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.
Do
we Really Want to Keep the Traditional Algorithms for Whole Numbers? John A. Van de Walle
