Using a Concept Cartoon to Gain Insight Into Children’s Calculation Strategies Introduction MATTHEW SExTON, Life in the twenty-first century requires an unprecedented level of mathematical ANN GERVASONI knowledge, visualisation and skill for full participation in community life, and for access and ROBYN to opportunities in education and employment. It is therefore concerning that some Australian BRANDENBURG students under-perform in mathematics despite consistent attempts to improve mathematics use a concept cartoon learning and teaching. For example, Gervasoni, Hadden and Turkenburg (2007) found as an effective way to that 30% of students beginning their final year of primary school in regional Victoria encourage students have underdeveloped arithmetic reasoning strategies, a key indicator of mathematical to communicate their competence. This highlights the importance of children developing reasoning strategies for preferred strategies calculating as opposed to using rote procedures (algorithms) or counting-based strategies. for solving addition Curriculum reform aimed at improving students’ reasoning strategies for calculating calculations. was the focus of a research project in which we participated along with two school communities in regional Victoria. The research involved trialling a curriculum in which the teaching of algorithms for Grade 3 and Grade 4 students was withheld in favour of emphasising mental computation, with students’ reasoning strategies recorded on empty number lines to enable monitoring of strategy choice (Gervasoni, Brandenburg, Turkenburg, & Hadden, 2009). Another focus of the research was assisting 24 APMC 14 (4) 2009 Using a Concept Cartoon to Gain Insight Into Children’s Calculation Strategies teachers to gain insight into how their students language or speech bubbles (Naylor & Keogh, approached calculations. 1999). The familiar settings and characters give This article explores one aspect of the relevance to the ideas that are being presented. research in which concept cartoons (Sexton, It is important that alternative conceptions, 2008) were introduced as an innovative way statements or questions pertaining to a for gaining insight into children’s strategies central idea are presented within the cartoon for addition calculations in a situation (Kabapinar, 2005; Naylor & Keogh, 1999). In that begs for the use of mental strategies most cases, alternative viewpoints are presented (24 + 99 = n). We examine some of the responses through the use of a group of characters of the 101 Grade 3 and Grade 4 students who engaging in a dialogue through the use of participated in the research to find out about speech bubbles with minimal use of written their calculation strategies, and then consider language. Due to the characters’ dialogue, the implications for subsequent learning and students have the freedom to make judgements teaching. that agree or disagree with the views expressed by the characters without feeling threatened by needing to express their own opinions Concept cartoons publicly (Kinchin, 2004). Concept cartoons are primarily intended to act as a teaching and A concept cartoon is a learning and teaching learning tool but they can also be used to assess tool used primarily in science education to student cognition (Naylor & Keogh, 1999). explore scientific concepts. However, we believe In some cases they have been used to assess they also have great potential in mathematics the affective domain (Kinchin, 2004; Sexton, education. The cartoons share some common 2008). Indeed, Sexton (2008) explored the traits with those used in comic strips, but rather use of concept cartoons in mathematics and than being designed to arouse hilarity, they found that they were a successful tool for aim to present students with the opportunity to gaining insight into children’s and teachers’ interpret and to understand concepts (Naylor perceptions of effective mathematics learning & Keogh, 1999). environments. In June 2008 the students were Concept cartoons involve the pictorial shown the Concept Cartoon in Figure 1. The representation of characters in settings familiar cartoon depicts four characters and dialogue ttoo ssttuuddeennttss aalloonngg wwiitthh tthhee uussee ooff wwrriitttteenn tthhaatt eexxppllaaiinnss eeaacchh cchhaarraacctteerr’’ss ssttrraatteeggyy ffoorr ssoollvviinngg 2244 ++ 9999 == n. This addition ccaallccuullaattiioonn wwaass sseelleecctteedd bbeeccaauussee iitt iiss uusseedd iinn tthhee EEaarrllyy NNuummeerraaccyy IInntteerrvviieeww ((CCllaarrkkee eett aall..,, 22000022)),, aanndd bbeeccaauussee wwee hhaadd nnootteedd pprreevviioouussllyy tthhaatt mmaannyy cchhiillddrreenn ssoouugghhtt ttoo ppeerrffoorrmm tthhee ccaallccuullaattiioonn uussiinngg aann aallggoorriitthhmm,, eevveenn tthhoouugghh iitt lleennddss iittsseellff ttoo bbeeiinngg ssoollvveedd uussiinngg mmeennttaall rreeaassoonniinngg ssttrraatteeggiieess.. OOvveerraallll,, wwee ssoouugghhtt ggrreeaatteerr iinnssiigghhtt aabboouutt tthhiiss pphheennoommeennoonn.. IInn ccrreeaattiinngg tthhee ccoonncceepptt ccaarrttoooonnss,, tthhee ssttrraatteeggiieess