Presented and produced by Seán Delaney. In this episode I speak to Professor Art Baroody from the College of Education at the University of Illinois at Urbana-Champaign about matters related to counting and early mathematical development. Professor Baroody shares insights from his extensive research in children's early mathematical learning with anecdotes from his life and work. Among the topics we discuss are: The word “count” is ambiguous; he prefers the terms verbal counting and object counting. Along with subitising, these are foundational for children’s sense of number. The rote portion of numbers (up to 12 in English) and the rule-governed portion of numbers (13 onwards in English) Being able to meaningfully count objects means understanding the cardinality principle How a teacher can assess a child’s competence in object counting. The “hidden stars” game. The importance of subitising (easily recognising, without counting, the number in a set). If a child can subitise small sets of objects and connect it to their verbal counting knowledge, the child can get insights into the structure of the count sequence and into our number system. The importance of children understanding the “increasing magnitude” principle of numbers. Subitising and learning addition and subtraction concepts The value of playing dice games. The successor principle: Each step in the counting sequence means you added one more. A child who starts out behind in kindergarten, typically gets further behind as school goes on, indicating the importance of informal mathematical knowledge for school readiness. Three components of a hypothetical learning trajectory: a goal, a learning progression, instructional activities that help children move from one level to the next. The relevance of a hypothetical learning trajectory for a teacher’s work: questions and instruction need to be developmentally appropriate for children. What number comes after 9? Whether you need to start at 1 or can answer this directly depends on your current level of understanding numbers. How schools typically target instruction at a level that is too low or too high for students. There are many published learning progressions and hypothetical learning trajectories available to teachers now, especially in number, arithmetic and counting development. A child’s mathematical power, routine expertise (learning something by rote – hard to apply it to a new problem and easy to forget) and adaptive expertise (learning something with understanding) Mathematical power comes from understanding, engaging in mathematical inquiry, to reason mathematically, to solve problems, having an interest in mathematics and using it. In short, conceptual understanding, mathematical thinking skills, and a positive disposition towards mathematics Example of applying knowledge to finding the area of a parallelogram Why memorising mathematics by rote is crazy. All children, even those with learning disabilities, can develop mathematical power up to lower secondary school level, if properly taught. Teaching mathematics by rote is cheating children. Things that can be discovered are the additive commutativity principle (3+5 = 5+3) Children are capable of much more than we give them credit for. Why getting children to learn off tables of number facts is cheating children. The importance of seeing patterns and relationships in the number tables – make it a thinking exercise and make mathematics learning fun. Working with his mentor Herb Ginsburg The use of manipulatives in teaching mathematics, even to college-level students. The value of children inventing procedures themselves. To understand fraction multiplication, the analogy of multiplication as repeated addition does not suffice. You need a more powerful analogy. A “groups of” analogy is more helpful. And it helps you understand why multiplication doesn’t always make something bigger. How to make sense of fraction division. How he conducts his research