Wo r d P r o b l e m So l vi n g a Schema Approach in Year 3 T his article outlines how a Brisbane independent school, Clayfield EDuARDA vAN College, improved the ability of its Year 3 students to solve addition and KlINKEN shows subtraction word problems by utilising a schematic approach. We had observed that how research on while students could read the words in the text of a written problem, many had difficulty reading and solving identifying the core information and were unable to derive a relevant number sentence. word problems may We drew on research that suggested a schematic approach to problem solving was be applied to help promising (Jonassen, 2003) and suitable for Year 3 (Griffin & Jitendra, 2008). In this year three students paper I briefly outline the most pertinent research and provide a short summary of how interpret and solve the unit unfolded. The word problem solving unit consisted of two 40-minute lessons per word problems. A week throughout an eight-week block and was undertaken in two co-educational Year set of questions is 3 classes. There is a difference between how provided so you may successful problem solvers go about their task when compared to weak problem solvers apply what you learn (Xin, Wiles & Lin, 2008). Students who solve problems successfully are able to look beyond from reading this the superficial surface features of a story and are able to analyse the underlying structure article to your own or schema of the problem. They recognise that although word problems may have different classroom. storylines, they are represented by a limited number of mathematical relationships. In contrast, students who are weak problem solvers are likely to be distracted by irrelevant APMC 17 (1) 2012 3 van Klinken information such as characters or setting. clearly reflect students’ increasing confusion They fail to recognise the schematic when the unknown quantity occurs in the similarities between different stories, treating more challenging positions (Figure 1). In our each new story as a completely isolated unit, the position of the unknown quantity problem (Schiff, Bauminger & Toledo, was one of the most important factors when 2009). simplifying or complicating problems. Research over a number of decades shows that weaker problem solvers can successfully be taught to solve problems in the same way as successful problem solvers by helping them to look at problems in a ‘top-down’ schematic way (Marshall, 1995). Carpenter and Moser (1983) classified word problems into three broad schemas: Change, Difference and Combine. In our unit we adopted this classification. The characteristics of each will shortly be defined. We limited the word problems in this unit to problems that required a simple one-step Figure 1. Graph comparing pre- and post-test results. mathematical calculation—the stories contained only one action. Nevertheless, During the eight weeks, we were able to even within this single-step parameter, a frequently observe the impact of children’s number of factors can contribute to number sense on their ability and efficiency difficulties for students. According to Reusser in solving word problems. Even the least able (2000), the reasons why some students fail to students were able to solve ‘result unknown’ solve word problems successfully are that: number sentences correctly because they (a) the scenario represented is outside the were able to use a ‘count all’ approach, experience of a student; (b) the vocabulary using fingers or hastily drawn illustrations to is unfamiliar; or (c) the story uses a difficult keep track. In ‘change unknown’ problems, sentence structure. Hannell (2005) noted many of our students counted forwards that relatively simple problems become or backwards from the starting amount, complicated when they include amounts or persevering with counting until the desired numbers that are larger than the student is number was reached. Although this strategy able to compute mentally. We often observed was effective for most questions because we this in our unit; as soon as numbers above utilised small numbers, the better students 20 were included in the word problem, used their knowledge of number facts to many students were unable to identify the solve these problems more efficiently. For underlying schema of the problem. the most difficult ‘start unknown’ problems, The position of the unknown quantity in many students used a trial and error method. the number sentence must also be considered The more able students were able to rework when analysing the difficulty of a word the problem into another form (for example, problem (Garcia, Jimenez & Hess, 2006). ? + 4 = 7 can easily be solved by reworking In one-step problems, the position of the it into 7 – 4 = ?). Despite the fact that our unknown quantity occurs in three possible students had been taught about ‘number positions. The easiest is ‘result unknown’ fact families,’ many were unable to apply this (7 – 3 = ?), followed by ‘change unknown’ when solving number sentences. (7 – ? = 4) with the most difficult being ‘start unknown’ (? + 