Monday, October 1, 2012

Traditional Algorithms VS Invented Strategies

There are significant differences between traditional algorithms and invented strategies.

The traditional algorithms are based on performing the operation on one place value at a time with transitions to the next position. They involve trades, regrouping, "borrows," "carries" or Dead Monkeys Smell Bad!  The procedures are rigid.

Traditional algorithms take the understanding out of place value. They are "digit oriented." They rely on procedures or steps that must be done in a specific order and usually start in the ones place. 

Look at theses examples of traditional algorithms for addition and subtraction. They follow a procedure.

Think about what is being said as you solve this problem: 7 plus 8 is 15. I put down the 5 and carry the 1.  One plus 5 plus 3 is 9. Three plus 2 is 5. The answer is 595.

Think about what is being said as you solve this problem: 3 minus 7, you can't do it so you have to go next door and borrow 1. Since they don't have any to give you they go next door and borrow one. The 3 becomes a 2 and the zero becomes a ten. Now they have one you can borrow so the ten becomes a 9 and the 3 becomes a 13. Now you can subtract! 13 minus 7 is 6, 9 minus 6 is 3 and 2 minus 1 is 1. The answer is 136.

Invented or flexible strategies develop a good understanding of the operations especially the commutative property and the distributive property of multiplication. Students start to see the relationships of addition to subtraction, addition to multiplication, and multiplication to division. What an important concept! 

Invented strategies involve taking apart and combining numbers in a wide variety of ways. They are "number oriented." Most of the partitions of numbers are based on place value and start in the largest place. 

Now look at the examples of the same problems. Although these may look complicated many invented strategies can be done mentally and just recorded on paper.

Think about what is being said as you solve this problem: 300 plus 200 is 500, 50 plus 30 is 80 and 7 plus 8 is 15. I add them together and 500 plus 80 equals 580 plus 10 is 590 plus 5 more is 595. The answer is 595.

Think about what is being said as you solve this problem: 300 minus 100 is 200. Since I know 67 + 33 gets me to 100 then 200 - 67 is 133. But I still have the 3 ones in 303 that I need to give back. 133 plus 3 is 136. The answer is 136.

Your Turn: 
Think about the traditional algorithms for multiplication and division. Are they rigid procedures? Are they "digit oriented?" Do they build understanding of number relationships? 

Be careful: Don't turn an invented strategy into a "procedure." 

Students must be allowed to develop their own strategies based on their own understandings!


  1. There is a clear distinction between the traditional and invented algorithms and this shows how learning has evolved. I will be recommending this site to our clients who are majorly students accessing our professional writing and editing services offered by Genuine Case Study Writers.