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<ul><li><p>Training elementary mathematics teachers in a one-semester courseAuthor(s): WILLIAM B. CRITTENDENSource: The Arithmetic Teacher, Vol. 21, No. 5 (MAY 1974), pp. 428-432Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190931 .Accessed: 10/06/2014 16:02</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.</p><p>http://www.jstor.org </p><p>This content downloaded from 195.78.108.112 on Tue, 10 Jun 2014 16:02:55 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/action/showPublisher?publisherCode=nctmhttp://www.jstor.org/stable/41190931?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>Training elementary mathematics teachers in a one-semester course </p><p>WILLIAM B. CRITTENDEN </p><p>An associate professor in education at Houston Baptist University, William Crittenden is the only member of the faculty teaching mathematics education. He has sought a model that would yield the best possible results in training elementary teachers in one semester. He has also been a leader in the Houston area in the evolution of performance-based training. </p><p>W hen the writer joined the faculty of Houston Baptist University in 1968, he was given the assignment of prescribing the mathematics education training experiences for prospective elementary teachers. Since the college is primarily for the liberal arts and the curriculum includes a large number of required cources, there was only one three-semester-hour course available in which elementary teachers could be pre- pared in mathematics. A "prepared" teacher was defined as one who had - </p><p>achieved a prescribed level of mastery of mathematics content; </p><p>accumulated a theoretical and empirical repertoire of teaching strategies, tech- niques, aids, and activities; and </p><p>exhibited a positive attitude toward mathe- matics as a field of study. </p><p>The challenge was obvious and almost overwhelming. How could all three vital objectives be achieved in 45 hours of class- work during a single semester while the students were involved in from four to six other courses? </p><p>No other mathematics courses were required for elementary education majors at Houston Baptist University. The variability in requirements from college to college in </p><p>America in this field was highlighted by Fisher (1967) who found that 40% of 78 teacher-training institutions graduated ele- mentary teachers with three semester hours' credit (in mathematics or mathematics education) or less, and 90% required six hours or less. </p><p>The problem at Houston Baptist Univer- sity became acutely clear as early attempts to solve it resulted in failures. If content were emphasized and methods ignored, students later complained (justifiably) that they had received little or no preparation in how to teach mathematics. On the other hand, when methods were emphasized and con- tent ignored, supervisors of student teaching complained that the students so prepared made embarrassing content errors in their teaching efforts. When scales were used on a pretest-posttest basis to measure positive gains in attitude toward mathematics, it was found that few students showed changes of attitude and, when they did change, the change was just as apt to be negative as positive. </p><p>It was decided that more time for the training was absolutely essential. Even though it was "not possible to award more than three hours' credit, a laboratory was scheduled for a second hour three times per </p><p>428 The Arithmetic Teacher </p><p>This content downloaded from 195.78.108.112 on Tue, 10 Jun 2014 16:02:55 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>week. Thus, the class time was increased from 45 clock hours per semester to 90. Simultaneously, arrangements were made with a nearby public elementary school to permit classroom teaching experiences col- laterally with campus seminars. </p><p>Since this methods course occupied a position in the training sequence between the classroom observations of the sopho- more education major and the student teaching of the senior education major, a descriptive name for this intermediate activity was needed. The program was referred to as "student preteaching in elementary mathematics," and the par- ticipants became known as "student pre- teachers." </p><p>Students in the program were given pretests in mathematics content, attitudes toward mathematics, and knowledge of mathematics teaching methods. Those who failed to achieve 90% level of mastery of content were given individually prescribed programs in the areas of greatest weakness. No formal lectures in mathematics content were delivered during the semester, but free time was allowed during seminars to provide opportunity for learners to ask for needed explanations. All inquiries were satisfied. Checks of students' work on the programs were made as a control measure. </p><p>Pretest results revealed that 95% of the students were deficient in their knowledge of elementary school mathematics; more than 50% of them disliked or feared mathematics and lacked confidence in their ability to teach it; and all of them had a limited knowledge of mathematics teaching methods. Most of them reported that their referent source for teaching techniques was to be found in the way their elementary school teachers had taught them. </p><p>With excellent cooperation from the laboratory school staff, the student pre- teachers were assigned to classes that had been divided into small groups on the basis of mathematics ability. The host school had reorganized schedules to permit the intra- classroom group teaching experiences to occur simultaneously, thus permitting the </p><p>preteachers to form carpools. The labo- ratory teaching proceeded two days per week with a two-hour seminar on the college campus one day each week. </p><p>The on-campus seminar was devoted to building and demonstrating teaching aids. The elementary school mathematics cur- riculum was summarized in ten broad concept areas, and a team report was prepared for each concept area. Games, activities, developmental techniques, mani- pulative and demonstrative aids, and evalua- tive methods were surveyed. Each team produced a 30-page handout booklet con- taining drawings and descriptions of the teaching aids associated with each concept area. At the end of the course, each student had a collection of ideas for aids and techniques in teaching elementary mathe- matics that could be useful as a reference source. </p><p>The student preteachers were given teaching responsibilities immediately, each one teaching a group of pupils from the beginning day of his field experience. This precipitant beginning was ill-advised. The cooperating teachers complained that the preteachers made arithmetic errors in their teaching, and they resented being made responsible for teaching mathematics con- tent to the preteachers. In addition, the cooperating teachers expected that the preteachers would use huge quantities of teaching aids, which they didn't; pre- teachers tended to rely predominantly on workbooks and duplicated worksheets, especially in the beginning of the course. </p><p>The student preteachers were panic- stricken at being given such heavy