I thought of something like Polish notation all by myself and asked the question https://cs.stackexchange.com/questions/111067/could-we-define-the-decimal-notation-of-a-natural-number-as-a-series-of-operatio. Then somebody wrote a comment on it saying there already is something like that which is Polish notation. I read some of the Wikipedia article Polish notation. I think Polish notation is good. It's more straight forward. I'm planning to later ask another question of whether students struggle less with interpreting mathematical expressions in Polish schools. However, I'm trying to get to the bottom of this and learn more about what notation they use in Polish schools and want my question more direct and answerable so I'm making it specifically about Polish schools. For example, here we just write something like $xy$ instead of $x \times y$. In Polish schools, do they even represent statements Canada uses variables to represent using variables in a similar way? If so, do they also write something like $xy$ instead of $\times x \text{ } y$? Do they instead of something like a more direct way of stating double or triple universal quantification? For example, the associative law of natural number addition is really the statement that all natural numbers have a certain property and for each natural number, it is defined to have that property if all natural numbers have another property, and each natural number is defined to have that property if all natural numbers have another property.

Here in Canada in Calculus, we write $f'$ to denote the derivative of the function $f$. I wonder whether in Polish schools, they specially denote operations so that they can introduce level 2 operations such as differentiation denoted $'$ ad composition denoted $\circ$ and stick them at the beginning. For example if $f$ represented one operation and $g$ represented another operation and $x$ was a variable, then $'fgx$ would mean you take $x$ then you apply $g$ then you apply $'f$ which is the derivative of $f$ whereas $'\circ fgx$ would mean that you take $\circ fg$ then differentiate it then apply it to $x$ or something like that.

I could maybe ask a question just about the lower level math or elementary school math that doesn't deal with level 2 operations like Calculus uses and then later ask a separate question about whether they do Calculus that way. Then it would be confusing what my question was. Do I want to know whether they do Calculus that way also?

That was the question. Now I'm just explaining why it might be useful. Maybe researchers will read this question and incorporate the following idea into their research. Maybe we will work towards a new and different system where everyone learns a more direct less confusing way of writing things in regular school and university, and the job market does their own hard work of giving work place specific training later including teaching short cuts in math research jobs but doesn't make it very advanced and confusing so that anyone will be able to learn how to do the job and not depend on people having learned stuff in school that not everyone can learn in school. Maybe people will sometimes use short cuts. However, there will be separate mathematical projects each of which is separately organized slowly and carefully and they will try and combing all the math research work into fewer organized projects. Then they will do all the slow careful thinking how to organize each project then once it's organized, they might just blindly follow rules that need to be applied so many times.

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    $\begingroup$ As the linked article notes, "Polish notation" is named after Jan Łukasiewicz and dates from 1924. So as an initial guess I'd say it's unlikely to be the standard in Polish schools or textbooks. $\endgroup$ – Daniel R. Collins Dec 15 '20 at 21:43

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