Chapter Four - High Tech Math and Science
In 1989 leaders of mathematics education and science education each published revolutionary reports calling for fundamental shifts in the way our students learn math and science. Technology promises to play a prominent role in the implementation of the curriculum standards of the National Council of Teachers of Mathematics (NCTM) and the recommendations of Project 2061: Science for All Americans. As this book went to press, the National Academy of Sciences was unveiling principles for science standards similar to the NCTM standards.
Full coverage of these two proposals lies outside the scope of this chapter, but there are several themes which are especially ripe for support from technology, and it upon those themes that this chapter will focus. This chapter also treats the two disciplines together, for that is a major recommendation of both groups - that students should experience the two in tandem, each serving the other to help explore and explain the world.
The themes of Project 2061 include an emphasis upon systems, models, constancy, patterns of change, evolution and scale. According to this report, teaching should start with questions about nature, engage students actively, concentrate on the collection and use of evidence, provide historical perspective, insist on clear expression, use a team approach, not separate knowing from finding out, and de-emphasize the memorization of technical vocabulary. It goes on to suggest that science teaching welcome curiosity, reward creativity, encourage a spirit of healthy questioning, avoid dogmatism and promote aesthetic responses. Science teaching should aim to counteract learning anxieties. Science teaching should extend beyond the school and should take its time (AAAS, 1989).
The NCTM standards likewise stress models, systems, the analysis and interpretation of data and the use of statistical inference as applied to real life situations. Mathematics are seen as delivering powerful support for the construction of meaning and the development of insight (Frye, 1989).
II. Models, Systems, Simulation and Statistical Inference
Most phenomena (physical or human) can be viewed as part of a system with elements woven together in a more or less complex web of relationships. Science and mathematics help us to understand how these elements interact. We build simplified versions of systems (called models) place them on computers (where they become simulations) and then we explore what happens when we change any of the elements (called variables).
When students first start learning about systems with such models, they explore on the computer with commercially designed or teacher invented simulations. Ultimately, we expect them to learn to construct their own models and simulations. The construction of a reasonably realistic model is convincing evidence of understanding on a very high level.
New technologies enable even young students to test the power of relationships between variables in order to explore cause-and-effect and to attempt fore-casting. These new technologies support hypothesis-testing, theory testing and model building by intermediate and middle school students as well as the older ones. They allow systems thinking to creep down into elementary classrooms.
For most research questions, we are hoping to find out how something works in order to make wise decisions and solve various problems. The more we know about a system and how its key variables interact, so the theory goes, the more likely we are to act in ways which are fruitful.
In the early stages of learning about systems, the computer provides students with simulations which allow them to see the effect of changing one variable or another. This is called parameter manipulation (Mandinach, 1992).
Intellimation's Physics Simulations
This package of physics simulations allows students to shoot off a piece of artillery over and over again without suffering powder burns, destroying the landscape or hurting anybody's ears. The student can adjust the parameters of four different variables (velocity magnitude, angle, friction and decay) and then fire off the ball to note the resulting trajectory which is etched across the screen each time until cleared by the student. After twenty trials, there will be twenty trajectories showing. The visualization builds intuitive understanding which serves as a foundation for more formal mathematical and scientific reasoning.
The Systems Thinking and Curriculum Innovation Network Project developed by ETS, has been exploring the use of a software program called STELLA, a simulation-modeling package, for six years with six high schools and two middle schools. The project results reported in studies have been dramatic (Mandinach, 1992).
Crawford (1992) describes a STACIN teacher-designed middle school wolf project which bridges language arts, math and science, involving students in "a computer program written about wolves and predator/prey relationships ."
Mandinach (1992) identifies four levels of systems thinking proceeding from the parameter manipulation described above to the construction of simplified models, called constrained modeling, and then the development of more complex models, called epitome modeling, and finally, the creation of learning environments. The higher up this ladder the student proceeds, the greater the understanding.
In addition to systems/modeling programs like STELLA, today's infotectives will employ user-friendly statistical packages like PEMD Discovery and DataDesk to support number crunching, picture making and inference. After all, we know that "a picture is worth a thousand words." A graph is often the best way to communicate the meaning lying hidden in a scattering of thousands of data points.
PEMD Education Group offers a CD-ROM disc called the Environmental Data Disc containing over 125 megabytes of data (such as temperature and precipitation data from over 800 US stations; worldwide food, agricultural and demographic data, economic and trade data, and much more) as well as a very user-friendly exploration tool called Discovery which allows students to explore relationships between variables by graphing various combinations. The graphing process requires a simple click on a menu to designate X or Y axis and then the selection of various data groups the student wishes to compare.
