“**Plan for the mess**” is one of my favourite teaching ideas. The idea is to bring students to a point where after a heavy messy work (most of the time calculations and symbol pushing) they say “why I didn’t see that earlier!” Here is how recently I used the idea in my linear algebra class.

The students had just solved a couple of two equations with two unknowns by whatever tools they had from highschool. Many of them had no idea about any geometric interpretation of such equations (i.e., solution, if there is one, is at the meeting point of two lines), and of course, none knew the vector interpretation in which, say \( \begin {cases} 2x + 3y = 4 \\x + 2y = 3\end{cases} \), can be thought as \( x\begin {bmatrix}2\\1\end{bmatrix}+y\begin {bmatrix}3\\2\end{bmatrix}=\begin {bmatrix}4\\3\end{bmatrix} \). Of course, one should be too naive to assume that students immediately appreciate the importance of having several interpretations for the same thing. Here is where you **plan for the mess. **Thus,** **I asked them to solve a system of three equations with three unknowns such as \( \begin {cases} 2x + 3y+4z = 4 \\x + 2y-z = -1\\3x-y+2z=2\end{cases} \). It was “fun” watching students messing around with the equations, substituting this into that, sometimes eliminating this unknown, the other time eliminating that unknown until they got the answer: \( x=0, y=0, z=1 \).

I am sure you can imagine how their faces look like when they realize that they could get the answer just by “looking” at the equation as

\[ x\begin {bmatrix}2\\1\\3\end{bmatrix}+y\begin {bmatrix}3\\2\\-1\end{bmatrix}+z\begin {bmatrix}4\\-1\\2\end{bmatrix}=\begin {bmatrix}4\\-1\\2\end{bmatrix} \].

Learning from a hard way is unforgettable!