Developing Math Sense for the 21st Century

June 12, 2008

Common uses of Computation

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

Algorithms

Algorithms

45 + 21 + 39More or less than 100?

45 + 21 + 39

43 x 26More 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.

1. 1004 – 97

2. 265 + 399 + 148

3. 68 x 5

4. 198 ÷ 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 = 40 + 16 = 56

7 + 5 = 12, so 70 + 50 = 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 1999 http://isc.bc.edu/)

• 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) 90056- 36

56 -36=___

• 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 21st 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 Changehttp://www.glumbert.com/media/multiply

Long Division

Friendly Numbers

Partial Quotients

Column Division

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