The Control Group A was not taught any mathematics subject-matter content during the four weeks of the experiment but was limited to instruction on the pedagogical strategies used in math teaching. Experimental Group B received instruction on the same pedagogical strategies as Group A, and, in addition, was taught selected topics from two strands (Geometry/Spatial Sense, and Data Management/Probability) of the Manitoba Grades 5-8 Math Curriculum; the Ontario Math Curriculum, Grades 1-8; and the Ontario Curriculum Exemplars - Mathematics, Grades 1-8. The "traditional" method (essentially lecture and solving of numerical examples/exercises) was utilized for instruction in Group B. Experimental Group C also received instruction on the same pedagogical strategies as Groups A and B and, additionally was taught the same subject-matter content as Group B, but using an enriched approach to math teaching. This enriched method consisted primarily of the frequent use of concrete materials (manipulatives), visual aids, student activities, "learner-talk" and the sharing of ideas, and less of "'teacher-talk". The manipulatives, examples, and illustrations used were drawn directly from real life situations and familiar everyday activities, and a deliberate effort was made to use these manipulatives in a meaningful way. By the "meaningful" use of manipulatives, I am referring to a situation where kids use the manipulatives as thinking tools to link the concrete actions and activities they perform in a math classroom to the math concepts underlying those activities. If manipulatives are not used meaningfully, then the students, especially the younger ones, would see the concrete materials as playing tools which have no bearing on the math content or concepts they are supposed to learn in a lesson.

Pilot testing of instruments

A pilot testing (trial run) of the research instruments was undertaken during a pilot study that utilized a population similar to, though very much smaller than the target population. The instruments tested were the (i) pretest questions, (ii) instructional (content) materials, (iii) continuous assessment exercises, and (iv) posttest questions. In light of the researcher's experience during the trial run, some of the instruments were modified for use in the actual experiment. For example, some questions in the Pretest and Posttest instruments that appeared ambiguous to some of those who participated in the pilot testing were reworded and clarified. Additionally, some of the activities originally designed for the activity learning sessions were not used in the actual study after they were tried out during the pilot study. It turned out that these activities consumed so much time and generated such a high level of "unproductive noise" that the researcher considered them distractions. Furthermore, during the pilot study, validity and reliability checks were made to establish the appropriateness of the instruments used for the research project.

Data Analysis

The summary of the experimental design of the study is given in Fig. 2, below.

In Fig. 2, X₁, X₂, X₃ are the three levels of treatment. Thus, X₁ represents the treatment given to Group A, while X₂ and X₃ represent the treatments given to Groups B and C, respectively. The study utilized a 3x2 factorial design, and the double classification Analysis of Variance was employed as the statistical tool. In addition, a post hoc analysis of group means, using Tukey's honestly significant difference (HSD) test (Sprinthall, 2000) was carried out for all significant sources of variation. Furthermore, "the Strength of Effect measures" (Kiess 203) were done to determine the strengths of all significant treatment variables.

Hypotheses

The study was designed to test the following three null hypotheses at the 5% level of significance.

H_O1: There will be no significant differences among the mean math achievement

scores of three equivalent groups of pre-service elementary school teachers
taught by three different methods for the same teaching duration.

H_O2: There will be no significant difference between the mean achievement scores

of subjects grouped by gender on the three methods of teaching.

H_O3: The interaction effect between treatment (teaching method) and gender of

the subjects taught will not be significant.

Findings and Results

Background and Attitude Surveys

From these surveys, it was found that a large majority (31 out of 36 or 86.1%) of pre-service elementary school teachers in the study came from non-science/mathematics academic backgrounds. They took courses in the Arts and Social Science disciplines in their undergraduate programs and majored in these areas. Most of them had sad memories and experiences with regard to their interactions with, and performance in math courses during their high school years. Many actually dropped mathematics as early as Grade 10 of high school. Generally, their attitude to math was negative and was laced with such self-deprecating comments as:

"I was never good at math".
"I never liked math".
"I could never master those endless equations".
"I'm glad I'm not a math major".
"Math is all about figures and symbols, anyway".
"I don't think I'm prepared or ready to teach math now, except the easy stuff in Kindergarten or Grade 1".

The results of the experiment are summarized in Tables 1-7 below.

Table 1
Distribution of Sample by Methods and Gender (N = 36)

Table 2
Pre-test total and mean scores by group

Explanation: In Table 2, the mean score of the subjects in Group A is 5.3 or 53%. Similarly, the mean scores of Groups B and C are 58% and 52%, respectively.

