Mathematical Creativity and Teaching Competence of Pre-Service Teachers
Danilo M. Parreño, Emellie G. Palomo
Abstract
This study determined the level of mathematical creativity of pre-service teachers as a whole and in terms of fluency, flexibility, and originality in their generation of ideas. It also looked into their teaching competence in terms of personal and social qualities, language proficiency and instructional competencies. On the inferential aspect, it ascertained the predictive ability of mathematical creativity and their teaching competence in terms of personal and social qualities, language proficiency, and instructional competencies.
Major findings revealed that the mathematical creativity of the participants was average as an entire group and in terms of fluency, flexibility, and originality.
The teaching competence was found to be high in terms of personal and social qualities, language proficiency, and instructional competencies. There was no significant relationship between mathematical creativity and its components and between mathematical creativity and problem-solving performance of the participants. The mathematical creativity of the participants as a whole and in terms of fluency, flexibility, and originality were not predictors of their teaching competence in terms of personal and social qualities, language proficiency, and instructional competencies.
Keywords: mathematical creativity, teaching competence, fluency, flexibility, originality
Funded by the University Rsearch and Development Center
©WVSU Research and Development Center
Creativity has been defined as a continuum of actions, as a process of bringing something new into being (Best & Thomas 2007) in (Mann 2006).
Mathematical creativity is difficult to develop if one is limited to rule-based applications without recognizing the essence of the problem to be solved.
The visionary classrooms described by leaders in the National Council of Teachers of Mathematics (NCTM) enable students to confidently engage in complex mathematical tasks and draws on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. (NCTM, 2000;
Mann 2006)
Creativity has been proposed as one of the major components to be included in the education of the 21st century (Mann, 2005). Therefore contemporary curricula emphasized the development of student’s creative thinking (Lamon, 2003). However, the conceptualization of creative learning varies due to the diversity of the proposed definitions and one of the problems that mathematics educators face is the lack of tool to identify mathematical creativity. Very often, mathematical attainment which is usually measured by a student’s score in a standardized test is also a good predictor of mathematical creativity.
Hadamard (1945) as cited by Mann (2005) believed that creativity in mathematics requires the intuitive mind with ample time for reflection and incubation of ideas. Because of this disconnect between time for reflection and measures of computational speed, many students who have the potential to make significant contributions become intimidated and conform to simply follow the crowd, and deny their creative nature (Csikszentmihalyi & Wolfe, 2000).
Mann (2005) suggested that there is a relationship between mathematical experiences (knowledge and skills) and creativity in mathematics as measured by the Creative Ability in Mathematics Test (Balka, 1974).
However, questions on limitations of this instrument in the measurement of mathematical creativity were also raised within the present study.
The competence of teachers entering the profession is a significant issue for providers of teacher education and the many education bodies who employ graduate teachers. At various stages of their pre-service and professional careers, teachers are formally judged on aspects of their competence. This appraisal begins when the performance of undergraduate student teachers is evaluated during periods of practice teaching and continues to post-graduation, when beginning teachers are appraised to enable their formal entry into the teaching profession.
As Salandanan (2006) said, methods influence a teacher’s entire performance in a given learning situation. What the teacher does and how he/
she does it; what he/she asks the students to do and how he/she wants them to do it depend on the method the teacher uses. All learning activities are in accordance to the method used. Thus it is necessary for pre-service teachers to be proficient with a variety of strategies/methods.
Teacher education programs have increased clinical experiences, offered guarantees and warranties that novice teachers are prepared to meet the needs of diverse students when they graduate, and established more rigorous admissions process to the teacher education program (Kent, 2007).
Teacher education programs are accused of graduating candidates that fall into the category of failing teachers or teachers who flee the profession when their career is only beginning (Haberman, 2005).
In the midst of the problems confronting any teacher education program, and the challenges it has to meet, it would be interesting to ascertain whether the teaching competence of the pre – service teachers is influenced by their mathematical creativity.
This investigation sought answers to the following questions:
1. What is the level of mathematical creativity of pre-service teachers in terms of fluency, flexibility and originality in their generation of ideas?
2. What is the level of teaching competence of the secondary pre- service teachers in terms of personal and social qualities, language proficiency, and, instructional competencies?
3. Is there a significant relationship between fluency and flexibility, fluency and originality, and flexibility and originality in the students’ generation of solutions and problems posed in the creative ability in mathematics test?
