Traditional Algorithms – Tools or Obstacles to Learning? By Dr. Alan Bates

Alan Bates is an Associate Professor of Early Childhood Education at Illinois State University. His research interests include children’s mathematical development, teacher’s math self-efficacy, and parent-school interaction.

Recently, Common Core math computation problems are circulating on Facebook and other websites. The problems demonstrate children’s problem solving using number lines and their knowledge of base ten. This type of problem solving is being criticized by many because it appears much more difficult than the traditional algorithms that many of us learned. It is very true that this type of problem solving is different, but really should not be more difficult for those that have a conceptual understanding of computation. Unfortunately, knowledge of traditional algorithms can undermine our ability to think about computation in different ways.


            Traditional algorithms, as many of us were taught, can be hindrances to learning about numbers and number relationships. Traditional algorithms require students to stack the numbers being worked with on top of each other and then start at the right or ones digits. These algorithms are often taught using vocabulary such as “borrow” and “carry.” Two words that aren’t true mathematic terms and often times the use of them deters from a true understanding of what is occurring in the algorithm.

            Teachers cannot avoid teaching students traditional algorithms. In fact, they are introduced under the Common Core but teachers should ensure that students have a strong understanding of number relationships and a strong sense of computational fluency – the ability to accurately and efficiently solve mathematical problems in flexible ways, before introducing them to traditional algorithms. Teachers should encourage children to be critical thinkers in all educational domains and traditional algorithms do not encourage critical, flexible thinking. Instead they focus on following steps and rules.

            More math methods books and math curricula in the United States are encouraging the invented strategy method of computation, which encourages children to think about the numbers in the problem and not just the specific steps involved in solving those problems. These strategies encourage base-ten thinking and use of numbers that are easy to work with such as rounding to the nearest ten. This type of thinking often occurs when doing mental mathematics. For example, consider the problem, 78 + 75. You may be inclined to put the 78 on top of the 75, add 5 plus 8, put 3 under the 8, “carry” the one, and then add 7 plus 7 then the 1 to get 153.” Sure, that may be easy for many of us but if you think of it as “75 plus 75 equals 150 plus 3 which is 153,” isn’t that just as easy if not easier? And of course children should be encouraged to use their previous knowledge of adding quarters so a problem like 75 + 75 is natural.

            Another example of an invented strategy, this time with multiplication, is taking a problem such as 7 X 39 and simply multiplying 7 X 40 to get 280 and then subtracting 7 to get the correct answer of 273. Those of us who were taught only the traditional algorithm may automatically stack the two numbers, with the 39 on top and the 7 under the 9, and begin multiplying as we were taught.. The National Council of Teaching of Mathematics supports of the invented strategy approach to problem solving, in that it encourages students to use their understanding of number relationships and properties of number such as place value to solve mathematic problems.

            With so many options of computation available to teachers, what should teachers do? The Common Core encourages teachers to teach children about base-ten and number relationships and allow them to use this knowledge in solving problems. Isn’t that what all teachers strive for? To provide students the knowledge they need to solve problems and allow them to use that knowledge the best way they see fit. When we teach algorithms, we show them exactly how to solve the problem, which does not allow any true problem solving to occur. We want students to apply knowledge and be able to explain their problem solving methods to demonstrate their understanding. A main goal in teaching students about computation is to provide them with a deep understanding of multiple methods of computation and their efficient use. Students should be able to not only use various methods, including the traditional algorithms but to explain their use. This more well-rounded understanding of computation and algorithms will serve students better in everyday life and in their future schooling.


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