Chunky vs. smooth
In mathematics education literature, the concept of an instantaneous rate of change has been described in terms of "chunky images of change" vs. "smooth images of change," borrowing the language of Castillo-Garsow et al.
Chunky images of change involve imagining change as occurring in completed chunks. Two features characterize chunky images of change: a unit chunk whose repetition makes up the variation, and the lack of an image of variation within the unit chunk. Therefore, ongoing change is generated by a sequence of equal-sized chunks, and this makes measuring change essentially about counting how many chunks have occurred.
As you state, students may interpret "8 meters per second" to mean that the smallest unit of variation is one second. This is a chunky image of change. In contrast,
Smooth images of change involve imagining a change in progress. Ongoing change is generated by conceptualizing a variable as always taking on values in the continuous, experiential flow of time.... Smooth images of change are not the same as chunky images of change cut up really small. Smooth images of change involve an entirely different conceptualization of variation.
In this paradigm, there is no unit of variation at all. An object moves continuously, and at each instant, it has a speed. One way to help students envision this is, as @guest philsopher also suggested, to think about the speedometer of a car. As the car accelerates, the speedometer measures the instantaneous speed of the car as it varies over time. "The instantaneous speed of the car is 8 meters per second" means that the speedometer is pointing at 8 m/s (or 28.8 km/h or 17.9 mph.)
Real phenomena and calculus
Here's a metaphor that the authors use to explain how images of change relate to "reality."
A child throwing a ball through a paper wall sees a continuous motion and knows that the ball collides with the wall. A traditional animator animating a ball being thrown through a paper wall draws frames, and it's possible that the ball never overlaps with the wall in any frame.
Now, imagine the child watching the animated movie. Drawing from her own physical experience, the child might imagine the ball colliding with the wall. However in the reality of the animator, the ball never collided with the wall, because none of the frames he drew show the ball in contact with the wall. This difference in the child’s reality (relating the ball to physical experience and perceiving the movie as continuous motion in progress) and the animator’s reality (individual frames in which there is no motion) is analogous to the difference between smooth and chunky images of change.
We also note that the “true” nature of the reality itself does not matter. The child uses smooth thinking in both the real ball and animated ball cases by imagining the ball in continuous motion. Similarly, the animator, who imagines the animated ball as individual frames, could also imagine the real ball in frames by imagining change at the molecular, atomic, or quantum level. Images of change are conceptualizations and are thus in the mind of the beholder.
The physical world is full of quantities which students perceive as varying smoothly. This experiential knowledge is a basis for the mathematical concept of instantaneous rate of change. Accordingly, Castillo-Garsow et al. give this suggestion for teaching calculus.
It was only with the advent of modern analysis that a fully chunky (epsilon-delta) calculus was possible—a calculus in which limits are characterized in measurable intervals (chunks) without necessitating an appeal to the non-chunky idea of “infinitesimal” or ideas of smooth motion such as “approaching” in proof. Still, during the development of this calculus, mathematicians relied on smooth thinking to develop intuitions and conjectures before writing the formal, chunky proofs. Because we ask students to reason about change and rate long before they study analysis, it seems reasonable that beginning with smooth images of change could create the foundations for students’ consideration of the difficult-to-learn concepts of limit, rate of change/differentiation, and accumulation/integration.
Units and functions
What to do if a student asks "Why isn't $f'\!(3)=0.008?$" is an entirely different question that could merit an entire article. My answer is that when we write
$$f(t)=t^2+2t+1,$$
we've omitted the units of the coefficients, obscuring the unit-dependence of the equation. To make the unit-dependence explicit, we can write
$$x=(1\,\text{m}/\text{s}^2)\cdot t^2 + (2\,\text{m}/\text{s})\cdot t+1\,\text{m}.$$
By changing the units of the coefficients, we can see that this is equivalent to
$$x=(10^{-6}\,\text{m}/\text{ms}^2)\cdot t^2+(2\cdot10^{-3}\,\text{m}/\text{ms})\cdot t+1\,\text{m}.$$
Now, we can calculate
\begin{align*}
\left.\frac{dx}{dt}\right|_{t=3\,\text{s}} &= 2\cdot(10^{-6}\,\text{m}/\text{ms}^2)\cdot (3\,\text{s})+(2\cdot10^{-3}\,\text{m}/\text{ms})\\
&= 2\cdot(10^{-6}\,\text{m}/\text{ms}^2)\cdot (3\cdot10^3\,\text{ms})+(2\cdot10^{-3}\,\text{m}/\text{ms})\\
&= 0.008\,\text{m/ms}.
\end{align*}
The velocity of the object at $t=3\,\text{s}$ is indeed $0.008\, \text{m/ms}$. Unfortunately, this does not mean that $f'\!(3)=0.008$.
Note that, interpreted as $\mathbb{R}\to\mathbb{R}$ functions, the two expressions for $x$ are not equal. We have
\begin{align*}
f(t)&=t^2+2t+1,\\
g(t)&=0.000001t^2+0.002t+1,
\end{align*}
and using function notation, the student's idea would have to be written as $g'\!(3000)=0.008$.
Thus we can see that $\mathbb{R}\to\mathbb{R}$ functions and modern function notation fail to capture how functions are often treated in the sciences: quantities that vary in response to other quantities, which is the original meaning of function. Using Leibniz notation, "the velocity at time $t=t_0\!$" can be represented independent of units as
$$\left.\frac{dx}{dt}\right|_{t=t_0}$$
However, if we use our two functions $f$ and $g$,
- $f'\!(3)$ is the velocity in meters per second at $t=3$ seconds.
- $g'\!(3)$ is the velocity in meters per millisecond at $t=3$ milliseconds.
- $g'\!(3000)$ is the velocity in meters per millisecond at $t=3$ seconds.
This is why many equations in physics are written with variables and parameters instead, i.e.
$$x=at^2+v_0t+x_0.$$
See also this post and discussion over on MathOverflow.
