Here is an arithmetic word problem.
What percentage of Indian students would correctly answer option D? What percentage of students in classes 3, 5, and 7 would answer 130 years?
Refer to the graph to see how close your guess is.
Response data of English-medium private schools
Note: The data is from about 750 – 2250 students in each of the classes 3 to 7 collected using Mindspark, a computer-based adaptive learning tool.
Only about 50% – 60% of students in each of the classes 3-7 answered correctly. After class 5, the percentage of students selecting 130 years as the answer seems to be decreasing. On possible reason behind this drop could be that students start realising that the age of 130 years may not unrealistic. However, 20% of students answered the age of Raghu as 130 years even in class 7. Also, the data surprisingly shows an increase in the number of students answering 25 years (option C) from class 3 to 7!
One may argue that students may be answering in haste. But based on our experience of conducting student interactions in three classes on a similar question, we saw that students did comprehend the question correctly and had rationale for the incorrect answer option picked by them.
In a study conducted by French researchers in 1979, on a similar question, more than three-fourths of class 1 and 2 students arrived at their answers by manipulating the numbers, such as 125 + 5 = 130 (adding numbers in the word problem given). Two years back Nanchong Shunqing Primary School in south-west China posed such a problem as a free-response question to 5^{th} graders to assess their critical thinking and it went viral in social media. Such studies have shown that it is useful to ask such non-traditional problems in open-ended manner. Though we don’t know the percentage of students who responded in different ways, some of the student responses received included: “I don’t know.”, “I cannot solve this.”, “We cannot be sure of the captain’s age. The number of the sheep and goats is irrelevant to the captain’s age.” The creative ones were: “The captain should be at least 18 years old because a minor is not allowed by law to operate a vessel.”, “The captain is 36 years old. He is quite narcissistic, so the number of animals corresponds to his age.”. Some people criticised the question makers for asking such a question. This inspired us to check out the extent of correct response among Indian students.
We saw students mechanically doing operations on given numbers in the non-traditional word problem described earlier. What about the case of traditional word problems?
The problem with keywords: We conduct student interviews to understand why students answer in the way they do. We go to a classroom in a school, pose a problem, invite student responses and ask them to articulate their reasoning to arrive at an answer. This way of uncovering student’s thinking also helps us understand misconceptions students have and their extent. We conducted one such student interview in an English-medium private school in Goa.
Gaurav (name changed) is a class 4 student in the school whose mother tongue is English. He is comfortable in English. The problem given to him was as follows.
Ram had 187 marbles. Shyam had 245 marbles. How many more marbles did Shyam have than Ram? |
Gaurav’s answer was 332. When asked to explain his answer, Gaurav didn’t have to think twice. Confidently he answered, “When ‘less’ appears in the question one needs to subtract and when the word ‘more’ appears the numbers are added.” So, when asked a simple arithmetic word problem in English, Gaurav follows keyword-based rules instead of trying to understand the problem and choose an appropriate arithmetic operation.
Our data on a large number of students on different types of addition and subtraction word problems shows that there are many students like Gaurav in classes 1-5 who tend to identify the operation (addition, subtraction etc.) to perform based on keywords like ‘more’, ‘less’ or ‘few’ in a word problem.
The ability to apply operations in real-life situations shows students’ conceptual understanding of these operations. Different contexts and situations are encountered in real life. Hence, it is important that learners demonstrate their understanding by solving various types of word problems based on their structures, involving a good range real-life situations. Students might get the answer right simply by mechanically choosing the operation to perform based on keywords in some types of word problems. But they will arrive at the incorrect answer if they apply key word strategy without comprehending correctly what is given in the question and what needs to be found out, in the case of other types of word problems. Hence educators, curriculum designers, and teachers need to be aware of all the types of word problems and enrich the meanings of operations through a variety of real-life situations. The 4 major types of word problems in addition and subtraction are join, separate, combine, and compare as classified by researchers based on their structures. Here are examples of some of these types of word problems in the table below.
