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Mabel Sykes: A Life Untold and an Architectural Geometry Book Rediscovered – Articles

Maureen T. Carroll (University of Scranton) and Elyn Rykken (Muhlenberg College)

In her four decades as an educator, Sykes wrote over a dozen articles on the teaching of geometry, algebra, and trigonometry. A list of these articles can be found in the bibliography. Spanning over three decades, Sykes’ sustained scholarship demonstrates her lifelong passion for teaching mathematics, and her willingness to freely share her thoughts about mathematics education beyond her own classroom. A common theme in Sykes’ articles is an emphasis on intentionality in the organization and presentation of mathematics within the high school curriculum, specifically algebra and geometry. For example, she quoted a portion of an address given by John Merle Coulter (chair of the University of Chicago's Department of Botany from 1896 to 1925) in both her 1903 article, entitled “Aims and Methods in the Teaching of Plane Geometry” [27], and her 1927 article, “Some Pedagogical Aspects of Geometry Teaching” [33]. The following is from the 1927 article:

Many years ago, Dr. J. M. Coulter, the botanist, read a paper on the “Mission of Science in Education.” Something that he said in this paper has a direct application to much of our geometry teaching today. He contrasted two types of instruction in the following words: “In the one case, the facts are presented in the helter-skelter fashion, solid and substantial enough, but a regular mob, with no logical arrangement, no evolution of a controlling idea. ... In the other case fewer facts are presented, but they are the important ones, and marshalled in orderly array, battalion by battalion. They move as a great whole towards some definite object. ... Instead of a level plain, there are mountain peaks and valleys. There is a perspective: there are vistas from every point of view.”

This quote encapsulates one of Sykes’ major tenets in her philosophy of teaching. In her 1905 article, “Radical and Conservative Elements in the Teaching of Mathematics” [28], she similarly affirmed that "what is of value in any course, is the point of view, not the mass of details, which must of necessity be forgotten.” To this end, she proclaimed that “the right of the teacher to think for himself and plan his own course is a divine right.” For her this included the right to change the order of the topics in order for the students to better understand their importance.

Sykes’ ideal geometry student would have the ability to solve real-world problems and follow logical arguments. Concerning the latter, as she wrote in 1903, “the ability to stick to the subject, to answer a question directly, to make or intelligently to participate in a logical argument, are necessary even in ordinary conversation.” It is in geometry class that “the pupil gets his first and often his only clear view of the logic of argumentation.” Regarding the former, in 1905 she wrote, “This does not mean that it is necessary for the pupil to retain text-proofs longer than the few recitations that they are under discussion. The important question is: Can the pupils apply theorems and general principles in inventing proofs and solving problems?” Sykes expanded on and illustrated what she meant by this in her 1906 article, “Some Practical Applications of Elementary Geometry” [29], where we find ten examples from land surveying and mechanical drawing. We show one of these examples in the "Sykes’ 1906 Persian Arch Construction" section of this article. She returned to this theme in her 1927 article [33] where she admonished textbook authors:

Consider almost any chapter in almost any geometry text with the following questions in mind: Is there a controlling idea in this chapter? ... Has each controlling idea its fundamental theorems or definitions, and are the other theorems and exercises grouped about these fundamental facts and made subordinate to them? In other words does the careful classification of material extend beyond the mere division into chapters, down to the last detail? Is this organization so clear and evident that the pupil is conscious of it?

Clearly she approached textbook authoring with these overarching ideals and found that others were not as intentional in their choices.

Other articles moved beyond this theme as she explored college entrance examination data and the wide range of abilities students bring to the mathematics classroom. In her 1917 article, “The Case Against High School Mathematics” [31], Sykes defended high-school mathematics teachers against attacks based on the high failure rates in mathematics for college entrance exams. She suggested that mathematics had been unfairly singled out since most other subject areas had similar failure rates. She also provocatively noted, “one cannot help wondering why all of the mathematics teaching in the country should be condemned from the results of examinations that are evidently intended to exclude young people from college instead of admitting young people to college.”

In her 1932 article, “The Problem of Individual Differences in Mathematics Classes” [36], Sykes described her study that “was undertaken in connection with sabbatical leave from one of the Chicago high schools and covers over fifty schools, twenty-five of which were visited by the writer.” In this and her follow-up article, “Differentiated Assignments” [37], she detailed methods that address the variety of abilities and learning styles teachers encounter in the classroom. The “Ability Grouping” method, where students are divided into classes that move at different paces according to their perceived abilities, is called tracking today. In the “Differentiated Assignments” method, students in the same class completed different assignments. Each student was allowed to chose between four “contract” levels, A, B, C, or D, each with a different set of problems ranging from easiest (D) to most difficult (A). The “Individual Instruction” method was suggested for “slow,” “failing,” or “repeater” pupils. Given the time and labor-intensive nature of individual instruction, it is not surprising that Sykes wrote in 1932, “the crowded condition of the programs of both teachers and pupils, however, limits the possibility of such work. One city conducts classes on Saturday mornings for pupils who need help on class work or who have work to make up. Attendance is voluntary. [36]” Regarding this research, Sykes did not draw any sweeping conclusions. Instead, she concluded by noting “further study might bring to light other variations of the general plans reported” [36].

Maureen T. Carroll (University of Scranton) and Elyn Rykken (Muhlenberg College), "Mabel Sykes: A Life Untold and an Architectural Geometry Book Rediscovered – Articles," Convergence (February 2020)