4 = 7). Our pre-test results 4 APMC 17 (1) 2012 Word Problem Solving: a Schema Approach in Year 3 The schematic diagrams We based our three schematic templates loosely on those suggested by Marshall In our unit, the use of the schematic diagram (1995). Each was introduced as a visual became integral to the teaching and learning reflection of class discussions as we explored process. A schematic diagram is a visual tool the essential mathematical relationship of to assist students in paring down information each schema. so that only the important structural The translation from word story to information remains. We found it to be the schematic diagram was challenging for our single most important means to assist weaker weaker students. They had to learn to think students to successfully analyse the schemas in a new way, looking beyond the superficial in the stories. story elements to identify the relevant Van Garderen (2007) has shown that the schema. We quickly became aware that these ability of students to represent information students needed considerably more time and visually is related to their ability to grasp individual assistance to make this critical the schema of the problem. Weak problem- conceptual step. solvers tend to include information about characters and setting, but only make superficial reference to the dynamic nature Introducing the schemas of the relationships and quantities in the story. Successful problem solvers, by contrast, We introduced the Change schema first are able to successfully represent the critical because of its dynamic nature and the many mathematical connections. Figures 2 and day-to-day examples that could be found 3 demonstrate how contrasting schematic in the classroom. Verbs are critical to the understanding is reflected in students’ visual action in the story and we explored them in representations. considerable detail. Students discovered that verbs such as “bought,” “found,” “baked” or “earned” were clues to increasing amounts, whereas verbs such as “lost,” “stolen,” “wilted” or “eaten” were clues to decreasing quantities. During this first phase of the unit we introduced the Change schematic diagram as a means of isolating the key information, describing it as a “picture that Figure 2. Visual representation of Question 1 in pre-test. “The shows only the really important information” girl was happy because she had money in the bank.” There is no evidence of schematic understanding. (Figure 4). Figure 3. Visual representation of Question 6 in post-test. The student represents the important information and is able to write a matching number sentence. See Appendix A for details of the questions. APMC 17 (1) 2012 5 van Klinken Figure 5. Difference problem: Mark is 8 years old and his cousin is 17. How Figure 4. Change problem: Jenny has $12 in the bank. She earns much older is $5 doing some jobs. How much money does she have now? his cousin? We began by presenting students with as comparing how long it took children to stories that had no unknown amounts; e.g., complete a task. “Chloe planted nineteen flower seedlings. The Combine schema was the last to be Over the weekend six died. Now there are introduced. Though similar to the Change only thirteen left.” We then worked with the schema, it is distinguished by the fact that students to help them derive both a matching it does not involve action, but is the linking schematic diagram and number sentence. We together of two static groups. By the time discussed what the original quantity was and we introduced this last schema, students whether it had been increased or decreased had considerable experience in identifying to give the final amount. Gradually we structures and we were interested to see introduced stories with a ‘result unknown’, whether they could apply their knowledge followed by ‘change unknown’ problems and to identify this third schema. We gave them finally ‘start unknown’ problems. After three a number of Combine word problems and weeks, during which time varying approaches asked them to identify the new mathematical and ideas were shared and discussed, most structure. Pleasingly, many students children were comfortably recognising, identified the ‘combining of two amounts’ solving and writing Change problems with as the essential feature of this schema. The the unknown amounts in all three positions. schematic diagram was then developed to Difference problems involve finding the reflect joining, the essence of this schema difference between two amounts (Figure 5). (Figure 6). Unfortunately, due to time They include those that seek to equalise two constraints, the Combine schema could not amounts by adding or subtracting an amount be explored in any detail. until the two quantities are the same (see question 10 in pre/post test). Of the three schemas, Difference problems are the most difficult for students to solve (Garcia et al., 2006). This was certainly the case in our unit where Difference problems generally Figure 6. Combine scored poorly in the pre-test. Because of problem: Robert has this, we spent four weeks