After reading an article by Richard R. Skemp discussing relational vs. instrumental understanding, in class we discussed how we had learned fraction operations (addition, subtraction, multiplication, division). For me and the people in my group, most of us had learned through instrumental understanding. We were taught a rule to memorize and apply in the given situation in order to solve the problem. A "rules without reasons" way of learning.
As we went over the different operations, for all of the first 3, addition, subtraction, and multiplication, we felt we had a good understanding of them. The answers we would get from problems made sense to us and we could explain to someone verbally why. However, when it came to division with fractions, we were a little stumped. Most of us were taught the rule of flipping the denominator and numerator of the fraction you are dividing by and then multiplying that with the other fraction. We knew the rule well and could explain to someone how to solve it. But why was the answer the way it was?
I thought this was sort of interesting. Why was it so difficult?
I think part of the difficulty comes from the fact that we often don't talk about dividing by fractions. Another part might be that we did not have a good idea of how to visualize what was happening or to describe the process. After looking into division with fractions, these two things coming together really helped.
Knowing how to read a division by fractions problem is really important. This helps you to understand what you are solving for. For example, given: 1\2 ÷ 1/4, what is being asked is how many fourths are in a half. This relates right back to division with integers, like 6 ÷ 3 is asking how many 3's are in 6. Once we see that this is what is being asked, we can better represent it with a picture and also understand our answer.
Keeping with the same problem (1\2 ÷ 1/4), it's helpful to begin with drawing what a whole is. After this, we can mark in what a half is and what a fourth of our whole will look like.Then it simply becomes finding out how many fourths will fit into the half.
We see that two will fit, so our answer is two.
3) Making sense of it all:
We can reason to see that the answer we got is correct. Remember, we're finding how many groups of a 1/4 go into a 1/2. We already know that a 1/4 is less than a 1/2, so we should expect our answer to be at least greater than one. Thinking in the same way, we can reason through other division by fraction problems. Like 4 ÷ 5/6 for example. We know that 5/6 is less than one whole, so we should expect our answer to be at least greater than one. We also know that we have four whole pieces, so we should expect for each whole that at least more than one 5/6 will fit into it, so our answer should be at least more than 4.
What if the dividend is smaller than the divisor? Let's look at the example of 1\3 ÷ 1/2. Since 1/2 is larger than 1/3 we should expect our answer to be less than 1, since not even one 1/2 will fit into the 1/3.
Having a better understanding of what fraction division problems are saying allowed me to think through the problem and reason where the answer should be, as well as to better understand how to visualize and model what is happening. Building a relational understanding of fraction division really helps to better understand it and now for me to help explain to others how to do it and incorporate the use of models effectively.