Looking broadly at how to introduce fractions I am following the Concrete-Pictorial-Abstract progression which is encouraged in the Waldorf movement. CPA is also prevalent in mainstream mathematics communities as well, most notably in the ‘mastery’ philosophy (if there is one!). I was first introduced to the concept by Bruno Reddy (@MrReddyMaths).
[Update (23/12/18): Thanks to @petergates3 on Twitter who pointed out that CPA seems to be build on the “enactive-iconic-symbolic” triad from Jerome Bruner. You can read more about Bruner’s key theories in this post by Jane Currell on the Maths No Problem website]
More recently, thanks to the work of Mark McCourt (@EmathsUK) and Bernie Westacott (@berniewestacott), I have come across the extended acronym CPAL with the L standing for language. The L highlights the importance of language which I touch upon herein but to be honest is something I need to learn a lot more about.
First follows a description of each element of CPA and some examples of their use and I finish by outlining a very rough three-week lesson plan for a main lesson block introducing fractions. This is very rough and I will be returning to this to refine it over time. Already I am thinking that I need to make more use of manipulatives such as cuisenaire rods and actually, I could slow down the speed with which the learning progresses to make sure that the relationships are as fully embedded as possible.
I am more than happy to receive feedback from anyone on this and consider this very much a working document so all suggestions very welcome.
The best ways to contact me are email: brendanbayew @ gmail . com (no spaces) or on twitter: @MrBayew
When a child in is in kindergarten and class 1 and 2 also, he or she is encouraged to play with numbers and objects and develop a ‘feel’ for number. Adding, subtracting multiplying and dividing all follow naturally from this organic process as do the times-tables.
Fractions in Class 4 is the first major threshold concept that students come across. It is often the first time students come across a maths topic so alien to their everyday life that the fear and the anxiety and the fixed mindsets begin to form. Some pupils are unlucky enough to have this passed onto them through either their maths-phobic parents at a very early age or poor teaching in the early years or KS1, but even otherwise robust individuals can be left by the mathematical way-side once fractions are involved.
So it is important that pupils are led sensitively into this unfamiliar world of fractions. Food is a very natural way into fractions as we share our food with our friends and family, and actually sharing biscuits or snacks out fairly is very important when you are nine and we are led into sharing equally between so many people.
So I would like to devise many ways to share things out equally between the class or groups within the class.
Dividing an apple up into 4 or 8 equal slices.
Dividing a cake up into 29 (!) equal slices.
Dividing a biscuit into 2 equal pieces.
Dividing a loaf of bread up or a pizza, the options are endless.
There is a heavy emphasis at this stage on the language that the teacher is very carefully choosing to use in order to describe the processes.
“So Johnny cut the biscuit into two equal pieces.
And he gave one piece to Jenny. How much of the biscuit does Johnny have now?
That’s right, Johnny has one half. And Jenny has one half.
And if we put them back together what do we have? We have one whole biscuit.
So how many half biscuits make a whole biscuit?
Two halves make a whole.
Oh look, on this table, Adam, Bernard and Charlie shared their loaf of bread equally between the three of them.
They shared one loaf into three equal parts
We call these thirds.
If we cut something into three equal parts each part is called a third.
And how many thirds are needed to make a whole again?
Three thirds make a whole.
What would happen if Bernard was feeling greedy and took Adam’s third, how many thirds would Bernard have now?
He’d have two thirds all to himself, Charlie would have one third and poor Adam would have none.”
So we have a physical process occuring where children are physically dividing objects into equal parts: halves, thirds, quarters etc.
This can occur outside also with chalk on the ground and small groups of children dividing up circles or squares into equal amounts.
Back in the classroom we can stay with the apples etc and ask questions like:
If Johnny has half an apple, what do I need to give him in order for him to have a whole apple?
If Adam has a third of a biscuit, what do I need to give him to make a whole biscuit?
If Bernard has two thirds of a biscuit what do I need to give him in order to make a whole biscuit?
So the concrete manipulation of resources is partly done directly by the children, inside and out, and partly demonstrated by the teacher with targeted questioning and ‘call and response’ type participation by the students.
There is a bridge between the concrete apples, biscuits and loafs and the purely pictorial (ie 2D representations on the board or on paper). That bridge is the non-specific manipulative. By this I mean the paper circle which can be cut into halves, thirds, quarters etc. This is a fantastic tool as children can manipulate with their hands something which will be used to represent fractions pictorially.
