**Main Content**

## Session Information

**03 SES 14 A, Curriculum Decision-Making in Three Parts of the World**

Paper Session

## Contribution

In educational settings, the goal of mathematicians and mathematics educators is to develop students’ mathematical competencies beyond conceptual, procedural, and contextual knowledge. Understanding that mathematics is founded on engaging in competencies such as communication, reasoning, argumentation, and problem-solving is an important message to convey to students. Research shows, however, that in reality mathematical competency demands is infrequent in many tasks and/or assessment items included in textbooks. Previous studies have pointed out that the general pattern in mathematics textbooks that the examples/exercises offered have been and still are quite uniform, with a strong emphasis put on the development of procedural knowledge.

Accordingly, the movement towards more competence-based resources requires changes in the written curriculum materials. Researchers have begun examining written curriculum materials to determine the extent and nature of problem-solving (Fan & Zhu, 2007), reasoning (Lithner, 2003), communicational approach (Park, 2016), justifications and explanations (Dolev & Even, 2015), and mathematical literacy (Gatabi, Stacey, & Gooya, 2012). In particular, these textbook studies have been conducted for the primary, middle, secondary, and university grades. Additionally, comparisons and analyses of textbooks have been made on, for instance, functions (Mesa, 2004), derivative (Park, 2016), continuity (Raman, 2002), and fraction division (Sun, 2011). Taken together, these studies confirm that there is a scarcity of opportunities for students to engage in or reflect on a variety of essential cognitive skills (e.g., reasoning-and-proving) within a specific content (e.g., geometry).

Although written curriculum materials are certainly not the sole influence on the development of students’ cognitive skills, they are an important one (Fan, Zhu, & Miao, 2013), which makes these textbook analyses a first step to understand the current state of mathematical competencies and informing efforts to infuse mathematical competencies throughout tasks and/or assessment items offered in textbooks: (i) These studies did not examine mathematical competency demands of tasks, examples, and/or exercises offered in textbooks, and (ii) Textbook studies that have been situated in calculus (e.g., Lithner, 2003; Park, 2016) did not focus on mathematical competencies generally. Moreover, the content used for calculus textbook analyses did not include integrals but, rather, involved all examples/exercises that call for a particular mathematical skill (e.g., Lithner, 2003).

Several mathematical competency frameworks, which refer to the structural plan for organizing the cognitive skills and abilities used in learning and doing mathematics (Kilpatrick, 2014, p. 85), have emerged. These frameworks described different cognitive skills and abilities that constitute mathematical competence (Kilpatrick, 2014), which in common, divide mathematical competence into a set of mental processes with particular emphasis on the fact that doing mathematics requires a host of competencies (Niss, Bruder, Planas, Turner, & Villa-Ochoa, 2016). One framework particularly had a considerable impact on the reform movements regarding the mathematics curricula and assessments in several European countries (e.g., Denmark, Sweden, Germany): the KOM (in Danish: Competencies and the Learning of Mathematics) framework putting forth eight mathematical competencies that overlap to a certain degree and have to be activated jointly while solving mathematical problems (Kilpatrick, 2014).

This study investigates how three widely used calculus textbooks in all around the world realize integrals as a potential to prompt mathematical competencies using Turner, Blum, and Niss’ (2015) rating scheme. For this purpose, we analysed examples about integrals – specifically, to assess the extent to which solving those examples calls for the activation of a particular set of mathematical competencies.

The research questions of the present study were the following:

- What are the mathematical competency demands of integral examples in calculus textbooks?
- What is the competency level that best fit the demand of the integral examples in calculus textbooks?

### Method

For selecting textbooks to be included in our study, we first looked at the textbooks’ market share. This led to the selection of three textbooks that together are used in approximately all universities where the official language of instruction is English. This estimation is based upon three standpoints: (1) teaching experiences of the authors at different universities at both national and international universities, (2) overall search of the homepages of universities (e.g., required textbooks in the syllabi of mathematics courses), and (3) information about the best-selling calculus textbooks from publishers. These textbooks are Calculus 7E (Stewart, 2008 – 2012), Thomas’ Calculus 12th Edition (Thomas Jr., Weir, & Hass, 1996 - 2019), and Calculus Single & Multivariable 6th Edition (Hughes-Hallett, McCallum, Gleason, Connally, Lovelock, Patterson et al., 1998 – 2013). The reason that we included these textbooks, namely Stewart Calculus (SC), Thomas’ Calculus (TC), and Hughes-Hallett et al. Calculus (HHC), in our study was not only because these textbooks are purposely put in the market to promote calculus courses in which mathematical thinking and computation reinforce each other but also because they encourage students with various learning styles to expand their mathematical knowledge. In all three textbooks, topics in general and the integral topic in particular are subdivided into numbered segments, mostly consisting of sets of worked examples and exercise problems. With the term ‘worked example’ we refer to the smallest unit that shows solution steps and final solutions to demonstrate the specific algorithms or to illustrate techniques and procedures (Renkl, 2002). To conduct the analysis of the calculus textbooks with regard to the competency demand, we examined all worked examples in three textbooks. Therefore, the data consisted of 262 worked examples (70, 91, and 101 of the examples in SC, TC, and HHC, respectively). The worked examples were categorized using the framework developed by Turner et al. (2015). Each worked example was coded so that it was included into one of the following six categories: Communication, Devising Strategies, Mathematising, Representation, Using Symbols, Operations, and Formal Language [Symbols and Formalism], Reasoning and Argument. The authors met in a training sessions to identify the competency demand on a scale from level 0 to 3 for each these six competencies and discuss the coding. Then, they individually coded all worked examples and discussed the different codes until reaching full agreement on each aspect of the categories.

