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

The students had just solved a couple of systems of two equations in two unknowns, using whatever tools they had from high school. Many of them had no idea about any geometric interpretation of such a system (i.e., that the solution, if there is one, sits where two lines meet), and of course none knew the vector interpretation, in which, say,

$$\begin{cases} 2x + 3y = 4 \\ x + 2y = 3 \end{cases} \qquad\Longleftrightarrow\qquad x\begin{bmatrix}2\\1\end{bmatrix} + y\begin{bmatrix}3\\2\end{bmatrix} = \begin{bmatrix}4\\3\end{bmatrix}$$

Of course, one would have to be quite naive to assume that students immediately appreciate the importance of having several interpretations of the same thing. This is where you plan for the mess. So I asked them to solve a system of three equations in three unknowns, such as

$$\begin{cases} 2x + 3y + 4z = 4 \\ x + 2y – z = -1 \\ 3x – y + 2z = 2 \end{cases}$$

It was “fun” watching the students mess around with the equations — substituting this into that, eliminating this unknown one moment and that unknown the next — until they got the answer: \(x=0\), \(y=0\), \(z=1\).

I am sure you can imagine what their faces looked like when they realised they could have got the answer just by “looking” at the system 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 the hard way is unforgettable!