Mathematical Approaches to Support K-12 Student Learning in Mathematics Rosalie A. Dance University of the Virgin Islands

Mathematical models of real world phenomena: investigations that o lead to mathematical concept learning and o reap understanding of phenomena of importance and interest to the student. Geometric illustrations of number, relations, numerical operations, algebraic operations, algebraic concepts, – including tangents, limits, area under curves, equilibrium values, rate of change, …. – Rolle’s theorem for 10 th graders. Play, experience, touch, see 2

Support for development of intuitive understanding of advanced concepts from earliest learning. 2 nd grade: What if I DID subtract a bigger number from a small one? What would happen if I tried to take 3 away from 2? 2 – 4? 2 – 5? 5 th grade?: What number could go in the box? 3 + █ = █ + █ = █ = 12 (given 2 █ means two boxes of the same thing) When students have studied graphical representations of linear functions and learned how to use them to solve linear equation: use a graph to solve non linear equation. Look for patterns. E.g., the differences between consecutive terms in the sequence of perfect squares is the sequence of odd numbers. why IS that? 1,4,9,16,25,36,… Notice limits. E.g., 1 + ½ + ¼ + 1/8 + … 3

Communication, spoken and written. Students talk about Mathematics to share their understanding of it, to explain real world phenomena using it.  Students enjoy discourse, and they profit from it.  It helps them develop their thinking, and it helps them recognize that mathematics is a human endeavor. Students should write mathematics, simply, clearly. Students should also write about mathematics they have done: letters, stories, essays. 4

Problem solving. Do it! Use Polya’s approach. Give instruction in how to – read problems, – take note of what information is given in the problem, – take note of what is asked, – connect what is given and what is asked, – work forward; and work backward, – think and work geometrically where possible, – express everything clearly, simply, in algebraic terms, – check reasonableness of a tentative result, – verify results. Give positive feedback generously. -Encourage pride in small steps and large ones. 5

Integrate history of mathematics and related history Especially include history from non-western cultures and from under-represented groups Include women Include men Include Africa, Asia, Europe, the Americas; the North, the South, East and West Connect to times and places they study in history and literature 6

Connections Connect mathematics to the students’ world AND to other mathematics CONTEXT. Sickle cell & malaria investigation. Alcohol, caffeine, heavy metals elimination from the bloodstream. Population issues. Life expectancy variations in different populations. Epidemics (the spread of communicable diseases.  Students should uncover significant information about topics that interest them by doing mathematics.  Investigations associated with the mathematics provide a hook for the math concepts in the learner’s memory. 7

Connections-2 CONNECTIONS TO OTHER MATHEMATICS :  Connect geometric thinking and algebraic language  Connect new concepts to old ones whenever possible.  Give hints about mathematics that lies ahead as it relates to mathematics students are doing now o Slopes in elementary algebra→what calculus is about. o Areas →what else calculus is about o Volumes in geometry class → volume problems in calculus course o Number patterns → function concept o Number patterns and intuitive notion of convergence of sequences 8

Technology Use technology to support and enhance conceptual understanding, thinking and problem solving. – Numerical methods – Graphical methods Use methods of operation that help students – to enjoy and appreciate their own skills and – to eschew any use of calculating devices that actually inhibits efficiency. 9

Overcome early obstacles Teach number sense – Fractions, decimals. (Eat 1/3 of your pie and give 2/3 to your mom to share with your dad.) – Signed numbers (Above and below sea level; gin rummy; in and out of debt.) Play – Sing the multiplication tables. – Use the zero concept to understand addition of signed numbers. – Act out un-doing addition and multiplication to develop inverse operation concept – Act out graphs; dance trig functions – Act out recursive functions to discover their rule (e.g., Tower of Hanoi) – Roll dice; what do you expect? Etc.! 10