Tuesday, January 3, 2017

The Truth Behind the Standard Algorithm

For as long as I can remember seeing in pictures, reading in books, and watching old TV shows, the standard algorithm has been the staple of mathematics during the elementary years.  These algorithms are burnt into our brain through images of the old school house, blackboards, and crummy movies.  However, they have maintained in instruction for an assortment of reasons. 

1.     They help most students calculate math.
2.     Our parents learned through them so therefore, children have also.
3.     Teachers tend to teach the way they are taught…thus the algorithms continue.
4.     The Common Core has them stated as necessary parts to instruction in grades 3-6.

Recent instructional pedagogy has produced strong data to support no longer using the standard algorithm as the main form of instruction.  The changes started in the late 1990’s and are now being pushed further by people such as Jo Boaler.  Their efforts are based off an understanding of mathematics rather than just calculation.  With all of our advancements, the United States continues to be one of the few remaining developed countries that use the standard algorithm as the main form of instruction. 

Personally, I couldn’t agree more with the changes being pushed in recent years.  Since starting as a K-12 Math Coordinator we have been discussing, developing, creating, and presenting alternatives to these algorithms that have more to do with understanding than calculating.  We have been working against traditional math trying to encourage students to do more than calculation.  I believe we can expect so much more from our children than rote mathematics.  I believe we need to focus on the “why” rather than giving students the “how.”

Lets go on a journey through some of the biggest reasons why teachers keep emphasizing these algorithms and why we as professionals need to make the decision to move on.

The methods in the algorithms are needed to learn the upper levels of mathematics
Forgive me but I started with my favorite reason most people give to keep the algorithms.  It is not enough to say that after grade 6 or 7 calculators are doing the vast majority of the dirty work in calculating math.  It is more important to understand that the methods used in the algorithms are not used in upper levels of math.  The only algorithm that reappears consistently is the division algorithm, which comes back when dividing polynomials.  Even that method for dividing polynomials is an inefficient method as compared to synthetic division or graphing solutions.  In all of these cases there are apps that can do much of the computation for us.  This doesn’t mean it isn’t important to know how to do these steps but that its’ importance is minimal as compared to the much larger picture of what the outcome of the division means.

The standard algorithm for multiplication is purely gone.  For some time area models have replaced the algorithm.  Even that is an incomplete comparison because we are comparing polynomials with multiple terms, not numbers.  Polynomial multiplication is closer to the partial products method than the standard algorithm.  Furthermore, the methods used emphasize the meaning of multiplication.  Not just calculating for a solution.

It is the methods parents know so we must teach it that way to help with home-school communication
It is ironic that statements similar to this one surface about math when they don’t surface about reading or writing.  I believe it has much more to do with the procedural drill and kill approaches taken when current teachers/parents were learning math.  In schools, we used to teach keyboarding at the high school level.  We taught students lattice multiplication in the late 1990’s and early 2000’s.  We went through phonics, to whole language, and back again. 

This goes to show that times change and we need to move with them.  With the technology of today we can communicate our methods of instruction with parents and more importantly the reasons why instructional methods are changing.  We need to emphasize instructing parents as much as instructing our students.

To be clear, we own this problem.  The problem is communication, not knowledge.

The algorithms work.  Why change what is working?
I would argue that the algorithms are not working.  In third grade, students learn to add multi-digit numbers together.  This addition should be fluent by the end of the year.  However, in fourth grade teachers are always re-assessing and arguing that the students don’t know how to add multi-digit numbers.   The same is true in fifth grade, sixth grade and so on. 

Is the problem that students don’t know how to do it or are not retaining the knowledge?  The answer, based on student performance is obvious.  Students are proficient at the skill in each grade level but when reassessed the following year no longer show the same level of proficiency.  The students don’t retain the process.  However, when using alternative methods such as partial sums they not only retain the ability to add they perform it at a fluency level doing much of the calculations in their head.  They learn that adding the hundreds, tens and then ones makes it easier to get the solution.  It also gives them a much better understanding of place value which means when the students transfer into multiplication it makes more sense.  

I leave this blog with a final thought.  Watch a student as they progress through Kindergarten to first and then second grade.  Students don’t naturally develop the traditional algorithms for addition or subtraction.  Instead, they focus on concepts that deal with place value.  The traditional algorithm must be learned through a teacher that directly teaches it.  That alone should tell us what we should be doing.  I believe there is a place for these algorithms.  However, only if they are taught after the sense making methods are discovered. 

As always, I don’t consider my opinion to be fact.  Because of that I have linked a few articles that support both sides of this story.  Enjoy the reads and come to your conclusion.  Please share it with me.  Hopefully we can learn together.