In my teaching, the concept of mass first arises as part of the "discovery"
of the Newtonian force laws:
The underlying task is to DEFINE a force concept which will be useful for
modelling the interactions by which particles influence each others'
motions.
1.) We seek to mathematically model force as a vector. We will try to find
a useful definition of the magnitude/direction of the net force on a
particle in terms of some measureable property of the particle's motion.
2.) First decide (ie. DEFINE) the measure of ZERO net force. What is the
motion of a FREE particle? Extended discussion leads to N1 as (at least a
tentatively) useful result.
3.) Since we have rejected velocity as the measure of net force, the
simplest next choice is acceleration. The simplest embodiment of this idea
is the linear relation F=ka. (Explore the unecessary complications
introduced by more complicated functions eg., F=ka^2)
4.) As far as possible (and useful) our force concept should relate to our
everyday "feelings" of force when eg., we push or pull an object. In this
regard we realize that we need to push much harder to accelerate a freight
car as compared to accelerating a child's wagon. This leads to the need to
define and introduce an inertial property of the body being forced (mass).
We are led (without much kicking and screaming to F=kma. (Willy Nilly, this
exercise quite naturally fosters the intuition that mass is a measure of the
amount of "stuff".)
5.) N3 is introduced by considering the impossibility of certain
interactions (in which an isolated body accelerates itself through internal
interactions alone).
6.) Bla bla bla . . .