J. B. J. Fourier is not the most famous mathematician in history. James Newman’s The World of Mathematics allows him two lines of citation in the index, while Gauss claims 43 lines. Nevertheless, Fourier is a constant, if unseen, presence in modern technology. Among the few lines devoted to Fourier is a succinct statement of his achievement: "[Fourier’s] theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves¾ in brief, every curve can be built up by piling up waves." (Jeans, in Newman, 1956, p. 2298). Like many succinct statements, this one is not quite right. But if we substitute for "every curve" the statement "most curves engineers would be interested in" and in place of "exactly reproduced" put "closely reproduced," then we have a good enough statement of the theorem for our purposes here. For mathematical accuracy combined with readability, Bressoud’s A Radical Approach to Real Analysis (Bressoud, 1994) is highly recommended.

Figure 1 shows how "piling up waves" approximates a curve. In this case the curve, chosen for simplicity, is given by the expression
y = x² - 1 . The approximating collection of waves is given by

Clearly, this piling up of waves requires some analytic machinery. Note: having made the point about "curves" and "waves," I will now generally refer to all these objects as "functions." Thus the caption for Figure 1 refers to functions, rather than curves and waves.

Until the middle of the 20th Century, Fourier’s name was probably most likely to be linked to his interest in heat conduction¾ which he began in 1807 and published in 1822¾ showing how a trigonometric series can be used to analyze heat conduction in solid bodies. But the theory is more general than a theory of heat. The above quotation by Jeans, for example, is in the context of an article on the mathematics of music and sound.

In 1965, James W. Cooley and John W. Tukey published their mathematical algorithm which has become known as the fast Fourier transform (FFT). FFT applications have co-evolved with computer technology and greatly expanded the applications of Fourier theory. Some examples of FFT applicability include: linear systems, antennas, optics, random processes, probability, and quantum physics. Most image processing techniques, including medical scanning systems such as ultrasound and CAT scans, and signal processing, for example the recent NASA-JPL pictures of the Martian landscape, rely on Fourier-based techniques.

A "Zero-Based" Curricular Approach

Glenda Lappan, president of the National Council of Teachers of Mathematics, has pointed out the degree to which technology allows students to tackle advanced mathematics:

"We also know that technology has shaped our world. In Colorado, seventh graders work on projects in a computer lab with technological tools that were unimagined in schools just 10 years ago…. In Wisconsin, students use calculators and computers with dynamic geometry drawing tools to "see" geometry in ways undreamed of when I was a high school geometry teacher. In Pennsylvania, students use symbol manipulation programs as tools to help solve problems I could not tackle until I was in graduate school. Many such mathematical experiences, all unheard of when the original Standards were published, are now happening for students all across the continent." (Lappan, 1998, World-Wide Web URL http://www.nctm.org/news-bulletin/1998/09/1998-09.lappan.html)

Although symbolic mathematical programs such as Mathematica have powerful Fourier applications built into them, these tools won’t make sense to students unless a careful curricular framework is created. The traditional textbook path to Fourier applications has recently been revised and rewritten by a group of international linguistics students. Their book, Who is Fourier? (Language Research Foundation, 1995), can be considered one of the first of the "zero-based" texts. The book contains a consistent set of concepts leading from simple right-angle trigonometry to the Fourier Transform. The authors have included exercises that use paper cutouts and simple paper-and-pencil problems. My goal has been to supplement these manipulatives with computer-based lab experiences. These are described in the next section.
 

Examples of Computer-Based Lab Projects

Using Mathematica, we can generate and plot interesting combinations of sine and cosine functions. Substituting the Mathematica command Play for the command Plot lets us hear these combinations, as shown in Figure 2.

Figure 2: Screen capture of Mathematica code for displaying and playing combinations of functions.
 
 

Listening to brain waves

In similar fashion, once we have collected data, we can experiment with the Plot-Play commands to listen to the data. We have done this with data collected using the LOGAL EEG probe described below. In a short video called "The Wired Fiddler Listens to His Brain Waves," we combined the sounds of an EEG scan with an improvised electronic violin solo [Figure 3].

