We follow the education theory of learning-by-doing, which in our case means using your computer and our R or Python code to interact with our book and modifying our computer codes and websites to analyze your own data and generate corresponding figures. Learning-by-doing is the core methodology of the project-based learning, which may be attributed to the progressive education of John Dewey (1859– 1952), an American philosopher and educator. Dewey felt that the experience of students and teachers together yields extra value for both. Instructors are not just to lecture and project authority, instead they are to collaborate with students and guide students to gain experience of solving problems of their interest. Although Dewey’s education theory was established initially for schoolchildren, we feel that the same is applicable to undergraduate and graduate students. Our way of learning-by-doing is to enable students to use R or Python code and other resources in the book and its website to reproduce the figures and numerical results in the book. Further, students can modify the computer code and solve their own problems, such as visualizing the climate data in a similar format and of their own interest, or analyzing their own data using a similar or a modified method. Thus, audience interaction is the main innovative feature of our book, allowing the audience to gain experience of practicing, thinking, applying, and consequently understanding. The ancient Chinese educator Confucius (551–479 BC) said, “I hear, and I forget; I see, and I remember; and I do, and I understand.” Although John Dewey and Confucius were more than 2,000 years apart, they had a similar philosophy of learning-by-doing.
As illustrated in Figure 0.1, our pedagogy has three stages: do, reflect, and apply. Coauthor S. S. P. S. practiced this pedagogy in recent years. He presents a question or a problem at the beginning of a class. Then he asks students to orally ask the same question, or describe the same problem, or answer his question in their own words. For example, why is the tiny amount of carbon dioxide in the atmosphere important for climate? Next, he and students search and re-search for data and literature, work on the observed carbon dioxide data at Mauna Loa using computer data visualization, and discuss the structure of greenhouse gasses whose molecules have three or more atoms. To understand the data better, they use more data analysis methods, such as the time series decomposition. Next, he encourages his students to share this experience with their grandparents, other family members, or friends. Finally, students apply the skills gained to solve their own problems with their own data, by doing homework, working on projects, finding employment, or making innovations. In this cycle, students have gathered experience and skills to improve their life and to engage with the community. In the short term, students trained in this project-based cycle of learning-by-doing have a better chance of becoming good problem solvers, smooth story narrators, and active leaders in a research project in a lab or an internship company, and consequently to become competitive in the job market. In the long term, students trained in this cycle are likely to become to life-time learners and educators. John Dewey said: “Education is not preparation for life but life itself.” We would like to modify this to “Education is not only preparation for a better life, but also is life itself.”
Dewey’s progressive education theory is in a sharp contrast to the traditional learning process based on the logic method, which aims at cultivating high-achieving scholars. The commonly used pedagogy of lecture-read-homework-exam often uses the logic-based approach. The climax of logic-based education is that the instructors have the pleasure of presenting their method and theory, while students are so creative that they will produce a new or a better theory. Many outstanding scholars went through their education this way, and many excellent textbooks were written for this approach. However, our book does not follow this approach, since our book is written for the general population of students, not just for the elite class or even scholars-to-be. If you wish to be such an ambitious scholar, then you may use our book very differently: you can read the book quickly and critically for gaining knowledge instead of skills, and skip our reader–book interaction functions.
Our pedagogy is result-oriented. If using car-driver and car-mechanic as metaphors, our book targets the 99% or more who are car-drivers, and provides some clues for the less than 1% of the audience who wish to become car-mechanics. Good drivers understand the limits of a car, are sensitive to abnormal noise or motion of a car, and can efficiently and safely get from A to B. Therefore, this book emphasizes (i) assumptions of a method, (ii) core and concise formulas, (iii) product development by a computer, and (iv) result interpretation. This is in contrast with traditional books, which are often in an expanded format of (ii) with many mathematical derivations, and challenge the mathematical capability of most readers. Our philosophy is to minimize the mathematical challenge, but to deepen the understanding of the ideas using visual tools and using storytelling experience. Our audience will be able to make an accurate problem statement, ask pointed questions, set up statistical models, solve the models, and interpret solutions. Therefore, instead of aiming our training at the few “mechanics” and thus incurring a high risk of failure, we wish to train a large number of good “drivers” with a large probability of success.