### Expected Outcomes

262 worked examples in three calculus textbooks were analysed with respect to the mathematical competency demands. The analysis of examples in each book has been conducted, and there were some considerable similarities as well as differences in their levels of competency demands. The three textbooks have a big proportion of features reflecting the level 0 description of the Mathematising, Representation, and Reasoning and Argument competency. 78.57, 79.12, and 74.26% of the examples in SC, TC, and HHC respectively demanded a low level of the Mathematising competency, since they were purely intra-mathematical or the relationship between the real situation and the model was not needed in solving the problem. Furthermore, the analyses indicated that the integral examples were rarely identified as highly demanding; level 3 was scarcely used. Specifically, in all three textbooks, the examples fit the level 3 description for Devising, indicating that the most of the examples comprise devising a multi-stage strategy, where using the strategy involves substantial metacognitive in the implementation of the strategy towards a solution (Turner et al., 2015). A similar pattern could be observed for the Communication competency. In addition, the majority of the examples were at levels 2 and 3, indicating that they demanded a high level of competency. For instance, 70 and 74.26% of them in SC and HHC respectively were high-level competency demand. More than 50% of the Symbols and Formalism competency were at level 2.

### References

Dolev, S., & Even, R. (2015). Justifications and explanations in Israeli 7th grade math textbooks. International Journal of Science and Mathematics Education, 13(2), 309-327. Gatabi, A. R., Stacey, K., & Gooya, Z. (2012). Investigating grade nine textbook problems for characteristics related to mathematical literacy. Mathematics Education Research Journal, 24(4), 403-421. Fan, L., & Zhu, Y. (2007). Representation of problem-solving procedures: A comparative look at China, Singapore, and US mathematics textbooks. Educational studies in Mathematics, 66(1), 61-75. Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: development status and directions. ZDM, 45(5), 633-646. Hughes-Hallett, D. (1991). Visualisation and calculus reform. In W. Zimmerman and S. Cunningham (Eds.), Visualisation in teaching and learning mathematics (pp. 121-126), MAA Notes 19. Hughes-Hallet, D., Gleason, A. M., & Mccallum, W. G. (2010). Calculus: Single and multivariable. New York: Wiley. Kilpatrick, J. (2014). Competency frameworks in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 85–87). Dordrecht: Springer. Lithner, J. (2003). Students' mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52(1), 29-55. Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: An empirical approach. Educational Studies in Mathematics, 56(2-3), 255-286. Niss, M., Bruder, R., Planas, N., Turner, R., & Villa-Ochoa, J. A. (2016). Survey team on: Conceptualisation of the role of competencies, knowing and knowledge in mathematics education research. ZDM, 48(5), 611– 632. Park, J. (2016). Communicational approach to study textbook discourse on the derivative. Educational Studies in Mathematics, 91(3), 395-421. Raman, M. (2002). Coordinating informal and formal aspects of mathematics: Student behavior and textbook messages. The Journal of Mathematical Behavior, 21(2), 135-150. Renkl, A. (2002). Worked-out examples: Instructional explanations support learning by self-explanations. Learning and Instruction, 12(5), 529–556. Stewart, J. (2010). Calculus. Mason, OH: Brooks/Cole Cengage Learning. Sun, X. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76(1), 65-85. Thomas, G. B., Weir, M. D., Hass, J., & Giordano, R. F. (2010). Thomas’ calculus: Early transcendentals. Boston: Pearson Addison-Wesley. Turner, R., Blum, W., & Niss, M. (2015). Using competencies to explain mathematical item demand: A work in progress. In K. Stacey & R. Turner (Eds.), Assessing mathematical literacy: The PISA experience (pp. 85–115). New York: Springer.

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## Programme by Networks, ECER 2021

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