Education is about supporting humans in their growth, with a special focus on exploring their intellectual potential. Learning to act following a given (even complex) pattern is losing its educational value very fast, because all well described activities can be automated. Education therefore should focus on developing those cognitive process dimensions of pupils where technology cannot compete with humans (Dagienė et al. (2020), Hromkovič and Lacher (2017), Hromkovič et al. (2020)). The contribution of this paper is conceptual. In the paper we show that starting with the algorithmic view on the historical development of number representations and calculations offers a natural, more understandable way for teaching mathematics in primary schools. We show that going consequently from concrete to abstract empowers pupils to be able to design own representations of numbers, rediscover the execution of arithmetic operations on their own, and even develop elementary calculations in own designed number systems. We show here how a successful process of rediscovery of arithmetic algorithms can be designed by using classical algorithm design methods as “induction” and “divide and conquer”. We show how that algorithmic thinking can essentially contribute to improving education in mathematics.