A gap in resources that should help students visualise mathematics

[Sudarshan_Hebbar]

I believe there is a significant gap in the availability of resources that emphasize the underlying logic of abstract mathematical concepts. While tools such as Desmos and GeoGebra are valuable for graphical visualization, they often fall short in fostering a deeper, intuitive understanding. Visualisation, in this sense, should go beyond plotting functions and instead aim to reveal the reasoning and common-sense foundations of the concept.

For example, on YouTube one can find an excellent intuitive visualisation of why the derivative of sin(x) is cos(x), using the unit circle. However, it is much harder to find equally intuitive explanations for the derivatives of the other trigonometric functions.

How might this gap be addressed, and what strategies or resources could be developed to extend such intuitive approaches to a broader range of mathematical and physical concepts?

 

[fresh_42]

This depends a lot on the examples you have in mind. Mathematics is often very counterintuitive. E.g., topology can easily trick you into misconceptions, let alone being hard to visualize on a background that is automatically a metric space. Mathematics as a whole is way too broad to respond in such a general way.

Those examples on some of the admittedly excellent YouTube channels are not helpful, in my opinion, for memorizing the concepts. They are fun to watch, you agree, and think you understood, but will you remember them the next day? I think you will finally have to sketch your own images and drawings to gain insight beyond consuming. It is always the what-if-not questions that drive mathematical understanding, not the what-if case. There is so much more behind your example about $\dfrac{d}{dt}\sin(t)=\cos(t)$ than the unit circle. E.g., what does it even mean? You may look at it as the image of the sine function under the derivation operator
$$
D\sin =\cos\,,
$$
or you may look at it pointwise
$$
t_0\longmapsto \left.\dfrac{d}{dt}\right|_{t=t_0}\sin(t)=\cos(t_0)\,,
$$
or as the linear approximation
$$
\sin(t_0+\varepsilon)= \sin(t_0) +\left(\left.\dfrac{d}{dt}\right|_{t=t_0}\sin(t)\right)\cdot \varepsilon+ r(\varepsilon)=\sin(t_0) +\cos(t_0)\cdot \varepsilon+r(\varepsilon)\,.
$$
The perspective changes the meaning. Any visualization is only one perspective.