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The need to foster student learning and to develop student critical thinking skills was a recurring theme at the 18th BCCE in Iowa. Critical thinking refers to those who can and do perform higher-order thought processes such as applying, analyzing, synthesizing, and evaluating information or methods in a variety of situations. The draft statement of principles (1) of the National Council for Excellence in Critical Thinking provides additional information on the scope of critical thinking. One of their founding principles is that "Knowing that something is so is not simply a matter of believing that it is so, it also entails being justified in that belief"(1).

Featured Documents
	sp3dn–Orbital Hybrids and Molecular Geometry
	Visualization of Wavefunctions of the Ionized Hydrogen Molecule
	Exploring Exotic Kinetics: An Introduction to the Use of Numerical Methods in Chemical Kinetics
	Heat, Work, and Entropy: A Molecular Level Illustration

Being justified in knowing may even be insufficient with respect to the ability to think critically about scientific content and science processes in response to the wider concerns of a highly technological society and a complex scientific landscape. One must distinguish between the lower order cognitive skills (LOCS) of knowing and understanding and the higher order cognitive skills (HOCS) of decision making and problem solving (not to be confused with algorithmic LOCS exercise skills) as described by Uri Zoller (2). Another way of looking at LOCS and HOCS is to distinguish them by recognizing that the former correspond to complex correct thinking and the latter form the essence of full critical thinking.

One way to help students develop higher order cognitive skills involves creating learning environments in which students can grow in their ability to reason and think, doing so within the context of the content and processes of science in a way that leads to solving real problems, decision making, and effective citizenship. The aim of environments leading to development of HOCS is excellence in thought processes within the discipline. Faculty must create experiences through which students are guided to explore real data in order to develop the desired knowledge base coupled to a thinking process that demonstrates excellence. Such excellence is observed when students write and compute beyond simple algorithmic methods. Thinking excellence is also shown when students demonstrate "rational, logical, consequential evaluative thinking in terms of what to accept (or reject) and what to believe in, followed by a decision (what to do (or not do) about it), followed by an accordingly responsible action"(2).

Symbolic mathematics templates provide teaching environments that promote development of HOCS. These documents contain selected content together with the ability to manipulate real data in a design that guides the student and requires the student to be an active partner in the learning process. The best templates provide ample opportunities for students to reflect upon the content and the outcomes of manipulation of data illustrating the content. There are no simple answers. There are explorations that students must make and must comment upon as they progress through the template. Each template provides the required challenge needed for HOCS development and the necessary support for students to reach the goal of excellence in thinking and learning within the discipline.

In this issue of the SymMath column we introduce three new Mathcad documents, one new Mathematica document, and three translations from Mathcad into either Maple or Mathematica. The new and translated documents demonstrate how higher order cognitive skills can be incorporated into teaching materials that both instruct and support student learning.

Hybrid Orbitals and Molecular Geometry

In the sp³dⁿ–Orbital Hybrids and Molecular Geometry Mathcad document, Mark Ellison guides students through a study of the angular components of spd-type hybrid orbitals. This document builds on his previously published Orbital Graphing document (3) that introduced sp hybridization via Mathcad. The Mathcad color graphics give students the opportunity to examine the orbitals interactively and thus construct a better understanding of the orbital directionality and its relationship to molecular structure. There are ample questions for students to answer and a faculty user can edit out the answers or create additional questions for the material presented. The notes in the document address the student directly. Summary questions at the end of the document require reflective thought from the student and thus foster higher order cognitive skill development. This document is appropriate for upper division chemistry majors in physical chemistry courses.

Ionized Hydrogen Molecule Wavefunctions

In the hyperlinked Mathcad document, Visualization of Wavefunctions of the Ionized Hydrogen Molecule, John Johnson provides a detailed development of the hydrogenic atom wave functions using spherical harmonics and radial wavefunctions. He then proceeds to explore the ionized hydrogen molecule and presents the LCAO solution and the exact solution using an oblate spherical coordinate system for finding the solutions to the Schrödinger equation and a prolate coordinate system for plotting the wave functions. Problems that focus student attention on important aspects of the material are embedded throughout the document. Through study of the material presented in this document students can compare the LCAO and exact solutions to the hydrogen molecule ion, see the limitations of the LCAO approach, and gain a better understanding of the molecular orbital nomenclature and the significance of orbital symmetry with respect to chemical bonding. Because this document is very comprehensive and requires a high level of mathematical sophistication, it should be used with the most capable upper division chemistry majors or graduate students.

Numerical Methods and Chemical Kinetics

In the Mathematica notebook, Exploring Exotic Kinetics: An Introduction to the Use of Numerical Methods in Chemical Kinetics, Michelle Francl develops the methods used to solve systems of differential equations for chemical kinetics. The simple rate laws used in typical physical chemistry courses fall short of the mark when one wishes to describe the variety of complex reactions that confront research chemists. These more complex reaction systems consist of many levels of differential equations that are difficult to solve analytically and thus there is a need for numerical integration methods. The notebook contains a detailed introduction to using Mathematica so that students can work through the document easily. The typical sequential reaction kinetics systems are explored first to build the confidence level of the student users. Questions, exercises, and hints are embedded throughout the notebook. More complex reactions are introduced systematically and the numerical solution approach applied in each case. The mastery exercise makes this notebook an excellent example of how higher order cognitive skills can be built into a mathematical document. The notebook concludes with an extensive set of references, some details of the Runge-Kutta method and a set of instructor notes. The notebook is appropriate for use in upper division physical chemistry courses or as a basis for a capstone student independent study project with a different chemical system.

Heat, Work, and Entropy at the Molecular Level

In the Mathcad template Heat, Work, and Entropy: A Molecular Level Illustration by Jeffrey Draves we find a compact integration of the macroscopic classical thermodynamics concepts of heat, work, and entropy to the same concepts from a statistical mechanics point of view. Students need to have a clear understanding of classical thermodynamics, have a working knowledge of the particle in a one-dimensional box from introductory quantum mechanics, and the fundamentals of statistical thermodynamics such as computing a partition function and using the Boltzmann distribution. The document uses two experiments to focus student attention. The first involves computing the change in internal energy for a gas when the temperature is raised at constant volume and w=0. The second asks students to compress the system obtained in the first experiment to achieve the same change in internal energy adiabatically and reversibly. The process is repeated for the distribution of particles over the energy levels of a one-dimensional box. This template is appropriate for upper division chemistry students toward the end of their first year of physical chemistry after they have completed the essential thermodynamics, quantum, and statistical mechanics topics in class. There are ample questions throughout the document to promote student learning. This document develops higher order cognitive skills in that it requires integration of separate parts of the curriculum in a way that develops deeper understanding of the component parts.

Literature Cited

1. The National Council for Excellence in Critical Thinking (accessed Aug 2004).
2. Zoller, U. J. Chem. Educ. 1993, 70, 195–197.
3. Ellison, M. J. Chem. Educ. 2004, 81, 158.

About the Feature Editor
	Theresa Julia Zielinski
JCE Citation
	Zielinski, T. J. J. Chem. Educ. 2004 81 1533.
Keywords
	Computer-Based Learning; Mathematics / Mathematical Methods; Physical Chemistry