Physical Models That Provide Guidance in Visualization

Article pubs.acs.org/jchemeduc

Physical Models That Provide Guidance in Visualization Deconstruction in an Inorganic Context Holly K. Schiltz* and Maria T. Oliver-Hoyo Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695 United States S Supporting Information *

ABSTRACT: Three physical model systems have been developed to help students deconstruct the visualization needed when learning symmetry and group theory. The systems provide students with physical and visual frames of reference to facilitate the complex visualization involved in symmetry concepts. The permanent reflection plane demonstration presents an explicit example of a reflection plane and provides visual indicators that students use to support or invalidate the presence of a reflection plane. The 3-D coordinate axis system provides an environment that allows students to envision symmetry operations beyond the basic geometry of bonds in a relevant molecular context while the proper rotation axis system is designed to provide a physical frame of reference to showcase multiple symmetry elements that students must identify in a molecular model. The 3-D coordinate axis and the proper rotation axis systems allow students to incorporate their own molecular modeling kits. All three model systems have corresponding worksheets designed after a modeling framework that takes into consideration three basic principles in “viewing” visualization as a problem solving tool: deconstruction, comparisons, and reuse of strategies. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Inorganic Chemistry, Hands-On Learning/Manipulatives, Group Theory/Symmetry, Molecular Properties/Structure

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master structural concepts.13 Each of the experiences a student has with a concept influences understanding when students are actively constructing knowledge.14 Until recently, a limited number of physical resources have been available to model symmetry and group theory. Mirrors have been used with pattern blocks,4 and modified molecular model kits8 have been used to demonstrate and reaffirm locations of reflection planes. Poster board models to represent specific point groups allow students to enact the operations on a specialized system that displays all the operations for that group.5 Students may be asked to identify symmetry elements in common objects, such as a tennis ball6 or a tire’s tread.7 The first physical system to allow students to explore C2 axes perpendicular to the principle rotation axis uses Tinkertoys with a specific modeling kit.8 An exercise in producing 2-D images from a collection of wooden cubes allows students to methodically practice 2-D to 3-D transformation, which may be a skill taken for granted.15 Computer simulations are available to illustrate what students should see as they manipulate molecular images, use calculated quantities and theory to support observations,16−18 and to demonstrate how 2-D images may be transformed to 3D.19 Though 3-D computer simulations are powerful instruments in visualization, physical manipulative models supply tactile information and allow students to choose their own point of view instead of a predetermined view programmed into the simulation. The model systems presented here incorporate both physical and visual frames of reference for students to

norganic chemistry covers a considerable number of visual topics that require mental visualization, such as symmetry and group theory. Mental visualization includes both the “perception and retention of form” and the “mental manipulation of visual shapes”.1 Students with good visualization skills are more flexible in their methods for problem solving.2 Possession of advanced visualization skills may also indicate advanced levels of thinking.3 This work presents three 3-D physical model systems that have been developed to allow students to explore and refine the visualization associated with inorganic symmetry topics. The models described here aim to illustrate specific problems students encounter and report when first introduced to symmetry and group theory and the development considered cost and simplicity of use. All three systems may be used to present multiple points of discussion beyond their primary purposes. Corresponding activity worksheets have been developed to promote visualization based on the feedback received from focus groups of undergraduate inorganic students while interacting with the model systems.

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PURPOSE AND LITERATURE Group theory usually begins with the introduction of symmetry. To help students understand and use the symmetry operations, examples and models may be given as physical models,4−8 computer simulations,9−11 2-D figures in textbooks and worksheets, and verbal descriptions in class. As students experience a concept in different ways, they grow to have a refined and sophisticated knowledge of the concept.12 More time spent working with molecular modeling helps students © 2012 American Chemical Society and Division of Chemical Education, Inc.

