Fractions are Numbers Too!
I grew up in the time of the "uptown" Numerator and "downtown" Denominator- math tricks that get students thinking about rote memorization instead of any conceptual understanding. None of my teachers knew of the common core standards, and my college days were over before it was first implemented. The way that fractions were taught left me, like so many other 30 somethings, with an implicit idea of fractions as unimportant. Fractions are different. Fractions are confusing so just cross multiply or invert. Fractions must not be that important because we only talk about them at the end of the year.
The Common Core, which has fractions as a separate domain of mathematics starting in 3rd grade, actually creates standards to help foster the conceptual understanding that seems to be missing in so many adults and children alike. Even still, when I became a teacher I had a decision to make. Either perpetuate the attitudes like those above which were passed on to me and continue to believe them myself, or break away and re-teach myself in order to create a new pathway of learning fractions for my students.
Learning Fractions for the 2nd Time
Thankfully, each year of teaching 3rd grade I have gotten a little wiser on how to seek out helpful resources. The National Council of Teachers of Mathematics magazine, Teaching Children Mathematics, has been my go-to for well-written action research on fractions and has given me great ideas for my unit planning. A major idea from these articles is that students must have tools and manipulatives so that they can explore with fractions. Pinterest has some great ideas for fulfilling this need, centered around the more artsy side, and some fraction resources I even created myself.
The first major takeaway on this journey is that fractions are not this completely separate thing that math teachers just have to deal with once and a while. Fractions are everywhere and EVERYONE use them all the time- when cooking, making purchases, sharing-and teachers need to emphasize the fact that often gets overlooked-fractions are numbers too! I even read about the importance of understanding fractions when ordering food-a certain fast food chain received complaints because the price of a 1/3 pound hamburger was more expensive than a 1/4 pound hamburger...
Repeating over and over again that fractions are numbers too places fractions in a completely different view for me. Now I'm sure that you too already knew that fractions are numbers, but have you explicitly stated this lately to your students? In previous years I wasn't. I just expected students to realize this on their own. Shame on me. But now I know better.
Have you used fractions in your informal conversations with students? How about showing where fractions are on the number line even when discussing whole numbers? Not to talk about fractions at that moment, but just for students to remember that they exist?
My second big takeaway is that there is always something new to learn about teaching and thinking about fractions. Last year, after three years of my reading articles and practicing problems for myself, I thought I had enough new knowledge to correctly focus on the difference between a numerator and denominator, but I was still doing it all wrong.
While I now had plenty of concrete and abstract visuals for students to use in their explorations and build what I thought was conceptual understanding, I was speaking with the "3 out of 4" language and not even realizing that I was hurting my students.
While vocabulary is necessary and an important practice for precision, leading an introduction to fractions with this interpretation can perpetuate the misconception that the numerator and denominator should be treated as separate numbers. The truth is that they must be viewed as a "unit"; they together represent one number on a number line.
It never even occurred to me until reading Van de Walle et al.'s text, Teaching Student-Centered Mathematics, that students would look at each digit as a whole in this way because I didn't see them that way.
After reading the fraction specific chapters of this text, it is now difficult (thankfully) for me to return to my old way of thinking and teaching. Of course students would use their schema of whole numbers and associate the numerator and denominator with what is already familiar to them.
Because of this I am an even bigger believer in mathematical standard of practice #6 - attend to precision. When thinking about a length model for working with fractions, I now make sure to replace my words for "3 out of 4" with the following:
"one whole has been partitioned into 4 pieces. This means that each piece has a length of 1/4 of the whole. We happen to have 3 of these 1/4's ." This way students are thinking of fractions as parts of the whole while at the same time developing their understanding of the iteration of unit fractions. It's a win-win for conceptual understanding and approaching the common core standards!
How are you re-framing your thinking about teaching and learning fractions?
P.S. If you haven't picked up a copy of Van de Walle's text, please consider doing so soon. This is a mind altering view on how to teach and learn mathematics.