Many older masters of various martial arts just encourage their disciples to practice more often in the open air and barefoot. They have learnt and inherited such a way of training from their teachers. Unfortunately, some younger masters consider this to be outdated and unnecessary.
They find that in the modern training there is no need for something like this. Most of them think that the exercisers who perform exercises barefoot, especially on a rough terrain or in winter, are taken there with a task to harden themselves. In the other words, many of them consider this to be an unnecessary fanaticism of the older masters and their basic desire to harden the body of the disciple as well as to strengthen the impact surface of the foot and make it tough.
Certainly, we could state this as one of the reasons for barefoot workout in the nature, but not the only one. Unluckily, today younger martial arts instructors do not accept the strengthening of foot muscles and elimination of the flat feet deformation as the main reason for such exercise in the nature. The old masters of martial arts obviously knew much about the human body as well as about the importance of strengthening the foot muscles.
More recently, there has been a large amount of different equipment in gyms (dojo) so that barefoot children often jump on mats, walk on ladders, various smaller benches, boxing bags, or the like. Instructors often create different games for their kids and make them interesting and entertaining. Plenty of these games are very suitable because they require a lot of movement, i.e. jogging and jumping, and all of that leads to strengthening the foot muscles.
Very often instructors also create exercises in which children have to walk on tiptoe or on their heels. Children, however, unconsciously choose one of the games that favour the development of certain leg and foot muscles, such as various jumps or skips over obstacles.
Children also sometimes play with the belt, so that they tie it around the leg, i.e. around the foot, pulling each other, or they play various games walking on tiptoe, and raise some smaller objects with their toes. Many of these games favour the development of the foot muscles.