Econometrics is often taught to economics undergraduates in an anecdotal way, anchored to real-life examples for intuition pumps. Much of the foundational econometrics intuition for economists often stems from that.
OLS is frequently introduced as "the line that minimises squared residuals" rather than the orthogonal projection of $y$ onto $\text{col}(X)$.
Gauss-Markov assumptions are often stated as a checklist of assumptions rather than a property of projections.
Mostly Harmless Econometrics (Angrist & Pischke's Econometrics Bible) anchors the intuition behind endogeneity (OVB) through the example of "people who go to college earn more, but they're also smarter/richer/more motivated".
Wooldridge (the other Econometrics Bible) illustrates:
Heteroskedasticity as "the variance of wages fans out with education level".
Multicollinearity as "experience and tenure move together because people who've worked longer at one firm have also worked longer overall".
Police spending and crime rates for simultaneity.
I have since realised that my friends over at the DPMMS and engineering departments are taught it completely differently, grounding intuition in linear algebra concepts. Honestly this is probably because the average economist isn't very good at linear algebra (although I often think that mathematicians and information engineers could conversely benefit from more data intuition!).
However, having relearnt all of it, I've come to appreciate that the geometric interpretation is elegant, and helps with a rigorous understanding of the econometrics concepts we appreciate anecdotally.
This series aims to develop that for other new economists.
I think mathematicians and ML researchers often dismiss the data intuition behind regression techniques, while economists often anchor too much to it and miss the geometric elegance. This series tries to bridge that.