5.3.- Método de medida

Conclusion and Recommendations

During the time period this study encompasses, trigonometry education in the United States underwent considerable changes. In the late eighteenth century,

trigonometry was taught as a topic in a larger mathematics course from a textbook that had few pedagogical tools and only the most basic of trigonometric formulas.

Trigonometry was taught exclusively using the line system. By the end of the nineteenth century, the ratio system had taken over completely and trigonometry was taught as its own course that covered the topic extensively with many applications to real life. Textbooks were full of pedagogical tools.

The path that the teaching of trigonometry took through the late eighteenth and nineteenth centuries did not always move in a linear fashion. Sometimes, trigonometry education stayed the same for a long time and then was suddenly changed, but other times changes happened more gradually. There were many international influences, and there were influential Americans and influential American institutions that changed the course of trigonometry instruction in this country.

This chapter brings together all the facts of history that have been laid out in the previous chapters and explains why and how the teaching of trigonometry changed over time.

Research Questions Answered

1. How did trigonometry textbooks change from 1776-1900:

a) in content? What topics were covered during this time period, and how do the topics change over time?

During the time period of this study, trigonometry textbooks changed dramatically in their content. In the late eighteenth and early nineteenth centuries,

trigonometry textbooks focused on computing trigonometric functions. Logarithms had a great deal of importance in trigonometry textbooks because they made trigonometric calculations much simpler, and the calculation of trigonometric tables was a main focus of the textbooks. Applications to trigonometry emphasized surveying and navigation. At this time, the line system was the only means of defining the trigonometric functions.

In the antebellum period, trigonometry textbooks grew in size and content. During this time, the discussion of trigonometric functions as periodically-changing functions became an important topic. Also, the ratio system appeared and grew in importance. Applications of trigonometric functions remained focused on surveying and navigation. Logarithms and trigonometric calculations were still given places of

importance as well.

In the late nineteenth century, trigonometric textbooks expanded even further. The ratio system completely dominated textbooks‘ definitions of the

trigonometric functions, although the line system was still found in some textbooks. Proving trigonometric identities and solving trigonometric equations became important

topics. The calculation of trigonometric tables decreased in importance as the Taylor series for sine and cosine were used to calculate tables to arbitrary accuracy, but logarithms remained important for calculations. Topics in trigonometry also became much more varied as trigonometry prepared students not only for surveying and navigation but also for calculus.

1. How did trigonometry textbooks change from 1776-1900:

b) in approach? Namely, in what order are the topics presented, and with what emphasis on each topic?

The major trend in the approach of trigonometry textbooks was increasing prevalence of algebraic topics and algebraic methods. In the late eighteenth and early nineteenth centuries, trigonometry was an extension of geometry. The proofs of the theorems were given by geometric methods, and the trigonometric functions were defined geometrically, as lines on a circle.

Over time, algebraic methods made an appearance. At first, the trigonometric ratios (the basis of analytic trigonometry) were given at the end of the textbook. As time went on, most textbooks began to move the trigonometric ratios earlier and earlier, and then they used the ratios to prove subsequent theorems, especially when the proofs were easier using the ratio system. Finally, the ratio system became the way to define the trigonometric functions, and many more theorems were proved using algebra rather than trigonometry.

Eventually, the ratio system and the line system changed places. At the beginning of the nineteenth century, the line system defined trigonometry and the ratio system was given at the end of the text or left out entirely. By the end of the nineteenth century, the ratio system defined trigonometry and the line system was given later or left out entirely.

1. How did trigonometry textbooks change from 1776-1900:

c) in pedagogy? Particularly, what types of questions and problems are posed to students within the textbooks, are answers and/or solutions given, and how many? What other pedagogical techniques are used?

From 1776 to 1900, the pedagogy in textbooks made great advances. In the late eighteenth century, overt pedagogical tools were basically absent from textbooks. Exercises for students to complete with either solutions given or answers only given (usually some of each) became common by the beginning of the nineteenth century. Setting off formulas and rules in special type that students needed to memorize was also common during this time. Moving forward in the nineteenth century, the numbers and different types of exercises grew substantially. Exercises that applied trigonometry to real-life situations appeared and became more prevalent throughout the nineteenth century.

By the late nineteenth century, trigonometric textbooks contained

abundant exercises, about ten times more than they did in the antebellum period, ranging from converting between different types of angle measurements and solving triangles to