Journal of Science Education and Technology, Vol. 7, No. 1, 1998
Mathematics For The Student Scientist A. Darien Lauten1 and Gary N. Lauten2
The Earth Day:Forest Watch Program, introduces elementary, middle, and secondary students to field laboratory, and satellite-data analysis methods for assessing the health of Eastern White Pine (Pinus strobus). In this Student-Scientist Partnership program, mathematics, as envisioned in the NCTM Standards, arises naturally and provides opportunities for science-mathematics interdisciplinary student learning. School mathematics becomes the vehicle for students to quantify, represent, analyze, and interpret meaningful, real data. KEY WORDS: Trigonometry; geometric size transformations; algebraic transformations; quantify; representation; analyze,
INTRODUCTION
stand assessment, 2) laboratory-based assessment of damage symptoms, and 3) image/data analysis of Landsat Thematic Mapper (TM) data for the area around their school. Forest Watch activities also give rise to important mathematical concepts that are part of the school mathematics curriculum. Students, in the context of their science investigations, generate questions that motivate their need to learn new mathematics. They become mathematically empowered as they develop ownership of the mathematics they need to solve their problems. In the spirit of the National Council of Teachers of Mathematics Curriculum and Evaluation Standards (referred to hereafter as the NCTM Standards), the students use problem-solving approaches to investigate and understand mathematical content (NCTM, p. 75) and learn to use and value the connections between mathematics and other disciplines (NCTM, p. 146). The Forest Watch program also provides meaningful opportunities for science and mathematics teachers to work in partnership to enhance student understanding of both disciplines. This article focuses on the mathematics that arises from student activities in the Forest Watch program.
Science and mathematics are a hands-on exploratory adventure for many students who learn science and mathematics and conduct research in partnership with their teachers and professional scientists. The Earth Day:Forest Watch Program (referred to hereafter as Forest Watch), provides an example of one such opportunity (Rock and Lauten, 1996). Developed at the University of New Hampshire, this NSF funded program introduces elementary, middle, and secondary school students to field, laboratory, and satellite-data analysis methods for assessing the health of a region-wide network of Eastern White Pine (Pinus strobus) forest stands. Students in the Forest Watch program participate in three types of activities, each patterned after activities conducted by scientists involved in the University’s ongoing research on assessing the health of coniferous species sensitive to various air and soil pollution factors (Lauten and Rock, 1992; Moss and Rock, 1991; Rock et al., 1993). The activities include: 1) white pine
1Oyster River
High School, Coe Drive, Durham, New Hampshire 03824. 2Complex Systems Research Center, Morse Hall, The University of New Hampshire, Durham, New Hampshire 03824. 3Correspondence should be addressed to Gary N. Lauten, Complex Systems Research Center, Morse Hall, The University of New Hampshire, Durham, New Hampshire 03824.
White Pine Stand Assessment: On-Site Investigations and Data Collection In the white pine stand assessment component of Forest Watch activities, students establish a 30 me45 1059-0145/98/0300-0045S15.00/0 © 1998 Plenum Publishing Corporation
46
Lauten and Lauten
Fig. 1. A photograph of students in the Forest Watch Program laying out their school’s study plot.
ter by 30 meter biology study site in a white pine stand near their school. The dimensions of the study site correspond to the 30 meter by 30 meter spatial resolution (pixel size) of the Landsat TM data. As they mark their site, students find themselves having to construct a square with sides aligned in a northsouth and east-west orientation. Students ask: What properties of a square will help us accomplish this
task?; How will we assure that the lengths of the sides of our study site will be 30 meters?; How will we be sure the sides of our square are north-south and eastwest? How will we be sure the square is in fact square? While developing answers to their questions, the students come to understand and apply geometric properties and relationships (NCTM, p. 112). Once they have established their study site, the students begin to investigate the land cover. Forest Watch activities encourage students to ask: How many trees are within the plot?; How tall are they?; How can we make meaningful measurements at our site?; What is the percentage of canopy cover? As the need arises, the students learn to use the tools of the forester. They measure the circumference of trees at breast-height. They ask questions such as: What is the standard of measurement called “breastheight”? How do we determine the diameter of the tree from the circumference? How should we represent our data? Would a stem-and-leaf plot or bar graph provide us with a better way to represent and analyze our data in order to compare it with data obtained by students at other sites in New England? These questions also reflect the NCTM Standards which suggest students develop formulae and proce-
Fig. 2. A scatter plot of student-collected data of tree height versus tree diameter.