Published: April 3, 2012 873

dx.doi.org/10.1021/ed200540p | J. Chem. Educ. 2012, 89, 873−877

Journal of Chemical Education

Article

that allows students to envision symmetry operations beyond the basic geometry of bonds in a relevant molecular context, instead of manipulating nonmolecular objects that students may not deem as important to visualize when solving inorganic problems in class.5−7 Model kits commonly used in general or organic chemistry are frequently recommended in inorganic courses as a supplement for students to practice symmetry operations. These model kits demonstrate the spatial geometry of molecules, but reflections, inversions, and improper rotations cannot be enacted on physical models due to the fixed construction of the model kits. Additionally, an extra hand may be needed to mark a plane, rotate the molecule, and point out differences while holding the model steady before visualizing multiple symmetry elements onto molecular models. This system allows students to extend their model kits to an inorganic context. It provides students with a 3-D workspace where the abstract visualization of symmetry elements is given tangible representation that students carry out themselves. This differs from other model systems where symmetry operations are demonstrated for students. The primary goal of this model system is to enable students to physically separate out each motion required in symmetry operations in a flexible, 3-D workspace. Modeling stepwise operations provides the unique perspective of seeing each step of the motion stopped and displayed simultaneously. Students observe the other halves of molecules to evaluate reflection planes and use dry erase markers to mark their progress (Figure 2). Though it is impossible to pull each atom on a model kit

track symmetry operations and were designed to let students approach complex visualizations in discrete steps. Multiple operations are explored for each model system in the accompanying activities.

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MODELS

Permanent Reflection Plane Model

The permanent reflection plane demonstration model (Figure 1) was constructed to explicitly demonstrate the difference

Figure 1. The permanent reflection plane model in the open position superimposes the mirror image of the front halves of the wood models onto the back halves. Water and ammonia molecules are shown.

between a mirror image and a reflection symmetry element while also providing points that students could use to support or invalidate the presence of a reflection plane. This model system demonstrates what constitutes a reflection plane by explicitly displaying a physical plane through two whole molecules so students may examine both sides simultaneously rather than manipulating half of an object with a mirror.4,8 Water and ammonia molecules were chosen because they are both based on tetrahedral electron domain geometries and can be manipulated to look alike to demonstrate the difference between a reflection plane and a plane arbitrarily placed through an ammonia molecule. It may be used when symmetry is first introduced to allow students to form their own construct of a reflection plane. Two molecule halves are oriented on a glossy physical plane, in combination with the reflected images, to display two “complete” water molecules. The opaque inserts open to reveal that the physical plane is a reflection plane for a water molecule and an arbitrary plane for an ammonia molecule. Students have used this distinction to discuss factors that are or are not involved in the reflection plane definition. This model system is used to promote discussion on identical and indistinguishable atoms, locating multiple reflection planes, atom labeling, and coordinate mapping. Presenting labeling and mapping in a simple model system introduces students to abstract ideas before they are actually required to use the concepts in more difficult problems. Students are instructed to examine the model from all sides in order for them to observe that a reflection plane separates two spatially independent and corresponding but equal sides of the molecule. They can distinguish between a mirror image and corresponding atoms related by a reflection plane. The physical aspects of this demonstration show students how to separate and compare the “sides” of other molecular models apart from this demonstration. This model specifically showcases the differences between a mirror image and a reflection element.

Figure 2. The 3-D coordinate axis model displays the steps of an improper rotation while dry erase markers label atoms and document motion. The allene molecule, C3H4, is shown.

through the center and invert it to the other side, students can still explore inversion centers. This system holds the initial and final points of a modeled inversion in place so students can explore the space around it and compare distances to the marked axes. Carrying out multiple operations stepwise and studying how a motion may affect the spatial orientation of the rest of the molecule aim to provide spatial understanding through physical manipulation. Proper Rotation Axis Model

The proper rotation axis model system is designed to provide a physical frame of reference for the symmetry elements that students must identify in a molecular model. This system can showcase multiple symmetry elements, but it is specialized to emphasize the aspects of proper and principle rotation axes with immediate feedback and a constant method to track rotational intervals. Proper rotations are the only symmetry