The student may use personally collected data (creating datastacks to fit an experiment, for example) may import data from already existing spreadsheets, or may rely upon PEMD's CD-ROM data. In order to support topics of particularly strong current interest, PEMD provides learning modules on AIDs, CO2 Emissions, Global Warming, Ozone and Petroleum, setting up datastacks for students with all key relevant variables already established and the design of the datastack already done for the student. All the student has to do is start clicking and the graphs will pop onto the screen with amazing speed and clarity. Once the graphs appear, the student may change their scale, their color and their size, stack them up or tile them so that as many as eight graphs can all be seen at the same time.
While PEMD Discovery is primarily visual, the data explored graphically in that program can also be exported to statistical packages like DataDesk if one wishes to obtain correlations and other measures.
DataDesk is a statistical package from Odesta with a visual emphasis. While it is capable of performing all of the statistical calculations you might wish to perform with a database, it goes much further by providing many different forms of display, many of which can be adjusted and manipulated to explore relationships visually.
Correlations (Pearson Product Moment, Kendall's tau, Spearman Rank Correlation, Covariance)
Cluster Analysis Frequency Breakdowns
Histograms Bar Charts
Pie Charts Scatterplots
Rotation Plot Dotplots
After students enter or import data from science or social studies projects, DataDesk allows them to explore mathematical relationships between whatever variables they select and see what they look like.
Technology Snapshot 4.1
One science teacher whose students had collected and recorded water samples from various sites along a community stream for over 20 years was able to enter into DataDesk more than a dozen measurements such as fecal coloform content and pH from a half dozen sites along the stream with the samples being repeated hourly through the night for ten times. Total datapoints exceeded 14,000 - a true data gold mine.
Once data for the twenty years were entered, each succeeding year was far less work, but the potential for insight was remarkable and the extracting of rich relationships was quite easy.
To see the pattern of fecal coloform content from site to site down the stream this year, the student highlights the right dates, clicks on the "time" field to designate it as an X axis and then on fecal content as a Y axis - the variable being graphed. The computer then draws the line or bar graph in just a few seconds, in color if wished. Repeat the process for each of the previous years and the screen fills with 19 more graphs. One can see the gradual pollution of the stream, pinpoint its zenith and locate the most likely origin.
Because of its emphasis upon visual displays and the many tools provided to adjust those visual displays, DataDesk supports student exploration of mathematical relationships without requiring understanding of graduate school level statistical formulas and concepts.
Correlations, for example, can make sense to upper elementary students without them understanding the mathematical procedures required to calculate them. In the case of the stream mentioned above, it takes a few moments to create scatterplots charting combinations of measures such as pH and fecal coloform content. If they are strongly related to each other, they will cluster close to the diagonal. One can see it clearly. But if one wants to know the actual correlation, it is a simple matter of clicking on a menu. Seconds later you have the relationship confirmed or denied. It is not difficult for a 5th or 6th grader to learn the difference between a correlation of .90 and one of .50.
III. Traveling the Electronic Highway
The KidsNet program on acid rain profiled in the chapter on social studies is an excellent example of an electronic highway linking students from around the world in information hunting and gathering that might have a positive impact upon social policies by spotlighting issues. Some problems like the disposal of hazardous waste were once sustained by the lack of exposure and disclosure, but students who are committed to a healthy planet may begin to employ new technologies to bring dangerous and harmful practices out into the open, identifying sources of pollution along community streams and joining with students in other towns, states and nations to establish data collections which will reveal significant regional patterns and help to pinpoint trouble spots.
Students across the world may become skilled environmental trouble-spotters. Students may employ their scientific technology skills in combination with their telecommunication skills to provide the globe with an environmental alert. Once they see the patterns, they may wish to act as infotectives, hypothesizing as to why certain variations may occur in their locations and then checking their hypotheses using the four tools of the hero mentioned in chapter one: creative powers, observation, questioning and an open mind. Trained in systems thinking, they are careful to look at the big picture, identifying all key elements which might have a significant influence on the patterns they are studying.
The electronic highway supports a new kind of global student community which trades stories, tricks of the trade and strategies as well as information. A cry for help posted on a bulletin board may attract several dozen helpful suggestions. The isolation of the smokestack science lab with its inclined planes and gas burners is replaced by a science program which is inclined to social action, employing science in the service of the community. Students stop asking, "Why do we have to learn this stuff?"
