Introduction
    Modern spreadsheets have the ability to make powerful mathematical concepts accessible to students at a younger age than ever before. Contours and 3-D graphing are topics that were previously reserved until well into the first year of college calculus. Three dimensional graphing now can be successfully taught to middle school students, with the assistance of a computer spreadsheet. This combination computer/hands-on activity exposes students to numerical and graphical representations of data on the same spreadsheet ìpage,î and forces them to make connections between the two forms of data. The spreadsheet creates an environment in which students have more of an opportunity to reflect and abstract than with older teaching tools and thus can provide greater potential to develop concepts (Piaget, 1970; Von Glaserfeld, 1995). Using a spreadsheet to teach mathematics can potentially provide a greater number of examples, more explanations by the teacher and/or students, and a greater variety of examples, thus leading to increased student understanding (Piaget, 1970; Von Glaserfeld, 1995; Vygotsky, 1986).

    This lesson will probably take several days. The lesson first emphasizes a conceptual understanding of 3-dimensional graphing, with the teacher guiding students through a discovery lesson using pre-made examples on the computer spreadsheet. Next, students do a hands-on activity with the computer spreadsheet working in teams at the computers. Finally, a hands-on small group computer activity using topographical maps that applies the concepts to a real world setting. For the final project, students will create a surface plot from a topographical map. I was introduced to using spreadsheets for 3-D plots by the Systemic Initiative for Montana Mathematics and Science project; more ideas for 3-dimensional graphing lessons can be found in The SIMMS Project Integrated Mathematics textbook series (MCTM SIMMS, 1995). Iíve done variations of this lesson with students from middle school through precalculus.

MAKING THE 3-DIMENSIONAL SURFACE CHART

To create a chart example such as the pyramid for use with the whole class, a matrix of data is entered into the spreadsheet, this simply involves typing the numbers into the spreadsheet cells. Next create a 3-D surface plot of the matrix using the built-in charting features of the spreadsheet (See Figure 1). In Microsoft Excel (1995), make sure to select the option for data series in columns instead of rows for a more accurate correspondence between the numerical data and the graph. If you use the default value of data series in rows , the graph will still look the same but the right side of the graph will be created from numbers on the left side of the matrix; the picture will be reversed. The original graph now can be rotated to visually correspond with the data (See Figure 2). My surface plot is at an ìangleî because I prefer it that way (See Figure 2a), some teachers or students may want to rotate it so the tip of the triangle points toward the top of the computer screen, as in the matrix (See Figure 2b). Most graphing software also has other controls available for setting the viewpoint and rotation of axes.

    Since I only have four numbers in the matrix-0,1,2,3, and three intervals, I only want three intervals in my chart legend and only three colors on the chart. This can be accomplished by changing the intervals between tickmarks on the z-axis, (See Figure 3 for directions). The cells in the matrix are then colored to match the colors of the chart (See Figure 4). This simple yet powerful technique of using corresponding colors for the graph and matrix soon makes it obvious to most students how the graph is connected to the numbers in the matrix.

    The chart and matrix in Figure 4, are now ready for the teacher to use for a classroom presentation. The spreadsheet provides an environment in which students can form mental links between the different representations of the same data. A constant flow of studentsí attention between representations becomes possible. The teacher can change numbers in the spreadsheet and the spreadsheet program will make the corresponding changes in the graph immediately upon entering the data (See Figures 5 and 6). Students can see connections and cause and effect. In Figure 6, two 0s were changed to 4s in the upper right-hand corner. Notice that colors do not change automatically in the matrix while the colors do change in the graph; the color coordination between the matrix and the graph must be performed manually. Making changes in the numbers of the matrix is an excellent way for teachers to help those students who cannot easily see the ties between the two data forms.

    If you are not happy with the default colors of the chart, they can easily be changed in Excel by formatting the individual legend keys. Individual legend keys are formatted by first getting into chart editing mode by double-clicking on the chart, and then selecting the individual legend keys while holding down the command key.

MAKING THE CONTOUR CHART

    After students are comfortable with how the numbers correspond to heights above the x-y plane, a contour map can easily be created by copying, pasting and then rotating the 3-D surface chart (See Figure 7). Even though a separate contour map can be created by going through charting process again, copying, pasting and then rotating seems to have a larger impact on students. When students can actually see one chart turned into another, the connections between representations of the data are retained, and most students immediately understand how to interpret the contour lines. Again, numbers can be changed by the teacher and the class can watch the corresponding changes in both the matrix, the 3-dimensional surface chart, and the contour chart. This type of exploration, manipulation, and multiple representation can facilitate the development of mathematical concepts (Bruner, 1962; Piaget, 1970; Von Glaserfeld, 1995; Vygotsky, 1986).