3-D Coordinate Axis Model

The 3-D coordinate axis model system is used to enable students to extend their own model kits into an environment 874



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elements that a student can enact upon a physical object without altering the structure of the molecular model and are important for inorganic students to understand because they dictate under which point group a molecule is classified. This system accommodates C2−C6 rotation axes of various objects and molecular model kits. Once students are able to identify the proper rotation axis, they use the system to hold the model in that rotation axis environment to look for other symmetry elements, such as additional proper rotation axes and reflection planes. Though Tinkertoy dowels have been used to locate perpendicular C2 axes,8 our model retains the principal rotation axis location so students may place themselves in a perpendicular position around the molecule to locate these difficult symmetry elements. A frame of reference, mental or visual, is needed to track simple rotations of a molecular model. Frames in the system outline the sites where a model takes on an indistinguishable configuration from its original orientation to ensure that students have a visual frame of reference as an intermediate step in the visualization (Figure 3). To help students make a tactile

Figure 4. Six frames around a molecule with a C3 rotation axis outline all the vertical reflection planes simultaneously.

understanding of the model.21 To find the elements of a model, students deconstruct the system by identifying the simplest parts of the image, problem, object, or situation. Studying relationships (which describe how elements are related to one another) and operations (which show how the elements interact), students practice how to look for patterns and rules of the model system that govern the relationships and operations. To help students extend their visualization skills to chemistry, the steps practiced in the model systems are compared to similar patterns in everyday life and similar previous instances in chemistry. Finally, students sum up their experiences with the model system by considering the strategy they used to answer the problem. Understanding their own problem-solving strategies provides the opportunity to regard their visualization steps as a problem-solving tool and model how to break down a complicated, abstract problem. These three principles are built into the worksheets designed for each model system. The worksheets are included in the Supporting Information. Students use both physical manipulations and traditional 2-D representations for practice in making 2-D to 3-D transformations. The systems aim to help students reflect on visualization processes by enacting each step involved and their use in the classroom has also revealed to us the visualization problems students encounter when solving abstract problems of this sort. The pedagogical aims of the activities accompanying each model system are listed in Table 1.

Figure 3. A molecule with a C3 rotation axis inserted into the model with the atoms aligned to the frames in the correct degree increments. Boron hydride is shown.

connection to the molecular model’s movement, the model system cues students with catches built under the circular base. As the system rotates around a stationary molecular model, catches corresponding to the degrees of rotation alert students that they have completed that amount of rotation. Students may match their molecular model to the preassigned system or check their expected degree of rotation by matching the system to their model. If students are not certain of the degree of rotation, this system provides a way to check if they are correct with a visual and physical affirmation of the operations they complete. Bookmarking a proper rotation with the system lets students find proper rotation axes in other locations around the molecule and any reflection planes, demonstrating how different symmetry elements are intertwined (Figure 4).

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STUDENT RESPONSE During the fall semesters of 2009 and 2010, help sessions were offered to undergraduates enrolled in first-semester inorganic chemistry to assist them with symmetry operations using the models described here. The help sessions were not mandatory and the work did not receive credit toward class grading. The students that came to the sessions generally described themselves as “just not able to see the symmetry operations.” Each semester, 20−25% of the students enrolled in the course attended the help sessions. The researcher was not the instructor but a supplemental source for assistance. The models were primarily used to explain the questions and problems students had difficulty visualizing during lecture. Student questions generated from these sessions and lectures were recorded and later used to develop and modify the

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ACTIVITY DESIGN The modeling framework by Lesh et al. was modified and used to design activities to accompany the model systems as it provides guidelines to elicit problem-solving thinking processes.20 The framework was simplified to three principles that the students must practice to view visualization as a problemsolving tool: (1) deconstruct the visual model into components (elements, relationships, operations, and patterns or rules), (2) compare any model components to reality, and (3) share and reuse the strategy they employed. Deconstruction of the visual components of a model has been shown to indicate deeper 875



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group on the opposite side of the molecule without the model system to mark that axis as a reference so the student could explore how the molecule would be affected. From the cumulative observations in these focus groups, these model systems have proven to be interactive and to promote meaningful discussions among students by showing them how to use language and observation of a tangible environment to not only verbalize their own confusion but to more accurately describe and practice symmetry operations. Currently (fall 2011), the model systems and activities are being used in coordination with the inorganic class during mandatory recitation sessions. The next phase for using these models systems consists of investigating the correlations of their uses and students’ achievement, promotion of visualization skills, and engagement level in class.