Here, we provide materials based on our Story-Picture-Observe-Review-Tell (SPORT) pedagogy to teach or learn mathematics effectively and efficiently, with depth and insight. Our approach is to use a single figure to explain all the definitions, fundamental identities, and their proofs of trig functions. This approach can be summarized as One-Figure-For-Every-Relation (OFFER).
SPORT is a pedagogy for mathematics education and is derived from learning-by-doing. SPORT assumes that students should first learn materials useful to them. Almost all the useful mathematics definitions, theorems, formulas, and algorithms have fantastic stories behind them. Each story must have a pictorial illustration. Hand-drawing such a picture can enhance the student’s understanding of the mathematical theory. Teachers can inspire students to observe the picture(s) and lead them to reinvent the theory in classrooms. Students can independently redraw the figures, observe them, and reach their own conclusions and extensions beyond textbooks. This S-P-O process can be reviewed or repeated many times to enhance learning. Ancient Chinese educator Confucius said that “To learn the new, review the old.” Finally, students can master the materials and tell stories to others. This completes the entire SPORT learning cycle.
The SPORT mathematics education, in nature, requires instructors to focus on the most fundamental and most useful materials. One-Figure-For-Every-Relation (OFFER) is a concrete approach to using the SPORT pedagogy. As an example of OFFER, this article uses only one figure to explain and derive all the fundamental trigonometric formulas. In another article, we will use a single formula to explain and derive all the fundamental theorems of calculus. In this way, we can liberate students from massive textbooks of more than 1,000 pages and lead them to see the insight, power, depth, and beauty of mathematics.
Trigonometric functions, particularly the sine and cosine, were invented by the Indian mathematician and astronomer Aryabhata (476 - 550 CE). He abstracted the bow-arrow shape in archery into a triangular geometry as shown in the above figure.
Aryabhata regarded the rigid bow of an archery set as a part of a circle, and the elastic string as a chord. When it is pulled, the upper half-string and the arrow form an angle θ, as shown in the figure. The original position of the upper half-string is called the half-chord, ardha-jya, the Latin spelling of the Sanskrit term ardhajyA. Here, ardha means half, and jya chord. In Sanskrit mathematics writing, jya was used, and ardha was omitted. Aryabhata included this in his 499 CE book named “Aryabhatiya” written in Sanskrit. In the book, he included a table of jya values, the first known sine table. The meaning of jya, i.e., the half-chord, got distorted, but it is still used today in our books, class- rooms, and computer programs. Sanskrit jya was phonetically transcribed to Arabic as jiba around 800 CE. In written Arabic, short vowels are often omitted. In this case, jiba became j-b. Later Arabic readers and the 1100 CE European translators mistakenly regarded the vowelless jb as the commonly used word jaib, since jiba was a rare Arabic word as a technical term. This misreading led to a very different meaning, jaib meaning “bosom” of a female body, “bay” in the bay window, “pocket,” “bend,”, “gulf,” or “fold,” whose Latin correspondence is sinus, meaning a curved surface. Later, sinus was anglicized into sine, i.e., the function sin(x).
REFERENCES:Nickerson, S., T. L. Tequida, and S.S.P. Shen, 2023: Teachers partner with scientists to learn the relevance of mathematics through climate research. Notices of the American Mathematical Society, 70, 614-618. DOI: https://doi.org/10.1090/noti2664.
Page, E., S.S.P. Shen, and R.C.J. Somerville, 2024: Can we do better at teaching mathematics to undergraduate atmospheric science students? Bulletin of the American Meteorological Society. DOI:https://doi.org/10.1175/BAMS-D-22-0245.1
