Awareness of philosophy of mathematics came to me late in life. (This is a confession, not a boast.) In retrospect, scattered questions, comments, and answers at math.stackexchange (MSE) demonstrated that others conceive of mathematics differently (in some respects) and often correctly (with logical consistency). As with any good learning experience, I understand more than before, and also less.
Two books I digitized for Project Gutenberg in 2012 and 2013 may be of interest to modern students and teachers, and are natural to showcase together. Each concerns the philosophy of mathematics, impinges on empirical and theoretical science, and has the readable tone of an essay.
An Introduction to Mathematics by Alfred North Whitehead explores topics of school and university mathematics, not as a textbook of methods but more as a guide to meaning.
Introduction to Mathematical Philosophy by Bertrand Russell covers material on sets, ordering, and limits, focusing on the infinite and on meaning.
One memorable interaction at MSE concerned a wish to multiply all real numbers in the open unit interval. The questioner was adamant that the product be non-zero, since each factor is non-zero. To the best of my ability to ascertain, the questioner viewed the answer as a matter of ontology rather than of definition. The fact that generalizing from finite products to infinite products all but enforces the value zero seemed to them deeply upsetting. The matter ended, but was not resolved.