical thinking, creativity, teamwork, communication skills and conflict resolution, which can be used across a range of occupational fields. Moreover, learners should be provided with opportunities to update their skills continuously through lifelong learning.” (p. 43) Similar are expressed in the National Education Policy 2019 “Students must develop not only cognitive skills – both ‘foundational skills’ of literacy and numeracy and ‘higher-order’ cognitive skills such as critical thinking and problem solving skills - but also social and emotional skills, also referred to as ‘soft skills’, including cultural awareness and empathy, perseverance and grit, teamwork and leadership, among others.” (p 25) “…More explicitly, the outcomes here include, among other things, increased critical thinking abilities, higher order thinking and deeper learning, mastery of content, problem solving, team work and communication skills besides general engagement and enjoyment of learning.” (pp 29-30) A brief internet search for ‘cognitive skills’ and ‘higher order thinking skills’ in education reveal similar specifications of educational goals. But what do we mean by ‘inquiry’? What do we mean by ‘problem solving skills’ and ‘decision making skills’? How is ‘research’ connected to ‘inquiry’ on the one hand, and ‘problem solving’ on the other? What follows is an attempt to address these questions, such that education policy makers, designers of syllabi, designers of learning-teaching resources (e.g., textbooks, MOOCs, lesson plans, ..), and learning facilitators (teachers, professors), as well as students and their parents have a clearer understanding of these closely related concepts.

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2 What is Problem Solving? If we use the term problem finding to denote the activity of identifying and formulating problems, problem solving can be defined as the activity that leads to solutions to problems. If so, problem solving involves identifying and formulating the problem, finding solutions, and choosing the best one. And in the case of those that require action, it also involves implementing the best solution such that the problem is removed, or at least, minimized. The first step calls for thinking, while the second involves action. This write up is about the first.

3 What is a Problem? Problem solving viewed this way leads us to ask, “What is a problem?” and “What is a solution?” We might say that a problem is a gap between a desirable state and an existing state, and a solution is an action or practice that results in an alignment of the two states. Let us take an example. Suppose a tap in our kitchen is leaking. A leaking tap is undesirable. The desirable state is that of a tap that does not leak. So the existing state is a problem. We call a plumber, who identifies the cause of the problem as a worn out washer. She replaces the washer. The tap stops leaking. The plumber has solved the problem by engaging in the action of replacing the washer. Let us take another example. Zeno, a primary school student, has been doing extremely well in geometry, but poorly in arithmetic. Doing well in geometry is not a problem, because it is desirable. But doing badly in arithmetic is a problem, because it is undesirable. The desirable state is that of doing well in both. The teacher talks to Zeno, and finds that he doesn’t understand the concept of multiplication. She gives him a number of examples to help him understand the concept, and gets him to do a few multiplication tasks. The result is that he starts doing well in arithmetic as well. The teacher has solved the problem. In a country called Calanka, there has been a rise in the number of suicides among the youth. Suicides are undesirable, and hence the situation in Calanka is a problem. Its government appoints a committee to investigate the problem. The committee concludes that the primary cause of youth suicides is the rise in anxiety among them, caused by the newly introduced multiple-choice-question-based examinations, combined with the highly competitive culture and parental pressure among the middle class. The committee recommends that (i) schools set up meditation practices in emotion regulation to counter anxiety and stress, and (ii) the nature of the exams be changed such that students are not ranked relative to one another. These recommendations are implemented in Calanka. The suicide rates come down significantly. The leaking tap is a problem in the household context. A students not doing well is a problem in the school context. And the rising suicide rates is a problem in the society/country context. These are all familiar pragmatic problems. There are also problems in the world of ideas, far removed from pragmatic considerations. Examples include research problems as such as the Konigsberg Problem in Mathematics (https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg) that resulted in the birth of graph theory; the mind-body problem in neuropsychology (https://en.wikipedia.org/wiki/Mind%E2%80%93body_problem); and the unsolved problems in economics (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_economics).

