The model of inquiry adopted for this study was a quantitative method described as a systematic empirical investigation of social phenomena via statistical, mathematical or computational techniques (Creswell & Plano Clark, 2011) within the blueprint of quasi- experimental design using pretest-posttest non-equivalent control groups (Bell, 2008). The quasi-experimental design allows identification of variables (Blaxter, Hughes & Tight, 1996) in the study. The quasi-independent variable-instructional strategy was manipulated at two levels (PBL & TM) and answering the research questions for the study required data that allowed assessment of the extent to which the PBL and TM influence students’ achievements in and beliefs about Further Mathematics. This study relied on interval (scores on Students’ Beliefs about Further Mathematics Questionnaire) and ratio (scores on Researcher-Designed Test and Teacher-Made Test) data as the strongest form of quantification (Okpala, Onocha, & Oyedeji, 1993). In this study, participants did not have control over which group (control or experimental) they belonged to or of receiving or not receiving the treatment based on quasi- experimental design.
One inherent advantage of this design is that it is typically easier to set up than true experimental designs (Shadish, Cook, & Campbell, 2002) but lacks randomisation of subjects to treatment conditions (Bell, 2008). Adopting quasi-experimental design in this study allowed the investigation of intact groups in real classroom settings since it was not necessary to randomly assemble students for any intervention during the school hours so as not to create artificial conditions. Students in control and experimental groups participated in the study in
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their natural classroom conditions. Additionally, using quasi-experimental designs minimises threats to external validity (Shadish, Cook, & Campbell, 2002) as natural environments do not suffer the same problems of artificiality as compared to a well-controlled laboratory setting (McKnight, Magid, Murphy, & McKnight, 2000). However, quasi-experimental designs may be weak in controlling for threats to internal validity (Robson, Shannon, Goldenhar, & Hale, 2001). External validity refers to how well the results of a study can be generalized to the population from which a sample was selected (Wimmer & Dominic, 2000). Internal validity refers to how well a study is able to control the variables that create possible (plausible) but incorrect explanations of results.
Threats to external validity include effects of selection, effects of setting, effects of history, and effects of testing (Wimmer & Dominic, 2000). The effects of selection were controlled in this study using homogeneous samples. Students in both the control and experimental groups were similar in terms of age (the mean age of students in the control group was 15.3 years while that of the experimental group was 15.4 years, meaning that students in both groups fell into Piaget’s formal operational stage). This is a stage where children develop abstract thought and can easily conserve and think logically in their mind. In addition, students in both groups had exposure to the same national curriculum for Further Mathematics (FME, 1985) and displayed similar cognitive abilities based on pre-tests scores. The National Curriculum for Further Mathematics and the National Policy on Education of Nigeria stipulated that the transiting students from Junior Secondary School year three (an equivalent of Grade nine) into Senior Secondary School year one (an equivalent of Grade 10) must obtain credit passes in English, Mathematics, Basic Science, Basic Technology and any other two subjects. Thus, credit passes in six subjects served as a baseline for all entrants into Senior Secondary School year one science. In this study, students in both groups fulfilled this condition.
History as a potential threat to external validity was controlled in this study in that students in both the control and experimental groups were pretested on the same day and within the same period and the posttest was equally administered on the same day (last day of the intervention) and within the same period. The present study was conducted in Ijebu Ode Local Government in Ijebu Division of Ogun State, Nigeria. Conducting the study in one location controlled for the effects of setting. Lastly, the three months time lag between the
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administration of the pretest and the posttest in this study ruled out the effects of testing in that test items in the pre-test instruments were re-shuffled to make the post-test instruments in order to reduce familiarity. Threats to internal validity such as history, maturation, statistical regression, selection, experimental mortality, testing, instrumentation and design contamination (Wimmer & Dominic, 2000) were controlled in this study as follows. First, there was no review of the National Curriculum of FM and there was no nationwide strike that could have brought about any change in the dependent variables of achievements in FM (RDT & TMT) and beliefs about FM (SBFMQ). Thus, both the experimental and control groups experienced the same stable environment in the education sector throughout the treatment period thereby ruling out the effects of history. Second, there was no visible vacation/coaching classes for students outside the normal school schedules such as during the holidays that could have led to the acquisition of any academic knowledge to the advantage of any of the students in either groups. The students in both groups having been brought up within the same society passed through similar social, cultural and physiological development that could have affected the dependent variables thus controlling any possible effects of maturation.
Third, students in both groups were from low income families thus ruling out the effects of statistical regression. The demographic section of the questionnaire filled by students reflected this (See Appendix 1). Fourth, students did not have the opportunity to choose which of the groups to belong to (experimental or control), which could have affected the dependent variables of achievements in FM and beliefs about FM. The schools that were selected for the study gave their students opportunity to participate in the study considering the fact that they were of similar characteristics in terms of age, exposure to national curriculum, class, criteria for placement, and language.
Fifth, no attrition of students in both groups during the treatments was recorded. Attendance showed that no student dropped out in both the control and experimental schools during the period of the study. This ruled out the effects of experimental mortality. Sixth, the time lag of three months between pre-tests and post-tests was adequate in ruling out the effects of testing in the study. More so, the test-items were re-shuffled in order to prevent hallo-effect that could result from familiarity with pre- test instrument into the post-test instrument. Seventh, the measurement methods of Teacher Made Test (TMT), Researcher Designed Test (RDT)
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and Students’ Beliefs about Further Mathematics Questionnaire (SBFMQ) did not change during the study as both groups supplied information on the same instruments. This controlled the threat of instrumentation in the study. Lastly, both the control and experimental groups were far apart from each other thus preventing interaction of students from both groups and the teachers at the control school had no information about the school used for the experimental. More so, either group had no reason to want to make the study succeed or fail thereby ruling out the probable effects of design contamination in the study.