In most Japanese lessons in English, when you are introduced to だ/です/である, it is simply explained as being like "is", or if they've already set a rule that sentences end in verbs, it's introduced as the verb "to be". That conveys, rightly or wrongly, all the things English speakers understand about "is".
This Japanese lesson starts with the assumption that part of the "だ" hiragana looks like "=", which I already don't agree with! Even the dakuten (゛) on the character looks more like an equals than the swooshy bowl they're comparing to "=".
It then says that だ is comparable to equality and mathematical equations, rather than like the lingual is/are/be, and then has to qualify that the equality is non-commutative. They could've saved time by not making that comparison, and instead comparing with is/are/be, which English speakers already understand is not commutative. "John will be early" != "Early will be John" (unless Yoda, you are)
I haven't studied many languages, but there's something about the flavor of so-called "foreign" language learning which is culturally anchored to the language of the teaching materials themselves. I wonder if these issues you have about comparisons that don't make sense and so on are about your own personal context and point of view, or if the issues reveal some differences between native Japanese teaching methods versus Japanese as she is taught to typical English language learners.
This Japanese lesson starts with the assumption that part of the "だ" hiragana looks like "=", which I already don't agree with! Even the dakuten (゛) on the character looks more like an equals than the swooshy bowl they're comparing to "=".
It then says that だ is comparable to equality and mathematical equations, rather than like the lingual is/are/be, and then has to qualify that the equality is non-commutative. They could've saved time by not making that comparison, and instead comparing with is/are/be, which English speakers already understand is not commutative. "John will be early" != "Early will be John" (unless Yoda, you are)