Number theory
While the word "algorithm" goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in "Elements", together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation formula_78,
or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called
"kuṭṭaka" ("pulveriser"), without a proof of correctness.
Mathetics
Seymour Papert, MIT mathematician, educator, and author, explains the rationale behind the term "mathetics" in Chapter 5 (A Word for Learning) of his book, "The Children's Machine". The origin of the word, according to Papert, is not from "mathematics," but from the Greek, "mathēmatikos", which means "disposed to learn." He feels this word (or one like it) should become as much part of the vocabulary about education as is the word pedagogy or instructional design.