Mathematics for the Student Scientist
47
Fig. 3. A photograph of a student using a densiometer to measure forest canopy cover.
dures for determining measures to solve problems (NCTM, p. 116) and construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations (NCTM, p. 167). To answer one of their questions, students make a clinometer and use it to measure indirectly the height of trees. Right triangle trigonometry empowers the students to collect data that at first seemed unobtainable. Students soon find themselves asking: How should we describe the “average” height of the trees? When is the median more appropriate than the mode? Is there a relationship between the diameter of a tree at breast height and the height of the tree? How could we find out? Students make inferences that are based on data they have collected and their analysis of the data (NCTM, p. 105). Asked to determine the percentage canopy cover, students make a tubular densiometer by placing crosshairs on one end of a cardboard tube. They then walk along the diagonals of their study site, stopping at equally spaced positions, and decide whether or not the intersection of the crosshairs reveals a leaf or the sky. Students then ask themselves: How can we use the data we have collected to estimate the percentage of canopy cover? In answering this question, the students extend their understanding of the concept of area and develop new procedures for determining measures to solve problems (NCTM, p. 118). Laboratory-Based Assessment of Damage Symptoms: More Precise Measurement and Data Analysis After students have investigated the more visually obvious features of their study plot, their teacher
Fig. 4. A photograph of a white pine needle-cluster exhibiting chlorotic mottle.
encourages them to look more closely. Students collect pine needles and observe the geometric patterns of needle growth. They compare the numerical patterns of needle clusters of different species of pine to aid in correct species identification. Looking more closely at the needles, the students observe mottling, an indication of plant stress. Students use randomization techniques to obtain needle samples. Precise measurement leads to determination of percentage of mottling on needles from north, south, east, and west facing needle clusters. Students take other precise measurements, such as the wet-weight/dry weight of the needles, which they can also use to determine the health of the trees on their study site. Again, the students must decide how best to represent their data for analysis and comparison with data collected at other sites. In these activities students make and interpret measurements, represent and draw inferences from data, and begin to understand sampling and recognize its role in statistical claims (NCTM, p. 167).
48
Lauten and Lauten
Fig. 5. A Landsat TM satellite image of Sandwich, MA.
Image/Analysis of Landsat TM Data: The Satellite Perspective
Soon after students begin their on-site monitoring of the health of the white pine in their region, they view their locale from a more global vantage point Forest Watch students use the image processing software program MultiSpec to investigate their region from a “satellite view”. MultiSpec, a freeware program developed by Dr. David Landgrebe at Purdue University, was developed as a NASA-sponsored learning tool. Using MultiSpec, students analyze 512 x 512 pixel Landsat TM data sets that include their school. Student excitement becomes contagious as they recognize buildings in their town and their school. The 30 meter spatial resolution of Landsat
TM images allows students to identify major roads, large buildings, bodies of water, tree-covered regions, and cultural features. Students view the image in “true-color”, a visually realistic presentation obtained when reflected red visible light is assigned to the red color gun of the computer monitor, reflected blue visible light is assigned to the blue color gun, and reflected green visible light is assigned to the green color gun. Most students in the 1990’s comfortably explore on computers. They quickly discover that they can continue “zooming” into the satellite image until all they see on the screen are larger and larger squares. What’s going on here?, they ask. “Zooming” provides motivation for the study of similarity. In a class activity, teachers ask students to set the zoom scale fac-
Mathematics for the Student Scientist
49
Fig. 6. Two satellite images with vertices and sides of triangles delineated. The image on the left has a zoom scale factor of “1”, and the image on the right has a factor of “2”.
