# The teaching of fractions

There are certain prerequisites for children to develop a solid understanding of fractions.  First, they must understand, through work on additive reasoning, that a whole can be split into parts and that the sum of those parts is the whole.  There’s a short step into multiplicative reasoning from here – that a whole can be split into multiple, equal parts and that the whole is the product of the size of each part and the number of parts.  Once this is understood, children can begin to think about the whole being worth one and the parts being fractions of one.  The ideas that follow are broadly sequential in terms of conceptual development.

Children will need to manipulate various representations of fractions, for example making them with fraction tiles (as both bars and circles); taking strips of paper and ripping them in to equal parts; and drawing bars and circles, dividing them into equal parts.  It is worthwhile to get children to do lots of judging by eye and marking equal parts of a whole as well as using squared paper to do so accurately.

Of course, there is a lot of language to work on whilst manipulating these models of fractions.  Children need to be shown clearly the link between the total number of parts and the language (but not yet necessarily the written form) of the denominator: two parts – halves; three parts – thirds; four parts – quarters etc.

With a secure start in the basics of splitting a whole into equal parts, children can work on the idea that fractions always refer to something.  A third, for example, doesn’t stand alone.  It might be a third of an apple or a third of twelve sweets or a third of one whole.  Modelling these full sentences and getting children to speak in this way should solidify their understanding of proportion.  Through the sharing out of objects, even very young children can work on the concept of fractions of numbers – sharing six sweets between three children means that each child has the same number of sweets and that two sweets is one third of six sweets.

Once children are comfortable with the idea that an object or a set of objects or a number can be split into equal parts, and that each of those equal parts can be described as a fraction of something, that object or that set of objects or that number, they can go on to work at greater depth.  By comparing strips of paper or bar models that are the same length yet are split into different fractions, children can look at the relationship between the size of each part and the number of parts.  That is, the greater the number of equal parts, the smaller the size of each part.  Children should be expected to think about how ¼ is smaller than ½ because ¼ of one whole is one of four equal parts whereas ½ of one whole is only one of two equal parts.  Then, questions like this should be relatively straightforward:

The understanding that unit fractions with larger denominators are smaller than unit fractions with smaller denominators will contribute significantly to work in comparing fractions later on.

Children could begin to look at improper fractions and mixed numbers next.  Using ¼ fraction tiles, they could make one whole and then see what happens if you add another ¼.

This lends itself to counting in unit fractions but we should exercise caution.  Children may be able to chant ‘Three quarters, four quarters, five quarters…’ but early conversion to mixed numbers as well should help to secure their understanding of the relationship between them.  Manipulatives like fraction tiles and multi-link cubes are great for representing improper fractions because they can trigger accurate mathematical talk to describe the improper fraction (the total number of cubes as the numerator and how many cubes in each whole as the denominator).  The same can be done to describe the mixed number (the number of wholes, then what is left over as a fraction of a whole).

Returning to additive reasoning, children could generate complements to 1 whole and record them as addition and subtraction statements.

A slight change to the representation used can support children to work with complements where denominators are different:

Placing two bar models of equal length one on top of the other is great scaffold for comparing fractions.  When the denominators of the fractions are the same, the bars should not even be necessary but when they are different, the image can help to structure thinking.

When dealing with fractions with different denominators, the practice that children had earlier of judging by eye to split a whole into equal parts and marking the divisions themselves becomes crucial, otherwise, things like this could happen:

A standard fraction wall is all that is needed to begin work on equivalence and the first step is of course shading one fraction and looking up or down the fraction wall to find fractions of equal size.  When children are comfortable with that, they can begin to look at patterns in the abstract representations, particularly the link between times tables, numerators and denominators.

Using the language of simplifying or cancelling fractions without first talking more generally about the concept is a mistake.  If children are well versed in using a fraction wall to find equivalents to a given fraction, it is only a slight tweak to talk about finding the equivalent fraction that has the fewest total parts.  It would be tempting to talk about finding the equivalent fraction that is ‘closest to the top’ of the fraction wall but this would be a mistake too.  The language of simplifying or cancelling can be used to attach to the concept of finding the equivalent fraction with the fewest total parts to get children thinking conceptually soundly.

One further aspect of thinking of fractions is to consider them as numbers.  To do this, plotting fractions on a number line directly beneath the bar model is a good way of linking the two representations.

Representing fractions as a proportion of one, as a part of a quantity and as a position on a number line significantly supports children’s development of proportional reasoning and ensures that future tricky concepts such as calculating with fractions can be built on a secure foundation.

