# Developing fractional understanding

Here’s another installment of my infrequent commentaries on my eldest daughter’s math development, this time on an unexpected result when we were talking about equivalent fractions.

I’ve been a firm believer in keeping challenging ideas, games, etc. around my children at all times, some of them intentionally beyond what they’re developmentally ready to handle.  Sometimes I’ll ask leading questions to see if there is any interest; sometimes they pick up an idea or toy again and create their own play rules or ask me to explain how it works.  My ground rule is that they should PLAY.  If my explanations ever bore them, they are welcome to drop it at any time and move on to something more interesting.  Not only does this keep with my mantra that learning should be fun (even if it involves work), but I believe it helps them see that mathematics (and anything else they learn) is about enjoyment and pushing yourself to discover more than what exists within the current boundaries of your understanding.

Several weeks ago

With that philosophy in mind, my 2nd grade daughter had been helping me make some fresh bread one afternoon when we needed a cup of one ingredient, but our cup measurer was dirty.  Whether from school or one of our earlier conversations, she responded something like, ‘No problem.  Just use 2 half-cups.‘  Maybe she actually said  ‘3 third-cups,’ but she clearly had the beginnings of fractions down, so I jumped.

So if I needed a half cup of something and my half-cup measurer was dirty, what else could I use?

It took some conversations, but eventually she drew a circle with a line through the center and shaded one side.  When she drew another line through the center roughly bisecting the original sectors, she declared, “Look, Dad.  Cutting each part in half doesn’t change what you have, it just cuts it into more pieces.  So, $\frac{1}{2}$ must be the same as $\frac{2}{4}$.”

That was a pretty cool moment of discovery for her, but then she upped the ante.  After some additional thought, she noticed that both parts of the fraction had been doubled, so she applied her rule again and asked if $\frac{1}{2}$ would also be the same as $\frac{4}{8}$.  Another drawing confirmed her discovery which led to gleeful proclamations that $\frac{1}{2}=\frac{2}{4}=\frac{4}{8}=\frac{8}{16}=\frac{16}{32}$. She knew she could go on, but what was the point?

I asked if doubling was the only way she could make equivalent fractions, which led to $\frac{1}{2}=\frac{3}{6}=\frac{9}{18}$. We didn’t go any further that day, but we had already “cooked up” far more than I had anticipated.  Off the top of my head, I don’t recall how elementary school curricula deal with the scope and sequence of teaching equivalent fractions, but it will be difficult for anyone to convince me that my daughter could have had a better experience or more fun learning.

Last weekend

I was putting my daughter to bed and for some reason she asked how much of a year 9 months was.  Being the teacher, I responded to her question with another question:  “Well, how many months are in a year?

So 9 months is $\frac{9}{12}$ of a year?” She asked.

Good job.  Can you think of any other smaller fractions that might be the same as $\frac{9}{12}$?” It was an innocent question, I thought, trying to get her to take our fraction doubling-trebling idea from earlier in reverse. If she didn’t get it, no big deal, but it was certainly worth asking.  That’s when I got surprised.

Almost immediately, she said, “Four and a half sixths.  Is that right?

Some purists out there might complain that $\frac{4.5}{6}$ isn’t “proper,” but I’ve seen far too many situations where rigid insistence on proper form served instead to stifle creativity far more than to enhance understanding or to encourage deeper exploration or creativity.  I praised the heck out of her solution, letting her know that she had just made a fraction “smaller” for the first time (that I knew of).”  I didn’t mention that her proposed numerator wasn’t whole.  It didn’t matter.

How did you do that?

Easy,” she said, as if her answer would have been obvious to anyone.  “If you can double the parts of a fraction, why can’t you halve them, too?

Why not, indeed?  No matter what they end up looking like.

I asked if there were any other smaller, equivalent fractions.  That took lots more thought and time than I expected, certainly more than her nearly instantaneous $\frac{4.5}{6}$ had required.  Eventually, she asked if I could hold out 2 fingers beside her 10 so that she could look at 12 fingers together.  A little more thought led to her grouping the 12 fingers into 4 equal groups and a claim that $\frac{9}{12}=\frac{3}{4}$.  I still don’t completely understand her long explanation, but it wasn’t as clear (to me, anyway) that she was saying that 3 of her sub-groups contained the original 9 fingers (or months), and so 9 of an original 12 individual months was the same as 3 of 4 equal sub-groups of months.

When she woke the next morning, she asked if we could play any more fraction games

Conclusion

As students of all ages are learning, we need to reserve space for them to think and be creative.  They should be allowed to give correct mathematical solutions, even if those answers aren’t the arithmetic solutions our “trained” minds expect.  For me, I expected my daughter to say $\frac{9}{12}=\frac{3}{4}$, if she answered at all.  I was far more delighted to hear $\frac{9}{12}=\frac{4.5}{6}$ than she will ever know, and I’m convinced that she’s becoming more confident and capable because she was allowed to do so.

### 4 responses to “Developing fractional understanding”

1. This entry was invaluable to me. I am a fourth grade teacher, and we teach equivalent fractions. I love how your daughter verbalized, “Cutting each part in half doesn’t change what you have, it just cuts it into more pieces.” I can use these words with my own class. I will be following your blog. I like “hearing” the thinking of a mathematician…all too often I feel I am not adequately explaining math concepts to the kids. And there’s not enough time to let them discover these things on their own. I am curious about the kinds of math games you have around the house.

2. jybuell

This is excellent.

3. Agree with you! Epic point of view