Mathematics is the study of idealised structures. All other sciences use to a greater or lesser degree these idealised structures to model the part of the world that they study. But for many Mathematics is a Science that is fun and challenging in itself. To study and explore the idealised structures, find out their properties, how they can best be understood and described, and which idealised structures are rich, natural and fruitful to study. Mathematics plays a crucial role in the society since it is core for almost all disciplines and it is applicable in day today life. The society has some sense of the importance of Applied Mathematics and Statistics, but they view Pure Mathematics as something with little use. In fact it is viewed as speculative Mathematics that is taught for its own sake. In this paper we raise important issues related to the study of teaching and learning of mathematics at the university level. We examine the challenges facing mathematics education by linking secondary education to university education. We demystify the fact that pure Mathematics is studied for its own sake by demonstrating that pure Mathematics is the logic and the vehicle for transforming the society. We give way forward to the concerns in mathematics education so as to pave way for improvements. The findings in this study are of interest to the teachers of mathematics, researchers, employers and many other people who are keen on the trends of mathematics.