The STR can be presented on three quite different levels:
You can write equations, connecting the components t,x,y,z in one reference frame with those t',x',y',z' of another. My book 'Epstein Explains Einstein' is typical for this approach. Epstein Diagrams or Loedel Diagrams are the adequate visualizations. I would call this approach the component notation.
At the other end of the scale, you can use tensor equations to formulate the physics of STR. This is of great advantage, it becomes unnecessary to proof the Lorentz invariance of such equations ! In his book 'Special Relativity' Albert Shadowitz (Dover 1968) uses this approach for the first half of Maxwell's 8 equations. His proof for the second half is less elegant - Shadowitz does not really introduce 'my' tensor M, but he mentions it as 'pseudotensor' Pij. However, the tensor notation is inadequate to be used at high school level.
The matrix notation lies somewhere in the middle of component and tensor notation. Of course, some basic knowledge in matrix arithmetics is needed. The matrix presentation turns out to be very powerful in calculations and elegant in notation. The following pages will give proof to these assertions. Suited diagrams are the Loedel diagrams with four-vectors and the Brehme diagrams with four-forms.
It would be nice to present the STR to high school students making two turns on a (endless ...) spiral: A first contact in component presentation, together with Epstein or Loedel diagrams, consolidating a profound 'Anschauung' ~ insight. The students will then appreciate the elegance and power of the matrix presentation in the second run, and problems as the Compton scattering and others are easily solved.
The tensor presentation of the STR is not useful but as an introductory chapter to a treatment of the GTR on university level. As far as the STR is concerned, the matrix notation can do as well - if not better. Let me start now with the proof of this statement.