Table 3
One-way ANOVA summary for pretest scores

Explanation: In Table 3, F is not significant at 0.05 l.o.s. (Critical value = 3.32).

Table 4
Posttest score totals by cell

Explanation: Table 4 reveals that the four male subjects in Group A had a total score of 25.0. This translates to a mean score of 25/4 or 62.5%. Similarly, the eight females in Group A got a mean score of 48.0/8 or 60.0%. Hence, by dividing the total score in a cell by the number of subjects in the cell, the mean score, and consequently the percentage score of each cell can be determined.

Table 5
Posttest means of treatment groups (On a scale of 1-10)

Explanation: In Table 5, a mean score of 6.1 stands for 6.1/10 or 61%. Similarly, scores of 8.1 and 9.0 stand for 81% and 90%, respectively.

Table 6
Two-way (3x2) ANOVA summary for posttest results

*p < 0.05

Post hoc tests

Tukey's HSD test

The Tukey's HSD test was run for the 'Methods of Teaching' hypothesis (H_O1) since the F-test was significant for that hypothesis from the factorial Analysis of Variance. Hence, using the values of the means in Table 5, a pairwise comparison of group means for Methods of Teaching (A, B, C) yields the results in Table 7.

Table 7
Pairwise comparison of group means

*p < 0.05 (Critical value = 0.62)

Explanation: Tukey's HSD test was used to determine precisely where the differences in the mean achievement scores of the three instructional groups came from. In Table 7, the pairwise comparison of group means indicate a difference of 2.9 between the mean score of Group A and that of Group C, a difference of 2.0 between the means of Group A and Group B, and a difference of 0.9 between the means of Groups B and C. Since each of these differences is greater than the critical value (0.62), the implication is that each of the differences is significant. In other words, there are significant differences among the performances of the research subjects taught by the three instructional methods. The greatest difference in performance (2.9) occurred between subjects in instructional groups A and C, while the least difference (0.9) was recorded between subjects in instructional group B and their counterparts in instructional group C.

Strength of treatment effects

The strength of a statistically significant effect in a factorial design is measured by 0². In the experiment performed in this study, there were two main effects. Since the two main effects were statistically significant, their strengths were calculated using values in Table 6. Thus, for the first main effect (Methods of teaching), we have: 0² = 0.71 or 71%. For the second main effect (Gender), 0² = 0.10 or 10%.

Summary and Discussion of Results

Summary

Pretest Results

The Analysis of Variance summary for the Pretest (Table 3) indicated that there were no statistically significant differences at the 0.05 level of significance among the mean Pretest scores of the three groups used in the study. This implies that there were no significant differences among the three experimental groups before treatment started. In other words, on average, the three groups were of the same standard in terms of their "entry level" for the study. No group was at an initial advantageous position over any other group before treatment started.

Hypothesis 1 (Hypothesis of no significant differences among the mean achievement scores of the three instructional groups).

From the results given in Tables 4 and 5, and the Analysis of Variance summary (Table 6), it was seen that there were significant differences among the performances of the groups as measured by their mean achievement scores. Hence, we reject the null hypothesis. Also, post hoc pairwise comparison of means showed that the difference between each pair of group means was significant (Table 7). Furthermore, the "strength of treatment effect" test revealed that the first main effect (Method of Teaching) had an 0² value of 0.71 or 71%. The implication of this is that "Method of Teaching" accounts for a large proportion (71%) of the variance in the mean achievement scores of the subjects in the three experimental groups. Putting all these together, the conclusion that could be drawn was that Method of Teaching was decidedly a significant factor in the performance of the three groups of subjects. The differences in the mean scores were over and above what could be attributed to chance factors alone. This conclusion is reinforced by the fact that the F-test was still significant even at the 0.01 level.

Hypothesis 2 (Hypothesis of no significant difference in the mean achievement scores of subjects grouped by gender).

This hypothesis considered the influence of gender as a variable in the performance of the experimental research subjects. From the results in Table 4, the 12 male subjects obtained an aggregate score of 100.5, which yielded a mean score of 8.4 or 84%, while the mean score of the 24 female subjects was 7.4 or 74%. Table 6 indicated a statistically significant difference between these means. So, we reject the null hypothesis. Hence, the gender of a participant was a significant factor in the study, with male subjects performing better, on average, than their female counterparts. The "strength of treatment effect" test showed that the second main effect (Gender) had an 0² value (strength) of 0.10 or 10%. This implies that gender accounts for 10% of the variance in the mean achievement scores of the subjects in the study. This value is relatively small when compared with the 71% contribution made by the first main effect (Method of Teaching). Therefore, based on these results, Method of Teaching could be said to have a more significant effect on the math performance of a subject, on the average, than the gender of that subject.

