I can only ask for success in early maths. Here we found that numerical activities in the home were a good predictor of early success. Benavides-Varela, S., Butterworth, B., Burgio, F., Arcara, G., Lucangeli, D., & Semenza, C. (2016). Numerical Activities and Information Learned at Home Link to the Exact Numeracy Skills in 5–6 Years-Old Children. Frontiers in Psychololgy, 7(94). doi:10.3389/fpsyg.2016.00094
This is a great question! Enjoying books/being read to is a strong predictor of school readiness and later school achievement. Students who enjoy books are more likely to become better readers and to do better in school for many years. I found this particularly interesting to read, although it is about older children: http://www.cls.ioe.ac.uk/news.aspx?itemid=2740&sitesectionid=27 Reading ability is so important for school because all of the subjects rely on it eventually (applied maths, history, science…).
My own research shows that re-reading the same books multiple times improves both preschool children’s enjoyment of books and their vocabularies. I was also involved in an 8-week intervention programme where we encouraged early years’ practitioners to re-read books multiple times. Each week they would have a “book of the week” that they would read daily and some would bring out items to play with that matched that week’s theme. The individual children who participated developed great school readiness skills, including listening and attention skills and being able to concentrate for longer. The boys also became more interested in looking at books, in general. Several settings decided to continue the programme on their own.
Self-regulation skills are now well-established predictors of academic success, in addition to certain early literacy and math skills. There is a very readable volume put out in the US by the Future of Children called Starting Early. It is available at https://futureofchildren.princeton.edu/news/starting-early-education-prekindergarten-third-grade. It has chapters reviewing the contributions of executive functioning to school readiness and academic achievement (by Blair and Raver), as well as chapters on early reading and mathematics. Many states include assessments of executive functions such as attentional control in their school readiness assessments (which are conducted by classroom teachers). The reading chapter by Snow and Matthews emphasizes the importance of vocabulary knowledge (as opposed to readily learned skills like letter names and sounds). The math chapter by Clements and Sarama illustrates how early math knowledge contributes not only to subsequent math achievement but also reading.