    The spreadsheet can automatically label the axes on the right and bottom edges of the graph, and the matrix was manually labeled on the left and top edges. One obvious method of making the correspondence between labeling on the different representations more straightforward would be to label both sides of the matrix (See Figure 8). It is important that students understand that the labels can be placed on either set of edges. However, Excel will not recognize labels on the right and bottom edges of the matrix and will not automatically put them in the chart. The type of labeling of the axes and the visual/mental conflict it can produce in students is a direct result of the technology used, and an example of how technology can affect communication and learning, in unanticipated but important ways. This is in contrast to a traditional blackboard approach to a 3-dimensional system when axes are always labeled in the same way. Hoyles, Sutherland and Healy (1991), have suggested that the computer can play a crucial role in generating the necessary cognitive conflict for individual conceptual development to occur.

    I like to compare the spreadsheetís representation of a three dimensional surface with other forms of axes, so students can be sure to have a grasp of the complete three dimensional world in which they are working. Figures 9a-c, show a similar graph, but with larger x,y and z axes ranges and different styles of axes, it is not obvious to all students that the previous graphs we have been working with are merely an octant from a complete three dimensional world. Figure 9d shows the same graph but from a different viewpoint. Unfortunately, changing the scale on the axes for 3-dimensional surface charts cannot be easily accomplished in some spreadsheets; Figure 9 was generated using Maple V Release 4 (Waterloo Maple, 1996).

THE FIRST STUDENT PROJECTS

    After an introduction and explanation of three dimensional systems, students are given matrices and asked to give a description of a graph. Then the problems are reversedóstudents are given graphs and asked to give the matrix. Examples of these types of problems are provided in Figures 10 and 11. This can be done as a whole class activity or in groups. The teacher and/or students can use the computer to check their hypotheses and revise them if necessary. Most students are quick to grasp the concepts and can make accurate predictions with very little practice.

    To check for student understanding, and to provide practice in generating these charts, the studentsí first assignment will be to generate charts of some simple surfaces, such as a rectangular pyramid, some stairs, and then a bowl (See Figure 12). These beginning problems will reveal to a teacher if students grasp the concepts of topographical views, contour lines, and their relation to the matrix. I prefer to have students working in pairs for these beginning activities. For younger students and maybe even some older students, a good preliminary activity would be to use Cuisinaire rods to build representations of surfaces, such as stairs or pyramids, before starting in the more abstract environment of the computer.

    The spreadsheet as used in this lesson, along with the teacher, can create an environment that reveals and develops studentsí intuitions. Researchers (Hoyles and Noss,1992; Noss, in press-b; Sutherland,1993; Sutherland and Rojano,1993) suggest that technology can act as a mediating factor between informal mathematical understanding and the development of formal mathematical concepts. Technology may create an environment in which students can use ideas before having fully discriminated the formal mathematical relationships involved. Developing mathematical understanding and meaning in the learner is still directed by the teacher, but may be mediated by the technology. Through these simple beginning projects such as graphing pyramids, bowls or stairs, students can begin to explore and discover, thus developing understanding and making connections between representations. When students have the chance to start on these projects, it is important to provide them with an ample amount of time for discovery, and to let them make mistakes without interference. Hoyles, Sutherland and Healy (1991), suggest that learning mathematics at the computer can provide an environment where a student can engage in private individual discovery and conflict, unheeded by concerns of exposing themselves to social judgments, as often occurs in a typical public classroom setting. Bruner (1962) asserts that students free to engage in this type of discovery are in a position to experience success and failure not as reward and punishment, but as information.

CREATING A 3-DIMENSIONAL SURFACE CHART FROM A TOPOGRAPHICAL MAP

    The final project may take several days; it applies the previously learned concepts and combines them with a more traditional hand-on approach. The objective is to make a three dimensional surface plot from a real topographical map. Students can choose their favorite geographical area or they can be assigned an area. Topographical maps are easily obtained from sporting goods stores, the world wide web or the Forest Service (See Figure 13). It is a good idea to laminate the original topo maps so they can be used over again. After students are provided with a copy of a topographical map to work with, they are given a sheet of graph paper on clear acetate, and a sheet of regular graph paper to record data. The acetate with the grid can be placed over the topo map and the height of the closest line for each cell can be recorded, either on paper or temporarily on the acetate with erasable markers (see Figure 14). It was originally a studentís idea to use the clear acetate and I've found it very effective. With older or more advanced students, I like to supply students with the maps and grids, tell them to make a 3-dimensional surface, and see what types of solutions they come up with and how they divide up the tasks. With younger students more direct instruction may need to be used. Iíve found that groups of at least three have worked well when doing the final project, with different students performing different tasks to complete the work. Once students have recorded the data, they enter it in the spreadsheet and create the 3-D surface of their topo map (See Figure 15). The 3-dimensional surface chart can be copied, pasted and rotated to provide a colored contour map, which can be compared to the original (See Figure 16).

    When using this activity in the classroom, it is a good idea to have the topographical maps enlarged before the data is recorded. This limits the problems of having multiple contour lines in a single grid square. For schools with more advanced technology Iíve found it fun to place the surfaces into 3-D rendering programs and apply different surfaces or use software with more advanced graphing and matrix capabilities. Computer algebra systems or 3-D rendering programs can create the 3-D surfaces from the matrix or from a color-coded contour map (See Figure 17).

