MATH CHAT Making Sense of Understanding, Grades 5-8

Understanding mathematics is central in contemporary thinking about mathematics. The idea that one can actually make sense of mathematics is a new one for many adults and children. The uneasy relationship that many have with mathematics is the result of never having made sense of it – rules and formulas were memorized and used before being quickly forgotten – and learning never “fit together. ” Math was a large collection of pieces. One repeatedly hears the refrain “I did OK in math, but I never understood what I was learning.” Mathematics education and cognitive psychology researchers have shed new light on what it means to “understand ” and the powerful effects of learning through understanding. Under-standing involves seeing how ideas fit together with new learning connecting with previous knowledge. It involves really thinking things through for oneself and building an “integrated grasp of ideas. ” Mathematical symbols and rules are linked to ideas. Multiplication, for example, is understood when children can link it to everyday situations and to their under-standings about addition. Basic area ideas about covering and counting lead to the generalization for the area of rectangles. Formulas for the area of parallelograms and triangles can be related to the work with rectangles. This approach helps students make sense of the formulas. Adding it Up: Helping Children Learn Mathematics describes the powerful effects that understanding mathematics has on learning. Key ideas include the following (pp.118-120): • Students with conceptual knowledge (understanding) know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful. • Students organize their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. • Because facts and methods learned with understanding are connected, they are easier to remember and use, and can be reconstructed when forgotten. • Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems. • Understanding helps students avoid many critical errors in solving problems by recognizing when answers aren’t sensible. • Conceptual understanding is a wise investment that pays off for students in many ways