In response to a recent article about the state of teaching of subtraction in schools vs teaching children to add the inverse or the negative: Mathematical computation is about change, movement. (Link at the bottom of the post.)
To move along the number line, in whatever direction, or plot one point to another on the Cartesian or imaginary planes, is to chart change. The direction, positive or negative depends on the starting place.
The example: Jonathan has two apples, but if you subtract one, how many does he have, is not about a negative apple (there us no such thing!) but about the change with respect to possessions; the movement from one amount to another. It is relational, depending on who holds the apples and who is receiving the apple.
Astronomy led to expression because of the movement of the heavenly bodies. Calculus is the expression of movement of all sorts of actions. The “rules” of math; why 2 acts the way it does, set definition, the differences of movement in a Euclidean world vs a spherical one- all of the rules are predicated on understanding and defining observed change, or predicting future change.
Somewhere along the way, the vision of math, the way we share this lovely process with our children, has been turned into some cookie cutter process. We lose the observation of this movement by disconnecting it from change and giving children math problems with no relationships to anything but counting. Showing that numbers can be broken apart and recombined, that they are fluid and can show change, (number sense we call it) is critical to math knowledge.
Subtraction is movement away from the center of one person or place- addition is movement toward a person or place. It is relative to the location of center.
As we get older, more mature, we begin to understand that we are not the center of the universe. Until then (and this is the teacher in me) we believe we are the center, that movement away from us is loss- subtraction, if you will. Take-away is a valid way to teach movement away from (possessions like apples moving from one position to another, or reducing numbers by other numbers, or defining the distance from ground to sub-basement, or of one planet’s orbit around the sun), especially to younger students.
As students get older, we can continue with the idea of movement: the concept of adding a negative number works well on a number line. It serves to explain the process of movement – in the multiplication of negative and positive, or negative and negative, numbers. It is a difficult idea for students who do not grasp the true nature of numbers. The words used to describe what is happening in math and the ensuing confusion are understandable- perhaps as suggested by a colleague who creates mathematical texts, we can simplify the terminology. Until then, we need to teach and talk about mathematics in as many ways as we can. There are so many ways to get to the accurate answer, one of them is sure to resonate with our students, or with each other as mathematicians, and each can be the correct way, no matter the language.