The Problem with Problems in Computational Science and ...



The Problem with Problems in Computational Science and Engineering Problem-Based Learning

D. E. Stevenson, 315 McAdams Hall, Department of Computer Science, Clemson University, PO Box 341906, Clemson, SC 29634-1906

steve@cs.clemson.edu

Partial support for this work was provided by the National Science Foundation's National Science, Technology, Engineering, and Mathematics Education Digital Library program under grant DUE-0435187 and STEP program DUE-0525474.

Abstract

Problem-based and case-based learning are well-known inquiry-based pedagogies. We propose a new inquiry-based approach called systems-questions-solutions (SQS), based on general systems theoretic approaches to problems. This approach naturally introduces verification and validation into the educational process.

Computational science, problem-based learning, schemata, verification, validation.

Central Issues

While computational science borrows methods, principles and values from many disciplines, it is evolving as a unique science in its own right.

Computational science applications require interdisciplinary teams with members who are often unfamiliar with the context and vocabulary of the problem. This lack of common background means that a significant effort is required to educate those members.

Because computational scientists continually face unfamiliar problems, they must be efficient learners and require knowledge organizational principles.

Applications, Algorithms, and Architectures

includegraphics{Shodor-Tetra.jpg}

Meeting the Education Requirement Head-On

Science is nothing if not organized.

Computational Sciences’ central theoretical principles are taken to be general systems and these ideas are used to develop a pedagogical technique called Systems-Questions-Solutions

What Counts for Knowledge in CS

|Focus Question: Questions that serve to focus the inquiry about events and/or objects to be studied |

|Declarative Knowledge |Implicit Knowledge |

|World View |Value Claims |

|Philosophy and Epistemology |Knowledge Claims |

|Theory | |

|Principles |Transformations |

|Constructs |Records |

|Concepts | |

|Events and Objects |

Verification and Validation as Epistemology

A model or simulation is verified if the model is in accordance with its specification (``Did we construct the program correctly?'')

Validated if the model correctly predicts behavior (``Did we construct the correct program?'').

|Consistency. Outside the strict mathematical meaning, consistency means ``freedom from contradiction.'' |

|Justification. According to the Routledge Dictionary of Philosophy, the term ``justification'' belongs to a set of terms that |

|also includes ``rational'', ``reasonable'' and ``warranted'' that do not have a commonly agreed definitions or relationships. |

|Law-like. Development and reasoning must use established laws. |

|Reasoned. Reasons are propositions that meet certain conditions such as ``being believed by someone.'' Suppose we have a proposition $P$ and we also have a statement ``If $P$|

|then $Q$'' then we have some ``reason'' to believe $Q$. But we need an external reason to believe $P$. So for someone to actually believe $Q$, that someone must believe $P$ |

|as a reason to believe $Q$. |

|Coherent. Areas of knowledge are coherent if all the elements ``fit'' together. This is a central tenet of modern science. |

|Credible. Scientific observation has an epistemic side, with sufficient meaning and credibility to contribute to knowledge. |

|Organized. Each science has an agreed-upon standard arrangement of ideas. Models and simulations should make use of this structure. |

|Relevant. Relevance is the ability to retrieve material that satisfies the needs of the user. There are many examples of people focusing on irrelevant things when attempting |

|to validate. |

Foundations of Learning

Our understanding of learning is based on How People Learn. Three fundamental components in learning:

deep factual knowledge (facts) of the subject,

deep knowledge of pragmatic issues (schemata), and

meta-cognitive exercises.

Problem-Based and Case-Based Learning

The author has used two popular active learning methods: problem-based learning (PB) and case-based methods (CM).

PBL is based on evidence that learners learn best when presented with a realistic problem to focus on while

CBM use collaborative analysis leading to judgments when there is no single, correctanswer.

Problem-based and case-based situations are the heart of in professional practice

The Challenge

Computational science education must refine knowledge and develop expertise in problem-solving at a general systems level yet be able to address learners' professional needs. Professionals should be learning how to learn professional methods, principles, and values from multiple disciplines and how to apply these concepts to problems

Systems, Science, and More

Science is about systems, questions about those systems, and solutions to questions posed on these systems

Systems thinking actually has ancient roots.

