Sustaining intellectual discovery
Correct approaches to teaching and philosophy of education are founded on the principle of developing a genuine love of the subject in which learning is a process of intellectual discovery and not simply the instrumentalizing of complex, interconnected facts.
This is especially true for extremely motivated and ambitious students who require stimulation at a level which they may not find in most educational establishments. When stimulation is provided, all too often it is simply in the form of the same types of problems just with increased complexity. Although this does of course exercise their cognitive muscles, it does not provide that which inquisitive minds seek; the expansion of their conceptual understanding and the truths that are consequently revealed about the natural order of things.
There are numerous and far reaching opportunities that can be exploited for this in the A level Mathematics, Further Mathematics and Physics curricula. Such opportunities are made numerous due to the prevalence of those topics traditionally taught and presented in textbooks as dictated rules, which are especially prevalent in Further Mathematics, as it is often wrongly felt that the concepts behind them are too difficult.
These topics can be identified and dissected into discrete conceptual steps from the very basics, to what is required for the curriculum, to complete understanding of the material and beyond to university level mathematics.
For example, not only should one give their bright students complicated problems involving inverses of 3×3 matrices, but more importantly they should guide them to an intuitive understanding of why the process dictated in textbooks for computing inverses works, and from that intuit how inverses of square matrices work for higher dimensions, and indeed n-dimensions.
In short, they should help them discover the elegance and beauty of inverses of matrices. I can state from personal experience that this approach is enthusiastically received by students.
A teacher should also develop multiple conceptual pathways for various topics within Mathematics. For example, the rules of calculus can be presented almost entirely through the lens of geometric principles, as well as from the traditional perspective of algebra and gradients of curves.
This is a most enriching experience for students of all abilities. More able students gain a greater resolution in their understanding and a greater appreciation for the interconnectivity of topics, whilst less able students are more likely to have a route to understanding that suits their individual preferences. Geometric interpretations of calculus for example, help those more visually inclined.
Good teachers always lead by example which is necessary to cultivate students’ respect and love of the subject as students feed off the authentic passions of their teacher. A teacher cannot, and indeed should not, feign a love for their subject. It is therefore advisable that teachers keep regularly engaged in the practice of further broadening and deepening their understanding of their subject to sustain and cultivate this passion. No teacher who truly loves their subject would object to this.
Nicholas Parkin is a mathematics and physics tutor.
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