Hypothesis 3 (Hypothesis of no interaction between Method of Teaching and Gender).

The results of the Analysis of Variance (Table 6) showed no significant interaction between teaching method and gender. Hence, we do not reject the null hypothesis.

Discussion

The finding that Method of Teaching was a significant factor in the performance of the experimental groups upholds the view widely held in research studies that the approach adopted in teaching mathematics to students plays a major role in what the students learn and their general attitude to math. For instance, in their argument for reform in the way math is currently taught, Ross, McDougall and Hogaboam-Gray citing several researchers in the field of math education, stated:

Reform in mathematics education is motivated by the finding that traditional teaching has produced low performance on basic competence tests...; the recognition that the world into which students will graduate requires greater ability to use mathematical skills...; and by advances in pedagogy that emphasize building on student prior knowledge, peer learning, and knowledge construction... (124).

My experience as a mathematics/science teacher educator, including work over the years with scores of both pre-service and in-service teachers, indicates that a good number of these teachers blame their weakness and lack of confidence in math on the way they were taught the subject in school. Pressed further, many of these teachers would state that during their secondary school years math meant nothing more than memorizing formulas and equations that seemed to have little or no relevance to real life. No doubt, the traditional approach to math (essentially, the "explain and solve" method), which these teachers were exposed to in their early school experiences left negative impressions on them. However, after exposure to the enriched approach (Method C) used in this study, it was observed that the pre-service teachers began to see math in a new light and made several highly positive comments:

"I really liked the activity on 'kitchen math'. I never realized I could do math in my kitchen."
"Math makes more meaning to me now because of the real life examples we used."
"Those probability games were really fun."
"I've drawn over 50 'tree diagrams' since I mastered the concept in our probability course. I'll surely use it to teach my students."
"How I wish I had learned math using manipulatives in high school."
"I feel more comfortable now with math."
"I'm still far from being an expert, but I can definitely teach Geometry in Grades 4 and 5 now."

These comments clearly suggest a new comfort level and confidence in math exhibited by the pre-service elementary school teachers in the study. In other words, after exposure to the enriched method of math teaching, the research subjects that benefited from this approach felt more confident and adequately prepared to teach math in elementary grades. They not only enjoyed the real life examples and manipulatives used to concretize the math concepts they learnt in the study, but also felt they could utilize similar approaches in their own teaching after their pre-service program. The comments and general positive attitude of the pre-service teachers in this study support the finding of Goodnough (2001) who carried out a project that enhanced the science professional knowledge of an elementary school teacher. As reported in Goodnough's study, the enhancement of the teacher's professional knowledge rubbed off on her students because the students' response to learning science dramatically changed from lethargy to excitement.

The second hypothesis of the study considered the influence of gender on the erformance of the research subjects. The results showed a significant difference in performance between the male and female subjects. Even though a fewer number of males than females participated in the study, and this might have created a situation that gave rise to a higher mean score for the male subjects, it is still known that the odds are generally against females in the field of mathematics, science, and technology education. By this, I mean that so many factors play against female students in the study of math and related disciplines. Examining this issue, Kenschaft enumerated "fifty-five cultural reasons why too few women win at mathematics" (1). These cultural reasons are classified into five categories, namely, "societal customs, family customs, customs in our educational system, customs specific to mathematics, and the effects of these customs on individuals"(1-2). I agree with the author that these cultural attitudes and patterns that tend to discourage girls and women from pursuing mathematics should be changed. The dire need to change the cultural attitudes stems from the fact that the majority of elementary school teachers are usually women. So, if these women were discouraged from studying math seriously when they were in school, then it follows that their foundation in math would be weak. When they enter the school system as teachers with this weak foundation, everybody loses: the kids they teach, parents, potential employers, and the society at large. It is crucial that a solid mathematics foundation be laid for young elementary school students because of the extreme importance of this level of education on the young learner's entire life. The "Early Math Strategy" report (Ontario Ministry of Education, 2003) emphasized this point, stating: "Because the early grades of schooling are an important period of educational growth, positive, successful experiences with mathematics during this time are crucial" (66). The report goes on to add: "The development of concepts and procedures during this period of cognitive growth is the foundation for future understanding and success in mathematics" (68).