4. Is mathematical creativity a significant predictor of the teaching competence of secondary pre-service teachers?
Methodology Research design
This study used the descriptive method of research. This study aimed to determine the level of mathematical creativity and teaching competence of secondary mathematics major students of a Teacher Education Institution in Iloilo City for Academic Year 2011-2012.
The independent variable in the study is the Mathematical Creativity expressed in terms of fluency, flexibility, and originality. The dependent variable is the teaching competence, expressed in terms of personal and social qualities, language proficiency, and instructional competencies.
Respondents
The purposively chosen participants of this study were the intact group of secondary mathematics major students of a Teacher Education Institution in Iloilo City for Academic Year 2011-2012.
Data Gathering Instruments
The BSEd student teaching evaluation instrument is used by Teacher Education Institution in evaluating the secondary mathematics major pre- service teachers. The same instrument measured the teaching competence of the participants in this study.
The BSED Student Teachers rating sheet is composed of three parts areas for the teaching competence: (1) personal and social qualities, (2) language proficiency and, (3) instructional competencies. Personal and social qualities are 20% of the total grade of a pre-service teacher and composed of 19- items. Language proficiency is 20% of the total grade of a pre-service teacher and composed of 5-items. Instructional competencies are 60% of the total grade of a pre-service teacher and subdivided into 4 parts: (1) planning and preparation skills 10%, (2) teaching skills 20%, (3) evaluation skills 15%, (4) management skills 15%.
The Creative Ability in Mathematics Test (CAMT) is designed to measure the participants’ mathematical creativity and deals with the three areas: fluency, flexibility, and originality.
The tentative draft of the CAMT was submitted to a panel of jurors composed of four (4) members, who were considered for their expertise in the field of mathematics. After the test validation and incorporation of all comments and suggestions, the CAMT was pilot tested to the Bachelor of Secondary Education in Mathematics students of another Teacher Education Institution in Iloilo City. The school was chosen because like the participants of the study, it is also a state educational institution offering teacher education program. Their pre-service teachers’ students have already covered all the mathematics subjects included in the instrument.
Construct validation of the instrument was done using factor analysis to identify 7 constructs for mathematical creativity. Reliability was determined using Cronbach alpha which showed an alpha value of 0.74. According to Fraenkel and Wallen (1988) for a research instrument to be reliable, it should have at least a reliability coefficient of 0.70
The instrument was reduced to the aforementioned 5-item instrument.
The revised instrument was distributed to the target respondents.
The mathematical creativity of the participants was determined by adding the scores in fluency, flexibility and originality.
The fluency score is given by the total number of correct mathematical problems or solutions posed or solved by the participants in the CAMT. Each correct response scores 1 point. To determine the level of fluency of the pre- service teachers, the researcher utilized the scale below:
Fluency Score Description
35-51 High
18-34 Average
1-17 Low
Creative Mathematical Ability Test (CMAT) Divergent Item Scoring Procedures (Balka, 1974) as cited in Mann (2005) was adapted in scoring the flexibility. Each category scores 1 point.
To know the level of flexibility of the pre-service teachers, the researcher used the scale below:
Flexibility Score Description
27-40 High
14-26 Average
1-13 Low
Creative Mathematical Ability Test (CMAT) Divergent Item Scoring Procedures (Balka, 1974) and Guildford’s Alternative Uses Task (1967) as cited by Mann (2005) was adapted in scoring the originality of CAMT.
Table 2 shows the original statistical infrequency stated in the scoring criteria of Guildford’s Alternative Uses Task (1967):
Table 1
Alternative Uses Task
Score Percentage of the total category expressed over the total number of respondents
2(Unique) x ≤ 1%
1(unusual) 1% < x ≤5%
0 x >5%
However, because of the small sample size, the statistical infrequency was adjusted. Therefore, for this study, the following statistical infrequency was adapted.
Table 2
Adjusted Criteria for Scoring Originality
Score Percentage of the total category expressed over the total number of respondents
2(Unique) x ≤ 5%
1(unusual) 5% < x ≤10%
0 x > 10%
To determine the level of originality of the pre-service teachers, the researcher uses the scale below:
Originality Score Description
11-16 High
6-10 Average
1-5 Low
Overall Mathematical Creativity Score
The total score of the three (3) components of the CAMT constitutes the overall mathematics creativity score of the participants.
