## In Search of The Standard Algorithms

There has been quite a bit of debate regarding the Standard Algorithms for Addition, Subtraction, Multiplication, and Long Division and whether they are or are not part of Investigations. Long Division is not a part of Investigations nor has it ever been. PWCS staff states that it intends to create a lesson teaching long division to students in Grade 5 but those materials are not available for review yet.

The debate regarding the other standard algorithms seems to evolve around the word teach and what that word actually entails. Here is a description of the “lessons” PWCS claims teach our children the standard algorithms.  PWCS staff have even intimated that the lessons described below are sufficient for students to master the processes.

Investigations has one single lesson on the Standard Algorithm for Addition. The lesson is in Grade 4, Unit 5, Session 2.4, and is entitled “Studying the US Algorithm for Addition”.

Students are sent home with an addition problem and asked to have their parents solve the problem. Upon returning to school the teacher gathers up the solutions and presents them to the class, making sure to include one example of the Standard Algorithm. The teacher then leads the class in a discussion demonstrating how the Standard Algorithm works, what the carried 1 means, and contrasting the process followed to other processes students are more familiar with. The class then breaks into groups and the students discuss the different strategies the teacher has presented.

After a period of time the teacher reconvenes the class and has the children comment on what they noticed about the problems. Teachers are instructed to ensure that the following points emerge from the discussion with the students (quotes are not exact due to copyright concerns):

• These solutions are Algorithms. Algorithms show a clear sequence of steps that can be used to solve a certain kind of problem. Some of you may have seen this solution before {indicating the standard algorithm}. It’s a way to solve an addition problem that many American adults learned when they were in school and may still use now. Adults from other countries may have learned a different way so this method is called the US Algorithm.
• The US Algorithm is a strategy where numbers are broken apart by place.
• The notation for the US Algorithm uses shortcuts to show numbers. For example the small 1 in the 10’s place represents the 10 in the 14. It is important to keep in mind the actual values represented by this notation when it is used.

Students are then sent home with a handful of problems to solve by whatever process they prefer.

That is the extent of the lesson.  There is no followup, no review, no practice of more than a few problems to ensure competence with the process.  Just a one day discussion followed by a handful of problems to be solved by whatever process the students prefer.

That same lesson is repeated later in Grade 4 for Multiplication and in Grade 5 for Subtraction.  Long Division, as stated before, is not covered in the Investigations series.

PWCS has stated that it is moving the lessons on Addition and Multiplication to Grade 3 and Subtraction to Grade 4, however, the lessons are unchanged from what is described here.

Certainly semantics plays a role in this discussion, however, in my estimation these lessons don’t come close to teaching students the standard algorithms, much less provide then with sufficient understanding or practice to master them.  Even the authors of Investigations refrained from using the word “teach” to describe these lessons.  The authors of Investigations state that the standard algorithms are “explicitly studied”, but they never claim they are taught, much less mastered by students in these lessons.

One specific bit of misinformation is conveyed to students in these lessons which is worth mentioning.

There is no such thing as the US Algorithm, except in constructivist circles.  The processes Investigations refers to as the US Algorithms are the Standard Algorithms.

The Standard Algorithms for arithmetic are believed to have been developed by Aryabhata in India in approximately 499 AD ; they first appeared in commercially developed textbooks as early as 1500 AD,  long before the US as a country was even imagined.  They are taught to students and used in countries throughout the world.  They are not unique to the US nor is the US responsible for introducing them.