Recommendations and Conclusion

Recommendations

Based on the findings of this study, I recommend that every teacher-training program be reinforced, with a view to further developing and sustaining the mathematics teacher's expertise. The reinforcement should involve not just dwelling on pedagogical knowledge but also emphasizing the need for a strong math content base for teachers. Strengthening the elementary teacher's mathematics content would be an effective way of building confidence in the teacher. As the results of this study show, those pre-service teachers who were actually taught math topics and concepts as part of the study (Groups B and C) got higher mean scores in the Posttest than the group of teachers who received instruction solely on math teaching strategies and techniques (Group A). The "Early Math Strategy" report (Ontario Ministry of Education, 2003) stresses that, "teacher knowledge of mathematics and skills in effective teaching are key to successful learning" (49). In agreeing with this position, I submit that students' mathematics success is one of the constituent legs of the tripod on which meaningful mathematics teaching and learning stands. This structure is represented in Fig. 3.

The program of effective math teacher education I am recommending, consequent upon the findings of this study, would entail not just an increase in the teacher's math content knowledge, and the thorough development of the teacher's pedagogical skills, but would also involve a conspicuous emphasis on the meaningful use of concrete materials (manipulatives) for teaching and learning. This program of effective teacher education should also emphasize the diversification of teaching approaches and classroom learning experiences that would suit the wide-ranging field of learners from diverse cultures and backgrounds in today's classrooms. Diversity is an important factor of modern society, and should be reflected in math teacher education. In addition, a viable program of math education should adopt the holistic approach suggested years ago by D'Ambrosio (1980). Published in the aptly titled book "Teaching Teachers, Teaching students," D'Ambrosio's holistic approach stresses the need for a broad, all-touching, multi-dimensional curriculum for the education of mathematics teachers so that these teachers would be able to transfer their in-depth knowledge to the students they teach. This study, in lending its support to D'Ambrosio's time-tested call, further emphasizes the urgency of strengthening the subject-matter content of mathematics teachers, especially at the crucial elementary school level, so as to stem the alarming dropout tide that currently engulfs mathematics programs in schools. A holistic approach to mathematics education would plug the students' prior knowledge and experiences into the mathematics teaching/learning equation. This is an essential factor in meaningful math education because current theory suggests that "the amount and quality of prior knowledge substantially influence gains in new knowledge and are closely linked to a capacity to apply higher order cognitive thought processes in constructing abstract knowledge" (Bischoff and Anderson 228). As is widely known, appropriate cognitive development in mathematics involves a good degree of abstract thinking (Backhouse, Haggarty, Pirie, and Stratton, 1992).

Conclusion

This study has drawn attention to the widespread dropout rate from math programs in school, and the poor performance of the few students who persist and complete math courses. Based on the findings of an investigation involving pre-service teachers, the study strongly recommends that the mathematics content base of these teachers be strengthened to make them not only more effective but also to build up their confidence in their ability to teach math. To maintain effectiveness after their training programs, and while on the job, math teachers should aspire and be encouraged to become life-long learners through well-planned in-service professional development programs geared toward the sustenance of teacher expertise, confidence, and efficacy. The all-important nature of a teacher's job is aptly captured in the summation of Darling-Hammond and Ball (qtd. in the "Early Math Strategy" report, Ontario Ministry of Education, 2003). Writing under the caption "Teachers make all the difference," the report states:

Teacher expertise - what teachers know and can do - affects all the core tasks of teaching. What teachers understand about content and students, for example, shapes how judiciously they select from texts and other materials and how effectively they present material in class. Teachers' skill in assessing their students' progress also depends on how deeply teachers know the content, and how well they understand and interpret student talk and written work (67).

Suggestions for further research

I acknowledge the small sample size used in this study may constrain wide generalization of its findings. It is, therefore, suggested that further research on the topic utilizing a larger sample, and spanning a longer time period be carried out. It is also suggested that, where possible, an equal number of male and female participants be used in any follow-up work, a desirable - though not absolutely necessary - condition that the present study could not meet due to prevailing circumstances and logistics.

This study looked at the immediate effects of strengthening the math content base of a group of pre-service elementary school teachers in terms of attitudes expressed, the level of achievement attained, and the confidence displayed by the teachers. A possible follow-up study could look at the actual classroom performance, over a longer stretch of time, of the teachers (whose content base was strengthened) and the performance of those teachers who did not benefit from the strengthening. When pitched against each other in a long-term research investigation, how would the two groups perform with regard to their overall teaching effectiveness?

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Academic Exchange Extra invites reader responses to any writings in this issue--especially articles advancing the scholarly debate of issues raised.