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4 What is Inquiry and What is Research Inquiry is the investigation of a question, relying on our own experience, observation, thinking, reasoning, and judgment, to look for an answer and arrive at a conclusion. We inquire because of our desire to find out something we don’t know, or don’t understand. The process of inquiry involves several closely connected parts. It often starts with an idea triggered by curiosity or reflection, and crystallizes into a question during the process. Rational inquiry is a specific form of inquiry. By ‘rational’, we mean ‘in accordance with reason.’ Collective rational inquiry requires us to: • identify and formulate the question to investigate; • think through appropriate ways to look for answers, and implement them; • arrive at conclusions, based on the answers; • critically evaluate the conclusions, our own as well as other people’s; and • justify the conclusions to the satisfaction of the inquiry community. Research is collective rational inquiry that aims to make a contribution to Academic Knowledge. By academic knowledge, we mean bodies of knowledge that come under categories like mathematics, physical sciences, biological sciences, human sciences, the humanities, medicine, engineering, and technology. Academic knowledge is generated and evaluated by researchers, and transmitted to learners through formal education in schools, colleges, universities and other educational institutions.

5 Problem Solving as Mechanical Application The kinds of questions that students are typically expected to answer in a classroom or examination do not call for inquiry, as they can be answered either by remembering pre-prepared answers, or by applying a mechanical procedure without having to engage in thinking. Consider the examples in (1): 1) a. Zeno bought two kilos of apples and five kilos of onions. The apples cost 120 rupee a kilo, and the onions cost 50 rupees a kilo. Zeno gave a five hundred rupee note to the shopkeeper. How much should he get back? b. What is the area of a circle whose radius is 10 meters? c. Given a circle, and a square whose diagonal is equal to the diameter of that circle, which has greater area, the circle or the square? Problems given in textbooks and in examinations are typically of this type. Undergraduate level problems are also largely of this type, though requiring more advanced knowledge.

6 Problem Solving as Thoughtful Innovative Application The questions in (1a-c) test the students’ ability to make mechanical calculations based on memorized formulae. Those in (2a-c) are not very different: 2) a. A goat is tied with a 10-meter rope that is attached to a stump on the ground. What is the area of the land that the goat would be able to graze on? b. A goat is in an enclosed circular meadow whose diameter is 10 meters, and there is a closed circular shed of 3 meters inside the meadow. What is the area of the land that the goat would be able to graze on? Such questions do not involve thoughtful, creative, or innovative thinking needed for dealing with novel (i.e., non-textbook) problems. What they need are skills acquired through repeated supervised practice and mechanically reproducible without being guided by . Compare them with the questions in (3a-d). They demand a moderate level of difficulty in modeling, but not sufficient to distinguish the talented from the ordinary.

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3) a. A goat is tied with a 10-meter rope to the corner of a closed rectangular shed that is 60 meters long and 40 meters wide. What is the area of the land that the goat would be able to graze on? b. A goat is tied with a 10-meter rope to the corner of a closed rectangular shed that is 6 meters long and 4 meters wide. The shed is on a field of grass, with the goat on the grass. What is the area of the land that the goat would be able to graze on? c. A goat is tied with a 10-meter rope to the corner of a closed rectangular shed that is two meters long and one meter wide. The shed is on a field of grass, with the goat on the grass. What is the area of the land that the goat would be able to graze on? d. A goat is tied with a 10-meter rope to the corner of a closed triangular shed whose sides are 5 meters each. The shed is on a field of grass, with the goat on the grass. What is the area of the land that the goat would be able to graze on? Question (3a) is the simplest. (3d) is the most challenging. This is not the kind of questions that students can solve in a Multiple Choice Question that allows only two minutes to tick the ‘correct’ option.

