The rapid development of present human societies has caused dynamical changes in educational systems, which, in turn, further accelerated this development. One of the side effects of modern education is that it has heavily confused “what” and “why”. Through the process of putting more and more “what” into school curricula we have got less and less “why”. “Why” is often left out and, at the end, “what” becomes irrelevant. What remains in our heads is what has had meaning; the stuff without meaning quickly dissipates and vanishes.
In my view “why” is fundamental for learning. These algebraic materials are made to emphasise it and complement “what” and “how” semantically in a meaningful way for secondary learners.
We need to be honest with ourselves. There are many ways to understand the reality in which we are immersed. All human models and representations draw on certain capabilities we have. Unfortunately for some students, the power of algebraic ideas is diminished by their condensed symbolic form. As a teacher I have learnt from students what is ambiguous for them and what generates frustration in their algebraic learning. I had to learn how to explain something that is often not self-explainable. My educational experience testifies that incorporating geometry into the explanation of algebraic processes and structures liberates students’ constrained intellectual experience based purely on numbers, letters and symbols. Algebraic visualisations are both tools for enabling teachers to overcome barriers in students’ algebraic learning and effective models for students to take a self-paced mathematical learning journey for themselves. Algebraic visualisations represent the personalisation of an impersonal algebra and a humanisation of its powerful principles.
Understanding abstract ideas requires the transmitter and receiver to tune into the same frequency. We teachers need to persist in this until we are sure that we speak the same language with our students. It is up to us, as educators, to make it happen. I have seen that meaningful connections generate engagement with the subject matter. When students are able to make sense of Mathematics, they not only show interest in what they do, but they start to value the subject and its accomplishments.
As a subject, Mathematics is much, much more than just doing calculations or solving equations. To comprehend its language, procedures and methods, and understand the connections between its different aspects is to feel what authentic, integral Mathematics is. At the same time, it is a process of discovering its internal logic, meaning, coherence and beauty. When students find these attributes in Mathematics, then it is not difficult for them to like, appreciate and love doing it. Subject engagement generates further engagement with the wider world in which we live and want to make sense of. We want to be effectively connected with the reality around us. When we work together we are overcoming the human limitations we have as individuals. Connected, we are not only more responsive, but more productive and more responsible as well.
Vladimir Miškovic