General systems theory is an approach to understanding individual systems through general principles that apply to all systems.

A system is any object with behaviors

Standard schemata

States and transitions,

components and couplings

time-invariant functions and relations.

A complex system is a system with multiple identifiable components that are coupled, making the definition recursive.

Systems Thinking to Systems Theory

Modern mathematical treatments of systems theory focus on questions that can only be posed in the context of a system in which the question and solution can be interpreted. Answers, meaning solutions, are developed in the context of a specific system and the question. The educational principle we want to develop is exploration of the triple of (systems, questions, solutions) as SQS.

Examples

(system, question), engineering fare.

(systems, solutions), Program debugging

(questions, solutions) synthesize one or more systems with given capabilities.

(solutions) ``To an engineer with a hammer, the whole world looks like a nail.''

SQS Concepts

Goal is Expertise

Expertise is the ability to decompose a problem in novel ways, ask novel questions, and to apply novel solutions: propositional knowledge is not enough

Computational science uses of three types of knowledge

propositional, implicit, and algorithmic

|Knowledge: Recall data or information |

|Comprehension: Understand the meaning, translation, interpolation, and interpretation of instructions and problems. State a problem in one's own words. |

|Application: Use a concept in a new situation or unprompted use of an abstraction. Applies what was learned in the classroom into novel situations in the work place. |

|Analysis: Separates material or concepts into component parts so that its organizational structure may be understood. Distinguishes between facts and inferences. |

|Synthesis: Builds a structure or pattern from diverse elements. Put parts together to form a whole, with emphasis on |

|creating a new meaning or structure. |

|Evaluation: Make judgments about the value of ideas or materials. select the most effective solution. |

CSE and Bloom’s Taxonomy

Much of computational science work takes place at the higher levels of Bloom's taxonomy

The Heart of Problem-Solving

Is that there is a schematic mapping from the semantics of the problem to the semantics of the solution.

Schemata in Problem Solving

Triggers and Patterns

Constraints and Criteria

Planning

Implementation

Triggers and Patterns

Pattern matching triggers. The setting of values, recognition of keywords, or recognition of patterns triggers the schemata.

Constraints and Criteria

Constraints and criteria are logical criteria and performance constraints that must be met. Criteria could be theorems or physical laws. A schema may be triggered and then ignored if the criteria or constraints fail to hold.

Planning

Planning is the organization of a computation or perhaps putting other schemata in play. Complex systems often have very large and complex decompositions that must be planned out, recursively activating other schemata.

Implementation

This is the computation of ``the answer.'' Even with all the available information, the ``answer'' may still be non-computable.

Examples

Physical principles are schemata. If the word{\em energy} occurs in a physics problem (pattern match), then the Law of Conservation of Energy must be involved.

Data structures in computational science are schemata. C++ provides some type abstraction capability; ML and OCAML provide much more. Specification of a type variable's value activates many criteria, constraints, and detailed code changes.

Problem-Solving the SQS Way

Problem-solving in computational science always has a complex step of converting the semantics of the problem into semantics of computation from which the program is written.} This particular step is often missing in normal classroom science and mathematics classes. The semantics-semantics link is provided by schemata.

The SQS Procedure

Linguistic Phase. A problem is received as words and images in the problem poser's vocabulary and context.

Concept Map Phase. During the concept map phase, the SQS-specific context is established through the lexicon, vocabulary, and concept maps

Schematic Map Phase. The issues are understood semantically. The system-question-solution classification takes place. Schemata relevant to the missing information are accessed. The initial SQS elements are formulated.

Initial mapping phase. The semantics of the problem are mapped to the computational semantics, which are primarily data

representation and algorithm ``snippets'' that will be used later in planning to suggest other coding schemata.

Completion. In order to produce a program, the semantics of the program from above are converted to programming language syntax.