    This method of producing three dimensional surfaces is not intended to produce professional quality maps, and the quality of the surface is limited by the software used. As with many computer activities, when we try to model continuous data such as elevation with discrete measuring tools, we must realize in advance that our accuracy is limited. Students will begin to intuitively question how to get smoother, more realistic graphs which can lead into discussions of modeling continuous phenomena with a discrete machine. This is not a drawback to using spreadsheets for these types of charts, but rather an opportunity to promote classroom discussion and understanding. A compromise must also be reached by the teacher and student as to how many points will be recorded on the final project. More points will result in a smoother graph, but will require more time and computer power.

    This activity is excellent for a multidisciplinary project. The matrices in the problems presented here represent elevation, but is certainly not limited to that. A 3-D surface plot or scatterplot could be used to represent many different forms of data such as temperature, air pollutants, or any function which is dependent upon two variables. Data collected from science experiments is well suited for this type of analysis and will further enhance students' understanding of three dimensional surfaces where they must interpret and analyze the surface plot in relation to real world variables.

    It has been my experience that it is important that students are fluent with the spreadsheet before trying to use it for analyzing real data. Otherwise too much attention is focused on how to use the computer and software, and the lesson deteriorates into a computer lesson, rather than a mathematics and science lesson.

BIBLIOGRAPHY AND REFERENCES

Bruner, J. S. (1962). On knowing: Essays for the left hand. Mass: Harvard University Press.

Cuisenaire Company of America, P.O. Box 5026, White Plains, NY 10602-5026; (800) 237-3142.

Hoyles, C. (1985).What is the point of group discussion in mathematics. Educational Studies in Mathematics, 16, 205-214.

Hoyles, C. (1992). Mathematics teaching and mathematics teachers: A meta-case study. For the Learning of Mathematics, 12 (3), p. 32-44. White Rock, BC, Canada: FLM Publishing Association.

Hoyles, C. & Noss, R. (1992). A pedagogy for mathematical microworlds. Educational Studies in Mathematics, 23, 31-57.

Hoyles, C., Sutherland, R., Healy, L. (1991). Children talking in computer environments: New insights into the role of discussion in mathematical learning. In K. Durkin & B. Shire (Eds.), Language in mathematical education: Research and practice (pp. 162-175). Bristol, PA: Open University Press.

Johnson, D. & Johnson, F. (1991). Learning together and alone (3rd ed.). Englewood Cliffs, NJ: Prentice-Hall.

Maple V Release 4, Student Edition (1996). [Computer software], Canada: Waterloo Maple.

Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277-289.

MCTM SIMMS, (1996). Integrated mathematics: A modeling approach using technology. Levels 1-6. Needham Heights, MA: Simon and Schuster.

Microsoft Excel, Version 5.0, (1995). [Computer software], Redmond, WA: Microsoft Corporation.

National Council of Teachers of Mathematics (1989a). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Noss, R. (in press-a). Computational environments in mathematics education [Special issue]. Educational Studies in Mathematics. Available WWW: Hostname: ioe.ac.uk Directory: www.ioe.ac.uk/rnoss/index.html

Noss, R. (in press-b). Meaning mathematically with computers. In P. Bryant & T. Nunes (Eds.), Children doing mathematics. Available WWW: Hostname: ioe.ac.uk Directory: www.ioe.ac.uk/rnoss/index.html

Noss, R. (1995). Computers as commodities. In A. DiSessa, C. Hoyles & R. Noss (Eds.), Computers for exploratory learning. Berlin: Springer-Verlag. Available WWW: Hostname: ioe.ac.uk Directory: www.ioe.ac.uk/rnoss/index.html

Noss, R. & Hoyles, C. (1993). Bob: A suitable case for treatment. Journal of Curriculum Studies, 25 (3), 201-218. Available WWW: Hostname: ioe.ac.uk Directory: www.ioe.ac.uk/rnoss/index.html

Piaget, J. (1955/1969). The language and thought of the child (D. Coltman, Trans.) . Cleveland, OH: The World Publishing Company.

Piaget, J. (1970). Science education and the psychology of the child. New York: Orion Press. (Original work published 1969)

Robertson, Bruce (1988). How to draw charts and diagrams. Cincinnati: North Light.

Sutherland, R. (1993). Consciousness of the unknown. For the Learning of Mathematics, 13 (1), 43-46.

Sutherland, R. & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behavior, 12, 353-383.

Von Glaserfeld, E. von (1995). Radical Constructivism: A way of knowing and learning. Washington, DC: The Falmer Press.

Vygotsky, L. (1986). Thought and language. Cambridge, Mass: The MIT Press.

Vygotsky, L. (1978). Mind in society. Cambridge, Mass: The MIT Press.
 
 

Objectives
The Student Will Be Able To: