Chapter IV
SUMMARY, CONCLUSIONS AND
14) Are the elementary math teachers’ beliefs and practices related to the degree of their exposures to seminars or workshops?
The sources of data were the 20 teacher-participants from five (5) high
performing schools, and 20 teacher-participants from three (3) low performing schools, and the 40 classes they handled at the time of the study. The teachers’ beliefs were determined by analyzing their answers in the belief questionnaire, while their practices were determined by analyzing their classroom interactions based on a classroom
observation checklist. Interviews were conducted primarily to determine the reason/s why a teacher cling to or changes his/her beliefs and practices.
Weighted means were computed to determine the tradition upon which the teachers’ beliefs and practices lean towards. Using Spearman rho, correlation coefficients were computed to determine if significant relationships exist between variables under investigation. Lastly, Wilcoxon W tests for independent means were used to compare the beliefs and practices of teachers from the two groups, and to compare the teachers’ IMT and SMT beliefs on the different belief components.
Findings
1.a.) Taken as a whole, the teacher-participants’ computed means on the SMT-Goals, SMT-Learning, IMT-Teaching and IMT-Learning are 4.58, 4.53, 4.68 and 4.56 respectively, indicating that they strongly agree on the four sub-components mentioned. Meanwhile, the teacher-participants’ computed means on the SMT- Teaching and IMT-Goals are 4.47 and 4.49 respectively, indicating that they agree with these two sub-components. Furthermore, their over-all SMT and IMT belief mean scores are 4.52 and 4.58 respectively, indicating that they strongly agree with both traditions. On the other hand, their mean score of 2.00 indicates that the participants’ content knowledge beliefs fall mid-way SMT and IMT.
When grouped accordingly, the following findings appear:
1.b.) The HPS teacher-participants’ computed means on the SMT-Goals, SMT-Teaching, IMT-Goals, IMT-Teaching and IMT-Learning are 4.64, 4.57, 4.64, 4.77, and 4.79 respectively, indicating that they strongly agree to the five sub-components
mentioned. Meanwhile, the participants’ computed mean on the SMT-Learning is 4.48, indicating that they agree to this component. Furthermore, their over-all SMT and IMT belief mean scores are 4.56 and 4.73 respectively, indicating that they strongly agree with both traditions. On the other hand, their mean score of 2.05 indicates that the participants’ content knowledge beliefs fall mid-way between SMT and IMT.
1.c.) The LPS teacher- participants’ computed means on the SMT-Goals, SMT-Learning, and IMT-Teaching are 4.52, 4.58 and 4.59 respectively, indicating that they strongly agree on the three sub-components mentioned. Meanwhile, the participants’
computed means on the SMT-Teaching, IMT Goals, and IMT-Learning are 4.37, 4.34 and 4.33 respectively, indicating that they agree to these three sub-components.
Furthermore, their over-all SMT and IMT belief mean scores are 4.49 and 4.42 respectively, indicating that they agree to both traditions. On the other hand, their mean score of 1.95 indicates that the participants’ content knowledge beliefs fall mid- way between SMT and IMT.
When the participants’ scores in IMT and SMT beliefs in each component are compared using Wilcoxon test for independent means, the following results are yielded:
1.d) Taken as a whole, the teacher-participants’ mean scores in the IMT and SMT beliefs are not significantly different, in terms of the following: goals of mathematics
education, W = 1557.5, z = -.61, p > .05; nature of math learning, W = 1586.5, z = -.37, p > .05. Similarly, their mean scores in IMT-Over-all and SMT-Over-all are not significantly different, W = 1539.5, z = -.78, p > .05. However, their mean scores
in SMT-Teaching and IMT-Teaching are significantly different, W = 1408, z = -2.07, p < .04. That is, the teacher-participants strongly agree with IMT-Teaching and just agree with SMT-Teaching.
When grouped accordingly, the following results appear:
1.e) The HPS teacher-participants’ mean scores in the IMT and SMT beliefs are not significantly different, in terms of the following: goals of mathematics education, W = 402, z = -.22, p > .05; nature of math teaching, W = 363.5, z = -1.31, p > .05.
