[I couldn’t find a good picture illustrating “division.” Niner came to my rescue and took this photo of her breakfast.]
I found an interesting question at Mathematics Education Research Blog. In the spirit of Liping Ma’s Knowing and Teaching Elementary Mathematics, Finnish researchers gave this problem to high school students and pre-service teachers:
We know that:
How could you use this relationship (without using long-division) to discover the answer to:
[No calculators allowed!]
The Finnish researchers concluded that “division seems not to be fully understood.” No surprise there!
Check out the pdf report for detailed analysis.
My Own Research
I wondered what my students would do with the problem. Chickenfoot has been working on geometry and algebra 2, so it took him a few minutes to drag his mind back to arithmetic, but then the question was easy for him. He falls in with the 30% of students who “produced either rigorous and complete solutions or correct solutions with missing elements in justification.”
Next up, Princess Kitten, who was working on Backwards Math division this morning. Unfortunately, she found that rather traumatic, so I hesitated to challenge her with another hard problem. We negotiated a trade: one more tough problem today, in exchange for no math requirement at all tomorrow.
Thinking It Through
Kitten has just finished up a unit about long division, so this problem looked easy to her, until I told her that long division was not allowed. “Can I do short division?” Nope, none of that, either. Short division is what we call long division when we do the subtraction steps mentally.
Looking at the problem again, she immediately recognized that 491 was smaller than 498, and she knew that subtraction would have something to do with the answer. She wrote down “7 diff,” meaning the difference between 498 and 491. And since the difference was not exactly 6, she told me there would be a remainder.
[I didn’t want to distract her by asking what would have happened if the difference was 12. I think she would have recognized that any multiple of 6 means no remainder.]
Kitten is not comfortable enough with fractions to handle the division completely. Since she’s only beginning 5th grade, I allowed her to answer in the form “___ R __.”
At this point, she stumbled. She couldn’t decide what the remainder should be or what number should come before the R. She knew there was a connection between the change in the division problem and subtracting 6, but she kept wanting to subtract from the answer: 
[A disturbing number of the students in the Finnish study did this, too, but they didn’t have the excuse of being in 5th grade!]
Words are Easier than Numbers
Numbers are confusing because they are so abstract. Word pictures are easier to imagine, so Kitten converted the numbers into a word problem:
On one side of the street, there are six clubs. On the other side of the street live 498 people, all of whom want to join the clubs. How many people will be in each of the clubs?
With this approach, Kitten was able to correctly answer and explain two related problems:
if 6 people moved away, 
and:
if 12 people moved away, 
But when 7 people moved away, her clubs came out uneven. One club lost an extra person, and she wasn’t sure what number to use for her answer. I added a new rule:
There have to be the same amount of people in each club. What happens when seven people move away? To keep the same number of people in each club, some people will have to be kicked out…
Aha! Finally, the remainder made sense to her.
What Does the Quotient Mean?
Still, Kitten wanted to subtract from 83, giving the answer “71 R 5.”
As long as everything came out even, she understood that the quotient (answer to the division problem) gave the number of people in each club. But she had trouble applying this concept when the problem involved a remainder. I’m not sure why that mental glitch caused her to revert to subtracting from 83, except perhaps that it was easier to subtract than to admit she didn’t know what to do.
I offered her one more problem-solving hint:
Smaller numbers are easier to work with. What if we had started with 18 people (3 in each club), and then 7 moved away…?
In the end, Kitten managed to come up with the correct answer. And when she finally saw it, she agreed that it made sense. In her own words: “491 is between 486 and 492, so the answer has to be between 81 and 82.”
What About You?
And now, my readers, it’s your turn:
- Can you answer the research question? Try to think of at least two different methods.
- How do your middle school or high school students handle the question?
- How would you explain it to a 5th-grader?
Denise Gaskins' Let's Play Math 
Kitten gets to preview all articles about herself. She noticed a link in my sidebar to a different post, How to Teach Math to a Struggling Student, and commented:
“I’ll tell you how to teach math to a struggling student. DO NOT GIVE THEM THIS PROBLEM!!!”
Hi,
I enjoyed reading about this as we were talking about this today in the 6th grade math class I observe for my college course. I am studying to be a middle school math teacher. Investigating blogs is one of our assignments this week. I enjoyed reviewing your site.
In the 6th grade class they were relating fraction, decimal and percent. We talked about 1/8 and got it’s decimal and percent version. From there we looked at 3/8 and tried to decide how we could solve this without division again. The one thing I observed is some kids really get the relationship between numbers and others don’t. They just do what is told to them in an algorithm. They don’t examine the relationship of the numbers and how to see them from different angles (ha ha). I think that has to start in the early elementary years. Don’t you?
Thanks,
Mary.
Hi, Mary, and welcome to my blog! Middle school math is my favorite topic, so I hope you find plenty of helpful ideas here.
I think you are right, that it’s best to start this sort of teaching as early as possible. That is why I emphasize number bonds and mental math in the early grades, and also why we do a lot of oral story problems, even in preschool. Still, I think there are quite a few things you can do with middle school students to build up this kind of thinking.
Mental math helps develop number sense and flexibility. The fraction calculation you mention and the division in the article above are both rather advanced topics in mental math. You may want to start your students on easier concepts, like addition and subtraction, and work up to the harder stuff. Here are a couple of links that might help:
Mental Mathematics Strategies
5 Tips for Using Mental Math in Your Classroom
The first thing I saw was that this posting was for middle grades, but I noticed at the end you asked the question about how the high school students would respond to the question. Sadly to say, I think the response would be the same no matter what age, but I struggled with it. I am an aspiring high school math teacher. Great job as I scroll though the blog. I will definitely keep up with it!
Alexseil
My “Grades 5+up” category is for any arithmetic-based post for middle school and beyond — even college level, as long as algebra or geometry are not required. I think I’ve tagged this article for both high school and middle school. The original researchers dealt with upper-grade high school students and college freshmen, so that is the target level for the question.
Since I have a strong interest in younger students, however, the bulk of my post deals with how my 5th-grader fought her way to some level of understanding. She did not fully master the problem, but I believe we made progress.
Denise, I really like this problem! Will have to have my kids play with it. Thanks for posting it.
Hello,
I’m investigating blogs as one of our assignments this week in my math ed class. I’ve been in the classroom for a few years and it’s really encouraging to find so many resources to help my students learn. Thanks.
In my 6th grade math class we are relating fractions, to decimals and percents. In converting fractions to percents I realized that, just like in the research, my kids didn’t realize the meaning of division. They could process fractions with small numerators and denominators but the larger the difference between the numerators and the denominators, the harder it was for them to visualize the problem. Like Denise, I observed is some kids really get the relationship between numbers and others don’t. How do you teach kids how to realize if their answers are reasonably or not?
For a first approximation, students should ask themselves, “Will this be more or less than 1/2?”
For better estimates, it helps to memorize a few “benchmark” conversions:
1/100 = 0.01 = 1%
1/10 = 0.1 = 10%
1/4 = 0.25 = 25%
1/3 = ~0.33 = ~ 33%
1/2 = 0.5 = 50%
3/4 = 0.75 = 75%
etc.
Then students can compare the fractions they are working with to these basic ones, to get an estimate of how reasonable their answers are.
Of course, the biggest problem may be to convince them to make an estimate at all. Many students just want to put down any sort of answer and don’t care whether it is reasonable or not. Somehow you need to convince them that an incorrect answer now means more pain for them later — because they will get extra homework, perhaps? But if they get these problems correct, they get a break on the homework? My kids always like to think they are getting a break. 😉