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Conceptual understanding of calculus is crucial in the fields of applied sciences, business, and engineering and technology subjects. However, the current status indicates that students possess only procedural knowledge developed from rote learning of procedures in calculus without insight of core ideas. Hence, this paper aims to assess students' challenges and to get insight on common barriers towards attaining conceptual knowledge of calculus. A purposive sample of 238 grade 12 natural sciences students from four different schools in one administrative Zone of Ethiopia were selected to participate in this study. An open ended test about limit of functions at a point and at infinity was administered and analyzed quantitatively and qualitatively. The findings reveal a number of factors about students' knowledge such as: lack of conceptual knowledge in limit of functions, knowledge characterized by a static view of dynamic processes, over generalization, inconsistent cognitive structure, over dependence on procedural learning, lack of coherent and flexibility of reasoning, lack of procedural fluency and wrong interpretation of symbolic notations. Students' thinking strategies influencing these challenges originate from their arithmetic thinking rather than algebraic, linguistic ambiguities, compartmentalized learning, dependence on concept image than concept definition, focus in obtaining correct answers for wrong reasons, and attention given to lower level cognitive demanding exercises.

A number of researchers in mathematical education assert that the instruction in geometry offered in South African schools is inadequate and that traditional teaching strategies do little to promote teachers understandings of their learners’ levels of mathematical thought. Van Hiele specifically states that the inability of many teachers to match instruction with their learners’ levels of geometrical understanding is a contributing factor to their failure to promote meaningful understandings in this topic.  This study investigated whether a sample of grade seven learners in previously disadvantaged primary schools met both the assessment criteria for geometry as stated by the South African Revised National Curriculum Statement and the implied Van Hiele thinking levels.  The data generated suggest that none of the 30 learners who participated in this study had attained these requirements and that language competency in general is a barrier to the attainment of higher levels of understanding amongst this group of second-language learners. It is suggested that not only Van Hiele Levels and Assessment Standards, but also learners’ cultural background and their specific use of words in the vernacular context, need to be taken into consideration by teachers when developing learning programmes. Possible strategies to meet these requirements are suggested.

This study examined the influence of the jigsaw cooperative learning strategy integrated with GeoGebra (JCLGS), on Ethiopian undergraduate statistics and chemistry learners' conceptual knowledge development and attitudinal change towards calculus. The post-positivism quantitative methods approach employed in a non-equivalent pre-and post-test comparison group quasi-experimental design. The samples had drawn using two-stage random sampling techniques. The sample size was 150 in both the experimental and comparison groups. Data were collected by using the calculus classroom achievement test and the five points Likert-scale attitude questionnaire. Data were analyzed using descriptive analysis, an independent-samples t-test and Two-Way ANOVA for repeated measures using SPSS 23.0. The results showed a statistically significant difference between the two groups of pre-post test scores on the Two-Way ANOVA, F(1,148)=106.913; 2 = .419; p<.01. The finding also implies that the blended learning strategy grounded in Vygotsky's social constructivism cognitive development learning theory had big practical significance on learners' conceptual knowledge development. Learners viewed the JCLGS as enjoyable and interesting. It was also a socially interactive and collaborative environment that allows learners' to be reflective, share prior experience and knowledge and independent learners. It encourages them to have a positive attitude towards calculus and GeoGebra. Because of this finding, mathematics and science educators are advised to model a similar blended learning strategy in a classroom instructional setting. It will benefit their learners to adequately construct conceptual knowledge and positively change their attitude towards mathematics.