3 Rephrasing word problems
Word problems have been around for a very long time. Consider these two specific examples:
- ‘It takes three men six hours to dig a ditch. How long does it take two men to dig the same ditch?’ (Traditional)
- ‘Suppose a scribe says to thee, four overseers have drawn one hundred great quadruple hekat of grain, their gangs consisting, respectively, of twelve, eight, six and four men. How much does each overseer receive?’ (Problem 68, Rhind Mathematical Papyrus, c. 1700 BC)
You probably found the second problem more difficult to understand because the context is less familiar. Many students find difficulties with this.
Difficulties with word problems arise because:
- the students are not yet fluent readers
- the language of instruction is not their mother tongue
- they do not understand the language used because the vocabulary is not familiar to them.
This can also mean that the students cannot imagine the context of the word problem (Nunes, 1993). Word problems are often essentially mathematical problems dressed up in everyday language. They can help students understand that mathematics can model the real world and they themselves become mathematicians when they do this. This is why students need to realise that the power of mathematics in real-world problems lies in its ability to model complex situations, from which they must extract the essential elements in order to solve these problems.
Focusing on the process of making sense of a complex situation and modelling it mathematically can also help students focus on making sense of word problems. Activity 3 looks at how to help students find out what they need to know more independently by rephrasing the word problems.
Activity 3: Making sense of the context and the mathematics in a word problem
Adapt these word problems so that they fit the level of learning of your own students.
The activity
Tell your students to read each problem and answer the following questions:
- Mandeep had 21 marbles. Simi had 18 fewer than Mandeep. If they wanted to share the marbles equally, how many would they each have?
- Rasheed’s mother baked three identical circular cakes for his birthday to share with friends and family. Fourteen adults and 20 children came to his party. The size of the slice of the cake the children got was half that of an adult size. What fraction of a cake was an adult portion and what fraction of a cake was a child portion?
- Savitri had to make a model of a cuboid kaleidoscope for her science project. She wanted to use chart paper to make the surface of the kaleidoscope. What area of chart paper would she require if she wanted to make a kaleidoscope with a length of 25 cm and a breadth of 4 cm ?
- Ramesh and Mahesh together can row a boat at a speed of 12 kph. At that speed it takes them 30 minutes to cross the lake. If they row at 10 kph, how long would it take them then to cross the lake?
- Lalita was awarded a 5 per cent increment in her annual salary because of her contribution to the water conservation project for her company. If her base salary was Rs. 3.5 lakh per annum, what is her revised monthly salary?
For each problem, consider:
- Do you know the meaning of each of the words or phrases highlighted in bold? Are there some terms or phrases that are new for you? Do you feel that these will be relevant to solving the problem?
- How would you go about learning what these words or phrases mean, or the mathematical ideas you would need to work with them?
- Rephrase the words and phrases highlighted in bold to simplify the word problem. If a particular term, word or phrase is not required, you may omit it. Which terms did you find difficult to rephrase? Why?
Your students will probably not all be at the same stage in their ability to make sense of the context and the mathematics in a word problem. This activity should provide you with an excellent opportunity to monitor their performance and provide them with constructive feedback. You may wish to have a look at Resource 2, ‘Monitoring and giving feedback’, to help you prepare for this aspect of the activity.
 Video: Monitoring and giving feedback |
Case Study 3: Mrs Chakrakodi reflects on using Activity 3
I am pleased that I used these three problems with my class. I have to say it was hard work getting them to focus at the start – they just seemed to find problems and say that they were stuck. However, I persevered. I told them to work in pairs, which I always think is helpful if the work is unfamiliar and likely to require a lot of thinking. I reminded them to note down anything that they did not know the meaning of and then think about how they could find out about these ideas.
After everyone had done some thinking, we shared what we could use to find out about the ideas. They first said, ‘Ask the teacher’, but I banned that for this exercise and asked them to be more imaginative. One said, ‘Use the internet’, another ‘Look it up in the textbook’, so I suggested that they look up what they could in their textbooks and that if they brought me a note of anything they couldn’t find, I would be their internet search engine for today! I made sure to be awkward and only give information on what was actually ‘entered in the search bar’ in order to make them think about what they really needed to know.
Once everyone felt they had the information they needed, they went onto the rephrasing exercise. It seemed that this was easy now because the class were working collaboratively and learning together by this time.
What I had not expected was that the students in my classroom who were multilingual really benefitted from the discussions of what the words meant. I told them to make sure to make a note of the meaning of the words in whatever language they are most comfortable with so they could refer to it later.
 Pause for thought
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2 Constructing stories to help make sense of mathematical concepts