# Penn Wood Professional Development – Language acquisition and reading comprehension

The York Reading for Meaning Project (Snowling, 2010) compared three interventions with a control to determine their effectiveness in developing reading comprehension. The interventions were led by teaching assistants and lasted for 20 weeks, each week comprising of three 30 minute sessions. The three interventions were:

• An oral language comprehension programme
• A text comprehension programme
• A combined oral and text comprehension programme

Their findings showed significant gains in reading comprehension scores for each intervention compared to the control group. Interestingly, the most effective intervention was not the text comprehension programme, but the oral language comprehension programme, which also resulted in greater gains in reading comprehension scores than the combined programme. These gains were still evident 11 months after the interventions ended.

With such an impact, it makes sense to attempt to turn this effective intervention by TAs into part of our day to day teaching. Perhaps we can adapt the programme to see even greater and longer lasting gains in language acquisition and reading comprehension if the ideas were embedded in our English lessons.

The simple view of reading identifies the importance of decoding and language comprehension in tandem to master reading for meaning. Hirsch would add to this the importance of domain knowledge, without which a reader would not make rapid connections between new and previously learned material. As such, explicitly teaching the general knowledge required to understand a text can support comprehension significantly. Of course, the challenge to this idea is that we can’t teach children the entirety of general knowledge. However, selecting great texts which reflect a variety of general knowledge schemas gives children the opportunity to develop key chunks of general knowledge on which further domain knowledge can be built through listening and reading.

Using great texts to teach language acquisition and reading comprehension is a perfect place to start. Once these texts have been selected, the first thing that teachers need to do is consider the following question:

Which words, phrases or concepts are children likely to find difficult to understand?

Jean Gross, in Time to Talk talks of three tiers of vocabulary. Tier 1 vocabulary includes words and concepts that children will come across first when they begin to communicate. Tier 2 vocabulary includes language that children will be able to understand the concept of and that is tricky yet functional. These words could be used in a number of contexts. Finally, tier 3 vocabulary includes language that is domain specific and only used in a small number of contexts.

When we’re looking at the bits of a text that children are likely to find difficult to understand, we’d need to be looking for those tier 2 words within a text. For children learning English as an additional language and for children in the early years, we’d also need to explicitly teach tier 1 vocabulary. Usually, these would be common nouns, verbs and concepts and these guidelines from Stories for Talking by Rebecca Bergmann are helpful when selecting them:

Nouns

• High frequency
• Functional
• Related to the story being studied
• Related by topic
• Feature around the classroom or school
• Easily supported with concrete objects

Verbs

• High frequency
• Functional
• Relate to the chosen nouns
• Easy to act out

Concepts

• High Frequency
• Functional
• Relate to the chosen nouns
• Most visually represented or repeated in the story
• Can be studied as a pair (big/little)
• Can be experienced practically around the classroom

Once that language has been identified, teachers can introduce it to children. By introducing it before children listen to or read a text, we can go some way to guarding against cognitive overload. Also, by increasing the number of interactions with this vocabulary, and by spacing those interactions, we increase the likelihood of long term retention of those ideas. Having said that, language is best learned in context so defining words for children will not suffice. The image below is an example of how language is introduced from the York Reading for Meaning Project programme materials:

This works well because the images provide contexts in which the word is used. The variety of images and contexts helps children to make connections between ideas. A slight amendment that includes the Talk for Writing approach would be to include the sentence from the text that the word is in.

Children will not internalise this language after one interaction with it. Children need to think hard about the meaning and application of the vocabulary over time if it is to be assimilated. The following question types come from Bringing Words to Life by Beck, McKeown and Kucan.

Where children have to differentiate between two scenarios, such as in the Example or non-example?’ question, the quality of the question comes from the two scenarios being minimally different and rooted in misconceptions about a word’s meaning.  With a set of questions like this for a number of focus words across a unit of work, children’s practice of thinking about and using language can be spaced over time in a variety  of contexts, giving children a great chance of adding permanently to their vocabulary.

Oral and text comprehension

By understanding the typical difficulties that struggling readers experience, we can plan to address those issues with some carefully panned practice. If we then consider the implications from the York Reading for Meaning Project, that the materials from both the oral and text comprehension can have such an impact on reading comprehension, then we can provide great lessons.

Developing Language Acquisition and Reading Comprehension at Penn Wood outlines those difficulties and what might be done. The York Reading for Meaning Project found that oral comprehension work is more effective than a text comprehension or a combined oral and text comprehension programme when measured using reading comprehension tests. All of the suggestions for addressing the profile of the struggling reader could be applied through reading a story but also through listening to one. Talk for Writing provides a great opportunity for this as children internalise and retell stories using text maps. Oral comprehension work can quite easily be introduced at the point of retelling. With the opportunity presented, the next step is to find ways of making oral language development work in the classroom on a day to day basis.