Foreword to the Instructor

Yoshiwara's Trigonometry

At its foundation, the textbook you are currently reading has been built from Katherine Yoshiwara's Trigonometry textbook. She has graciously offered her text to the education community as an OER (Open Educational Resource) to be used and modified in any fashion the educator sees fit. I chose to start with her textbook as a basis because of the measured approach and her wonderful descriptions.

Unfortunately, there were many decisions in the design of the Yoshiwara text for which I was not comfortable, and unraveling these design choices without rewriting the textbook was impossible. Some examples include:

• Waiting until halfway through the textbook to introduce radian measure
• Graphing trigonometric functions using only degree measure
• Waiting until the middle of the textbook to introduce proofs in Trigonometry
• Introducing the Laws of Sines and Cosines immediately after the students' first experience with Trigonometry (via right triangle trigonometry)
• Never introducing half-angle identities, power reduction formulas, or the sum-to-product and product-to-sum identities

Despite these shortcomings, I truly like the style the Yoshiwara text brings to the mathematical landscape and, as such, it is the basis of this textbook.

OpenStax Algebra and Trigonometry

The folks over at OpenStax have done an admirable job at playing a leadership role in the OER community. I chose not to use their Algebra and Trigonometry textbook as a basis for this book due to voice and rigor (I have used their text in the past for Trigonometry and found that the levels and expectations were too low); however, I did supplement this text with some of their examples when it came to the trigonometric identities material—the Yoshiwara text was lacking in this area (literally not covering some of the major identities).

Stitz and Zeager's Precalculus

The Stitz and Zeager Precalculus is an OER textbook that treats mathematics instruction in a very classic, and rigorous, style. Since the Stitz and Zeager is a Precalculus textbook, the only content from that textbook used to augment the textbook you are currently reading comes from the final two chapters on Trigonometry. I could not use those two chapters as a basis of a single-course Trigonometry textbook due to the terseness of the text and the assumption of a pre-exposure to Trigonometry; however, I did incorporate some of their examples into this textbook. Additionally, I used their textbook to fill in the gaps that were missing from the Yoshiwara textbook (namely for sinusoidal applications).

McKeague and Turner's Trigonometry

Trigonometry by McKeague and Turner is completely copyrighted and, as such, there is nothing in this textbook directly from their textbook; however, of all the texts I have read covering Trigonometry, the McKeague and Turner textbook is the best (by far). As such, this textbook is heavily influenced by their style and the flow of this text is close to their approach.

The Result

If you have read the previous information, then you might be concerned about the "voice" of the text sounding as if it has been written by five authors—this is not the case. I have rewritten the vast majority of the text, aligning notations, pedagogical styles, voice, and approach to my own. Moreover, at least a dozen sections that were missing (or not adequate) from the Yoshiwara text have been written solely by me.

What are you looking for?

• Are you looking for a textbook that reads like a "how-to" book for solving math problems?
• Do you search long hours into the night for that one textbook that contains all the "recipes" so that students don't have to understand the math?
• Are you satisfied with the vast majority of algebra textbooks at the higher education level that are watered down?

Then, go no further! Literally... GO NO FURTHER... at least, not with this textbook. Close it now and move on to anything else. If I still have your interest, then please read on.

The decline of modern mathematics education

In modern day undergraduate math textbooks, it has become an increasing issue that content is being watered down and treated superficially so that more students can "get it." Unfortunately, this is having the opposite effect. Students have become professionals at mimicking the shallow procedures given them and are subsequently losing out on the logic-building skills mathematics has to offer. Content has been made to be rapid fire, with little-to-no depth of investigation. The effect of this shallow-depth treatment is that students give up quickly and are less likely to focus on truly understanding the process. Contrary to building a society of critical thinkers, we are engineering a culture of followers insistent upon knowing the shortcuts. Philosophical questions of why have been replaced with utilitarian queries of how. This is not to say that the application of mathematics is secondary to theoretical mathematics; however, instead of an integrated focus between the two approaches, many authors have chosen to throw out the theory for fear of overwhelming the student.

Adding fuel to this inferno of the uneducated is the lack of time we, as professors, have in the classroom with our students. It seems that every few years an hour or so is lost from our schedules while another topic is introduced to the curriculum. It has become too easy to give students the "recipe" for solving a problem and move on to the next topic all-the-while hoping that they can figure out why it works on their own.

A proposed reinvention—the 30-foot-deep pond

If the United States is to compete in the global market, we must somehow increase the level of creativity of our students. Rewarding perseverance and reconsidering how we have approached the treatment of mathematics is paramount. The days of holding back on number theory because it is "too abstract" must fall by the wayside. The frenetic pace of shoveling one chapter per week into a student's brain needs to be slowed; thus, allowing the student to begin a meaningful meander through the beautiful realm of mathematics. The development of concepts can no longer be piecemeal and staccato. Topics must be introduced in a logical fashion and a thorough exploration of the subject should be allowed. Chapters should not be designed to be "consumed" in a certain, short period of time. Instead, they should organically continue until their focus has been thoroughly explored.

We stand at a point in history where information flows to us in immense quantities and the possibilities for an educated society are staggering. However, in these days of "big-box store education," students are treated more like cattle than the possible leaders of our future. The days of sitting in the round and discussing topics from a philosophical standpoint are gone and the term "professor" has lost its meaning. We no longer profess—we regurgitate. Professing is left for late night YouTube channels dedicated to "educating" in 10-minute videos. The Socrates of today is a one-way conversation brought to you in high definition with limited commercial interruption. The word "educate" in media circles is synonymous with "entertain."

We must invest time in honoring the theory and history of mathematics while satiating a student's desire to see math in action. We should treat mathematics as an exploration in understanding rather than just a lesson in computation. We must do what we ask our students to do and think critically about why particular topics have been taught in certain courses. Only after detailed consideration of such issues should we redesign our curriculum in an intelligent manner that properly conveys the importance of mathematics in our daily lives.

The core concept behind the design of this textbook is encapsulated in the opening statement of this Foreword—"welcome to my 30-foot-deep pond." I firmly believe that the "deep pond" approach is the absolute best method to develop true understanding of the theory and usefulness, as well as the beauty, of mathematics. In this text, I treat each topic thoroughly and with such great detail that the student could be trusted to know not only how to compute trivial "math textbook" problems, but also how to apply their knowledge to the real world.