tor on the image to “1”, place a transparency over the screen, and mark three easily distinguishable points on their screen. Students then “zoom in” to a scale factor of “2” and mark, with a different color, the new location of the same three points. Connecting the three points for each color reveals two similar triangles on the transparency. What is the relationship between the lengths of the sides of the triangles? What is the area of each triangle? If the length of the side of a triangle is doubled, how does that affect the area of the triangle? What if the activity is repeated with different scale factors? Students explore transformations of geometric figures (NCTM, p. 112) and deepen their understanding of relationships between length and area. After students notice the change in area of the region enclosed by the marked points on the transparency, they are asked to focus on the area of the land enclosed by the triangles. The actual area on the earth of the region between the marked points on the image has not changed even though the region appears larger on the computer screen. Students are asked to characterize this change in perspective. How does this change in perspective relate to a distance above the earth from which the region is viewed? Here students visualize the change in perspective while developing spatial sense (NCTM, p. 112).
As the students work with zooming, questions arise concerning the actual area of regions they observe on their image. Students begin to ask: How
Fig. 7. Materials for a student activity—gray-scale converter, scene of a suburban house, and a blank grid.
50
Lauten and Lauten
Fig. 8. A selection graph for the marked pixel on a Landsat TM image.
much land does the wooded region near our school cover? How large is the “footprint” of our school? They use their knowledge about the number of pixels in their image (512 x 512) and the spatial resolution, or size of these pixels, (30 meters by 30 meters) to devise various ways for determining area. Some students “zoom-in” and count pixels; others place a transparent grid over the computer screen; and others average the areas of rectangles that over and under-estimate the area. The mathematics in these efforts is rich—spatial sense, unit conversion, distance measurement, area of regular and irregular regions, and estimation and approximation skills, all in a real-world context (NCTM). Becoming more comfortable with the satellite image by now, students begin to question how satellite sensors represent what they “see”. Student subsets of Landsat TM data contain reflectance levels in 5 bands, or discrete regions, of the electromagnetic spectrum; red, green, and blue reflected visible light, which our eyes can see, and two bands of reflected infrared energy, which our eyes cannot see. The data from each
pixel of the students’ image contains a relative brightness level from 0 to 255 (0 is no reflectance and 255 is total reflectance) for each of the five bands. Students return to the question, Why does the image become just a collection of colored squares when we zoom in? Questions such as these lead to investigations centered on how satellites collect and transmit data. In these investigations they digitize brightness levels to develop an understanding of how satellite sensors digitize reflectance levels. Students draw a “gray-scale” view of a scene and then draw a grid over the depicted scene to simulate pixels. Using their “gray-scale converter”, they assign a reflectance value to each “pixel”. Students can be heard asking and saying, Which shade is this?; How do I decide what grayscale value to assign if there are several shades in the grid square?; Are grid squares like the squares I saw on the computer image when I zoomed-in? So this is what image resolution is about—now I get it! Students then “decode” the image on a new grid. More advanced students can pursue how transmitted digital data can be error corrected. The mathematics involved
Mathematics for the Student Scientist
51
Fig. 9. A selection graph for the marked cluster of pixels on a Landsat TM image. in coding, decoding, and error correcting digitized information is mathematics for and of the twentieth century (NCTM, p. 176). When students do the previous activity, they are, in effect, looking at the reflectance values for only one band of the electromagnetic spectrum. However, their data subset contains information for five bands. They return to the computer, select one pixel, and choose “Selection Graph”. A graph appears on the screen that provides the reflectance value for each band in that pixel. Interpretation of graphical data becomes the mathematical issue here. What do the reflectance values for the pixel tell us? After the previous Forest Watch activities, students have a deeper understanding of reflectance levels and the digitization of data. Students then learn that band 5, the mid infrared band, can be used to analyze the water content in vegetation. Band 4, the near infrared band, can be used to analyze the biomass or amount of plant material present in vegetation. With their scientist partner and their teacher, students investigate the selection graph associated with objects they recognize. They find that cultural
features such as bridges have high reflectance levels in all three visible bands and water has low reflectance levels in all bands (both the visible bands and the infrared bands), indicating that water absorbs visible light and infrared energy. Vegetation regions have high reflectance levels in band 4, near infrared energy. When students select a region of their image, the Selection Graph gives maximum, minimum, and average values for each band and the standard deviation. Students learn to make sense of the concepts represented on the graphs as well as the graphs themselves. As the students become better able to read and interpret Selection Graphs they learn to use that information to identify various features on satellite images (NCTM, p. 105). For example, one student visually inspected balsam fir on the North face of Mount Moosilauke as part of a summer environmental program project that used Forest Watch materials. Back at the laboratory, working with a satellite image, he found he needed to classify the types of trees on the south face. He described his analysis of the data in the paragraph
52
Lauten and Lauten
Fig. 10. Selection graphs that appear in student’s report.
