One way to approach this is to teach a formula:
The position vector of P, where P divides AB in the ratio m:n, is given by p=(na+mb)/(m+n)
If you know
- how to convert a position vector to a coordinate
- the convention that capital letters represents points and bold lower case letters represent corresponding position vectors
- how to multiply or divide a vector by a scalar
- how to add vectors together
Given that the point P divides S(3,4,-1) and T(5,8,11) in the ratio 3:1, find P.
At this point, a student knows how to find a point which divides a line segment in a given ratio. They may have no idea why this rule works. They may have no idea what a position vector is. They may have no idea what a ratio is. They may have no idea about the 3D coordinate system. Do they understand the rule?
In maths, it seems to me, knowing means being able to recall a particular mathematical result, such as the formula given above. Understanding means grasping to some extent the chain of previously established facts and causal links which lead to the given mathematical result being true. In this sense, knowing and understanding are different in maths.
You might say the "understanding" I describe is just more knowing. I disagree, because there is a categorical difference between knowing a particular fact, or set of facts, and understanding why that fact is true. The litmus test, for me, is that anyone could remember the formula given above. Most people could be trained to apply the rule to solve problems using the formula, provided the problems were stated in a fairly standard way. Only those who have already mastered a sufficient body of knowledge and understanding in maths would be capable of understanding what the formula is actually about, and why it works.
I suppose I am saying that understanding is about the connectedness of one's knowing. Maths is a domain in which this distinction is particularly evident, because it is relatively easy to learn a fact which has no connection to anything else you know. For example:
A Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space.I could memorise that, and regurgitate it. I have no idea what any of the mathematical words in the definition mean, apart from "complex" and a very vague recognition of "manifold". I don't understand this definition because it does not connect to anything else I already know (and understand through connections to other knowledge) in maths.