This is a common meme in martial arts training. And martial arts training is pretty much the only realm of physical training where someone might say this. And let's be clear, we mean Asian martial arts when we say this. A boxing coach isn't going to say this. A wrestling coach isn't going to say this. That fact is the first thing I brought up in the discussion thread where this popped up.
On the thread from the original image source we see this comment:
Here we get to one of the two key components of this: the idea of how muscle memory works. Muscle memory is the creation of motor programs in our brain: connections of all the various nerves involved in a particular action are formed and strengthened. Timing components are tied together so that everything happens at just the right moment. The timing of doing something slowly is clearly not the same as the timing for doing it fast. So the motor program for fast is not the same as the motor program for slow.
Each individual motor unit in our muscles is supplied by a single nerve from our brain, through the spine, to the muscle. Each motor unit is comprised primarily of either Type I or Type II fibers. Type I are the slow, weak fibers and they are used to move slowly. Type II fibers are the fast, strong fibers and they are used to move at fighting speed. As such the neurons for fast and slow movement are not even the same neurons in the brain. And, so the motor program for fast is not the same as the motor program for slow. It cannot be.
Running is just fast walking
One of the other points brought up in this thread is the assertion that "running is just fast walking."
This statement is false. We can see it merely in the definitions of the words.
One cannot get good at running a race simply by walking a lot. They aren't the same activity.
And we see the same thing in martial training, especially with weapons. If I perform a descending cut slowly versus quickly there is a clear difference. Slowly I have to battle gravity, I have to keep my weapon from falling faster. Quickly and I outpace gravity by a significant margin.
Cut slowly and my muscles that pull upwards are actually doing most of the work with an eccentric (lengthening) activation. Quickly and the opposite muscle groups are doing most of the work with a concentric (shortening) activation. Different muscles used differently. So the motor program for fast is not the same as the motor program for slow.
Furthermore, as I train a given motor program the myelination of the neurons increases. Myelin is the insulation on the "wires". It makes the signals go faster. And it means that one motor program influences another less and less the more it is trained.
Doesn't this contradict "slow" strength training?
Sure. If I had ever said that all we need is strength training. But I have repeatedly said otherwise. The key neurological benefit of strength training is that it teaches the motor units near each other to fire in sync. Also, muscles can better synchronize when they are better myelinated.
There is a speed below which strength training becomes less useful but that's only when it you are aiming to move at less than one or two seconds for the motion. That timing is for each of up and down. So two to five seconds (with a pause at either end), total per rep, is still good training for a 300 ms strike.
But slow training is useful!
"My dance instructor said so." Sure, but it's really a matter of how much slow training and how slow. What slow training can do is teach us the proprioception of the correct movement. That is: what does it feel like to move through the correct motion. This feeling is built from the movement sensors in the joints.
But we also have movement sensors in our muscles and they are clearly not getting the correct propriocetive training.
Proprioceptive training doesn't take that long. A few dozens of repetitions max. You can do that in the first few days of learning a new action. Taking a long time to get good at doing an action slowly is a waste of time.
How slow is also worth noting. Jogging can be part of training for running a race, since it is the same gait. But walking is not. Jogging really is slow running. So it can be a (small) portion of your race training.
I've just started learning a new style of dance. And the instructors will have us do new actions fairly slowly. But only a minute or so. Then they turn the music on.
Martial arts training has buckets of old traditions many of whose origin is lost to time or simply forgotten. (Slow training may have been intended for meditative purposes.) And there arises a problem of people who do not critically analyze these traditions. Of people who do not update this "wisdom" based on new facts.
The motor program for fast is not the same as the motor program for slow.
Now there's a part 2.