Modification and Judgment. There are often a large number of judgment calls in program design, made concrete by performance constraints and logical criteria. Design decisions are often based on what can be safely ignored or approximated, again often a judgment, not a knowledge based move. Along the way, there are many judgment calls based on algorithm complexity, which is a measure of performance. Very large problems are often placed on networks of computers; some networks being composed of thousands of processors. These networks are too complex to have a complete understanding of state.

Iterate. A major difference between experts and novices is that experts will iterate or perhaps attempt to solve the problem in a totally different way. Large, multiprocessor systems generally require several iterations just for the first version.

Verification and Validation of Models and Simulations

Under the discipline-centric view of models, models are accepted or rejected based on successful experience without any planned effort to justify the model or simulation. In the 21st Century, this will not work as the time available to design, implement, test and use models is too short to gain ``justification by experience.''

What’s the issue?

The justification steps require two different processes: verification and validation, which are often confused the modeling and simulation community. Computational science project teams cannot assume that all team members understands discipline-specific terms, methods and contexts nor can they be expected to accept what to them is ad hoc reasoning.

Standard Terminology

Verification and validation terminology has been standardize by the Defense Modeling and Simulation Agency and the American Institute of Aeronautics and Astronautics

Verification is …

the demonstration that the model is logically correct and follows from the physical and mathematical laws used. For a computer simulation, verification shows that the specifications are fulfilled.

Validation is …

the demonstration that the model correctly predicts the phenomena modeled. This can be a lengthy and expensive process particularly if new experiments are conducted.

Fundamental Idea is Explanation

The approach outlined here was find characteristics of verification and validation that would resonate with scientists and engineers. The characteristics chosen were those of scientific explanations. These were further divided into verification and validation characteristics.

Verification

Verification must show that a model or simulation is true by demonstration or evidence. The use of true is problematic because we need to know what true means. In mathematics, we are speaking of the proof, and in computer science we are generally speaking of testing. The four characteristics we consider are (1) consistency, (2) justification, (3) lawfulness, and (4) reasoned.

|Consistency. Outside the strict mathematical meaning, consistency means ``freedom from contradiction.'' |

|Justification. According to the Routledge Dictionary of Philosophy, the term ``justification'' belongs to a set of terms that also includes ``rational'', ``reasonable'' and |

|``warranted'' that do not have a commonly agreed definitions or relationships. |

|Law-like. Development and reasoning must use established laws. |

|Reasoned. Reasons are propositions that meet certain conditions such as ``being believed by someone.'' Suppose we have a proposition $P$ and we also have a statement ``If $P$|

|then $Q$'' then we have some ``reason'' to believe $Q$. But we need an external reason to believe $P$. So for someone to actually believe $Q$, that someone must believe $P$ |

|as a reason to believe $Q$. |

Whereas verification is concerned about rules, validation (See Figure \ref{figure-validation}) is concerned with the agreement between the model or simulation and the evidence produced from observations.

|Coherent. Areas of knowledge are coherent if all the elements ``fit'' together. This is a central tenet of modern science. |

|Credible. Scientific observation has an epistemic side, with sufficient meaning and credibility to contribute to knowledge. |

|Organized. Each science has an agreed-upon standard arrangement of ideas. Models and simulations should make use of this structure. |

|Relevant. Relevance is the ability to retrieve material that satisfies the needs of the user. There are many examples of people focusing on irrelevant things when attempting |

|to validate. |

Conclusion

The Systems-Question-Solution methodology has been developed to address the issue of which problems are suitable for PBL/CBM use. SQS is based on a universal approach to STEM subjects: general systems theory. We propose the meaningful problem solving approach MPS for expanding the SQS paradigm to specific problems.

Represents the situation faced in professional practice.

Represents a framework for developing both professional methods and professional practice.

Addresses the issue of semantics-to-semantics relationships.

Is teachable.

We assert that the method is assessable, primarily because of it has been used in STEM education for many years. The problem is to assess the method at a level that would be consistent with current education assessment standards.

Computational science considerably complicates the question of correctness because of the mixed disciplinary nature of the team. Some workers equate verification and validation, but these are quite different ideas. The SQS approach emphasizes the correctness issue.

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