The mean mathematical creativity of the entire group was obtained by:
total correct mathematical score of respondents
Mean mathematical creativity score = _______________________________________________________
(number of repondents)
To find out the level of mathematical creativity of the pre-service teachers, the scale below was used:
Mathematical Creativity Score Description
49 and above High Mathematical Creativity 24 - 48 Average Mathematical Creativity Below 24 Low Mathematical Creativity Data Gathering Procedure
In conducting the study, the researcher sent letters of request to the Dean of the College of Education of the teacher education institution where the test was administered, to teacher validators of the test items, to the respondents, and to the President of another teacher education institution in Iloilo City where the test was pilot tested. Upon approval of the President, the test was pilot tested to the fourth year Bachelor of Science in Teaching Mathematics students of another teacher institution in Iloilo City.
The researcher also sent a letter of request to the Chairman of the Division of Secondary Teacher Training (DSTT) to use the BSED Student Teaching Evaluation sheet. After the evaluation forms were filled out, the data was evaluated and then analyzed.
Statistical Data Analysis Procedure
The data gathered where subjected to appropriate statistical treatment.
The Statistical Package for Social Sciences (SPSS) was used to analyze the data gathered.
Descriptive statistics. The descriptive statistical tools employed were means and standard deviation.
Mean. This was used to determine the level of mathematical creativity in terms of fluency, flexibility, and originality, and the level of teaching competence of the secondary pre-service teachers classified in terms of personal social qualities, language proficiency, and instructional competencies.
Standard deviation. This was used to determine the homogeneity or heterogeneity of the secondary pre-service teachers towards mathematical creativity and their teaching competence.
Inferential statistics. The inferential statistical tools used were Pearson’s Product Correlation Coefficient, and Multiple Linear Regression Analysis.
Pearson’s Product Moment Correlation Coefficient. This was used to find out significant relationship existing between fluency and flexibility;
fluency and originality; and flexibility and originality in their generations of solutions and problems posed in the creative ability in mathematics test and mathematical creativity.
Multiple linear regression analysis. This was utilized to test if mathematical creativity is a predictor of the teaching competence of the participants. All inferential tests were set at 0.05 alpha level of significance.
Results and Discussion
Level of Mathematical Creativity and Teaching Competence of Secondary Pre-service
Results show that the level of mathematical creativity (M = 36.93;
SD = 13.16) is average. The relatively large standard deviation indicates that the respondents’ mathematical creativity varies which implies that there were respondents who were very creative and there were those who were not creative enough and were lazy to think and perform tasks assigned to them.
The average mathematical creativity of the respondents is assumed to be based on the creative potentials of every individual. These creative potentials can be achieved by exposing the respondents to rich mathematical problem solving experiences which can develop style of creativity which then could lead to creative mathematical talent. The study reveals that the secondary mathematics pre-service teachers have an average level of fluency, flexibility and originality in their generations of solutions and problems posed in the Creative Ability in Mathematics Test including the over-all mathematical creativity because they lack time to think creatively.
To encourage mathematical creativity, mathematical assessment needs to provide an avenue for students to generate creative responses so that teachers can better determine who are the mathematically creative students and move toward personalized instruction and individualized learning.
Appropriate assessment tools should provide opportunities for students to solve mathematical problems in multiple ways, pose problems, generalize patterns, and make connections with various aspects of mathematics (Sheffield, 2009).
Hadamard (1945) as cited by Mann (2002) believed that creativity in mathematics requires the intuitive mind with ample time for reflection and incubation of ideas. Because of this disconnect between time for reflection and measures of computational speed, many students who have the potential to make significant contributions become intimidated and conform to simply follow the crowd, and deny their creative nature (Csikszentmihalyi & Wolfe, 2000).
The level of teaching competence of the secondary pre-service teachers with mean and standard deviation of (M = 9.77; SD = 0.43), which means that the teaching competence of secondary pre-service teachers were high.
When classified in terms of Personal and Social Qualities (M = 9.84;
SD = 0.37), Language Proficiency (M = 9.55; SD = 0.81) and Instructional Competencies (M = 9.81; SD = 0.40). This indicates that in terms of the secondary pre-service teachers’ Personal and Social Qualities, Language Proficiency and Instructional Competencies, their level of teaching competence in general tends to have high performance.
The teacher is the key in the teaching process (Bustos & Espiritu, 1985) and with the right kind of teachers, the school is a good one (Huggets, 1950).
The prospective mathematics teachers are prepared to teach mathematics as their major field of specialization, having received adequate training and experiences from the teacher training school. These pre-service teachers are then expected to have gained the necessary knowledge, skills, and attitudes to perform with confidence and efficiency inside the classroom. The present study reveals that the secondary mathematics pre-service teachers have a high level of teaching competence.