IV. Opening the Information Window: The Play is the Thing
Students now have extraordinary information resources available to them. In addition to sources such as the videodiscs, CD-ROM discs and online databases mentioned in the social studies chapter, young scientists can also test their "what if?" questions with hardware and software designed to be used in the classroom science lab. This software may serve to increase the amount of hands-on science in many schools by eliminating dangers, difficulties and time constraints.
IBM's Personal Science Laboratory (PSL), for example, offers probes which can measure such variables as temperature, light intensity, voltage, pH, distance, and force. As the students work with these probes, the software collects data, stores it in a data table, and plots data-points on a graph. Students are provided with "real time graphing," seeing the results of their experiment on screen as it is conducted. The software also offers data manipulation and analysis tools to help students discover relationships among the data after it has been collected and stored.
Teachers have reported that because this technology relieved students from the tedium of data collection, it freed them to concentrate on the meaning of the data. They praised the speed and accuracy of the data collection process and indicated that students were given a greater opportunity to explore relationships between variables (IBM, 1990).
A number of Level III videodisc programs (those using computer software to drive the videodisc player), involve students in problem solving scenarios. In ErgoMotion, for example, a program offered by Houghton Mifflin, students learn how the laws of physics affect everyday life, redesigning a roller coaster in one activity and seeing how physics affects its design. In Science Sleuths, a Videodiscovery program utilizing barcodes, students solve open-ended scientific mysteries such as why the lawn mowers are exploding in one housing development, navigating through a variety of documents and visual resources on the videodisc.
The availability of videodisc-based science programs keeps growing as Optical Data Corporation won state approval in Texas for its Windows program and TLTG (Texas Learning Technology Group) is close behind them with a middle school program.
In order to make power learning a reality in the science classroom, we must take a careful look at the tools we place in the hands of our students and ask to what extent they mirror the kinds of tools being used by practicing scientists. Meeting the national educational goals in science requires a substantial shift in practice to bring school science into alignment with real world science.
V. Learning Teams
While many educators are concerned about the isolating effects of students working long hours on computers, collaborative problem-solving can thrive in a high technology context so long as students are assigned to work in teams. The experimental nature of the activity keeps curiosity and task commitment high while the shared screen seems to help provide focus and structure. One benefit of these new technologies may be the cooperative learning which will result from teachers moving off stage to support more student-centered learning and research.
VI. Seeing is Believing (and Understanding)
Those science programs which hitherto relied almost exclusively upon textbooks often met with difficulty trying to inspire student enthusiasm for science while explaining complex concepts. With the advent of videodisc-based science programs, the teacher can have thousands of illustrations, slides, charts, graphs, drawings and models close at hand to make science more appetizing and more comprehensible.
Since each videodisc can hold some 54,000 pictures on each side and since the computer software knows how to reach any one picture in a second or two, the teacher can conduct lessons with great flexibility and power. Because the software programs driving such videodiscs usually permit word searches, teacher and class can navigate through the 54,000 pictures with efficiency and save the results of the search in whatever sequence they wish for later viewing.
In addition to still pictures, most videodisc programs also offer brief video segments to illustrate such phenomena as a heart beating under a variety of conditions (e.g., the use of stimulants). Unlike the VCR, the videodisc player supports very careful analysis of such material. Each frame (moving normally at 30 frames per second) can be frozen in place without damaging the disc. One can adjust the speed, backwards and forward, to half a dozen speeds. Because each frame has a number assigned to it, it is possible to return to a section of the video in less than 2 seconds without rewinding. Slowing the video down and looking frame by frame radically changes what students see and understand. It is also possible to leap into a different unit's pictures to explore relationships between shared concepts, moving back and forth between earthquakes and volcanoes, for example.
In recent decades the percentage of students showing interest in math or science careers has declined dramatically (Jones, 1990). Given the extensive impact that math, science and technology will all have upon the society of the next century, we cannot afford a generation which is illiterate or alienated from either discipline. As with social studies, new technologies promise to set math and science instruction free from their smokestack limitations, making them both more enjoyable, enlightening and helpful to students trying to make sense of their worlds.
We expect that the infotectives emerging from this kind of math and science experience will possess the ability to achieve insight as they proceed through life - insight to guide voting, career selection and a contribution to society. Rather than being befuddled by infomercials and propaganda about global warming and the ozone layer, they will puzzle their way out of the forest, using systems thinking to climb a tree and see the forest and its pathways. Shown a series of graphs, they will ask what models lie behind the pictures. What assumptions are hidden below the surface pictures? At a time when technology promises much, either in the way of destruction or progress, the path we end up selecting is more likely to be a healthy one if we have citizens capable of asking such questions while employing scientific and mathematical reasoning.