College Algebra: Synthetic Division

Author introduction: Both are mathematicians writing a new College Algebra textbook to address the changing needs of their students.  Both are concerned about keeping a rigorous textbook despite the changes being made.  Both wrote several mathematics textbooks, including Calculus.

James Taylor was a Mathematics professor at Colgate University from 1869; it was one of 24 American math textbooks – written by American and published in America. Ross Middlemiss taught at Washington University (in Missouri) from 1929 – 1969 and had a background in engineering mathematics.

Taylor (1892) and Middlemiss (1952)

I chose these College Algebra textbooks because the authors seemed to have similar motivations (see below) for writing, but in the 60 years between the texts, learning theories had changed and new recommendations (for secondary mathematics anyway) had been published:

  • NEA. (1894).  Report of the Committee of Ten on Secondary School Studies: With the Reports of the Conferences Arranged by the Committee
  • MAA. (1923). The reorganization of mathematics in secondary education: a report of the National Committee on Mathematical Requirements under the auspices of the Mathematical Association of America, Inc.
  • NCTM. (1940).
    Fifteenth Yearbook: The Place of Mathematics in Secondary Education.

Secondary education had opened to a more diverse population across that 60 years, and so a more diverse population was entering universities. The chart below illustrates that as the number of high school graduates in the United States increased, so did the number of enrolled college students.
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Students with Diverse Backgrounds

Both authors had different definitions of “changing population”, but both wrote a new College Algebra to address similar issues.

Taylor (1892) mentioned the number of new sciences being added as fields of study and how mathematics must compensate by paring down content; that is, mathematics must be concise without losing rigor.

Middlemiss (1952) described the his new students: from the sciences, engineering, liberal arts, sociology, etc. and how he hoped to teach them all mathematical thinking and values of the mathematical discipline even if they wouldn’t need to remember the details of the content. He tried to appeal to them without losing rigor.

Both were careful to include most proofs, although both recommended the reader look at other references for complicated proofs.  Both asked student questions that are proof-like, or led to general conclusions.

His book seems written more to the student where Taylor seems written more to a teacher.  Middlemiss (following recommendations) used multiple representations, many illustrations, many application problems (connections to real world) as well as historical connections and very detailed explanations.

Middlemiss also gave a sample time table for the teacher and argued that it should be easily divided into 48 lessons of one hour each (in class) and two hours each (out of class).  Taylor argued that he carefully wrote his sections so that the teacher could change ordering without loss of students’ understanding.

Multiple Representations

The difference in use of multiple representations between the books is clear: Taylor (1892) provided one illustration in his textbook (support of his development of differentiation) while Middlemiss (1952) included multiple illustrations (triangles, rectangles, graphs) to support mathematical ideas. Middlemiss first included graphical representation with his introduction of Functions and Graphs, but then required students to provide graphical and/or tabular support for their arguments throughout the remainder of the text. Middlemiss also sometimes asked students to solve equations using two different methods, which seemed new compared to Taylor.

Synthetic Division, or Horner’s Method

William George Horner (1786 – 1837) was a British mathematician.  Both books refered to a numerical method for solving equations as “Horner’s method.” Taylor called synthetic division “Horner’s Method”.

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Taylor presents synthetic division using an example of “standard” long division for a general cubic polynomial and then shows the process using the general coefficients.  I’ll admit – I’m not entirely sure I understand his explanation, but the process seems the same. He then gives three examples using numeric values before his student question section.Algebra-03

Middlemiss also began with the standard long division example. He then removed the x terms and showed only the coefficients. He then showed a simplified version, removing repeated terms and the coefficient “1” for x^2.

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Finally, Middlemiss gave a step-by-step rule referring back to the original method and provided two two examples before moving on. As a curiosity, notice that both Taylor and Middlemiss had their divisor on the left. Taylor had divisor on left for synthetic and standard division, while Middlemiss had it on the right and explained only that we move it over for no other reason but convention. It seemed that the main difference was that Taylor was trying to be brief, while Middlemiss tried to be “friendly” to teachers or students who did not have a strong mathematical background.