re- sponsibilities so quickly when they were well aware of their state of unpreparedness. Since most of them feared or disliked mathematics anyway, the required teaching of it in the presence of experienced teachers, principals, or college observers, with no preliminary preparation acted as a crystalliz- ing agent on their attitudes, rather than as a stimulus to positive change. </p><p>Through trial and error, the student preteachers did learn to adjust their teaching </p><p>May 1974 429 </p><p>This content downloaded from 195.78.108.112 on Tue, 10 Jun 2014 16:02:55 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>methods to the levels of their pupils. They did learn to manufacture and use teaching aids. And they did learn more mathematics content as a result of the seminar and laboratory experiences. Posttests showed significant gains in knowledge of mathe- matics content and teaching methods. However, the attitudes of the preteachers toward mathematics regressed, being more negative at the completion of the course than at the beginning. </p><p>The effect of the program upon the pupils in the host school was indeterminate. Subjectively, everyone involved in the program agreed that pupils responded positively to the use of aids and games, but no evidence concerning their achieve- ment gains or losses was available. The cooperating teachers complained that there was a lack of continuity in the program, caused by the presence of preteachers only two days per week. They were not eager to continue the program in other semesters, even with revisions calculated to eliminate mistakes. </p><p>Even though the student preteaching program as described achieved two of the three essential objectives, it was declared a failure. To be considered "prepared," an elementary teacher must have knowledge of mathematics content, mathematics teach- ing methods, and a wholesome attitude toward the subject. The latter objective was still unachieved. </p><p>Determined to press forward in the search for a model which offered promise in achieving all three objectives, the writer and his colleagues in the Education Division analyzed the previous efforts and submitted a new proposal which they believed would eliminate the earlier flaws. It was decided to keep all factors in the training model con- stant except the entry to the full respon- sibility of teaching. What had proved to be a frightening experience because of the abruptness of entering the teaching process with no preparation might be turned into a constructive, gratifying experience by first preparing the preteacher. Thus, a phase-in process in four steps, with each new step </p><p>requiring a higher level of responsibility, culminating in full teaching during the last part of the program was designed. By so delaying the ultimate burden of teaching until the students felt themselves to be better prepared, it was believed positive attitude changes could be effected. </p><p>The Research Council of the University approved a grant of funds for the new project in the spring semester of 1971. The new proposal called for the controlled phase-in of the student preteachers to the actual teaching responsibility, and it pro- vided for an evaluation of the program effects on the pupils of the host classrooms. </p><p>Another school (Lakeview Elementary School, Fort Bend Independent School District, Sugarland, Texas) was enlisted to serve as a laboratory for the project. Careful orientation to the new program was provided the faculty of the new school, and volunteers were selected to serve as cooperating teachers. In addition, an equal number of control teachers were selected to provide data for comparisons of pupil achievement and attitude gains. </p><p>The actual experiences provided for the student preteachers were approximately the same as described in the earlier experiment, except for the phase-in sequence. The 16-week semester was divided into four phases of four weeks' duration each. (See fig. 1.) Phase I was the Preparation phase, and the group remained on the college campus meeting in seminars six hours weekly. Intensive work on mathe- matics content and teaching methods was accomplished during this period. The students were trained in construction of teaching aids. Practice in intraclassroom grouping and achievement of the "restaurant effect" (Creswell and Crittenden 1971) were undertaken. A conscious attempt to "shore up" the confidence of the preteachers was initiated. </p><p>Phase II of the project, labeled Observa- tion, involved the assignment of preteachers to classrooms where they observed the cooperating teachers and pupils during the mathematics classes. The preteachers were </p><p>430 The Arithmetic Teacher </p><p>This content downloaded from 195.78.108.112 on Tue, 10 Jun 2014 16:02:55 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>First training model (not recommended) </p><p>Student preteaching two days weekly for entire semester Seminar on content and methods one day weekly </p><p>0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Weeks </p><p>Second training model (recommended) </p><p>Phase I Phase II Phase HI Phase IV Preparation Observation Tutoring Preteaching </p><p>Campus Seminars Laboratory school Laboratory school Laboratory school full observations tutoring and teaching responsibility </p><p>teacher-aid, activities 6 hours weekly Weekly 2 days weekly 2 days weekly </p><p>Content and Content and methods Continuation Continuation of seminar methods seminar of seminar </p><p>1 day weekly 1 day weekly 1 day weekly </p><p>0 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Weeks </p><p>Fig. 1 </p><p>trained in the use of office machines, audiovisual equipment, and various teach- ing aids. Their seminars on the college campus were devoted to a continuation of skill development. </p><p>Phase III, the Tutoring period, called for four weeks of tutoring pupils on a one-to- one, one-to-two, or one-to-three basis. By this time preteachers had been assigned their groups and they began to become intimately acquainted with the pupils they would eventually teach. Also, the pre- teachers had molecules-of-learning packets (Crittenden 1971) ready for try out. During Phase IV, the Preteaching phase, the students had full responsibility for teaching their groups for two days per week for four weeks. </p><p>Results of the pretest-posttest data re- vealed that all three objectives for the course were attained. Students gained significantly in their knowledge of mathe- matics and mathematics teaching methods, and in addition a majority of them moved from negative to positive attitudes toward mathematics. </p><p>The actual classroom experience with grade school pupils is essential to the positive attitude change, but to assign the </p><p>subject students the full teaching respon sibility before they have developed self- confidence risks incurring the opposite result. </p><p>The ultimate dependent variables of grade-school pupils' (1) attitudes toward mathematics and (2) achievement while undergoing teaching by student preteaching teams were observed by comparisons be- tween control and experimental samples. The Lakeview project involved seven sec- tions of two matched samples of pupils i...</p></li></ul>