Augustus De Morgan — logician, historian of mathematics, and the man who named mathematical induction — believed arithmetic should be understood, not memorised. His Elements of Arithmetic (1830) begins with a company of horsemen and the bare idea of counting, and builds numeration, the four operations, fractions and proportion from first principles, explaining at every step why the rules work.
It taught generations of Victorian students that even ‘simple’ arithmetic rests on real ideas, and it remains a model of mathematical exposition. Download the free EPUB and relearn what you thought you already knew.