Fostering Professional Growth: A Criterion-Referenced Format for Student Teachers in Mathematics

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Fostering Professional Growth: A Criterion-Referenced Formatfor Student Teachers in Mathematics

Alan ZollmanC. Leiand Smith

Department of Curriculum and InstructionUniversity of KentuckyLexington, Kentucky 40506-0017

"The goal of teacher education is to ’light the path’ for thosewho follow, providing directions on how to plan and teachmathematics" (National Council of Teachers of Mathematics[NCTM], 1991, p. 160). Lighting the path for middle and highschool student teachers in mathematics is a formidable task.Communication, knowledge, and rapport are essential for theparticipants in the student teaching experience for thedevelopment of teachers of mathematics.

To aid in this sharing of communication and knowledge,most universities utilize a student teaching assessmentinstrument; however, for the mathematics student teacher,assessment instruments are frequently very nebulous. Neitherthe format nor the process is precise or developmental (Smith& Stevens, 1984). One major goal for assessment of studentteachers is to nurture the inexperienced teacher. Yet in middleand high school mathematics teaching, generic formats telllittle about the specific teaching behaviors for teaching schoolmathematics(NCTM, 1991). Thusasubject-specificassessmentfor school mathematics would be useful. The purpose of thisarticle is to provide a paradigm for the development of acriterion-referenced format to meet this exigency.

To help nurture (and help assess) mathematics studentteachers, one approach is for the supervising classroom teacherand the university coordinator to use a criterion-referencedformat instrument. A criterion-referenced format can serve sixfunctionsby: (a) providing areference; (b) setting expectations;(c) guiding the supervising teacher in systemic observation; (d)encouraging diagnostic feedbackforimprovement; (e)providinga base for dialogue among the student teacher, the supervisingteacher, and the university coordinator; and (f) promotingprofessional growth of all parties. In criterion-referencedformats a student teacher’s performance is observed on taskswhich sample the critical competencies of teaching the subject(Smith & Stevens, 1984). In light ofNCTM’s Curriculum andEvaluation Standards/or School Mathematics (1989), such acriterion-referenced evaluation format is beneficial. Manytimes during the student teaching experience, the supervisingteacher does not know how to specifically give advice, thestudent teacher does not know what to ask, and the universitycoordinator does not know what has been understood.A criterion-referenced format also has logistics merits as it

can reduce the time for complex reporting and analysis forassessment of student teachers. Thus, it is likely to be usedcorrectly in terms of its objectives by the supervising teacher

and university coordinator.

Criterion-Referenced Format Design

For any student teaching instrument design to be useful innurturance and assessment, all participant;, namely the studentteacher, the supervising teacher, andthe university coordinator;must be involved. At the same time, the format design formathematics studentteachersmustdeal with threedomains: (a)knowledgeofteachingpedagogy,(b)knowledgeofmathematics,and (c) knowledge of school mathematics (NCTM, 1991).Teaching pedagogy entails generic teacher competencies notspecific to the content area. Knowledge of mathematics dealswith content knowledge. Knowledge of school mathematicsinvolves the skills of communicating school mathematics toone who does not understand. Examples ofteaching pedagogyknowledge would be questioning strategies or classroommanagement techniques. An example of knowledge ofmathematics would be solving simultaneous linear equations.Todistinguish, an exampleofknowledgeofschool mathematicswould be demonstrating factoring trinomials using rectangularareaswhilerelating this understandingtothepreviousknowledgeof factoring whole numbers.

As shown in Figure 1, the design for a criterion-referencedformat could be viewed as two triangles intertwined with eachother, onefortheparticipantsandoneforthe areasofknowledge.All threeparticipantsmusthavestrong linesofcommunications.In addition, each participant must have a consciousness ofhowthe three areas of knowledge are related. The most growthwould be expected for the student teacher in each of the threeareas of knowledge. Still professional development may, andshould, occur for the supervising teacher and the universitycoordinator in these three areas. As Berliner (1986), Berliner etal., (1988), and Shulman (1986) discuss in their expert/noviceresearch, all participants can be learners from the strengths ofeach of the participants.

