The purpose of this study was to investigate the effect of the use of an Alternative-Solution Worksheet (ASW) on American ninth-grade students' problem solving performance, and to determine the extent to which instruction in alternative solutions promotes "look back" strategies. "Look back" strategies are based on Polya's (1973) problem solving steps, and they are an examination of what was done or learned previously. The ASW was designed to encourage students to utilize "look back" strategies by generating alternative solutions to the problems. This mixed-methods study was conducted with two existing groups of ninth-grade Algebra I students. An experimental group of 18 students received instruction in utilizing the ASW for two 55-minute class periods a week for a period of four weeks. A comparison group of 14 students did not receive any instruction. Data for this study were collected by pre- and post-testing, ASWs, focus groups, and one student's "think aloud" process. For the quantitative analysis, a one-way ANCOVA was conducted to determine if there was a significant difference in the mean post-test scores between the experimental group and the comparison group. The students' pre-test score was the covariate. The findings indicated that the experimental group scored slightly better on the post-test, and R2=.345, a medium effect size. There were no significant correlations between the ASW scores and the pre- and post-test scores, but the ASW scores were significantly correlated with the students' EXPLORE9 math and reading percentiles. The qualitative findings indicated that "look back" occurred at all six levels of Bloom's Revised Taxonomy, but it is the "look back" that occurs at the upper three levels, in the context of higher order thinking skills, that results in better mathematical problem solving abilities. In addition, positive affective changes were evident despite little improvement in students' mathematical problem solving abilities. The results of this study indicated that higher order thinking skills need to be practiced regularly so students can use them effectively.

In this investigation, two schools - a control school and an experimental school – were compared in terms of learner performance in two traditional grade 11 mathematics tests, namely the pre-intervention test and the post-intervention test. Both schools completed the two tests simultaneously. Educators saw both tests before intervention. In the experimental school, four grade 11 mathematics classes were studied. The four classes were given worksheets that complied with an open-ended approach (OEA) to mathematics teaching and learning for learners to work independently on, with the teacher only facilitating. The learner-centredness expressed in the OEA complied with learner-centredness as envisaged by the National Curriculum Statement (NCS), and was predominantly constructivist in character. Throughout the five-month intervention, the author observed proceedings in two of the four classes in the experimental school, ensuring that questions the teacher asked complied with the OEA. The two classes would be referred to as monitored classes. The other two classes at the experimental school worked on the worksheet, with the teacher having been briefed about what was expected of the learners using the worksheet -basically that the learners would have to take own initiatives in solving the mathematics problems with minimal teacher intervention. The two grade 11 mathematics classes were monitored, but not as frequently as the monitored classes. The classes will be referred to as unmonitored classes. At the control school the educators followed their usual (traditional) teaching approach. Both the experimental and control schools followed the same grade 11 mathematics work schedule. The educators in the control school taught without any interference from the author, but the classes at the control school were occasionally observed by the author. In addition to the intervention comparison, the author also gathered qualitative information about participating educators' and learners' experiences and opinions about the OEA at the experimental school by using interviews. The results of the pre-intervention test showed no statistical difference between the experimental and control school performance, meaning that the learners from both schools were of comparable pre-requisite knowledge. In the post-intervention test, learners from the two monitored classes meaningfully outperformed those from the two unmonitored experimental classes and those from the control school. However, there was no significant difference in performance between learners from the two unmonitored classes and those from control school, The study concludes that the appropriate OEA intervention was responsible for the good results of the monitored classes., and then uses the gathered qualitative information to design a model for the successful implementation of' OEA in mathematics classes. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2010.

In this investigation, two schools - a control school and an experimental school – were compared in terms of learner performance in two traditional grade 11 mathematics tests, namely the pre-intervention test and the post-intervention test. Both schools completed the two tests simultaneously. Educators saw both tests before intervention. In the experimental school, four grade 11 mathematics classes were studied. The four classes were given worksheets that complied with an open-ended approach (OEA) to mathematics teaching and learning for learners to work independently on, with the teacher only facilitating. The learner-centredness expressed in the OEA complied with learner-centredness as envisaged by the National Curriculum Statement (NCS), and was predominantly constructivist in character. Throughout the five-month intervention, the author observed proceedings in two of the four classes in the experimental school, ensuring that questions the teacher asked complied with the OEA. The two classes would be referred to as monitored classes. The other two classes at the experimental school worked on the worksheet, with the teacher having been briefed about what was expected of the learners using the worksheet -basically that the learners would have to take own initiatives in solving the mathematics problems with minimal teacher intervention. The two grade 11 mathematics classes were monitored, but not as frequently as the monitored classes. The classes will be referred to as unmonitored classes. At the control school the educators followed their usual (traditional) teaching approach. Both the experimental and control schools followed the same grade 11 mathematics work schedule. The educators in the control school taught without any interference from the author, but the classes at the control school were occasionally observed by the author. In addition to the intervention comparison, the author also gathered qualitative information about participating educators' and learners' experiences and opinions about the OEA at the experimental school by using interviews. The results of the pre-intervention test showed no statistical difference between the experimental and control school performance, meaning that the learners from both schools were of comparable pre-requisite knowledge. In the post-intervention test, learners from the two monitored classes meaningfully outperformed those from the two unmonitored experimental classes and those from the control school. However, there was no significant difference in performance between learners from the two unmonitored classes and those from control school, The study concludes that the appropriate OEA intervention was responsible for the good results of the monitored classes., and then uses the gathered qualitative information to design a model for the successful implementation of' OEA in mathematics classes. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2010.

The author shares some of the findings of the research he conducted in 2007 on grade 11 mathematics learners in two schools, one experimental and the other one control. In his study, the author claims that an open-ended approach towards teaching and learning of mathematics enhances understanding of mathematics by the learners. The outcomes of the study can be summarised as follows: 1. In the experimental school, where the author intervened by introducing an open-ended approach to teaching mathematics (by means of giving the learners an open-ended approach compliant worksheet to work on throughout the intervention period), the performance of the learners in the post-test was better than that of the learners from the control school. Both schools were of similar performance in the pre-test. The two schools wrote the same pre-test and same post-test. Both schools were following common work schedule. 2. Within the experimental school, post-test performance of the learners in the class where the intervention was monitored throughout the intervention period (thus ensuring compliance of the teacher to the open-ended approach) out-performed those in which monitoring was less frequent. 3. There was no significant difference in performance between learners from the unmonitored experimental class and those from the control class.

The author shares some of the findings of the research he conducted in 2007 on grade 11 mathematics learners in two schools, one experimental and the other one control. In his study, the author claims that an open-ended approach towards teaching and learning of mathematics enhances understanding of mathematics by the learners. The outcomes of the study can be summarised as follows: 1. In the experimental school, where the author intervened by introducing an open-ended approach to teaching mathematics (by means of giving the learners an open-ended approach compliant worksheet to work on throughout the intervention period), the performance of the learners in the post-test was better than that of the learners from the control school. Both schools were of similar performance in the pre-test. The two schools wrote the same pre-test and same post-test. Both schools were following common work schedule. 2. Within the experimental school, post-test performance of the learners in the class where the intervention was monitored throughout the intervention period (thus ensuring compliance of the teacher to the open-ended approach) out-performed those in which monitoring was less frequent. 3. There was no significant difference in performance between learners from the unmonitored experimental class and those from the control class.