Figure 3: Screen capture from The Wired Fiddler Listens to His Brain Waves
 

We extended this project one additional step by having a musician compose a score based on the brain-wave and violin themes[Figure 4]. The result provides a good proof-of-concept for making interdisciplinary connections.

Figure 4: Portion of score created from sound track of The Wired Fiddler Listens to His Brain Waves.

The LOGAL EEG Probe

The LOGAL company of Cambridge, Massachusetts, produces a variety of equipment and curricular materials for interactive exploration in science. Figure 5 below shows a student using the EEG probe and accompanying computer software. Figure 6 shows a photograph of the computer screen. The probe has an adequate sampling rate (200 Hz) for analyzing the EEG wave. Most classroom computers will allow a memory buffer sufficient to capture at least 10 seconds of a wave, which is enough for demonstrating effects such as evoked potentials. The sample lesson used with the software is included in Figure 5.

Figure 5: LOGAL EEG probe and Beginning Assignment

Comparing Sound Waves and Brain Waves

The significant difference between using probes for sound waves and for brain waves lies not so much in the relative complexity of the respective sources, but in the fact that we have no "gold standard" for brain waves. We can generate, with tuning forks or computers, mixes of sounds. We can also verify certain types of biological data, such as heart rate, through systems independent of the CBL. The best we can do to verify the EEG probe is to look for evoked potentials, usually through eye blinks, and we have been able to do this. We could obviously try to compare results from our simple EEG probe with those obtained by a certified technician with clinical equipment, but at this stage we have not sought access to such resources.

Figure 6: Photograph of computer screen showing EEG pattern obtained from LOGAL probe.

We can show students the results of using Fourier analysis to find the underlying frequencies that make up a complex signal. We can synthesize such a signal, both for sound data and for hypothetical brain wave data, based on what is known about the types of waves that make up the EEG signal. Figure 7 shows a synthesized wave, formed by adding fundamental frequencies of about 14 Hz and 32 Hz, with some random noise included. Figure 8 shows the power spectrum of the wave, using the MathematicaFourier function. Figure 9 shows the Fourier function applied to data obtained with the LOGAL EEG probe.
 
 

Figure 7: Screen capture of the plot of a combination of fundamental cycles with random noise.

Athough we may not be able to generate a consistent "standard EEG" as we can, for example generate a standard 220 Hz. tone with a tuning fork, we believe that it can be valuable for students and teachers to get familiar with problems inherent in studying noisy data. In this regard, we find that physics teachers, being used to calibrating instruments and separating signals from noise, may be somewhat ahead of mathematics teachers, as shown by cartoon in Figure 10.

Conclusion

Using real EEG data gives students a chance to look at an open-ended scientific question. The problems inherent in analyzing EEG data can provide convincing motivation for developing the mathematical machinery that comprises Fourier analytic methods.

Our pilot project "Making Waves in the Mathematics Classroom" has recently received funding from the  Eisenhower Program for Mathematics and Science Education to do the following:

1. Assemble and train four action teams, including in-service teachers, a university methods teacher and a UN-L graduate assistant.

2. Action teams will present the "Making Waves [Anticipating Fourier]" package to college mathematics methods students.
The work includes translating Mathematica notebooks into TI-92 code.

3. Follow-up and evaluation on related activities in high school classes.

4. Summative evaluation and revision for further investigation.

We will be reporting the outcomes of this project in a future conference paper. We will generate interim reports through our Making Waves Website.

References

Bressoud, D. (1995). A Radical Approach to Real Analysis. Washington, D. C.: MAA Press.

Jeans, J. (1956). Mathematics of Music. In J. R. Newman (Ed.), The World of Mathematics (p. 2298). New York: Simon and Schuster.

Language Research Foundation (1995). Who is Fourier? A Mathematical Adventure. Boston: Language Research Foundation.

Misulis, K. E. (1994). Spehlmann’s Evoked Potential Primer. Newton, MA: Butterworth-Heinemann.