Numberless word problems: why and how to use them

Jane and Emre lose one pen a week each. Their friend Mila loses two pens a week. If the teacher has a box of 27 pens, how many weeks will it last? This is the kind of word problem you…

Jane and Emre lose one pen a week each. Their friend Mila loses two pens a week. If the teacher has a box of 27 pens, how many weeks will it last?

This is the kind of word problem you often find in maths textbooks. But when pupils are presented with a question like this, how many skim-read it, without actually considering what they’re being asked to do?

According to Camilla Gilmore, a professor of mathematical cognition at the University of Loughborough, it’s more common than not. “What some children do is read the problem, or perhaps they don’t even read part of the problem in detail, and then they number-spot – they pick the numbers out of the problem and then they’ll make some kind of guess about what operation they need to do,” she explains. “So, they might look at the numbers and work out – oh, I think I need to add these. They might spot a word, so if they see the word ‘more’ somewhere in the problem, they will think – I need to add these.” What we want children to do, she says, is to read the problem, visualise the situation, identify the different features of it, and work out what’s been asked. They should then identify what kind of operation they would need to do and finally do the calculations. Why you need to know if a pupil was born preterm Does underlining really matter? Forget ‘balanced’ literacy; we need balanced learning So how can teachers ensure that their pupils do the former, rather than the latter? According to researchers in the US, the answer may lie in taking the numbers out of the problem. In 2019, Professor Karen Givvin and her colleagues from the University of California tasked participants with explaining word problems: one group worked on typical word problems, the other on the numberless version. Afterwards, both groups were asked to solve four typical word problems. The group that had been previously working on numberless word problems outperformed the other group in this final task. In the research, the authors hypothesise that removing numbers from word problems prompted participants to approach the problem differently, developing an understanding of what the situation asked them to do instead of just jumping to calculations. While the study was conducted on college students, the authors argue they have no reason to believe this intervention would not also work with younger students who struggle with maths – or even with those who don’t. So let’s go back to the original question. Taking out the numbers would mean it was presented like this: Jane and Emre lose a number of pens each week. Their friend Mila loses more pens a week. If a teacher has a box with a certain amount of pens, how many weeks will the box last? Instead of thinking about numbers, the children are now encouraged to think about relationships. In a classroom setting, children could discuss that they need to add the number of pens both Jane and Emre lose a week, with the number of pens Mila loses, and then consider how this quantity relates to the number of pens in the teacher’s box and to the number of weeks it will last. Maths specialist Martin Noon is a firm advocate for this approach – and actually, he says that this work is already happening in maths classrooms across the country, with teachers removing the numbers from word problems to scaffold problem-solving skills. “We need to get students to think about the information they have got first, before they jump into messing around with numbers,” he says. “That’s how I see numberless problems. Knowing what to do is more important, whatever the numbers.” Importance of literacy Removing the numbers from questions also highlights the important role literacy plays in maths when it comes to pupils’ understanding, he adds. “It’s all about understanding the English ability of the students first, and making sure they understand what the question is asking, before getting them to process the mathematics. The literacy side of it is really important,” he says. Professor Gilmore agrees. She says that some children seem to jump to the numbers when reading a word problem because of language difficulties. So, when thinking about maths problems children are struggling with, teachers should ask whether it’s challenging because of the maths, or because of the language in the question. “Doing this scaffolded approach of numberless word problems, supporting children through understanding the language, might be a nice step to help all children, but in particular those that find [the language] difficult,” she says. Introducing numberless word problems So, how should teachers go about introducing numberless word problems successfully into their lessons? For Gilmore, the key is around fostering discussion. “It’s really about generating lots of discussion. When you are presenting a problem with no numbers in it, ask questions like: what can we know about this situation? What kind of operation could we use?” she explains. Numbers, she says, can be reintroduced into the problem gradually. Each time a piece of numerical information is added in, teachers should ask the children how that piece of information changes what they know about the problem. According to primary maths advisor Tom Oakley, teachers also need to factor in the unfamiliarity that can confuse students at first. Writing in a recent tweet, he said: “While it might seem simple (replacing the numbers with words for untold quantities), it can be alien to the children at first. Therefore, you will need to think about how and when you incorporate these into your lessons, as this unfamiliarity may cause teething problems.” So, the next time your class is confused by three friends who are always losing pens, apples, or any other object, why not take the numbers out, and let the mathematical language speak for itself? 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