in this issue A Picture Is Worth a Thousand Words (and Equations) Teachers, as well as educational researchers, have obsewed that an abstract concept cannot be understood until the learner has formulated some concrete picture to associate with the idea. On reflection this seems logical: abstract concepts themselves are based on observed facts and relationships (laws) that are then winnowed-sometimes over many generations--to their inherent, essential properties. he& abstractions are so useful as a s u m m a j ofa knowledge that those already conversant with the discipline sometimes forget that-the concept is not the rial phenomemon it is trying to represent. If a teacher disan abstract concept or theory without reference to its concrete underlying phenomenon, the resulting confusion in students (and sometimes in their colleagues and themselves) can lead to frustration and the perception that science has no relation to evervdav life. While it mav not be necessary for teachersto pive a complete historicainarrative when thev teach a new conceDt. thev would do well to present the ievelopment of theidea and introduce it with wncrete illustrations. These illustrations can range from merely showing a graph relating the phenomena under discussion for more sophisticated students to a classroom demonstration of chemical reactions for the novice learners. Many articles in this issue have a component somewhere in this spectrum; all try at some level to help students "get the picture". One of science's most fundamental ways of expressing abstractions is in the form of eauations. Brief lines of svmbols can explain the production of the energy that powers giant stars, predict how fast a falling object will being going when it strikes the earth, describe the relationships among pressure, volume, and temperature of a gas, or relate income to education level. While thev " aooreciate the density of information available in the equation format, scientists and mathematicians also have felt the need for a concrete picture of the relationships and early on invented the eranh as wav to make the abstract concept visible. To the person who is still basically concrete-operational a graph still looks abstract, but to those who are comfortable with manipulating abstractions, a graph provides all the "concreteness" thev need. Peckham and McNaught (page 554) wncern t6emselves with helping students make
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Journal of Chemical Education
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this transition, specifically so that students can use Maxwell-Boltzmann distribution diagrams. They suggest that, for beginning students, it is best to show a diagram of a distribution that they are already familiar with, for example, the weight of 30-year-old males in a given population. Since students can intuitively understand this diagram, they can then relate it to those showing particle speed and energy. The more sophisticated student can be introduced to more subtle uses for graphs than simply plotting data and deriving values such as rate constants. Palasota -physical . and Deming (page 560) present the use of three-dimensional plots for studying reactions that have more than one independent variable. They show how to perform multifactor experiments using a central composite design. By finding out how variables simultaneously affect the reaction, better understanding and predicitablity is achieved. Graph theory itself is becoming a frequently used tool in chemistrv. The most common annlications deal with chemical structure; in fact, next month's issue will contain the ~roceedinesof a svm~osiumon this to~ic.However. m a ~ h theory has other appiications and ~elhkinand ~ & h k (page 544) illustrate its usefulness in determining the kinetics of complex reactions. They define howukinetic graphs are constructed for a variety of reaction types and then show how to use them to determine reaction routes and rates. To a working scientist a graph provides a snapshot of reality. To the student or layperson, this connection is not apparent. The novice needs a more concrete illustration of the abstract wncept under discussion. Two of the favorite modes used by teachers are models and demonstrations. Sawyer and Martens (page 551) combine these two modes in their equilibrium machine, a physical model using . Stvrofoam balls that demonstrates the concents of equilibrium, activation energy, and catalysis. They gve comolete construction details for the device and its use. Models are most freauentlv used to illustrate strucutral concepts; as the size df the holecule under wnsideration mows. however. both the cost and the effort to produce accurate models is prohibitive. Beneditti a n d - ~ o r o s e t t i (page 569) present a wav to build inexpensive models of &cieic acids that, while not as detailed as the space-filling models commericallv available, still retain the to~ological features that detendine the molecules'biological a>ti&es. ~
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