I'm sure this is meant as joshing, but I don't really get it. I reduce abstract concepts to geometry all the time in my classes-- every math person does. For instance on the first day of Calc, I reduce the rate of change of a fcn to the slope of a tangent line.
It is true though that at least in the US many students see determinants for the first time in Calc III where they are used for Jacobians and are introduced as computational gadgets (many instructors would say that they did not have the extra day to describe them in another way, such as the linked-to article does). That's too bad, and I could definitely understand it leaving a bad taste.
The difference is calc 1 is not higher level math and the derivative of a real function is not an abstract concept. Professional mathematicians/grad students/high level undergrads don't think of the determinant via some weird geometric intuition, as that won't really provide enough information or rigor to do anything useful.
I wonder how he would explain rhombuses with negative area.
In uni we defined determinants starting with systems of linear equations and showing that hey, all these formulas for solving m-by-n systems can be generalised using this one monstrous equation, that has a simple recursive definition. We'll call it a determinant.