Teaching is a profession that deals basically with helping people learn and cope with life. Thus, it is not surprising that the developments of these skills are given emphasis among teacher training institutions.
Competent teachers are creative teachers; and to become competent teachers, one must be creative. This study shows that the participants have an average level of mathematical creativity in generating their solutions and posing their mathematical problems. This only show that the participants are not solemnly exposed in solving open ended problems and time constraints is a necessity. Maybe these participants were not given sufficient avenue to generate creative responses in their mathematics classes.
The participants have a high level of teaching competence. They seem to show well-developed personal and social skills, language proficiency, and instructional competencies. This may be because that teaching brings about desired learning and behavioral changes in them and enhances their development. Another reason is mentors have equipped them with extensive training and skills that are essential in making them productive teachers in the future. Moreover, this study also supports the findings of Clifford and Green (2004) when they view the mentor-mentee relationship as a significant factor in pre-service teacher education. They pointed out the positive rapport of a good relationship can foster pre-service teachers’ development of teaching competence and self-efficacy beliefs.
The high teaching competence of the se pre-service teachers is also an indication that they were systematically taught and prepared for the task of teaching. The result of the present study on the level of teaching competence of the secondary mathematics pre-service teachers is supported by other studies. Angostura (1995) states that mathematics teachers are very competent in evaluation although least competent in actual teaching. Refuerzo (1994) adds that the poor quality of graduates in teacher education was brought about
by the poor quality of faculty in most teacher training schools, both private and public. Pendon (2003) concludes that the high teaching competence of respondents is an indication that they were thoroughly taught and prepared for the task of teaching.
It was noted, however, that the same findings differed from Ibes’s (1995) statement, that teacher education has been attracting only mediocre students, thus, failing to maintain high standards and turn-out, too many would-be-teachers of schools subjects for which there is no demand in the field. Furthermore, it confirmed Refuerzo’s findings that the outputs of teacher education institutions do not possess the expected moral qualities, social traits and personality traits that would equip one to become a good and effective teacher.
This is a contradiction to Pascual’s (2001) study which revealed that teacher education institutions prepare teacher education students to a much extent for field practice as perceived by the respondents group. In addition the competitive would be of teaching, and were even performing better than their non-WVSU counterparts. The predictor of this is the actual teaching performance.
Another study to support teaching competence is that of Valli and Angostenelli (1993) regarding the importance of professional preparation for teachers. It was revealed that through training on the different strategies on classroom management, difficulty in the classroom control is no longer a problem, thus, making a classroom a better environment for learning.
The result of this study contradicts Haylock (1997) who found that students who are equal in mathematical achievement may have significant differences in performance on measures of mathematical creativity.
Feldhusen and Westby (2003) asserted that an individual’s knowledge base was the fundamental source of their creative thought. Students who have not yet attained sufficient mathematical knowledge and skills may be unable to demonstrate creative mathematical thinking.
It was also found out that no relationship existed between mathematical creativity and teaching competence among secondary pre-service teachers.
Today concerned educators are continuously initiating changes that will help strengthen the country’s teacher education program. The general perception that quality education is built upon effective teacher education serves as a motivating force for educators to give more emphasis on the
curriculum program, content subject, and pre-service program for teachers.
Now and then, officials of teacher education Institutions hold conferences, seminars and other trainings aimed at formulating new policies, educational innovations, current institutional practices, and research in education.
This investigation showed that the level teaching competence of secondary mathematics pre-service teachers were found to be high in terms of their personal and social skills, language proficiency, and instructional competencies.
To produce very good prospective mathematics teachers, the present admission procedure of teacher education students is being followed by teacher training institutions. The incoming freshmen should possess positive attitude towards teaching, have good moral, intellectual, and social characteristics to become effective teachers. This policy supports Yanos, (1994) idea that one way to improve quality teacher is to institute selective admission procedure.
Hutchinson and Johnson (1990) likewise report that with the proper attitude and desirable career expectations at the start of the teacher training, success will surely be achieved.
The findings of the study revealed that college mathematics teachers greatly influenced the pre-service teacher’s mathematical creativity, and teaching competence; therefore, they must be equipped with effective teaching strategies, great personality, and great passion in teaching mathematics for they are the role models of these
These was the eagerness of these pre-service teachers to create multiple solutions or pose multiple problems in the Creative Ability in Mathematics Test but time does not permit them to do so; this may be a possible reason that these prospective teachers have low score in their originality. Yap (2009) reveals that these higher academic-ability male students in Primary 5 involved in the study are fluent in generating responses to open-ended mathematical problems, but their responses lack flexibility and originality. Therefore, they cannot be considered as highly mathematically creative.