7 Problem Solving as Inquiry Let us move on to problem solving as inquiry, which requires combining imagination, intuition, and insight with rigorous thinking and reasoning: 4) a. In Euclidean geometry, every finite line, however short, has infinitely many points. And every finite line can be bisected, that is, divided into equal parts. Consider another geometry, called ‘discrete geometry’, where every finite line has a finite number of points, such that the length of a line is the number of points it is made of. Is every finite line bisectable in this geometry? State your argument(s). b. Consider the conjecture that every circle is a polygon. Is this true in Euclidean geometry? Is it true in discrete geometry outlined in (4a)? Prove your answers. c. If a line segment AB rotates around point A, then (a) right angle is defined as one fourth of a complete rotation, (b) an acute angle is defined as less than a right angle, (c) a straight angle is defined as two right angles, and (d) an obtuse angle is defined as more than a right angle but less than a straight angle. A vertex is defined as the interior angle of a polygon. A circle circumscribes a polygon iff all the vertices of the polygon lie on the perimeter of the circle. Given these definitions, can a circle circumscribe a triangle? Prove your answer. d. In the Aristotelean theory of the motion of inanimate bodies, inertia is absence of motion (the state of rest) and force is that which results in motion. In the Galileo-Newton theory of motion, inertia is uniform velocity and force is that which results in a change in velocity. Construct a theory of falling bodies using Aristotle’s concepts of inertia and force. e. One of the general principles of reproduction and heredity in biology is that acquired characteristics cannot be inherited. Do a google search to find out what this implies. Then do a google search to find out the facts of the transmission of HIV from mother to offspring, and critically evaluate the textbook claim. To do this, you will find it useful to read the article “We know that roundworms inherit knowledge” at https://thewire.in/the- sciences/we-know-roundworms-inherit-knowledge-now-were-starting-to-find-out-how. f. Consider the following scenario. Xeno has been lost in a dense forest for days, and has been starving. He then finds a human body, killed by a tiger just a few minutes ago. Would his eating the flesh of that body be ethically wrong? Provide an argument to defend your answer.

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These questions call for the following inquiry abilities: (4a)/(4b): deducing the logical consequences of a novel axiom; (4c): identifying a potential logical contradiction in standard textbooks; (4d) constructing scientific theories; (4e) critically evaluating scientific theories; and (4f) constructing arguments in support of one’s position. Notice that these problems are formulated as questions. As we said, inquiry is the process of looking for and finding an answer to a question through one’s own thinking and observation. Hence, the activity of looking for and finding solutions to these problems is an inquiry activity. To summarise, there is a wide spectrum of problems that can be used in both teaching and assessment, ranging from mere mechanical high speed application all the way to inquiry. At the high end of the spectrum, the distinction between problem solving and inquiry disappears. The problems listed in the Wikipedia entries on the unsolved problems in mathematics (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics), physics (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics), biology (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_biology), and philosophy (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_philosophy) move from the terrain of inquiry that undergraduate students can practice, to the domain of research that can be undertaken by graduate students and professional researchers.

8 Problem Solving in Knowledge Application There are different kinds of knowledge application, implicit in domains like applied mathematics, applied physics, applied linguistics, and applied philosophy. One of them is the application of one domain of knowledge to another domain. For instance, applied mathematics can be the application of mathematics in physics (e.g., the use of calculus in theories of motion), and applied physics can be the application of physics in engineering and technology (e.g., the application of quantum theory to build nuclear bombs, and of relativity theory to build the GPS.) The second kind of application is the application of knowledge in solving the problems of personal, public, or professional lives. An example of the second kind would be one described in the Nature article, “How a decision-analysis tool helped one scientist couple make some tough career choices” (https://www.nature.com/articles/d41586-019-02106-5), where a couple “…used a structured decision-making (SDM) process called PrOACT (short for: frame the problem, identify objectives, explore alternatives, predict consequences, evaluate trade-offs) …” to solve a career problem.

9 A Recommendation It would be valuable to construct an undergraduate MOOC on Problem Solving, to sensitize students to a wide variety of problems, and the strategies to deal with them, and solve them.

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