Hello Sasha A dichotomy is a distance between two opposites e.g. "there is a dichotomy between what politicians say and what they do". It is often thought that work (or learning) and play do not go together. They are considered opposites. Children are increasingly expected to perform. Instead of allowing children to play, and therefore to be children, their early years are increasingly used as a preparation to better performance later. The distance between playing and learning is therefore increasing. In this context, a false dichotomy means that play and learning are presented as the only two options, as opposites not to be mixed. But children often learn a lot while playing. And learning can also be fun. I hope this helps.

Only meant as a comment (and my exercise). It’s such an interesting topic… As already mentioned before, a false dichotomy is a kind of constructed or assumed dilemma. This thinking model excludes a long row of possibilities between two extreme points or alternatives. In reality, between the colors white and black are countless shades of grey! If you observe playing children, you can often see them learning WHILE and BY playing. In fact, children like to learn as long as playful teaching meets their interests and needs. But often we have to see that the enthusiastic "I want to learn" gets replaced by "I must learn, and I don't like what I have to learn!" Consider for a moment, you would learn a foreign language only by doing playful things... Nothing against funny language games as long as they challenge the learners to utter more than a single word... But, for example, a puzzle-game excludes the necessity of playing on more than this one field of single words. You have fun and entertainment, but you will forget the words you added to a long, long list due to lacking communication.

Here, a false dichotomy means a distinction or a choice between two things that aren't actually either/or situations. In other words, with a false dichotomy, you think that you can only have one or the other, but in reality, you can actually have both. It doesn't mean that you always will have both, but it can happen. An example of a false dichotomy might be as follows: "You can be successful or you can be happy." Here, I'm telling you that you cannot be successful and happy at the same time. You can be successful OR you can be happy. This, of course, is not true. Many people who are successful are happy, and many happy people are successful. In the paragraph you gave, the false dichotomy is between learning and play. This means that there is a false choice between learning and play–you either learn or play, and you can't have both. The paragraph is really saying that playing and learning can be the same thing, but instead, students are taught to prepare for tests instead of understand how the world around them works. When children play, they understand the rules of their society and environment better, so a child can definitely learn and play at the same time. Just like you can be successful and happy at the same time, you can learn and play at the same time too. Society makes it seem like you can only play or only learn, but you can have both at the same time. That, my friend, is a false dichotomy. Надеюсь что я помогал.)

The dichotomy in the mathematics meaning is a solving method which consists of deviiding a field on small fields hence the search of the unknown element will be easier.