below. His work reveals that he had learned to make inferences and present convincing arguments based on the analysis of real data (NCTM, p. 105).
fir. Once compared, these two reflectance graphs demonstrate great similarity. One can say, all right this spectral signature was Balsam Fir so other areas with the same signature are also most likely fir.
The two graphs above) show the areas where the samples were taken from. From [on-site] observations, we know that the area [on the north face] were balsam
After work such as the preceding with satellite images, the students are ready for more sophisticated
Mathematics for the Student Scientist
53
Fig. 11. A diagram that depicts an algebraic transformation.
analysis in order to diagnose the health of trees. The scientist partner and teacher help the students establish a new set of data that they call "band 6". Band 6 data contains the ratio of band 5 data to band 4 data. Normal healthy plants have a low ratio of water content to biomass: The 5/4 ratio is below .5. Band 6 ratio values closer to one are an indication of plant stress. Students learn to interpret the ratio in complex, real situations, as illustrated in the following student report.
When the image comes up on the computer, there are all these colors that appear as different things. You have to know what they represent in order to learn anything from them. Basically, in my project, the 6, 4, 3 band combination has three colors. The green represents the healthy vegetation. The pink, or real light red represents mineral substances. This can either be roads, buildings, beaches, rocks, or other manmade things that come from minerals. The last color is the most important to my project, which is red. Red represents damaged vegetation [high 5/4 ratio regions]
By comparing these two bands, which is called a five over four ratio, you can determine the healthiness of the vegetation. When we looked at the graphs we could determine if the [trees in the area were] healthy or not depending on the height of the peak at channel four and also depending on the 5/4 ratio. After looking at the images, the graphs, and visiting the area, we concluded that the area directly on the reservoir is stressed, but as the vegetation gets farther from the water, it becomes more healthy.
The student then went on to analyze a site he had visited and had viewed on the computer image.
Students also use color to enhance visually their understanding of the satellite image and the implications of the ratio values. In the following project, the student assigned band 6, the 5/4 ratio band, to the red color gun of the computer monitor; band 4, near-infrared, to the green color gun; and band 3, visual red light, to the blue color gun.
The Mount Washington site had a considerable amount of damage, mostly around the higher elevations. Damage was visible somewhere around four to six thousand feet. This is the alpine garden area. The damage isn’t severe, but it is definitely noticeable. I think that the damage may have to do with the delicate vegetation that lives there. There is a Jot of bare rock on and around the peak, but this is a lighter pink, not to be confused with the deep red damaged vegetation.
Students in the Forest Watch program are soon ready to consider how data sets are transformed to enhance analysis. Students are presented with an image that appears almost monochromatic, regardless of what color guns students assign to the bands.
54
Lauten and Lauten
Fig. 12. Unstretched (left) and stretched (right) Landsat TM images of Beverly, Massachusetts.