early in the unit six rocks. Together discussing practical scenarios that compared Robert and Max have fifteen rocks. amounts. Activities included the comparison How many rocks of ages, temperatures and height, as well does Max have? 6 APMC 17 (1) 2012 Word Problem Solving: a Schema Approach in Year 3 Conclusion References Carpenter, T. P. & Moser, J. M. (1983). The acquisition Students and teachers enjoyed this unit of addition and subtraction concepts. In R. Lesh & immensely because it provided an opportunity M. Landau (Eds), Acquisition of mathematical concepts and processes (pp. 7–44). New York: Academic Press. to look at mathematics in a deeper way, Garcia, A. I., Jimenez, J. E. & Hess, S. (2006). other than that provided for in the regular Solving arithmetic word problems: An analysis of classification as a function of difficulty in children curriculum. The schema approach gave with and without arithmetic learning difficulties. teachers an alternative and successful path Journal of Learning Disabilities, 39(3), 270–281. to approach a difficult curriculum topic. It Griffin, C. C. & Jitendra, A. K. (2008). Word problem- solving instruction in inclusive third-grade provided numerous opportunities to link mathematics classrooms. Journal of Educational ‘real life’ with ‘classroom’ mathematics. Research, 102(3), 187–201. Additionally, it engaged students constantly in Hannell, G. (2005). Dyscalculia: Action plans for successful learning in mathematics. London: David Fulton mental calculation. The schematic diagram, Publishers. which was a new concept for our teachers, Jonassen, D. H. (2003). Designing research-based proved to be a pivotal tool in revealing instruction for story problems. Educational Psychology Review, 15(3), 267–296. the ‘bare bones’ of a problem. Throughout Marshall, S. P. (1995). Schemas in problem solving. the unit there were many opportunities for Cambridge: Cambridge University Press. Reusser, K. (2000). Success and failure in school students to articulate, compare and clarify mathematics: Effects of instruction and school their thinking through class and informal environment. European Child and Adolescent discussions. Psychiatry, 9(2), 17–26. Schiff, R., Bauminger, N. & Toledo, I. (2009) Analogical The success of the unit in quantitative problem solving in children with verbal and terms is easily measured by comparing pre- nonverbal learning disabilities. Journal of Learning and post-test results. We were encouraged Disabilities, 42(1), 3–13. van Garderen, D. (2007). Teaching students with by the fact that our students made the most learning difficulties to use diagrams to solve dramatic gains in the most difficult problems, mathematical word problems. Journal of Learning solving word problems that most were unable Disabilities, 40(6), 540–553. Xin, Y. P., Wiles, B. & Lin, Y. (2008). Teaching conceptual to attempt at the start of the unit. model based word problem story grammar to Finally, this small research project enhance mathematics problem solving. Journal of Special Education, 42(3), 163–178. supports the results of other research showing that a schematic approach to teaching word problems can help Year 3 students to conceptualise word problems Eduarda van Klinken in a schematic way, thereby leading to a Brisbane Girls Grammar School deeper understanding as well as greater <[email protected]> flexibility and accuracy when solving word problems. Further details of the unit can be obtained from the author ([email protected]) APMC 17 (1) 2012 7 van Klinken Appendix A. Pre- and post-test questions Number Question Word problem Problem type sentence Jenny has $12 in the bank. She earns $5 doing some 1 Change 12 + 5 = jobs. How much money does she have now? There are 14 girls and 10 boys in Year 2 this year. How 2 Combine 14 + 10 = ? many children? There are 25 children in this class. Yesterday five 3 children were away sick. How many children were in Change 25 – 5=? the class yesterday? Mark is 8 years old and his cousin is 17. How much 4 Difference 8 + ? = 17 older is his cousin? Mum baked 12 cupcakes on Monday. On Tuesday she 5 baked some more. Now she has 20 cupcakes. How Change 12 + ? = 20 many did she bake on Tuesday? Kim had some lollies in her pocket. Her father gave 6 her four more lollies. Now she has 14 lollies. How Change ? + 4 = 14 many did she have in her pocket to start with? Robert has six rocks. Together Robert and Max have 7 Combine 6 + ? = 15 fifteen rocks. How many rocks does Max have? Joe had 19 marbles. He gave some to his brother. Now 8 he only has 13 marbles left. How many did he give to Change 19 – ? = 13 his brother? Anna had a bunch of flowers. Four died and she put 9 them in the bin. Now there are only six flowers left Change ? – 4 = 6 over. How many were in the bunch to start with? Len has fourteen fish. Emma has only six fish. How Difference 6 + ? = 14 10 many more fish does Emma need to have the same (equalise) 14 – 6 = ? number as Len? Charlie is 12 years old. He is 8 years older than his 12 – 8 = ? 11 Difference brother. How old is his brother? 8 + ? = 12 8 APMC 17 (1) 2012