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.
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.