"We're Into the Third and Final Quarter..."

So once said a sports commentator on British TV. Yogi Berra (the famous New York Yankees catcher) once was asked if he preferred his pizza cut into four pieces or eight. He replied, four as eight pieces would be too many! Fractional misconceptions are everywhere as seen in this advert:

The chart above actually gets my vote for The Most Pointless Legend Ever (look closely at the bottom left-hand corner).
As I and my colleagues bounced ideas off each other as to why fractions are so problematic, it became clear that what worried us most was that students might not have a concrete grasp of the basics of fractions. That maybe when we teach fractions we rush too quickly to procedures and algorithms (and end up saying things like "'reduce' this fraction" or "there's no need to wonder why, just invert and multiply").
So our line of inquiry going into a Grade 2/3 class was based on these two wonderings:
Do the students really understand fractional names?
What is the best way of getting students to work with these fractional names?

We showed the following screen to students and asked them to discuss in pairs where the labels should go:

 

Good class dialogue led to some consensus:





But when they reached the last label there were more than a few puzzled looks:

Some students wanted to put the fourths label on the last circle simply because it was the only one without a label. Others argued that the last circle was in fact split into fourths, in fact there were three fourths:

This puzzled me at first until I realised that what the student saw was three fours and not three fourths.
So to address this we tried to connect it to the language of fifths, sixths, eights and tenths. By getting them to see the pattern that five fifths make a whole as do six sixths, eight eighths and ten tenths, then you must need four fourths to make a whole. They then realised that the fourths label would go on the same circle as the quarters label. Of course the language of thirds and halves (why halves and not twoths?) doesn't make this linguistic pattern easier to spot: sometimes the English language gets in the of learning Math. All of this certainly challenged any notions that we might have had that these students all knew how to label fractions; to assume otherwise will lead to a host of problems. 

We then gave the students this Yogi Berra-inspired problem:
Would you  prefer five half-pizzas or nine quarter-pizzas?
We were able to give three pairs of students iPads and they used an app called Virtual Manipulatives. It was noticeable how much easier this made it for these students to model and communicate their thinking:

Other students relying on pencil and paper sometimes struggled to see how to compare the two quantities and some were still stuck on the idea that nine pieces must be better than five pieces:

There was also one pair who drew five pizzas split in halves and nine pizzas split in quarters: they had misinterpreted what five half-pizzas and nine quarter-pizzas meant. I wonder how many other students make this error?

All of us left realising that students need a wealth of concrete experience with fractions, much more than they are probably currently getting. We also agreed that this experience shouldn't just be confined to a two-week unit; it is something that should be experienced across (and even beyond) the Math curriculum throughout the year.

Fraction immersion, as it where.

A word of caution though: learning about fractions involves so much more than the 'pizza' fractions seen above. Here fractions are being used to describe an area, but fractions can also be used to describe a set (in a class of 13 girls and 15 boys, what fraction are girls?) and fractions can be used as a number (what number is halfway between three-quarters and two?) Students must  experience fractions in each of these situations.