[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: , and later, .
[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, ,
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?