This paper tries to explain children’s varying concepts of zero and infinity and its fundamental operations which have been and continue to be a source of controversy and difficulty among the students in all basic education levels: primary, elementary and secondary. This was done using both qualitative and quantitative methods of research. It was qualitatively done using semi-structured interviews and quantitatively with the use of paper-and-pencil test. Interview transcriptions show that inadequacy of our schoolchildren to make sense out of their concepts of zero and infinity and relate this knowledge to fundamental operation procedures. Respondents manifested three knowledge typologies: schema-related, fragmented and void knowledge of their conceptions of zero and infinity. Their associations of these concepts were either literal, metaphorical or non-associative. Quantitatively, F-ratio test results show that significant differences do exist in their mean scores in addition and subtraction with zero, that is, there was a remarkable improvement in the performance of these fundamental operations as students go up the educational ladder. Yet their mean scores on division with zero and all fundamental operations on infinity did not differ significantly indicating their view of these concepts did not change over the years.