What is the best way to conceptualise a 'division' This may seem like a strange question, but my question is more along the lines of trying to figure out how people were able to 'discover' the formulas which have shaped our understanding of the universe we live in.
I think I can understand why Distance = Speed * Time,
effectively 'times' will 'grow' one variable by another variable
but I can't conceptualise why the speed is found by dividing distance by time, it seems like saying: 

I can discover the speed an object was/is moving by shrinking the distance an object has moved by the amount of time it took to move that distance.

It doesn't seem to make sense to me, but Galileo(?) must have had a way to conceptualise these variables in a way which enabled him to realise they are able to be joined this way.
I am a strong believer that if teachers are able to conceptualise these things in a 'common sense' kind of way, it removes the strain on students to learn formulas parrot fashion, which never really helps them learn WHY something happens the way it does.
Any helpful way to conceptualise this would be really helpful, maybe the 'why' is only taught to smart people at phd level!
 A: Divisions should be conceptualized as ratios. For example, speed is a ratio between how much distance you move and how long it takes you to move that distance. When you go 60 miles per hour, you don’t necessarily go 60 miles, or travel for an hour, but the distance you go and the time you take have this ratio.
A: Consider $\frac{a}{b}$. 
You aren’t really shrinking while dividing. You are $\textbf{dividing}$ $a$ into equal parts of $b$. So speed is defined as distance covered per unit time. For example we know that we have moved a distance of 1km in 10 minutes and we are interested in knowing how much we covered in the first minute, provider we went in the same speed, we’d say that we covered 100m. This we found by dividing 1km into equal portions of 10 (and by multiplying by 1 minute). 
In conclusion, we are dividing the total distance into equal parts of time. 
A: Here's a conceptualization that you may find useful. Let's start with your example of speed, $v = d/t$; the amount of distance traveled per unit time. For whatever unit you've chosen to measure time, associate a unit box to it, with the inside associated to the distance that can be covered in that measure of time. To get the total distance after some elapsed time, you just need to stack that amount of boxes (or fraction of boxes) on top of one another. For example, say we have an object that moves at $4\:m/s$ for $3.5\:s$, this visual looks like

This type of conceptualization of division can be extended to pretty much any ratio you like. Construct a little unit box of whatever the divisor is, and put inside however much of the numerator is associated to it. To see this process with another quantity, let's look at specific heat: $c = Q/m\Delta T$, the amount of heat required to raise a unit of mass by a unit of temperature. Here our divisor now consists of two quantities, so our unit box will be associated to a unit of mass and of temperature. To get the total heat needed to raise an object of mass $M$ by some change in temperature $\Delta T$, we'll need $M\times\Delta T$ of these boxes. For example, suppose some material has $c = 4\:J/(kg\cdot K)$. If we have $M = 3\:kg$ of this and want to raise its temperature by $2\:K$, then the visualization of the total heat needed looks like

