Developing Math Sense for the 21 st Century June 12, 2008

Common uses of Computation

Algorithm An Algorithm is a systematic, step-by-step procedure used to find an answer, usually to a computation problem.

Algorithms

More or less than 100?

43 x 26 More or less than 1000? 43 x 26

Do the following problems mentally, jotting down intermediate steps if necessary. Share your methods with your group. Your group should be prepared to share at least 2 different methods with the class – x ÷ 17

Developing Number and Operation Sense Look at numbers as whole quantities Look at problems as a whole before choosing a strategy Think from left to right about numbers Use landmarks in the number system Reason from number relationships that you know 5 x 8 = 40, so 7 x 8 = = = 12, so = 120 Use any operation that makes sense for the situation

Students learn new algorithms faster when they first experience an algorithm through alternative visual models and discuss their logic, than when the teacher tells about the algorithm and then drills students on its application Students who are skillful with a particular procedure are very reluctant to attach meanings to it after the fact. Algorithms

The traditional, rote approach to teaching algorithms fosters beliefs such as: Math consists mostly of symbols on paper Following the “rules” of math is most important Math is mostly memorizing facts and rules Speed and accuracy are more important than understanding There is one right way to solve any problem Math symbols and rules have little to do with common sense, intuition or the real world.

our advanced calculus and AP physics students rate last in the world. (Wm. Schmidt, MSU, The Widening Achievement Gap) Our 8th grade advanced students rank in the middle of the pack in the world.(See TIMS and TIMSS-R 1996 and In one study only 60% of 10 year olds achieved mastery of subtraction using the standard “borrowing” algorithm. A Japanese study found that only 56% of third graders and 74% of fifth graders achieved mastery of this algorithm. Traditional math instruction isn’t working as well as we might think.

Research tells us that between 3rd and 6th grade the equivalent of one year is spent teaching long division. Yet, when 17 year olds were tested for division with a 2 digit divisor, fewer than 50% answered it correctly. Children learn to perform the steps (rules) of the algorithm without thinking about the math behind it

900 ÷30 = __ 30 ) =___

Problems can, and should, be solved in more than one way Studying several algorithms for an operation can help students understand the operation Providing several alternative algorithms for an operation affords flexibility Presenting several alternatives gives the message that mathematics is a creative field.

If you consider the traditional instructional model to be handing the kid a shovel and making him expert in using the shovel, then the 21 st century approach is handing the kid a fully-stocked tool shed and training him to select the most appropriate tool from the shed to apply to the math task at hand. The kid who only knows the shovel can only respond if the task is digging, but the kid who owns the tool shed can respond flexibly to any task. Bruce Harrison

Standard Algorithm Partial Sums Column Addition Lattice Addition Opposite Change

Standard Algorithm Trade-First Partial Differences Counting Up Same Change

Standard Algorithm Partial Products Lattice Russian Peasant Same Change

Long Division Friendly Numbers Partial Quotients Column Division

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