Similarly, their mean scores in IMT-Over-all and SMT-Over-all are not significantly different, W = 360, z = -1.36, p > .05. However, their mean scores in SMT-Learning and IMT-Learning are significantly different, W = 334, z = -2.13, p < .04. That is, the HPS teacher-participants strongly agree with IMT-Learning and just agree with SMT- Learning.
1.f) The LPS teacher-participants’ mean scores in the IMT and SMT beliefs are not significantly different, in terms of the following: goals of mathematics education, W = 384, z = -.72, p > .05; nature of math teaching, W = 351.5, z = -1.59, p > .05.
Similarly, their mean scores in IMT-Over-all and SMT-Over-all are not significantly different, W = 405, z = -.14, p > .05. However, their mean scores in SMT-Learning and IMT-Learning are significantly different, W = 334.5, z = -2.07, p < .04. That is, the LPS teacher-participants strongly agree with IMT-Learning and just agree with SMT-Learning.
When the teacher-participants’ scores in the different belief components are inter- correlated, the following results are yielded:
1.g.) Taken as a whole, the teacher-participants’ scores in the following pairs are
significantly related: SMT-Goal and SMT-Teaching, r = .54, p < .01; SMT-Goal and SMT-Learning, r = .69, p < .01; SMT-Teaching and SMT-Learning, r = .42, p < .01;
IMT-Goal and IMT-Teaching, r = .52, p < .01, IMT-Goal and IMT-Learning, r = .48, p < .01, IMT-Teaching and IMT-Learning, r = .64, p < .01. All the correlations
computed indicated that the teachers’ beliefs are coherent in terms of SMT and in terms of IMT.
Similarly, the following pairs of variables are significantly related among the teacher- participants: SMT-Goals and IMT-Goals, r = .67, p < .01; SMT-Teaching and IMT- Teaching, r = .76, p < .01.
When grouped accordingly,
1.h.) The HPS teacher-participants’ scores in the following pairs are significantly related:
SMT-Goal and SMT-Teaching, r = .62, p < .01; SMT-Goal and SMT-Learning, r = .77, p < .01; IMT-Goal and IMT-Teaching, r = .52, p < .02, and IMT-Teaching and IMT-Learning, r = .65, p < .01. Similarly, the following pairs of variables are significantly related among the teacher-participants: SMT-Goals and IMT-Goals, r = .69, p < .01; SMT-Teaching and IMT-Teaching, r = .85, p < .01.
1.i.) The LPS teacher-participants’ scores in the following pairs are significantly related:
SMT-Goal and SMT-Learning, r = .67, p < .01; SMT-Teaching and SMT-Learning, r = .49, p < .03; IMT-Goal and IMT-Teaching, r = .53, p < .02, IMT-Goal and IMT- Learning, r = .69, p < .01; and IMT-Teaching and IMT-Learning, r = .69, p < .01.
Similarly, the following pairs of variables are significantly related among the teacher- participants: SMT-Goals and IMT-Goals, r = .66, p < .01; SMT-Teaching and IMT- Teaching, r = .58, p < .01; SMT-Learning and IMT-Learning, r = .55, p < .01.
1.j.) Results of the interview reveal that the teachers’ pre-existing beliefs were acquired through their experiences when they were still students, and while assuming their roles as teachers.
When the beliefs of the HPS and LPS teacher-participants are compared using Wilcoxon test for independent means, the following results are yielded:
2.a.) The HPS and LPS teachers’ beliefs are significantly different in terms of their Over- all IMT beliefs, W = 297.5, z = -3.06, p < .01; and IMT-Learning, W = 253.5, z = -4.29, p < .01. Their beliefs are not significantly different in terms of Over-all SMT, W = 378, z = -.87, p > .05; SMT-Goal, W = 363, z = -.87, p > .05; SMT- Teaching, W = 346.5, z = -1.74, p > .05; SMT-Learning, W = 393.5, z = -.45, p > .05;
IMT-Goal, W = 340, z = -1.92, p > .05; and IMT-Teaching, W = 353.5, z = -1.58, p > .05. Meanwhile, the difference in their content knowledge beliefs scores is not
significant, W = 392, z = -.69, p > .05.
