Solving word problems is a key component of math curriculum in primary schools. One must have acquired basic language skills to make sense of word problems. So why do children still find certain word problems more difficult than others characterized by similar linguistic demands? (Briars & Larkin, 1984; Carpenter & Moser, 1984; De Corte & Verschaffel, 1987; Kintsch & Greeno, 1985; Nunes & Bryant, 1996; Riley et al., 1983; Verschaffel et al., 2020)? Here I will share some views on this issue from a developmental perspective with a focus on additive reasoning problems.
Preschool children can develop initial thinking about addition and subtraction based on their everyday experiences (e.g., their own physical actions or observations) of putting something in a set (addition) and taking away something from a set (subtraction) (Piaget, 1952). Children often use these “schemes of action” to solve math word problems. Therefore, Combine problems (e.g., “John has four pencils and Steven has three. How many do they have altogether?”) are easy for children because they can solve the problems by imagining two groups of pencils joined together.
However, the difficulty of Change problems differs by where the unknown quantity is located in the question. Take the following problem as an example – “Susan had eight oranges and then she gave five of them away. How many did she have left?” This question should not be challenging for children because they can use the “take things away” action scheme to solve the problems.
By contrast, a Change problem becomes more difficult if it involves an unknown starting quantity (e.g., Jerry had some cookies; he gave Alice seven and he has five left. How many did he have before he gave cookies to Alice?). This problem describes a situation where the quantity decreases, whereas it has an unknown initial state that should be solved by an addition, so there is a conflict between the decrease in quantity and the operation of addition. Children have to understand the inverse relation between subtraction and addition to solve the problem, which is a concept difficult for some children to master (Bisanz et al., 2009; Bryant et al., 1999; Canobi et al., 2003; Ching, 2023; Ching & Nunes, 2017; Gilmore & Papadatou-Pastou, 2009; Nunes et al., 2015; Robinson, 2017; Verschaffel et al., 2012).
Gérard Vergnaud (1982) contends that the three types of meanings represented by natural numbers can also influence the levels of difficulty of word problems. These meanings include (1) quantities, (2) transformations, and (3) relations. Consider the following two problems. The first problem involves a quantity and a transformation, while the second problem concerns a combination of two transformations.
- Sophia had seven stickers (quantity). She played a game and lost three stickers (transformation). How many stickers did she have after the game?
- Alice played two games of marbles. She won seven in the first game (transformation) and lost three in the second game (transformation). What happened, counting the two games together?
Research showed that combining transformations is more difficult than combining a quantity and a transformation (e.g., Brown, 1981; Vergnaud, 1982). When children are about seven years old, they achieve about 80% correct responses in the first problem, but they only achieve a comparable level of success two years later in the second problem. According to Vergnaud, children’s thinking has to go beyond natural numbers when they need to combine transformations.
Natural numbers are counting numbers. In a Change problem with an unknown end state, for example, children can count the number of stickers that a person had before he or she started the game, count and take away the stickers that he or she lost in the second game, and find out how many he or she had left in the end. In the case of the Alice problem, if children count the stickers that Alice won in the first game, they need to count them as “one more, two more, three more” and so on. Therefore, they are actually not counting stickers, but the relation of the number that she now has to the number she had to start with – the transformations are now relations, which are more difficult for children to grasp compared with simply counting quantities.
Findings that Compare problems are difficult for children than Combine and Change problems may also be explained by the same reason that these problems require children to quantify relations. Consider this example, “Jason has five tickets. Harry has nine tickets. How many more tickets does Harry have than Jason?” The question in this problem concerns neither a quantity (i.e. Jason’s or Harry’s tickets) nor about a transformation (no one lost or got more tickets). Instead, it is about the relation between the two quantities.
Most preschool children can rightly point out that Harry has more tickets, but the majority cannot quantify the relation or the difference between the two. Therefore, learning to use numbers to represent quantities and learning to use numbers to quantify relations are not the same, even when the same numbers are involved. Relations are more abstract and more difficult for children. Thompson (1993) argues that the ability to think of numbers as measures of relations at young age serves as a foundation for understanding algebra.
In summary, word problems that can be solved by the same arithmetic operation but belong to different problem types have varying difficulty. Here I have reviewed two kinds of problems that are challenging for children: those that involve the inverse relation between addition and subtraction, and those that involve thinking about relations. Teachers should recognize the intellectual demands of each type of problems from a psychological perspective, and design assessments and organize teaching activities that help children handle the relations involved in each problem, such as schema-based instruction (e.g., Fuchs et al. 2010; Jitendra et al., 2007; Jitendra & Hoff, 1996).
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