A simple visual trick for multiplication for students when they first encounter this is to rotate the first matrix by 90degrees clockwise.

You can then overlay the columns in question and simply multiply the numbers of the column entries and add them.

I wouldn't teach it that way, you're teaching 2 non-standard matrix operations, and matrix multiplication is simple enough on the face of it that there's no need to build up to it with additional layers.

You also lose an intuitive property of matrix multiplication, that you can figure out where the resulting element value goes in terms of which row and which column you're iterating over. If you're iterating over the second row of the left-hand side matrix, and the third column of the right-hand-side matrix, that implies the resulting value must go in the second row, third column, of the result matrix.

Matrix multiplication can easily be visualised and taught in terms of dot products, without using additional transformations. An example: https://www.mathsisfun.com/algebra/matrix-multiplying.html

It is the dot product. It’s just that kids age 12 who learn this from my experience get confused easily and fail to remember what they need to add. This helps them visually. Of course you don’t tell them this ‘instead’, but ‘in addition’.

Just when you do it, you can mentally rotate it by 90degrees and line them up, this is the same thing.

Also I don’t see what’s so non-standard about a transpose, or a column reversal, but okay

Yeah come on. Just tell the kids that you are taking the dot product of every row vector in A and every column vector in B. The weirdness comes from the fact that nobody ever told them that you can write the dot product as the multiplication of a transposed vector aka a row vector and a column vector as in a^Tb = <a,b>.

Someone who is being told to multiply this entry with that entry will get confused if they don't have the high level overview. They will miss the forest for the trees.

Yes. So I tell them that, and as purely visual trick to remember what it means I tell them to just rotate it by 90 degrees and they can see what to add if they get confused.

It’s the same as with the unit circle. I teach them about pi, but also speak about Tau in case they get confused, so it makes more visual sense to them.

Same with small kids when a/b=c. I draw little arrows to show them how you can rewrite it, as purely visual trick.

Oh well, what works for one might not work for another i suppose.