3.) Taken as a whole, the teacher-participants’ mean score in the COC is 1.99 indicating that their teaching practices fall midway the SMT and IMT. When grouped accordingly, the HPS teachers’ mean score of 2.20 implies that the HPS teachers’
teaching practices lean towards IMT, while the LPS teachers’ mean score of 1.78 implies that the LPS teachers’ teaching practices lean towards SMT. Results of the interview reveals that the teachers’ practices are mainly influenced by the DECS policies, colleagues and recommendations from seminars.
4) Using Wilcoxon test for independent means, there is a significant difference between the COC mean scores of HPS teachers and LPS teachers, W = 269.5, z = -3.81, p < .01. While the HPS teachers’ practices lean towards IMT, the LPS teachers’
practices lean towards SMT. Furthermore, qualitative analysis shows that there are differences in the teaching practices of the HPS and LPS teachers.
5.) Considering all the teacher-respondents, the computed correlation coefficient between their Over-all IMT beliefs and practices is r = 0.31, while between their IMT-
Learning and practices is r = 0.39. Both are significant at .05 level.
When grouped accordingly, all the computed coefficients of relationships between practices and beliefs are not significant.
6.) The participants’ length of math teaching years is not significantly related to any of their belief components. When grouped accordingly, similar results were yielded among the HPS teachers. On the other hand, the LPS teachers’ length of math teaching is significantly related with SMT-Over all, SMT Goal and SMT Learning.
Meanwhile, the participants’ length of math teaching is not significantly related to their practices, r = 0.24, p > .05. Similar results are yielded, when the participants are grouped accordingly. For the HPS’ teacher-participants, the computed coefficients is r
= -.32, p >.05; while in the LPS’ teachers, the computed coefficient is r = .38, p >
.05.
7.) The participants’ degree of exposures to seminars is not significantly related to any of their belief components. Similar results were yielded when grouped accordingly.
Moreover, the participants’ degree of exposures to seminars is not significantly related to their practices, r = 0.27, p > .05. Similar results are yielded, when the participants are grouped accordingly. For the HPS’ teacher-participants, the computed coefficients is r = .04, p >.05; while in the LPS’ teachers, the computed coefficient is r = .29, p > .05.
Conclusions
In the light of the aforementioned findings, the following conclusions are derived:
1) The participants, as a whole, adhere almost equally to both SMT and IMT traditions.
While they consider the IMT where students are engaged in problem solving, they could not leave behind the SMT where mastery of skills is emphasized. Whereas, problem solving can be considered on top of the hierarchy in mathematics skills, mastery of facts and skills are considered to be at the lower part of the said hierarchy.
The participants of the study are elementary teachers handling elementary students who are just beginning, or have not even started, to make decisions on their own. This being the case, elementary teachers might just be trying to suit their beliefs on the capability of their students.
Meanwhile, when grouped accordingly, the HPS teachers’ beliefs on the nature of math learning lean towards IMT, while that of the LPS teachers’ lean towards SMT. The higher achievement of the HPS students could be attributed to this factor, since their
teachers believe that the students should be given more opportunities to explore than just to follow procedures. On the other hand, LPS teachers believe that their students should be given more activities such as drills to help their students master the facts and skills, than to allow them to explore math concepts on their own.
2) The differences between the HPS teachers’ and LPS teachers’ beliefs on the Over-all IMT and IMT-Learning are significant. Since, HPS and LPS teachers are handling elementary students, teachers from both groups recognize the importance of lower level mathematical skills. Hence, their SMT beliefs are not significantly different.
However, in terms of the IMT-Learning, the HPS teachers are more confident that their students, despite being in the elementary grade level, can be responsible for their own learning as they are given time to explore things on their own, make conjecture if possible, and create their own solutions when necessary.
3.) The teachers’ practices taken as a whole fall midway between the SMT and IMT.
When grouped accordingly, the HPS teachers’ practices lean towards IMT while those of the LPS teachers’ lean towards SMT. Teaching activity involves a lot of learning activities. Since the HPS teachers’ beliefs on the nature of math learning lean towards IMT, then IMT learning activities are so evident in their practices. On the other hand, since LPS teachers’ beliefs on the nature of math learning lean towards SMT, then SMT learning activities are so evident in their practices.
