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Student reflections on learning with challenging tasks: ‘I think the worksheets were just for practice, and the challenges were for maths’

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Abstract

The current study considered young students’ (7 and 8 years old) experiences and perceptions of mathematics lessons involving challenging (i.e. cognitively demanding) tasks. We used the Constant Comparative Method to analyse the interview responses (n = 73) regarding what work artefacts students were most proud of creating and why. Five themes emerged that characterised student reflections: enjoyment, effort, learning, productivity and meaningful mathematics. Overall, there was evidence that students embraced struggle and persisted when engaged in mathematics lessons involving challenging tasks and, moreover, that many students enjoyed the process of being challenged. In the second section of the paper, the lesson structure preferences of a subset of participants (n = 23) when learning with challenging tasks are considered. Overall, more students preferred the teach-first lesson structure to the task-first lesson structure, primarily because it activated their cognition to prepare them for work on the challenging task. However, a substantial minority of students (42 %) instead endorsed the task-first lesson structure, with several students explaining they preferred this structure precisely because it was so cognitively demanding. Other reasons for preferring the task-first structure included that it allowed the focus of the lesson to be on the challenging task and the subsequent discussion of student work. A key implication of these combined findings is that, for many students, work on challenging tasks appeared to remain cognitively demanding irrespective of the structure of the lesson.

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Notes

  1. This quantitative analysis was intended to be exploratory. Consequently, the analysis is only included in the text of the ‘Results: students reflection on their work’ section when deemed noteworthy, in particular, when it was statistically significant or approached statistical significance. The absence of any commentary indicates that the association between gender, year level or mathematical performance and the relevant theme was not statistically significant (p < 0.05) nor approaching statistical significance (p < 0.10).

References

  • Baxter, J. A., & Williams, S. (2010). Social and analytic scaffolding in middle school mathematics: managing the dilemma of telling. Journal of Mathematics Teacher Education, 13(1), 7–26.

    Article  Google Scholar 

  • Boston, M. D., & Smith, M. S. (2011). A ‘task-centric approach’ to professional development: enhancing and sustaining mathematics teachers’ ability to implement cognitively challenging mathematical tasks. ZDM, 43(6–7), 965–977.

    Article  Google Scholar 

  • Cheeseman, J., Clarke, D.M., Roche, A., & Wilson, K. (2013). Teachers’ views of the challenging elements of a task. In V. Steinle, L. Ball & C. Bardini (Eds.), Proceedings of the 36th annual conference of the Mathematics Education Research Group of Australasia (pp. 154–161). Melbourne, Australia: MERGA.

  • Clarke, D. M., & Clarke, B. A. (2003). Encouraging perseverance in elementary mathematics: a tale of two problems. Teaching Children Mathematics, 10(4), 204–209.

    Google Scholar 

  • Clarke, D. M., Cheeseman, J., Roche, A., & van der Schans, S. (2014). Teaching strategies for building student persistence on challenging tasks: insights emerging from two approaches to teacher professional learning. Mathematics Teacher Education and Development, 16(2), 46–70.

    Google Scholar 

  • Darragh, L. (2013). Sticking with it or doing it quickly: what performances do we encourage in our mathematics learners? In V. Steinle, L. Ball & C. Bardini (Eds.), Proceedings of the 36th annual conference of the Mathematics Education Research Group of Australasia (pp. 218–225). Melbourne, Australia: MERGA.

  • Glaser, B. G. (1965). The constant comparative method of qualitative analysis. Social Problems, 12(4), 436–445.

    Article  Google Scholar 

  • Glaser, B. G. (1969). The constant comparative method of qualitative analysis. In G. McCall & J. Simmons (Eds.), Issues in participant observation: a text and reader. Reading: Addison-Wesley Pub. Co..

    Google Scholar 

  • Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1994). Emotional contagion. New York: Cambridge University Press.

    Google Scholar 

  • Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.

    Article  Google Scholar 

  • Hollingsworth, H., McCrae, B., & Lokan, J. (2003). Teaching mathematics in Australia: results from the TIMSS 1999 video study. Camberwell: Australian Council for Educational Research.

    Google Scholar 

  • Jackson, K., Garrison, A., Wilson, J., Gibbons, L., & Shahan, E. (2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class discussions in middle-grades mathematics instruction. Journal for Research in Mathematics Education, 44(4), 646–682.

    Article  Google Scholar 

  • Ke, F. (2014). An implementation of design-based learning through creating educational computer games: a case study on mathematics learning during design and computing. Computers and Education, 73, 26–39.

    Article  Google Scholar 

  • Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.

    Article  Google Scholar 

  • Krapp, A. (2005). Basic needs and the development of interest and intrinsic motivational orientations. Learning and Instruction, 15(5), 381–395.

    Article  Google Scholar 

  • Lambert, R., & Stylianou, D. A. (2013). Posing cognitively demanding tasks to all students. Mathematics Teaching in the Middle School, 18(8), 500–506.

    Article  Google Scholar 

  • Leikin, R., Levav-Waynberg, A., Gurevich, I., & Mednikov, L. (2006). Implementation of multiple solution connecting tasks: do students’ attitudes support teachers’ reluctance? Focus on Learning Problems in Mathematics, 28(1), 1–22.

    Google Scholar 

  • McClain, K. (2002). Teacher’s and students’ understanding: the role of tools and inscriptions in supporting effective communication. Journal of the Learning Sciences, 11(2–3), 217–249.

    Article  Google Scholar 

  • Middleton, M. J., & Midgley, C. (2002). Beyond motivation: middle school students’ perceptions of press for understanding in math. Contemporary Educational Psychology, 27(3), 373–391.