Figure 1. Concept cartoon for gaining insight about children’s mental calculation strategies. APMC 14 (4) 2009 25 Sexton, Gervasoni & Brandenburg used by each character were chosen to enable “would make sure that it is right with blocks.” teachers to determine whether children: This reliance on materials is problematic when 1 perceive the calculation as too hard the addends become so large, and suggests that (Patrick); Jake is not using reasoning-based strategies. Eva 2 can solve the calculation in different ways, also chose Yen but explained that to solve the but recognise that a mental strategy is most story she “counted with my fingers. I go to the efficient (Samson); tens and then I do my ones and I trade.” This 3 would solve the calculation using a use of fingers suggests that Eva is following a visualised addition algorithm (Yen), or; procedure and using counting-based strategies 4 can remember a strategy, but would need to calculate. to write to perform the calculation (Jana). Jason chose both Jana and Samson Once shown the concept cartoon, the explaining, “I find it easier to write it down students were asked to identify the name of or I could do it in my head.” Jason explained the character that best matched their personal he would actually “add the 9 and 4 then the strategy choice for this calculation and provide 90 and 20. Next I would put them together.” a reason for choosing this character. Finally, Chris would also do it like Samson, but would the students were required to explain how “add on 1 to 99 to give me 100 and (then add) they would calculate 24 + 99 = n. These 23. That gives me 123.” explanations provided further insight about Matt chose Patrick because “it is hard,” but children’s approaches, and enabled students wished that he “could be like Samson because to describe alternative strategies to the ones he is so good at working things out.” Let’s used by the four characters. Some illustrative hope that both Matt and his teacher believe examples follow. that this wish can come true. Table 1 The number of students who chose each character (n = 101) Class Yen Samson Patrick Jana Samson & Yen Patrick & Samson Jana & Samson Missing Total June 2008 A 3 2 5 6 16 B 0 9 2 6 1 1 19 C 5 10 0 4 1 1 21 D 2 4 5 6 1 1 19 E 6 3 1 5 1 16 Total 16 28 13 27 1 1 1 4 101 Kate chose Jana (who remembers how to A summary of the students’ character do it but needs to write it down) because “I like choices is provided in Table 1. to write it down on paper because it is easier.” The characters chosen by the students However, when explaining how she would lead to some important reflections. perform the calculation, Kate explained, “I We could assume that most students would turn 99 into 100 and then plus 24 would recognise that this calculation and take away 1 from the 100 and make it (24 + 99 = n) might easily be performed 123.” This is a mental reasoning strategy such mentally, just as Samson did. In contrast, fewer as Samson used. Tania also chose Jana, but than one third of the students did so. Further, explained that to work it out she “would get a about one sixth of the students selected Yen friend to help me and grab a calculator.” This who imagined solving the problem using strategy suggests a lack of confidence in her an algorithm; for these students, imagining ability to perform the calculation. the steps of a learnt procedure was more Jake chose Yen (who imagined the addition useful than recognising the potential of using algorithm) because “I know the sum,” but he compensation to add 23 to 100. Another third 26 APMC 14 (4) 2009 Using a Concept Cartoon to Gain Insight Into Children’s Calculation Strategies of the students chose Jana who remembered impacts negatively on students’ development a way to solve the problem but would need of number sense and mental calculation to write it down. We suspect that many in this strategies. Despite this, it is quite common group might also use the standard addition for Australian teachers to introduce Grade 2 algorithm. About one sixth of the students and Year 3 students to algorithms for addition selected Patrick who did not know what to do and subtraction and this may have impacted because the problem was too hard. on our students’ strategy choices for solving The fact that about two-thirds of Grade 3 24 + 99 = n. and Grade 4 students find this problem either We propose that a more effective approach too hard to solve, or would be likely to solve is to delay the introduction of algorithms it using an algorithm suggests that we need until grade 5 and instead focus on assisting to provide students with more opportunities students to construct a range of powerful to develop mental reasoning strategies. We mental calculation strategies. One approach argue that an aim of mathematics education that teachers in our research used was to is for students to learn a range of strategies for encourage students to record their calculation performing mental calculations, and to make strategies on empty number lines, so that wise choices about the best strategy to use for a monitoring and reflection on strategy choice given calculation. could occur. This approach is widely used in the Netherlands (Beishuizen & Anghileri, 1998). For example, some children in a project Teaching implications class were asked to calculate mentally the answer to 26 + 17 = n, and then represent Mental computation and arithmetic reasoning their thinking on an empty number line. An strategies have been the focus of many studies illustrative example from Leroy, a student in a (e.g., Clarke, Cheeseman, Gervasoni, et al., project class is shown in Figure 2. 