I became very interested in the addition at the top. It is easy to see why someone might think 1/4 + 1/4 = 2/8, either by having an image such as this, or by simply adding tops and bottoms.
Upon seeing this, a teacher might search for a better way to represent adding fractions using blocks. But perhaps the problem lies with using blocks at all, as a metaphor for adding fractions. But if not blocks, then what? A common response is to say we just need to be clear what is a whole, what the unit is. But I'm not sure this is useful when adding fractions. It seems to be an added complexity arising from the desire to present fractions in this way.
These images are certainly not what comes to my mind when I add fractions. So what is happening when I add (say) 1/4 + 1/4? I am going to try to speak from my experience, which is all I can do, but aware all the time that I am an 'experienced mathematician'.
I read it as 'one quarter plus one quarter'. I am immediately aware of the multiple meanings that the symbol 1/4 can have: one quarter is simultaneously, for me, a type of number and a part of a whole. [how is it perceived by this learner? how about that one?] I am going to add two things, two numbers, together.
Here I am stressing the fact that a quarter is a thing in its own right. The whole, of which it is a part, recedes into the background. So I am adding a thing - a quarter - to another thing - another quarter. And I can do this, because they are the same type of thing: "one quarter plus one quarter is two quarters".
Adding requires the two things being added to be the same type of thing, as described beautifully by Gattegno in his farewell address to the ATM (available here):
Once at my desk in Addis Ababa in 1957, I blushed, I was so ashamed of myself. 1957, twenty years after I got my doctorate in mathematics, I understood what we do when we add two fractions... I did not know that to add two fractions involves addition. I said it but I didn’t know it. I could write it, I could get the answer, but I didn’t know what it meant to add two fractions. And suddenly, I realised that, whenever I have pears and apples, two pears and three apples, I don’t have five apples or five pears. I have something altered, I have five pieces of fruit. So why did I do that? Because I wanted to find how to get them together, I had to raise myself to another level where the pears, pearness and appleness are replaced by fruitness. And at that moment, I can say five. And I never realised that ‘common denominator’ meant ‘give the same name’ to both. And in the middle of the word ‘denominator’ I see a French word ‘nom’ which I knew very well.
It didn’t strike me, ever, that it is addition that forces me to get denominators, common denominators, not fractions. That was my shame...
I wonder if this desire to represent fractions using blocks/images is a result of a fascination with the 'concrete-pictorial-abstract' (CPA) model? In the search for concrete and pictorial representations, the blocks, or pictures of blocks, come out. There is no doubt that physical apparatus and images are useful/essential in teaching mathematics, but are they helpful here?
More generally, what if the CPA model itself - or rather how it can be interpreted - is not that helpful? In his paper When is a Symbol Symbolic (available here), John Mason describes what I interpret to be reticence about the CPA model. In describing Bruner's Enactive-Iconic-Symbolic modes - from which CPA was derived - Mason suggests that:
People seem to have identified
ENACTIVE with physical toys
ICONIC with drawings and pictures
SYMBOLIC with words and letters
... and missed the essential qualities which I describe as:
ENACTIVE - confidently manipulable
ICONIC - having a sense or image of
SYMBOLIC - having an articulation of
Notice too that SYMBOLIC expression must ultimately become ENACTIVE if the idea is to be built upon or become a component in a more complex idea. Thus to a child 1, 2, 3 are truly symbolic, having little or no meamng. With time and extensive encounters a sense of one-ness and two-ness develops which underpins the symbols and provide a source of meaning when 1, 2 and 3 are encountered in a new context. To proceed with arithmetic it is essential that 1, 2, 3 become enactive elements, become friends. If they remain as unfriendly symbols then arithmetic must be a source of great mystery.
CPA seems to me an oversimplification of Bruner's theory. CPA is about things, EIS is about physical and mental ways of working on mathematics. CPA is often interpreted as a path that learners must follow, from playing with stuff, to drawings, then to symbols. I find that learners (including myself) move backwards and forwards through the modes of EIS, most notably creating symbols through articulation, that then become manipulable (mentally of physically).
What is more, learners subjected to the CPA path may be deprived of the chance to create and then manipulate symbols. It is widely accepted that children of a very young age are involved in abstract processes (i.e. formation of language), and that 'abstract' mental activity should be a part of most if not all mathematics lessons.
Back to 1/4 + 1/4, then. When a learner adds 1/4 + 1/4 and gets 2/8, they have not been able to access a useful meaning for these symbols. Only when I can select what is useful from the multiple images that I hold of these symbols can I confidently manipulate this expression.
Let's look at another example of adding fractions. What do you see when I write 1/4 + 1/5? What does this or that learner see? Now, work it out...
I would suggest that what you did when calculating 1/4 + 1/5 was a series of what Dick Tahta called 'metonymic' shifts, i.e transforming the symbols into other equivalent symbols. You might have transformed 1/4 + 1/5 into 0.25 + 0.2, or into 5/20 + 4/20. These are not metaphors, but rather other names for the same thing - metonyms.
In the article Counting Counts (reprinted in the most recent issue of MT), Dick Tahta suggests:
One of our problems in teaching arithmetic is the move from a stress of metaphor to a stress on metonymy. We offer children counters and rods and so on, in order to mimic processes which we eventually want them to transfer to written or spoken numerals. There are various accounts of the sometimes unexpected difficulties that this may cause.
So good for me and you, experienced mathematicians, but how did we get to this stage? I am not saying here that learners can or should be expected to perform these metonymic shifts immediately, but rather that we as teachers must consider very carefully the metaphors we use to 'help' learners - and whether we use metaphors at all. Elsewhere in this article, Tahta states:
Mathematics itself seems to need to get away from metaphor as soon as it can...
So, what might be a helpful way of introducing the addition of fractions? Perhaps there is a way to avoid metaphor completely, and present the addition of fractions as a series of metonymic shifts?
Assuming that the children encountering adding fractions for the first time can not yet confidently manipulate fractions using equivalent fractions or decimals (which of course might not be the case), an alternative might be to use words (i.e. starting from symbols they are confident in manipulating). Also, I might not start with 1/4 + 1/4, but rather something like 3/5 + 2/5.
Perhaps the metonymic chain might go something like this:
What is three dogs plus two dogs?
What is three blocks plus two blocks? (picking things up around the room...)
What is three wishes plus two wishes?
What is three fishes plus two fishes?
What is three fifths plus two fifths?
What is three plus two?
Perhaps I might include an example of trying to add two different things, to draw attention to adding being about a collecting together of same things. I might also slip into metaphor, perhaps showing some carefully chosen images, but always stressing the fact that we are adding one thing to the same type of thing.