Table 1. Pedagogical Aims of the Activities Accompanying Each Model System Model System Permanent Reflection Plane

3-D Coordinate Axis

Proper Rotation Axis

Aims Students use discussion to develop a detailed working definition of a reflection plane. Students articulate the reasons for the presence or absence of a reflection plane. Students gain practice in: discussing atoms in terms of 3-D spaces, labeling atoms, and differentiating atoms within and on either side of a reflection plane. Students construct and work with physical representations of symmetry elements. Students operate proper and improper rotations of varying degrees. Students conduct multiple operations and compare each step and the final outcome to the original orientation. Students provide reasons for completing operations in discussion between group members or in writing. Students construct visual and physical frames of reference around molecule-specific proper rotation axes. Students practice connecting visual and physical cues to the geometry of molecules. Students locate and argue the presence of multiple rotation axes simultaneously. Students locate reflection planes after defining the rotation axes.

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CONCLUSION Inorganic chemistry courses offer opportunities for students to use visualization in developing a sophisticated understanding of concepts. These concepts are better illustrated if students have a model system with which they can interact, track the steps of the visualization, and incorporate their own molecular models or models used in previous chemistry courses. Specially designed activities clarify the model systems by asking students to deconstruct the model components, compare the concepts to reality, and examine their processes. The physical model systems presented here add another instructional dimension to the resources students have available by addressing the components and steps of symmetry operations. They give students a physical frame of reference to help them explore all visual−spatial aspects of the transformations to become more familiar with these skills.

accompanying worksheets so the more problematic examples and points of discussion were emphasized. Students worked through the examples provided individually or in groups of two to four students. If students complete and thoroughly discuss all questions in detail, each worksheet may take up to 45 min. During help sessions, students utilized the models to observe molecules from different views and to explore different operations. Students came together to argue and discuss the results of various operations and showed their engagement via unsolicited use of their hands, gestures, and writing utensils to point out atoms or monitor changes around all three model systems. Though students were able to recite the definitions of all the symmetry elements, a recurring mistake was the confusion between a reflection and a 2-fold rotation. When enacting an improper rotation operation on staggered ethane (problem 1 on the 3-D coordinate axis activity), almost all students rotated the two halves of the molecule to complete the operation until they were prompted to confirm where each atom should end up after the reflection operation. After making that mistake, students were observed checking that the operations they enacted matched their definition of the operation and frequently gesturing where additional reflection planes and rotation axes were located around the molecular model and the symmetry elements framed by the 3-D coordinate axis model. A considerable number of students reported difficulty describing what they looked for when identifying a proper rotation axis. A common problem for many students is locating the 3-fold axis on an octahedron, even when they know to look down one of the faces. They explained that they could find the same face indicated in lecture, but doubted that the bonds not included in that face would also follow the 3-fold rotation. Students used the proper rotation axis system to line up three bonds and observe that the remaining three fell between the system’s frames. When looking for different rotation axes on a model of adamantane (problem 6 on the proper rotation axis activity), several students could locate a C2 axis but could not see that it contained the carbon on the bridging methylene

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ASSOCIATED CONTENT

S Supporting Information *

Model construction information with additional photos, assembling instructions, and corresponding activities with suggested answers. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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ACKNOWLEDGMENTS This work has been generously supported by NSF TUES Grant No. DUE-1043529 entitled “VIPs: Visualization to Invigorate Problem-Solving”.

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Physical Models That Provide Guidance in Visualization

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