In setting up thecriterion-referencedinstrument,thereshouldbeadivisionbetweenabehavioroccurringandthequalityofthebehavior for each criterion (Smith & Stevens, 1984). To thispurpose, the criterion-referenced format could ask for: (a) thefrequency of the teaching behavior; (b) the quality of thebehavior; and (c) comments, supporting evidence, strengths,concerns, and/or suggestions on each criterion. The followingis a possible blueprint for each referenced criterion:

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Professional Growth

Figure 1. Design for a criterion-reference format.

PARTICIPANTSAREAS OF KNOWLEDGE

CRITERION-REFERENCED FORMAT

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Column Q: Quality0 = not appropriate

Column F: Frequency0 = not appropriate1 = seldom observed2 = sometimes observed3 = frequently observedComments: supporting

suggestions

1 = concerns about performance2 = standard performance3 = outstanding performance

evidence, strengths, concerns,

The comments section should not be underestimated as animportant aspect in the development process. In fact, anycriterion judged as a concern requires a suggestive action bediscussed with the student teacher. The major objective is toallow the student teacher to construct a self-identity as ateacher.Thus. mistakes in the teaching ofmathematics need not be seenas fatal; they can be viewed as opportunities for growth for thestudent teacher (ZoUman & Alien, 1990).

Inknowledgeofteachingpedagogy,therearc several availablegeneric teaching assessment instruments. Most of these havecriteria that can be listed under the areas of:

1. planning and preparation,2. managing for instruction,3. implementing instruction,4. evaluating student learning,5. administration, and6. additional professional responsibilities (ETS, 1990).In the area of knowledge of mathematics, NCTM in

Professional Standards for Teaching Mathematics (1991) andthe Mathematics Association of America (MAA) in A CallforChange: Recommendations for the Mathematical PreparationofTeachers of Mathematics (1991) have criteria can be listed inthe categories of:

1. understanding mathematical concepts, structures, andprocedures;

2. identifying and interpreting representations ofmathematical concepts, structures, and procedures;

3. reasoning mathematical problems;4. communicating mathematics;5. understandingandappreciating the nature ofmathematics

and the role of mathematics in culture; and6. developing a willingness to do mathematics.In the area of knowledge of school mathematics, the

Professional Standards for Teaching Mathematics (1991)describe instructional activities that can be listed under thecriteria of:

1. creating mathematical environments,2. teaching mathematical concepts and procedures,3. teaching mathematical connections,4. teaching problem solving,5. teaching mathematical communication,6. teaching mathematical reasoning, and7. promoting mathematical disposition, and

8. assessing students’ understanding of mathematics.Obviously, criteria in one area of knowledge overlap with

the other two areas. For example. Good, Grouws, and Ebmeier(1983) report that classroom organizational structure interactswith the effects of instructional treatment. The specific criteriapresentedherearetohelpfocus analysisandsuggestimprovementfor the novice teacher. Each ofthe three areas are necessary butnot alone sufficient. All three areas of knowledge must bebalanced.

Format Examples

The following three paragraphs present examples ofreferenced criteria from each of the three knowledge areas.They model how other competencies could be developed. Anumber of referenced criteria could be used in support of eachmajor area. Readers should examine NCTM’s ProfessionalStandards for Teaching Mathematics (1991) for furtherelaborations and examples of the knowledge of schoolmathematics, MAA’s A Call for Change: Recommendationsfor the Mathematical Preparation of Teachers of Mathematics(1991) for knowledge of mathematics, and the EducationalTesting Service’s Inventory of the Professional Functions ofMiddle School Teachers (1989) for knowledge of teachingpedagogy.

