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THE IMPORTANCE OF MODELLING IN PRIMARY
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LEARNING THE LANGUAGE OF MATHEMATICS
Reasoning in mathematics depends on a sound understanding of mathematical language. In this article, Shannen Doherty explains how she advocates teaching this to students throughout the Maths curriculum.
By Shannen Doherty
We live in a world full of TV, tablets, and screens galore. This must be affecting children’s vital early years experiences. Anecdotally, I have seen children starting reception far behind their peers when it comes to language. We’d be fools to ignore the fact that some children do not have a linguistically rich early experience. Adding covid to the mix will surely have had a knock-on effect, too. Children have missed out on the chance to socialise and talk and practise conversing, all of which support their development. So, when it comes to using mathematical language in school, they need support. It doesn’t just happen overnight!
Mathematical language is the means through which we communicate our ideas and our thinking. It’s crucial to the learning of our students. The National Curriculum mentions mathematical language in their aims, “reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language”. As far as I’m concerned, teaching mathematical language is the key to unlocking reasoning. I am a firm believer that children need explicit vocabulary teaching and structured language in mathematics. We need to take a structured approach. Learning vocabulary can’t be left to chance.
I was recently asked, “Why did we start using ‘addend’ to talk about the numbers in addition?” But why wouldn’t we?
Whenever we have this debate, I hear two justifications:
1) If children are learning phoneme, grapheme, digraph and trigraph in phonics then why shouldn’t they learn addend, minuend, subtrahend, etc?
2) Children remember the names of all the dinosaurs so they can learn the parts of an equation, too.
Both are obviously true. Children clearly do have the capacity to learn new words, but they also have a thirst for it. They love learning new and interesting vocabulary.
However, I can’t help thinking we’re missing the point when we give these justifications. We aren’t teaching mathematical language just because children can learn new words. We’re teaching it because they should learn these words. Mathematical language strengthens understanding and facilitates mathematical thinking. If we want them to be fluent with the mathematics, they need to be fluent with the language, too. We’d be doing our pupils a disservice if we didn’t teach them the words that could be the difference between solidifying their understanding and their explanations or not.
In Thinking Deeply about Primary Mathematics, Kieran Mackle says, “When mathematical vocabulary is taught to pupils in advance of their use and they are given the opportunity to familiarise themselves with their essence over time, we give them permission to not only increase the accuracy of their explanations but to solidify their understanding of the concepts they are explaining.”
So how do we do it?
Explicit teaching of mathematic vocabulary is essential. It’s not enough for new and technical language to be learnt through exposure. They need to say the word, repeat the word, clap out the syllables, look at the etymology or root of the word, find similar words, read it in context, use it in context, revisit and retrieve the word. It’s not a two-minute job at the start of a lesson, it’s part of an extended journey that we guide our pupils through in a careful manner.
But before you can do any of that, you need to decide which vocabulary to teach. Less is more. Think about the language that is going to have the biggest impact on understanding and reasoning and start there. You also need to consider when you’re going to teach it. When in their mathematical journey will they learn a word? When will they hear it again?
And then you need to ensure that the language being used across the school is consistent. If one teacher is only using factor and product but another is only using multiplier, multiplicand and product then there’s work to be done. Teachers need to be taught which words to use, and everyone needs to know when they’ve been introduced and when they come up again.
Once the vocabulary is taught, we need to structure the language beyond that. Stem sentences are the way to do this. They are integral to teaching mathematical reasoning. A stem sentence provides the bones of verbal and written explanations. It’s a mantra for a concept.
The National Centre for Excellence in the of Teaching Mathematics (NCETM) have worked on stem sentences for years and these can be found throughout their Professional Development materials, or spines. We are time poor as teachers so I would highly recommend using these spines to plan and to find high quality stem sentences.
Stem sentences serve several purposes. They lay the foundations for mathematical thinking and reasoning; they provide the structure to the language so our pupils can focus on the mathematics; they emphasise using the correct language; they provide a pathway to making generalisations; and they support students in seeing the underlying structures of the mathematics at hand.
