I mention this when I teach number theory - and the fact that the proof in Euclid is entirely in words. People complain about mathematical notation sometimes, b

I mention this when I teach number theory - and the fact that the proof in Euclid is entirely in words. People complain about mathematical notation sometimes, but the imperfect notation we have is better than just words for everything. (We could use more pictures, however.) Likewise, if I define normal subgroups, then simple groups - and it would be a shame, at that point, not to mention the classification of the finite simple groups. I think from the students' reactions that these things are interesting to them. People interest people. So I mention Euclid, and maybe I take 1 minute. The finite simple groups, maybe 2 minutes. The class is 50 minutes long. Is that "teaching the history of math"? Really, the original question is ill-posed. Let us say I have a 50-minute class in a content course. How much of that time do I have to spend talking about history before I'm "teaching the history of math"? One minute? Ten minutes? Do I have to give an assignment on history? But a little bit of history, or culture, or a random story - I think that's part of learning the subject, broadly understood (as you said, humanely understood). Nothing but definition-theorem-proof would be pretty deadly.