4. The HPS and LPS teachers’ practices are significantly different. While the HPS
teachers’ practices lean towards IMT, the LPS teachers’ practices lean towards SMT.
Qualitative analysis shows that the HPS students are given more time to explore, discuss among themselves and explain their solutions than the LPS students. In some HPS classes, students are even allowed to pose their own problems. If these were the trends in the HPS classes, then the higher achievement of HPS students than the LPS students in the past could be attributed to the greater involvement of the HPS students in their learning activities, as compared to the LPS students.
5. Considering all the teacher-respondents, their Over-all IMT beliefs and IMT-Learning are both significantly related to practices. However, when grouped accordingly, all the computed coefficients of relationships between practices and beliefs are not significant. The non-existence of significant relationship between the teaching
practices with any of the belief sub-component of IMT and SMT could be explained by teachers’ lack of philosophical bases for their action in the classroom, since they
neither think of the IMT nor the SMT when they plan and implement their lesson plans. Or, they consider both the IMT and the SMT when they plan and implement their plans.
Moreover, since no relationship was found when the participants were grouped according to the performance level of their schools, it is a possibility that the school culture has a direct influence on how the classroom activities should be conducted.
Hence, the teachers’ practices are influenced by some external factors, and not entirely by their own beliefs.
6.) The participants’ length of math teaching is not significantly related to any of their belief components. When grouped accordingly, similar results were yielded among the HPS teachers. On the other hand, the LPS teachers’ length of math teaching is significantly related with SMT-Over all, SMT Goal and SMT Learning. As an LPS teacher stays longer in math teaching, the more he/she believes that the goal of math education is for the students to master mathematical facts and algorithms (instead of the IMT-counterpart which aims that the students generate their own solutions to problems); and that the learning activities are confined only to constant and repeated practice of important mathematical skills (instead of the IMT-counterpart that includes learning activities encouraging the students to explore, discuss, negotiate, pose and solve problems).
7.) Whether taken all together or grouped accordingly, the participants’ degree of exposure to seminars is not significantly related to any of their belief components.
Similar results were yielded when the participants’ degree of exposures to seminars is correlated with the teachers’ practices. Results reveal that either the participants did not fully internalize the discussions in the seminars they attended, or the seminars themselves did not deliberately include discussions on philosophical underpinnings of mathematics education, math teaching and math learning. The deliberate inclusion of such topics could give the participants an opportunity to reflect on their current beliefs and practices, and to examine those that need to be strengthened and revised.
Recommendations
On the basis of the findings of the study, the following recommendations are hereby drawn:
1. Conduct in-service trainings that would deliberately include opportunities for teacher-participants to reflect on their beliefs and practices, since some teachers are not fully aware of the tradition that they adhere to when teaching. It is also important to include activities that will provide opportunities for teacher-
participants to experience the IMT-related activities, since these activities are very evident and associated with the teaching practices of teachers from high
performing schools (HPS). While the teachers’ beliefs were acquired through their experiences as students, and later as teachers, there seems no other factor that could change their beliefs unless they experience something new and more meaningful.
2. Actively involve teachers in staff development that will make them fully aware of their beliefs and teaching practices, since passive attendance in seminars hinders the teachers from re-examining their beliefs and practices.
3. Include a course in the pre service and in-service programs that would deal with philosophical aspects of mathematics education so that prospective and current teachers are provided with opportunities to get exposed to the different traditions and philosophies underlying the teaching and learning of mathematics. This will give them awareness on where to anchor their beliefs and practices concerning mathematics education.
4. Expose the teachers, particularly those in the LPS, to learning materials and activities that are inclined towards IMT. The IMT activities are very evident and
associated with the teaching practices of teachers from high performing schools (HPS). This being the case, their practices could explain the higher achievement of the HPS students than LPS students.
5. Conduct studies that would explore in-depth the practices of those teachers who adhere strongly to the IMT only, and not to the SMT; and of those teachers who adhere strongly to the SMT only, and not to the IMT.