2002; Fuson, 1992; Gervasoni, 2006; Steffe, Cobb, & von Glasersfeld, 1988). The data presented here suggest that approximately half of the students would use an algorithm (either writing it down or performing the steps mentally) to solve this type of problem. Figure 2. Leroy’s empty number line representation of his strategy for calculating 26+17=43. However, we argue that this type of problem is best solved mentally. These results support Leroy’s strategy involved some flexible the conclusions of earlier research, suggesting number splitting and re-grouping. His that learning algorithms can reduce students’ explanation using the empty number line capacity to draw upon number sense to solve representation provided a good opportunity calculations that are best performed mentally for discussion and reflection on his strategy (e.g., Narode, Board, & Davenport,1993). Our choice, comparison with the strategies other data also suggest that not all children have students used, and consideration of which mental strategies available or select calculation students’ strategies involved the fewest steps strategies according to how they best fit the or were the most elegant. Interestingly, Leroy demands of a task (Griffin, Case, & Siegler, initially found it simpler to add 16 to 20, than 1994). Further, Narode, Board, and Davenport to add 10 to 26. Monitoring strategy selection (1993) suggest that introducing algorithms too by written recording on empty number lines early in schooling is detrimental to students’ and through reflection on strategy choice in developing arithmetic reasoning strategies. class discussion is important for stimulating Clarke, Clarke and Horne (2006) also argue progress towards higher-level strategies that the early introduction of written methods (Beishuizen & Anghileri, 1998). APMC 14 (4) 2009 27 Sexton, Gervasoni & Brandenburg Gervasoni, A., Brandenburg, R., Turkenburg, K., & Conclusion Hadden, T. (2009). Caught in the middle: tensions rise when teachers and students relinquish algorithms. AAsssseessssiinngg CChhiillddrreenn’’ss In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds). The use of this concept cartoon with Grade 3 Proceedings of the 33rd annual conference of the International and Grade 4 students demonstrates that it was Group for the Psychology of Mathematics Education (Vol. 3, UUnnddeerrssttaannddiinngg ooff LLeennggtthh pp. 57–64). Thessaloniki, Greece: PME. successful in providing teachers with insight Gervasoni, A. (2006). Insights about the addition about a range of strategies that students use to strategies used by grade 1 and grade 2 children perform an addition calculation that begs to who are vulnerable in number learning. J. Novotná, MMeeaassuurreemmeenntt:: H. Moraová, M. Krátká, & N. Stehlíková (Eds) be solved using mental reasoning strategies. Proceedings of the 30th Conference of the International Our findings highlight that few students Group for Psychology of Mathematics Education, (Vol. 3, actually recognise that using mental strategies pp. 177–184). Prague: PME. Gervasoni, A., Hadden, T., & Turkenburg, K. (2007). is the most powerful and efficient way to solve Exploring the number knowledge of children to a calculation such as 24 + 99 = n and instead inform the development of a professional learning plan for teachers in the Ballarat diocese as a means rely on written methods or the visualisation of building community capacity. In J. Watson & K. of written methods. The concept cartoon also Beswick (Eds), Proceedings of the 30th annual conference provides a useful context for discussing the of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 305–313). Sydney: MERGA. advantages and disadvantages of the strategies Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: used by the various characters in the cartoon. Providing the central conceptual prerequisites for first This type of reflection leads students towards formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: using more powerful strategies for similar Cognitive theory and classroom practice (pp. 25–49). calculations in the future. Our challenge for Cambridge, MA: MIT Press/Bradford. you is to use this concept cartoon to find out Kabapinar, F. (2005). Effectiveness of teaching via concept cartoons from the point of view of constructivist how your students approach this calculation. approach. Educational Sciences: Theory & Practice, 5(1), Perhaps they have also become over reliant on 135–146. Kinchin, I. M. (2004). Investigating students’ beliefs about written methods. 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