However, among them, are a few highly mathematically creative students who are able to generate ingenious responses to the problems.
Table 3
Relationship among Flexibility, Fluency and Originality
Flexibility Fluency Originality
Flexibility 1.00 0.844** 0.651**
Fluency 1.00 .400*
Originality 1.00
Note: **Correlation is significant at the 0.01 level (2-tailed).
*Correlation is significant relationship at the 0.05 level (2-tailed).
The Pearson’s Product Moment Correlation Coefficient was employed to find out the significant relationship existing between Mathematical Creativity and Teaching Competence of the Secondary pre-service teachers. Based on the results, there are significant relationships between flexibility and fluency, flexibility and originality, and fluency and originality in the Creative Ability in Mathematics Test of the secondary pre-service teachers with r = 0.844**, p-value = 0.000; r = 0.651**, p-value = 0.000; r = 0.400*, p-value = 0.026 respectively.
Table 4 shows that mathematical creativity is not a significant predictor of teaching competence of the secondary pre-service teachers.
Table 4
Result of the Significant Predictors of the Teaching Competence Dependent**
Variable Predictors Mean SD R2 β T p-
value Teaching Competence**
Mathematical
Creativity 23.69 8.39 0.061 0.090 -1.227 0.625
Conclusions
Study shows a significant relationship between fluency and flexibility, fluency and originality, flexibility and originality of the participants. Those who got a high score in their fluency are also high in their flexibility and originality .In contrast; those participants who got low score in their fluency are also low in their flexibility and originality.
Those participants who got a high score in the mathematical creativity are not the one who got the high score in the problem solving test. Finally, this study revealed that mathematical creativity of the participants are not predictors of the teaching competence of the participants.
The high or low score of a participant in a mathematical creativity test does not affect the teaching competence of the participants. This may be due to the fact that the teacher education institution where these participants belong developed an established curriculum program that equipped participants with every potential knowledge and skills they need.
Consequently, teachers of these participants are probably highly competent to train, motivate, and prepare these participants in the actual teaching world. The findings of the study can shed light on how the teacher training program in particular schools can be employed to teacher training school administrators, faculty members, students, and future researchers.
Recommendations
To enliven math classes, teachers are encouraged to expose the participants to various open-ended mathematical problem solving which can develop style of creativity that may lead to creative mathematical talent among students. Teachers should provide creative applications of mathematics in the exploration of problems and in the teaching of mathematical content in the classroom. The importance of mathematical creativity in combination with computational accuracy must be emphasized among students to further develop their mathematical ability and understanding. Improvement in some parts of their mathematical competencies should be given consideration. Giving ample time and extensive exposure of mathematical creativity problems to the participants would contribute in making them brilliant, creative and globally competitive mathematics teachers of the future. Mathematical assessment needs to provide an avenue for students to generate creative responses so that teachers can better determine who the mathematically creative students are and move toward personalized instruction and individualized learning. Teacher Education Institution should maintain the curriculum and instructions they provided to these participants to maintain their high level of problem solving performance. Mathematics teachers of these participants should maintain their superb and effective teaching strategies. Observing the excellent screening procedure of this Teacher Education Institution can help maintain their high level of problem solving performance.
Being one of the best Teacher Education Institution in the country, this teacher education should maintain every, guideline, policy, curriculum, and instructions to maintain the high level of teaching competence their graduates have. Maintaining the performance of the high caliber teaching force of this teacher education institution can help maintain the high level of teaching competence of their graduates. A highly professional selection procedure that this institution is using be maintained for the preservation of the high level of teaching competence of their graduates. Mathematics curriculum should emphasize mathematical creativity in all the mathematical competencies of these participants.
Future analyses of mathematics instruction should include mathematical creativity in all mathematics subjects to improve the mathematical creativity of the participants.
It is suggested that similar studies on the use of model approach in solving mathematical problems be conducted, with more subjects and longer duration of the experimental treatment. Experimental designs ranging from eight weeks to a semester are more favorable.
It is also recommended that in-depth secondary analysis of similar study will be done through qualitative research methodology. Seminars, workshops, and symposiums about mathematical creativity be conducted to the secondary pre-service teachers to enhance their teaching styles and to eliminate the boredom of students in mathematics. It is likewise recommended that similar and more intensive studies should be conducted by other researchers to shed further light on this particular field of investigation especially on the mathematical creativity and teaching competence of pre- service teachers.
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