Why Is Long Division So Hard?

 

Over the years, I've noticed that high school students struggle with performing long division in classes such as Algebra II.  One of the elementary teachers I know, told me her students struggle with dividing two digit into four digit numbers.  This comment was confirmed when I ran across an article published back in 1942 in which the author discussed why long division is so difficult.

I realized if students struggled with it in elementary school, it makes sense they will struggle with it in high school because they haven't mastered it. If students are unable to divide a four digit number by a two digit, how are they going to divide a polynomial by another polynomial?

If you take time to look at the steps used, long division is quite complex.  Yes, I said steps because that is usually how it is taught.  Long division uses estimation, multiplication, subtraction repeated multiple times until the answer is found.  If any mistakes occur, the answer will be incorrect so each step has to be performed perfectly.

One study indicated that the more steps a division problem requires, the higher the chances are that the student will get the wrong answer due to the increased possibility of making a mistake.  In addition, a student synthesizes all the mathematical knowledge in order to solve long division so if a student is weak in multiplication, they will struggle with the process.

Since most students learn to solve long division using an algorithm, they acquire little knowledge of what is actually happening and lack a conceptual understanding.  I admit, I could do long division in elementary but I never understood I was splitting the dividend into smaller groups of so many.  I couldn't have explained it other than taking you through the algorithm.

Another issue in long division of numbers or polynomials is that students do not understand that zero's perform as place holders indicating that the divisor cannot go into the dividend and another digit must be brought down so it becomes big enough.  It is something that occurs in both numerical and polynomial division.  Furthermore, many of the "hints" they learned such as the big number goes into the house does not apply when discussing polynomials

I will say that long division using polynomials is a bit easier than using only numbers because it's easier for students to make the first term of the divisor "match" the first term of the dividend rather than looking at the whole "number" and estimating. Unfortunately, the basic algorithm is still the same so if students did not get long division in elementary school, they often struggle with it in high school. 

No matter whether you are using numbers or polynomials, long division is a struggle for most students.  I'm still trying to figure out how to make a visual so students can "see" what is happening at each step.  Let me now what you think, I'd love to hear.  Have a great day.