Why Division is Difficult

Division, particularly long division, is often a stumbling block for many students. This difficulty can be attributed to a combination of factors.

First students struggle with a conceptual understanding of division. Unlike addition, subtraction, and multiplication, division is often seen as a more abstract concept. Division can represent different real-world scenarios, such as sharing, grouping, or measuring. Understanding these interpretations can be challenging for students.

Second, they need to develop the procedural knowledge rather than relying on shortcuts.  Long division involves a series of steps, including division, multiplication, subtraction, and bringing down digits. Each step requires careful attention and precise calculations. Any mistake in a single step can lead to incorrect answers, making it crucial to master each step.

Third students often have specific misconceptions associated with division.  A common misconception is that the remainder should always be converted to a decimal. While this is sometimes appropriate, it's important to understand that the remainder can also represent a fractional part or a leftover quantity. Students may not fully grasp the concept of division by zero and its undefined nature.

Now it's time to look at strategies designed to improve division skills. It is important to take students from concrete to abstract.  Use physical objects like blocks or counters to model division problems. Connect division to real-life situations, such as sharing snacks or dividing money.

Utilize a gradual release of responsibility through guided practice and independent practice Provide step-by-step guidance and support as students learn the division algorithm. Gradually increase the level of independence, allowing students to practice on their own.

Take time to help students learn to do error analysis. Help students identify common errors, such as incorrect placement of digits or miscalculations. Provide targeted instruction to address specific misconceptions and reinforce correct procedures. In addition, work on estimation and basic facts. Teach students to round numbers to estimate quotients. Practice basic division facts to improve fluency and mental math skills.

In addition, establish an algebraic connection. Ensure that students have a strong understanding of arithmetic division before moving on to algebraic division. Use diagrams and models to help students visualize the steps involved in algebraic division. Provide ample opportunities for students to practice algebraic division problems.

By addressing the underlying causes of division difficulties and employing effective teaching strategies, educators can help students develop a strong foundation in division and set them up for success in future math courses, including algebra.  Let me know what you think, I'd love to hear.  Have a great day.

Teaching Division

  Division seems to be as difficult for students to master as subtraction.  I have students who struggle with it if its a division problem.  If I change it into a multiplication problem, they struggle less but its still a challenge.

According to two different articles I read,  division is introduced to young children by using the idea of sharing, such as there are 12 pieces of candy shared between 3 children. How many pieces will they each get. 

The second stage of teaching division has students looking at division as grouping rather than sharing.  It also incorporates relating it to division.   Add into the mix the fact that the division algorithm makes less sense to most students than any other operation because its a left to right rather than the usual right to left process.

Most of the students who are successful using the division algorithm have memorized the steps without understanding the concepts behind each step.  I am the first to admit, I am one of those.  I was told how to do it, I followed the steps and got the answer,  What more did I need to know?

The students who struggle are the ones who do not know their multiplication tables well and who missed the step of placing a zero in when they cannot divide the number.  These same students  struggle when division is taken from using straight numbers such as 28/7 to dividing polynomials such as x^2 + 5x + 6/x+2.  The algorithm is the same but it uses variables.

In my opinion, division is really just a way of finding the second factor of a number when you know one factor.  Yes, sort of the reverse of multiplication.  When my students struggle with 28/7, I'll often ask them "7 times what gives you 28? because it is sometimes easier for them.

It has been suggested students should not even attempt the division algorithm until they have their multiplication facts down cold, otherwise it is difficult for them to learn.  Its also suggested students have division down as both sharing and grouping because they need both interpretations.  With grouping if you have a problem like 900/30 it could be interpreted as 900 items split into 30 groups or 900 items split into groups with 30 items.

Add into this the idea of remainders.  I don't think the elementary teachers or books have taken time to explain remainders properly.  My students see it as remainders, not as so many items left over which are not enough to make a group itself.

I'd love to hear from you all on ideas you have to help make division easier for high school students who struggle.  Hope you are having a good break.

Then comes the question of when will I ever use polynomial division?  It is used in most crypto and error correcting algorithms which are used in most modern digital devices.