Why aren't lower aromatic organic compounds taught?
Hueckel's rule (i.e. that aromatic compounds have $4n+2$ $\pi$ electrons) places limits on aromaticity in cycles with less than 6 carbons.

As there are only 3 cases to look at, let's enumerate them:

5 carbons: cyclopentadiene ($\ce{C5H6}$) is not aromatic, but the cyclopentadienide anion ($\ce{[C5H5]^{-}}$) is. This species is well known in organometallic chemistry, due to its propensity to act as an $\eta^5$ ligand and form sandwich complexes with transition metals. The most famous of these is no doubt ferrocene, where two cyclopentadienide ligands sandwich an $\ce{Fe^{2+}}$ ion. This ion is didactically problematic and in my experience not covered in high school or early undergraduate curricula because most of its interest lies in its sandwich bonding mode with metal ions, which cannot be easily reconciled with the simplified VB/VSEPR/hybridisation theory used to teach introductory organic chemistry. The same goes for the under-loved tropylium cation ($\ce{[C7H7]^+}$), which has 7 carbons.

4 carbons: cyclobutadiene is antiaromatic, and as such only exists at very low temperatures. If I recall correctly, it possesses two unusually long bonds and two unusually short bonds, quite at odds with the bond equivalency found in benzene. This molecule would have to lose 2 e- to become aromatic. It does so happen that the cyclobutadienyl ligand is known in $\eta^{4}$ metal complexes such as $\ce{(C4H4)Fe(CO)3}$ (1) - I was not aware of these complexes until an expert in the field pointed them out to me.

3 carbons: I was not sure about this, but apparently the cyclopropenyl cation is a thing and is aromatic. Cotton (2) also mentions this being known as a $\eta^3$ ligand. I suspect that it is not taught because it's too weird.

In conclusion:

The smaller aromatic rings are principally known as polyhaptic ligands to metals due to their non-neutrality. This places them outside of the comfort zone of the theories of molecular structure and bonding used in introductory organic chemistry, and as such they are usually left until later.

(1) Cotton, F.A., Wilkinson, G.; Advanced Inorganic Chemistry; Wiley Interscience, 5e, (1988), p. 1170

(2) Ibid, p. 1168