So the pupils can have manipulatives on their desks and the teacher can use large ones with which to demonstrate. Circles (squares etc) can be divided along pre-printed lines and put back together and stuck into main lesson books. Pupils can cut out halves, thirds and quarters, but also 2-thirds and 2-quarters and 3-quarters and a ‘feel’ for the relationships between these things will be growing within the children.
Once the link between half-apples and half-circles (be sure to include other shapes like squares and hexagons) is secure we can move to pictorial representations. Start with actual pictures but I would encourage the conversation about how difficult it is to accurately draw an actual apple or loaf of bread, let alone divide it equally. So the need will arise naturally to represent things with circles or squares or bars.
At this stage, providing pre-printed worksheets with shapes for the children to divide up is going to save a lot of time and frustration.
Also, worksheets with problems on such as the following are lovely: Highlighting the question “are they equal?”
It is really important at this stage to introduce non-examples (like the square bottom right above) as well so students can see when something is or is not a fraction.
Also, moving into a combination of pictorial and abstract is going to be needed quite swiftly as students will need to be able to read a question on a worksheet without a narration by the teacher
The work in a main lesson book and/or the work on a worksheet will want to combine both the abstract representation of a fraction and the pictorial representation to begin with.
Worksheets like the following from John Corbett (@Corbettmaths) allow for a blending of the pictorial and abstract:
This one too, from TES author BBallard (https://www.tes.com/member/bballard) has a nice blend of pictorial and abstract and actually takes the concept quite far in terms of allowing for those discussions about equivalent fractions. If the questions arise from examples such as these then the concept of equivalent fractions may seem a more natural progression.
Once pupils are comfortable with working with fractions in this way then they can begin to work with the abstract fractions and problems such as
can be displayed on the board as part of a starter activity or can start to appear in practice sheets.
Mapping out a main lesson:
When I was working on introducing the sound-letter link in Class1 (https://hospitablewanderer.com/2018/04/04/creating-the-sound-letter-link-in-class-one/) I was struck by the power of the three-day rhythm and I am wondering whether it can be applied here in the same way.
The following pages are a first attempt at making something of this idea.
Movement runs parallel to the rest of the activities and every day the pupils will be embodying fractions.
Throughout the three weeks they can begin to get a feel for fractions in their bodies through movement, in their hands with the concrete manipulatives, visually through their pictorial representations and they will be using the abstract representations alongside all of this from early on in the process.
I don’t think we should shy away from the abstract form as they were invented for a reason, and that reason was to make communication about fractions much easier. So pupils are introduced to the abstract form on day 3 and continue to build their understanding of fractions through all three representations (CPA) throughout the three-week block
Jamie York’s daily practice materials look great: https://www.jamieyorkpress.com/wp-content/uploads/bsk-pdf-manager/2018/09/Facts-Review-Sheets-4th-Grade-Computer-Formatted.pdf
There are only 100 days of this though so another resource will be needed.
For the practice lessons, all the previously learnt content should be put into a retrieval practice element at the start of each lesson.
A practice lesson might look like this:
10 mins – retrieval practice – 4 ops up to 100 and times tables + fractions
15 mins – careful modelling by the teacher of a new skill such as adding fractions same denominator, and independent intelligent practice (minimally different questions) of that skill by students plus in-class marking
5 minutes – Exit ticket which is a very short mini-assessment – up to 5 questions which encapsulate the key skill(s) from the lesson which the teacher can mark before next lesson to inform the planning for the next practice lesson.
Assessment would take many forms.
In-lesson assessment is often called Assessment-For-Learning or student-teacher feedback. Really we are looking for real-time data on how the students are progressing.
Targeted questioning is an excellent tool for this. Cold calling (see Teach Like a Champion 2.0 by Doug Lemov) students gives us a better picture than just asking those with their hands up, and asking a range of students.
Diagnostic questioning is another excellent tool for getting a clear picture of where the class are at and for wheedling out misconceptions.
Mini-white boards are a good tool as well.
I would probably use exit tickets once a week in main lesson to see how students are progressing.
And at the end of each half term I would give the class a paper assessment, the data from which would feed into the practice lessons for the next half term and any topics which the class were weak on would come back round during the retrieval practice section of the practice lessons. To read about how I use this in class visit my blog post “moving from no-stakes to low-stakes”.
Finally, I would love to hear from you if you have any comments about this piece, good or bad! the best way to contact me is twitter: @MrBayew or leave a comment below.
Thanks for reading and happy teaching!