Academic risk factors and deficits of learned hopelessness: a longitudinal study of Hong Kong secondary school students

Preschoolers' conceptual understanding and procedural skills were examined so as to explore the role of number-words and concept-procedure interactions in their additional knowledge. Eighteen three- to four-year-olds and 24 four- to five-year-olds judged commutativity and associativity principles and solved two-term problems involving number words and unknown numbers. The older preschoolers outperformed younger preschoolers in judging concepts involving unknown numbers and children made more accurate commutativity than associativity judgements. Children with conceptual profiles indicating a strong understanding of concepts applied to unknown numbers were more accurate at solving number-word problems than those with a poor conceptual understanding. The findings suggest that an important mathematical development during the preschool years may be learning to appreciate addition concepts as general principles that apply when exact numbers are unknown.