When teaching a new concept, the stem sentence should appear throughout the learning sequence. For example, if you are working on the concept of ‘same difference’ then you will want something like ‘I have increased my minuend by ___ so I must also increase my subtrahend by ___ to keep the difference the same.’ or ‘I’ve subtracted ____ from the minuend and the subtrahend so the difference stays the same.’
It’s important that the stem sentence continues throughout your lesson. Each time you move onto a different example, the whole class should say the sentence together while filling in the gaps. This repetition ensures each child is hearing and saying the reasoning behind a concept again and again.
As you move through a concept and want to challenge the children further, you can begin to gradually remove certain parts of the stem sentence, so the children become more independent in their thinking and reasoning. I have found that colour-coding parts of the sentence and gradually removing significant words but leaving a coloured line behind is a good way to scaffold their independence.
Sometimes teachers will bemoan pupils whose explanations go around the houses, but just like anything else we teach, mathematical thinking and reasoning needs careful modelling and scaffolding to support our pupils through their learning journey. Explicit teaching of vocabulary and stem sentences is essential to this. It can’t be left to chance.
LET’S TOARK ABOUGHT SPELLING: PALT TOUGH
For many, the teaching of spelling can seem like a mammoth task, with no clear way to begin. Neil o ers his suggestions about how spelling can be taught methodically and e ciently in Part 2 of his series on spelling.
By Neil Almond
In the November 2021 edition of HWRK Magazine, I offered readers an extremely brief tour and overview of modern English, tracing it back to its ancestral roots of Proto Indo-European and the cocktail of geopolitical and social influences that have impacted the language over many years to its modern-day incarnation. In this article, I want to talk more about the structure of the English language, and start to appreciate how it is that we may begin teaching spelling so that it sticks.
As mentioned in the previous article, English is such a hard language to read and spell is because it has a deep orthography or complex code. Depending on your accent there are approximately 44 sounds that make up every word in the English language In English there are multiple ways to spell the 44 sounds of the English system - around 176 common spellings to be precise. In primary schools, this process begins formally in Reception through phonics lessons (instruction in phonics, whether systematic and synthetic or not, is any attempt to explain to anyone what sound is being represented by a singular or collection of abstract symbols) and, depending on which phonics scheme your school uses, will depend on when this type on instruction ends and how many spelling alternatives are taught.
Phonics is also the main method by which students are beginning to learn to decode; I use this word purposefully as it encompasses what is rarely understood outside of education: writing systems are codes where the act of taking abstract symbols off the page and converting those into the correct sounds is the act of cracking the code. The act of writing a string of abstract symbols (which we call letters), with spaces between them when required, is the act of writing the code. Understanding this simple principle is one of the keys to understanding how to spell effectively and how to support student in their spelling. This means that to spell (and read) most words in the English language a mastery of the 176 common spellings is required to represent the approximate 44 sounds used within the language.
Along with understanding letters are used to code sounds, other structural knowledge about the English language that is important to know to teach and learn to spell effectively include the following:
1) A sound can be represented by 1,2,3 or 4 letters.
2) One sound can be represented with different spellings.
3) The same spelling can represent different sounds (known as code overlap).
Let’s look at this in practice by looking at the incorrect spellings of the title of this article and the previous one. For this, words between two slashes represent the sound and the letters between two angle brackets will represent spelling.
Let’s tork abawt spelling
Let’s Toark Abought Spelling: Palt tough
Before we analyse the words in to see how I have made use of the structural knowledge, let’s look at the number of sounds in the words that have been spelt incorrectly. ‘Talk’ is comprised of 3 sounds, ‘about’ spelt with 4 sounds, ‘part’ with 3 sounds and ‘two’ with 2 sounds. Across all the words within the title, you can see how a sound has been represented by a different number of letters - /t/ has been represented by one letter <t>, /or/ has been spelt with 3 letters <oar> (as in ‘boar’), /aw/ has been represented by 4 letters <ough> and /ar/ is represented by the spelling <al> (as used in the word half).
