The Problem With Remainders

I am returning to the topic of division, specifically remainders because so many of my students do not seem to understand what it represents.  I’ve observed many students who took a remainder and made it the decimal portion of the final answer.  For instance, if they have 24/5, they do the division and instead of 4 4/5 or 4.8, they put 4.4 because the remainder is four. I have no idea where that misunderstanding comes from.

I’ve tried to research why this happens but have not been successful as most every article only discusses the process of division and how difficult it is.  It is hard finding articles discussing why students have issues with writing down the correct form of the remainder. I wonder if they do not connect the remainder with a fraction and  a decimal equivalent. I'm not sure they even understand the concept of a remainder . 

Most articles state that the process of long division is extremely difficult and students struggle to learn the multi-step process but few address misunderstandings associated with remainders. I found one author who indicated that higher order thinking skills are needed to interpret the type of remainder. The remainder requires students to determine its context in order to figure out the type of remainder that is needed.

There are four types of remainders that students will run into when doing division.  The first type of remainder is the one you leave alone as a fraction such as in 25/4 gives 6 and 1 left over as the answer.  The numerator tells you how many were left over out of the groups.  In other words, the part of a whole.  This is the case where my students tend to write 6.1.  The second type of problem looks at only the remainder so the problem might tell you that you have 25 quarters and wants to know how many you have left over once you turn the quarters into dollars.  In this case, the answer is one quarter. 

The third type of remainder is one that is either rounded up or down depending on the circumstances.  For instance, if your can of paint will cover 300 square feet with two layers and you need to cover 1100 square feet with a two coats, how many cans should you buy.  If you do it with a calculator, you’ll end up with 3.666666667 or 3 ⅔ cans.  In this case, we need to round up because we cannot buy ⅔ of a can.

The final type of remainder is the sharing remainder where you want a fractional answer such as you are sharing 25 cookies among 4 people, how many cookies will each person have?  It is 6 ¼.  The answer tells us that each person is going to get 6 ¼ cookies.

Based on my own experiences teaching, students have issues with remainders because they have not fully learned how to use long division in addition to not knowing how to interpret remainders.  Furthermore, they are also weak on connecting remainders with fractions and changing those fractions into decimals.  Let me know what you think, I’d love to hear.  Have a great day.