So… How does it work?
I suspect the diagram isn’t the whole story. Maybe there was a table, since erased, where the students would pick a random multiplier from the center and a number from the circumference, i.e. somewhere else was written “2x 8 = 16”. Maybe the students had to go around the circle for each multiplier for practice, or the teacher would randomly point at a multiplier and a circumference number and the class would call out the product.
Notice that the outside is up to 12x, as in the old fashion of learning “times tables”.
Notice that 10x is not included, its just too easy.
Notice that the inside has 2x to 8x, so the class has not yet rote learn the 9x, 11x or 12x…
What is wrong with doing the times tables in order ? you could just do it by ADDITION. If you were doing 7 times tables… the last one was 42 ? the next one is 49… because that is 7 more…
To be sure you know the times tables… have at least one of the two numbers in some mixed up (as if random ) order.
The teacher could use the blackboard for verbal or written excercise …
The way these usually work is that you have a number in the center, and then a ring of numbers around that. The students then take the number in the middle and multiply it by each number around the circle and write the answer in another ring outside of the first.
Since the wheel on the chalkboard has multiple numbers inside and no space to write the answer outside, it was probably used verbally. The teacher would likely point to one of the numbers inside (say, 3x, for example) then one of the numbers along the ring (like the 8 at the top) and the students would be expected to give the answer to 3 x 8.
Here’s an example:
Here’s a variation where some of the answers have been filled in, so to complete the wheel you have to find some of the answers for the blanks in the outer ring but also you have to find what number multiplied by the center number yields the outer answer so that you can fill in the blanks in the inner ring:
Ah. I thought it was a variation on a multiplication table, where you look up the multiplicand and the multiplier to find the product. I couldn’t see how to do that on the wheel – because it is actually an exercise instead of a table!
Alternatively, the children could be expected to copy the wheel to their own slates and complete it. Good way to keep them quiet for an hour. 
Just like to add that besides the mystery of the multiplication wheel, I think that finding something like this is so cool*!* It’s sort of like an antique, chalk & slate ‘black box’ recorder for an old school*!!* This is what was happening on that last day in that classroom a hundred years ago*!* 
If it is just an exercise, then why are the numbers in the outer circle written multiple times, why not just write them once?
I’ve seen something like that, where the teacher would point to the second piece (or, in the case of language lessons, the Nth one) instead of saying it; the students said out loud what she pointed to. Having multiple randomized instances means the students can’t start answering until she’s finished pointing: less guessing, less wrong guesses.