The last few years has seen a huge jump in my content understanding when it applies to fractions. During my elementary years and high school I couldn’t think of an area in math that I hated more or that I had a more difficult time understanding. Come to think of it this was all areas in math. Hated is not understanded here either, I despised math. So I find it funny now that teaching and learning math has become my passion. I love all aspects of it. Now I am not going to get into my feelings about the way I was taught math (a whole other blog post, maybe even a book!) or the fact I suffered because of the way I was taught math. I am just going to take a snapshot of an area (representing fractions/partitioning fraction models) that we have to be very careful about how we introduce to students.
A few years back I had the pleasure to take part in some learning involving the the fraction pathways that Cathy Bruce, Tara Flynn and Shelley Yearley are developing for the Ontario Ministry of Education. I supported some of the teachers I am working with as they took part in the collaborative inquiry around the fraction pathways. For more info check out this link on edugains. http://www.edugains.ca/newsite/DigitalPapers/FractionsLearningPathway/index.html
To say this moved my understanding forward as far as fractions is concerned is the understatement of the year. I now fully understand fractions and the key understandings behind the concepts. I have done a lot of other reading to, the book a Focus on Fractions Bringing Research to the Classrooms by Petit, Laird, Marsden and Ebby is also a go to resource.
During the learning for the fraction inquiry they spoke about how in other countries such as Japan they use fraction models that have longevity across grades. They reference the number line and rectangular area model as two such models. One area model that research has shown can be difficult for students to partition or use accurately is the circle model. This is a quote from the fraction pathway resource “In particular, partitioning circles equally is much more difficult with odd or large numbers whereas rectangular area models and number lines are more readily and accurately partitioned evenly for odd and large numbers (Watanabe, 2012).” They also told us that once this model is shown to students at young age it is very hard for them to access other more efficient models once they are shown them in later grades (more so even for girls). Students will tend to always revert to using a circle as the model of choice which then potentially creates more errors. Please read their research because there is a lot more information and I am only using bits for this blog. All this background leads to the whole point of my blog.
We have been working with representing and partitioning in the grade 2/3 classroom at one of my schools. The pizza model or circle model has already come up. Students are using it from prior knowledge, because it has not even been addressed in the current classroom this year. We have had some discussions about using consistent models like the number line or the rectangular area model.
I brought one student who has pretty strong understanding in regards to equipartitioning. The student knows the fractional parts have to be equal but in class has shown they really like to use a circle model and is struggling with representing the fractions as equal parts when she partitions using the that model.
Here are some videos of me working with her to help her potentially discover why these problems are arising and address the issue I spoke of above.
I prompted the student to see if they could think of a different way to partition the circle. Here is what the student tried next.
As you can see the student was struggling to find a way to partition the circle accurately.
This was happening consistently in class when this student was asked to represent fractions using her own model. Using 1/2 or 1/4 as the fraction to be represented was fine but as soon as the fraction was an odd number like thirds or fifths it became very difficult using this model. I then demonstrated how a circle can be partitioned into thirds and then allowed the student to try again after I modelled it. I wanted it to really hit home that this model is difficult to use and that there is potentially a better one to use.
The last video shows the student trying to partition the circle after having it modelled for them.
You can see my modelled example on the page the student is using and they still struggled to partition the circle into thirds.
I had to edit the length of the videos so some parts are missing. The student did go back and colour in one third on the rectangle. I also edited out the last question I asked (not on purpose a tech failure on my part trying to edit! Opps), “What model made it easier for you to represent one third?” The students answer was immediate, “The rectangle!” This also another good example of how a quick guided session can help move a students thinking forward rather quickly. I know as teachers in primary its easy to go to the pizza as a model for context, but maybe it is just as easy to use a rectangular brownie or cake model! There are also fine motor issues to worry about in primary when they are drawing models, I used pre-drawn circles ans rectangles so imagine how much harder it would be to partition the circle if they had to draw the circle too! I look forward to hearing some thoughts on this and please if want to read more about this check out the Fraction Pathway research on edugains.ca.