Different types of word problems
Word problem type and sub-type | Example | Semantic equation | |
Join | Result unknown | Raju has 9 pencils. Geeta gives him 3 more. How many pencils does Raju have now? | 9 + 3 = __ |
Change unknown | Amit has 9 pencils. He gets some more pencils from his sister. He has 12 pencils now. How many pencils does he get from his sister? | 9 + __ = 12 | |
Start unknown | Bholu has some pencils in a box. He puts 3 more pencils in the box. There are 12 pencils in the box now. How many pencils were there in the box in the beginning? | __ + 3 = 12 | |
Separate | Change unknown | Ram has 12 pencils. He gives some pencils to his sister. He has 9 pencils now. How many pencils did he give to his sister? | 12 – __ = 9 |
Combine
(part-part-whole) |
Whole unknown | Geeta has 3 red pencils and 9 blue pencils. How many pencils does she have? | 3 + 9 = __ |
Part unknown | Asha has 12 pens. 9 of her pens are red and the rest are blue. How many blue pens does Asha have? | 9 + __ = 12 | |
Compare | Difference unknown | Geeta has 12 pencils. Raju has 9 pencils. How many more pencils does Geeta have than Raju? | 12 – 9 = ___ or 9 + __ = 12 |
Compared quantity unknown | Raju has 3 fewer pencils than Geeta. Geeta has 12 pencils. How many pencils does Raju have? | 12 – 3 = ___ | |
Referent unknown | Farah has 2 more pencils than Alex. Farah has 5 pencils. How many pencils does Alex have? | ___ + 2 = 5 |
Note: The above table does not show all the 14 types of word problems. There are 2 more separate types of word problems like join and 3 more compare type based on keywords.
Once the real understanding of a concept is achieved, it is expected that a learner would also be able to apply the concept learnt in both familiar and unfamiliar situation. As per the National Curriculum Framework (NCF 2005), students are expected to learn word problems on addition and subtraction with numbers up to 1000 by class 3. As per our assessment of this skill, we found the performance to be varying on the 14 different word problem types in different classes. Among compare type problems there were some sub-types like ‘compare quantity unknown’ and ‘referent unknown’ where by the beginning of class 4, almost 50% of the private school students were unable to correctly solve such word problems. Here’s a graph showing performance of Indian private school students on all the compare type word problems.
As can be seen, there is a considerable variation in the performance on the different compare word problem types. The keyword strategy fails for the referent unknown problems. A teacher needs to be aware which types reinforce the ‘keyword strategy’ and which don’t. The types where students struggle, for her classroom instructions to be more effective.
Students should be proficient in solving word problems on basic operations as that is an important skill connecting Maths with real-life and characterizes the conceptual understanding of the operations. The student response data demonstrates that is not the case. Students are often mechanically solving word problems performing operations on given numbers. Often, they don’t see if their answer makes sense and relate to their real-life experience. This is apparent when they use keyword strategy to solve word problems. The strategy often works for typical problems, but breaks down in certain cases which are also important in real life situations. So, it is important that teachers and textbooks give students exposure to all the types of word problems. School mathematics should have connections to real life. Word problems are supposed to help with exactly that.
Our findings are in lines with the findings of the studies documented in the book Making Sense of Word Problems^{[1]}^{.} .The findings of the studies indicate that while solving arithmetic word problems in a school environment, a majority of students demonstrated the tendency to apply one or more arithmetic operations to the assigned data without a realistic consideration of the context. For many students, school mathematics has no connection with their real-life experiences. When solving an arithmetic word problem, they simply apply the arithmetic operations algorithmically with neither realistic consideration nor the use of common sense. The main reasons for such behaviour are the stereotyped way in which word problems are typically presented in school: they are often instructed to follow rules for the word problem solving as mentioned in the book.
As students progress to higher classes, the connection of school mathematics with their real-life experience seems to stagnate as seen from the response data of the ‘age of Raghu’ question being discussed. So, when students find word problems difficult, should it be attributed to their language ability alone? Or does this point to the need for more emphasis on real-life connections in school mathematics apart from developing problem solving skills? Further, an analysis of textbook content shows not all these types of word problems are covered and adequately. This would be fine if students were able to apply their learning to unfamiliar types of word problems. Since this is not the case, it calls for curricula to create learning opportunities by promoting adequate exposure to all the types of word problems, especially the difficult ones (compare and start-unknown problems).
Word problems must involve realistic and relatable contexts to enable students draw from their real-life experiences. Textbooks and classroom instructions must cover all the types of word problems and increase exposure to the more difficult types. It may be desirable to occasionally pose non-traditional and open-ended problems like the ‘age of Raghu problem’ without definite answers. Students should be encouraged to show the information given in the word problem visually (bar models or tape diagrams really help) and in their own words for better sense-making. The response to question of sheep and dog clearly demonstrate the focus of learner to calculate the answer rather than see if the question provides appropriate inputs to decode the question and derive at the correct response. So, instructions should focus a lot more on the process of decoding a word problem, guess the answer before actually arriving at the answer systematically. Whether it is a word problem or an abstract mathematical problem, more students must also develop a habit of reflecting on their answer obtained and check if it is sensible. Instructions in classrooms need to emphasize this.
[1] A range of findings have shown how students consistently answer them in ways that fail to take account of the reality of the situations described. This book (monograph) by Eric de Corte, Brian Greer and Lieven Verschaffel reports on studies carried out to investigate this “suspension of sense-making” in answering word problems.
Maulik Shah
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