Example One: An example of a referenced criterion inknowledgeofteachingpedagogyundermanaging forinstructioncould be:F Q_ _ The student teacher appropriately manages student

behavior by:a. communicating standards of performance and

expectations to students;b. monitoring student compliance with established

standards;c. managing student misconduct consistently and in a

positive manner;d. using genuine, specific praise to reinforce desired

conduct;e. maintaining instructional momentum while

constructively managing disruptive behavior;f. redirecting students who are off task;g. using scheduled reinforcement to affect behavioral

change;h. usingnon-verbalcommunications tomanagestudent

behavior;i. demonstratinggoodjudgmentinactionsandreaction;j. possessing tolerance, patience, anda sense ofhumor

to interact effectively with students. (FCPS, 1988)Comments:

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Example Two: An example of a reference criterion inknowledgeofmathematicsundersolvingmathematicalproblemscould be:F Q_ _ Thestudentteacherappropriatelymodelsmathematical

problem solving by:a. demonstrating attempts to solve new problems;b. working problems through the four stages of

understanding, planning strategy, carrying out theplan, and looking back;

c. interacting with others, both students and teachers,to pose and solve new problems;

d. applying strategies of pervious problems to a newproblem;

e. connecting a variety ofproblem-solving strategies,e.g., pictorial and algebraic;

f. exploring new problem-solving strategies inmathematics;

g. communicating new problem-solving strategies inmathematics;

h. summarizing problem-solving strategies.Comments:

Example Three: An example of a referenced criterion inknowledgeofschool mathematics underteaching mathematicalconnections could be:F Q__ _ The student teacher appropriately facilitates student

learning of mathematical connections by:a. presenting problems with applications of other

subject areas;b. presenting problems with applications of other

mathematical topics;c. presenting problems with applications in students’

daily life;d. relating various representations of concepts or

procedures to one another;e. linking conceptual and procedural knowledge;

Comments:

Evaluation ofteaching is a process ofgathering informationand providing a basis for interpreting the information about thevery complex task of teaching mathematics. The processshould be longitudinally used throughout the student teachingexperience, a formative not summative evaluation. The goal ofevaluation of student teaching is to assist student teachers toimprove their teaching and develop professionally (Zollman &

Alien, 1990). A criterion-referenced format can be a valuabletool in thisrespectasitnaturallyestablishesalineofprofessionalcommunications among the student teacher, the supervisingteacher, and the university coordinator.

As with any evaluation instrument, a criterion-referencedform should only be one tool for the development process. Acriterion-referenced format is in no way comprehensive, as nopractical instrument could be, but it can be a basis ofcommunicating valued ideals in the teaching of mathematics.

References

Berliner, D. C. (1986). In pursuit of the expert pedagogue.Educational Researcher, 75(7), 5-13.

Berliner, D. C.. Stein, P., Sabers, D., Clarridge, P. B., Cushing,K., & Pinnegar, S. (1988). Implications of research onpedagogical expertise and experience for mathematicsteaching. In Grouws, D. A. & Cooney, T. J. (Eds.),Perspectives onresearchoneffective mathematics teaching(pp. 67-95). Reston, VA: NCTM.

Educational Testing Service. (1989). Inventory of theprofessionalfunctions ofmiddle schoolteachers. Princeton,NJ: Author.

Fayette County Public Schools (1988). Teacher evaluationsystem. Lexington, KY: Author.

Good, T., Grouws, D., & Ebmeier, H. (1983). Activemathematics teaching. New York: Longman.

Mathematical Association of America. (1991). A call forchange: Recommendations for the mathematicalpreparation of teachers ofmathematics. Washington, DC:MAA.

National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for schoolmathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991).Professional standardsfor teaching mathematics. Reston,VA: Author.

Shulman, L. S. (1986). Those who understand: Knowledgegrowth in teaching. Educational Researcher, 15(2), 4-14.

Smith, C. L., & Stevens, J. T. (1984). A criterion-referencedevaluation of student teachers in science. School Scienceand Mathematics, 84(2), 125-135.

Zollman, A., & Alien. N. (1990). Standards for the evaluationoftheK-4 teaching ofmathematics: Theevaluationprocess.In S. Beal (Ed.), The seventh mathematics methodsconference papers (pp. 42-45). Chicago, IL: Saint XavierCollege Press.

Volume 93(3), March 1993