That one sound can be represented with different spellings can be seen when comparing the two titles above. /or/ is represented by <or> and by <oar> and /aw/ is represented by <aw> and <ough>.
Finally, that the same spelling can represent different sounds can be seen by the spelling <ough> where it is used to represent the spelling of the sound /aw/ and /oo/ in ‘about’ and ‘two’.
From this, any activity used to promote learning to spell should enforce those principles. Therefore, I am personally sceptical about activities that force students to look at words as ‘whole’ units as if each word is a distinct picture themselves. Such activities include the ever popular ‘look, say, cover, write, check’ and looking at word shapes. In these activities
students would have had a group of words to learn and then use place the correct word into different height boxes that matches the ‘shape’ of the word.
It is unclear, at least from a popular spelling program that recommends this activity, how it is supposed to aid with spelling.
Before explaining a spelling activity that I think is worthwhile, it is important to know that I think the main purpose of spelling lessons is not to teach students how to spell certain words (though naturally this is a consequence of such lessons) but to make the complexities and the structural knowledge of English spelling code as transparent as possible.
It is not possible to teach students how to spell every word in the English language; it is folly to try and do so as it would take up for too much time. Instead, lessons should be giving students tools by which they are likely to spell unknown words correctly.
What activities do I recommend that students do to help them become better spellers? As Daniel Willingham has told us, we only remember what we think about. Any activity needs to get students thinking carefully (and ideally multiple times) about spellings. Phonemes (sounds) are the basis of our language, so it makes sense to get students thinking about the sounds that they hear and relate this to the spellings that they know.
Students in Year 2 and above will be expected to read and spell polysyllabic words. So help to syllabify words and say these with a ‘spelling voice’ where you avoid saying the words with unstressed vowels. For example, if I wanted to students to spell the word amusement, I would show them how we can split the word into three syllables by focusing on the number of vowel sounds (a|muse|ment) and ensure that I stress the initial /a/ sound instead of sounding it as an /uh/ sound (or schwa, as linguists call it). Getting students to finger clap each syllable and saying it in a spelling voice further helps students as they break down longer words into more manageable chunks.
Once students have practiced this several times, when they come to try to spell a new word independently, the teacher now has a few tools at their disposal. Instead of just providing the spelling for the students, teachers can ask the students which syllable and which sound within that syllable they are struggling with. This has the benefit of students seeing that they simply do not know how to spell a certain sound contained within the word rather than the whole word itself.
Getting students to think about spelling in terms of the syllables within a word, and the number of sounds within those syllables is a good bet for getting students to think hard about the structure of the words at a meaningful level when compared to ‘word shape’ or ‘look, cover, write, check’ activities.
One way to get students to practice the skills above is using an activity I have called ‘word inspection’. The activity involves a list of words that, importantly, are arranged by sound. In the case of the example below, they all contain a different way to spell the /ae/ sound. Instead of just writing each word out several times, first students are asked to identify the number of syllables and write the word again marking each syllable. Next, they are asked to identify the number sounds in each word. Having already identified the number of syllables previously, this step is naturally scaffolded by the proceeding one. The final step is to identify which letters spell the target sound. In the
word ‘neighbour’, you can see that it is the <eigh> spelling of /ae/. Once the sheet has been completed, the students will see all the common ways that /ae/ can be spelt along with some rare ways to spell /ae/ but in common words (<aigh> is only used in straight and <ea> is used in only a handful of words). It is useful for students to know this, as it limits the ways that a sound would be spelled.
1. This example would be used in UKS2. it can be adapted in a number of ways by including more than one target sound or by providing fewer spelling alternatives depending on the year group. As mentioned earlier, the purpose of the above is not necessarily to get students to spell that particular set of words correctly, but to get the student thinking carefully about what they can do the next time that they want to spell a word. A student who wants to spell ‘investigate’ will no longer ask a teacher how to spell the word in its entirety, but instead they will ask the teacher how to spell a particular sound within the word. Taking this one step further would be if the student asks if it’s the <ai> or the <a-e> spelling. That is when you know that students are truly cracking the spelling code, and I will look at how we can potentially get to that point with students next time in Part 3.
