A grandmother attempts to teach her four-year-old granddaughter the multiplication tables using simple repetition, but they repeatedly start over at 'three fives'; the child keeps coming up with 'thirty-five'. We consider three possible explanations: self-perpetuating frequency of behavior, saliency of memory and Vygotsky's next or proximal zones of development (ZPD). Instead of interpreting the ZPD as learning (scaffolding, self-regulation or metacognition), we propose a strictly developmental view, based on Vygotsky's late and as yet untranslated pedological lectures. Because the lectures are unfinished, there are three apparent contradictions in his system of zones. Using the child language studies of the late Michael Halliday, we propose some possible solutions. In conclusion, we propose three alterations of the grandmotherly teachings that might orient the child towards the next, or proximal, zone of her development.