When the question arises, What’s wrong with this image?, the teacher or scientist partner explains that the image had not been “stretched”. In reality the raw satellite data for each of the bands is clustered within a small portion of the total range of brightness values (0-255). In order to enhance the feature definition in an image, the data must be algebraically stretched. Stretching an image involves a linear algebraic transformation of the data, mathematics accessible to first year algebra students. These students are familiar with the slope-intercept equation of a line, y = a + bx. Using the maximum and minimum reflectance values of the data values in each band on the image, students are able to “stretch” the data so that the reflectance values are distributed in a wider data range. The magnitude of the transformation becomes the slope or b value. The b value is defined by the ratio (new maximum—new minimum)/(old maximum—old minimum). The initial value, a, or “y-intercept” is the new minimum. The difference between each reflectance value to be transformed and the old minimum is the x value. The transformed x value is y. For example, if reflectance values between 10 and 70 are to be stretched to values between 5 and 245, the magnitude of the transformation is (245-5)7(70-10) or 4. To stretch a reflectance value of 35, a student would first subtract 10 (the old minimum) from 35 to obtain 25. The student would then
multiply 25 by 4 (the magnitude of the transformation) and obtain 100. Finally the student would add the 100 to 5 (the new minimum) to obtain 105 (the new reflectance value). The algebra equation is y = 5 + 4x . Forest Watch students learn the algebraic transformation by participating in a game-like activity in which they manually stretch values. When they understand the process, the students assign values to MultiSpec and have the program stretch the image. In learning to stretch data values, students are helped to make connections between transformations of geometric figures and algebraic transformations of numbers—all within the visual context of satellite images (NCTM, p. 112, p. 150). Figure 12 illustrates an “unstretched” and “stretched” image of a region surrounding Beverly, Massachusetts.
SUMMARY
This article is intended to illustrate the school mathematics that arises naturally in the context of investigating and monitoring the health of a local forest. This mathematics is accessible to most middle and early high school students and represents school mathematics as envisioned in the NCTM Standards. Through their inquiry-based science investigations, Forest Watch students learn to represent and interpret data and to make convincing arguments based
Mathematics for the Student Scientist on careful data analysis. The students use proportional reasoning as they work with ratios, similarity, size, and scale. They develop deeper understanding of the concept of area in the context of finding the area of irregularly shaped regions at their study site and on the computer image. Geometric size transformations encountered in “zooming” are extended to algebraic transformations that relate directly to student work with linear functions, an important first year algebra concept. The Forest Watch program provides many opportunities for mathematics-science interdisciplinary student learning while solving problems essential to planet earth. The mathematics provides a vehicle for students to quantify, represent, analyze, and interpret meaningful, real data.
ACKNOWLEDGMENTS The authors wish to acknowledge the National Science Foundation (NSF) (ESI-9452792) for providing the funds to expand and enrich the Forest Watch materials and the participating Forest Watch teachers and students who have made this program possible. Darien Lauten, one of the authors, would especially like to thank the Consortium of Mathematics and its Applications for inviting her to write with the NSF
55 sponsored ARISE 9’11 Mathematics Core Curriculum Project. Through this work she deepened her interest in the mathematics that arises in the context of student investigations in real world situations. Some activities similar to those described in this article appear in ARISE curriculum materials. Finally the authors wish to thank the NASA Space Grant Program for its continuing support of Forest Watch (NGT’40030).
REFERENCES Lauten, G. N. and Rock, B. N. (1992). Physiological and spectral analysis of the effects of sodium chloride on Syringa vulgaris. Proceedings of I GARS S’92 1: 236’238. Moss, D. M. and Rock, B. N. (1991). Analysis of red edge spectral characteristics and total chlorophyll values for red spruce (Picea rubens) branch segments from Mount Moosilauke, New H a m p s h i r e , USA. Proceedings of IGARSS’91 3: 1529’1532. National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Rock, B. N. and Lauten, G. N. (1996). K-12th Grade Students as Active Contributors to Research Investigations. J. Science Education and Technology 5(4): 255’266. Rock, B. N., Skole, D., and Choudhury, B. (1993). Monitoring vegetation change using satellite data. In Solomon, A. and Shugart, H. H. (eds.), Vegetation Dynamics and Global Change, Chapman and Hall, New York/London, pp. 153’167 plus plates.