    Article  Google Scholar 

  • Muis, K. R., Psaradellis, C., Lajoie, S. P., Di Leo, I., & Chevrier, M. (2015). The role of epistemic emotions in mathematics problem solving. Contemporary Educational Psychology, 42, 172–185.

    Article  Google Scholar 

  • NRC (National Research Council Staff). (1989). Everybody counts: a report to the nation on the future of mathematics education. Washington: National Academies Press.

    Google Scholar 

  • Pekrun, R. (2006). The control-value theory of achievement emotions: assumptions, corollaries, and implications for educational research and practice. Educational Psychology Review, 18(4), 315–341. doi:10.1007/s10648-006-9029-9.

    Article  Google Scholar 

  • Ridlon, C. L. (2009). Learning mathematics via a problem-centered approach: a two-year study. Mathematical Thinking and Learning, 11(4), 188–225.

    Article  Google Scholar 

  • Russo, J. (2015). Teaching with challenging tasks: Two 'how many' problems. Prime Number, 30(4), 9–11.

  • Russo, J. (2016a). Teaching mathematics in primary schools with challenging tasks: the big (not so) friendly giant. Australian Primary Mathematics Classroom, 21(3), 8–15.

  • Russo, J. (2016b). Teaching with challenging tasks: baskets and boundaries. Prime Number, 31(3), 7–9.

  • Russo, J. (2016c). Teaching with challenging tasks: hopping with fiona the frog. Prime Number, 31(2), 10–11.

  • Russo, J., & Hopkins, S. (2017a). Task-First vs Teach-First: Does lesson structure matter for student mathematical performance when teaching with challenging tasks? Manuscript submitted for publication.

  • Russo, J., & Hopkins, S. (2017b). Class challenging tasks: Using cognitive Load theory to inform the design of challenging mathematical tasks. Australian Primary Mathematics Classroom, 22(1).

  • Russo, J., &amp Hopkins, S. (in press). How does lesson structure impact teachers’ willingness to teach with Challenging Tasks? Mathematics Teacher Education and Development.

  • Sherin, M. G. (2002). When teaching becomes learning. Cognition and Instruction, 20(2), 119–150.

    Article  Google Scholar 

  • Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.

    Article  Google Scholar 

  • Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.

    Article  Google Scholar 

  • Stein, M. K., Smith, M., Henningsen, M., & Silver, E. A. (2009). Implementing standards-based mathematics instruction. New York: Teachers College Press.

    Google Scholar 

  • Strauss, A., & Corbin, J. (1994). Grounded theory methodology. In N. Denzin & Y. Lincoln (Eds.), Handbook of qualitative research (pp. 273–285). Thousand Oaks: Sage.

    Google Scholar 

  • Sullivan, P., & Mornane, A. (2013). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education Research Journal, 25, 1–21.

    Article  Google Scholar 

  • Sullivan, P., Tobias, S., & McDonough, A. (2006). Perhaps the decision of some students not to engage in learning mathematics in school is deliberate. Educational Studies in Mathematics, 62(1), 81–99.

    Article  Google Scholar 

  • Sullivan, P., Clarke, D. M., & Clarke, B. (2009). Converting mathematics tasks to learning opportunities: an important aspect of knowledge for mathematics teaching. Mathematics Education Research Journal, 21(1), 85–105.

    Article  Google Scholar 

  • Sullivan, P., Cheeseman, J., Michels, D., Mornane, A., Clarke, D.M., Roche, A., & Middleton, J. (2011). Challenging mathematics tasks: what they are and how to use them. Paper presented at the Maths is multidimensional (Mathematical Association of Victoria), Melbourne, Australia: MAV.

  • Sullivan, P., Aulert, A., Lehmann, A., Hislop, B., Shepherd, O., & Stubbs, A. (2013). Classroom culture, challenging mathematical tasks and student persistence. In V. Steinle, L. Ball & C. Bardini (Eds.), Proceedings of the 36th annual conference of the Mathematics Education Research Group of Australasia (pp. 618–625). Melbourne, Australia: MERGA.

  • Sullivan, P., Askew, M., Cheeseman, J., Clarke, D. M., Mornane, A., Roche, A., & Walker, N. (2014). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education, 18(2), 1–18.

    Google Scholar 

  • Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load. Educational Psychology Review, 22(2), 123–138. doi:10.1007/s10648-010-9128-5.

    Article  Google Scholar 

  • Sweller, J., Kirschner, P. A., & Clark, R. E. (2007). Why minimally guided teaching techniques do not work: a reply to commentaries. Educational Psychologist, 42(2), 115–121.

    Article  Google Scholar 

  • Teddlie, C., & Tashakkori, A. (2009). Foundations of mixed methods research: integrating quantitative and qualitative approaches in the social and behavioral sciences. Thousand Oaks: Sage.

    Google Scholar 

  • Tzur, R. (2008). A researcher perplexity: why do mathematical tasks undergo metamorphosis in teacher hands? In O. Figuras, J. Cortina, S. Alatorre, T. Rojano & A. Sepulveda (Eds.), Proceedings of 32nd conference of the International Group for the Psychology of Mathematics Education (pp. 139–147). Morelia, Mexico: PME.

  • Valås, H., & Søvik, N. (1993). Variables affecting students’ intrinsic motivation for school mathematics: two empirical studies based on Deci and Ryan’s theory on motivation. Learning and Instruction, 3(4), 281–298.

    Article  Google Scholar 

  • Westwood, P. (2011). The problem with problems: potential difficulties in implementing problem-based learning as the core method in primary school mathematics. Australian Journal of Learning Difficulties, 16(1), 5–18.

    Article  Google Scholar 

  • Wilhelm, A. G. (2014). Mathematics teachers’ enactment of cognitively demanding tasks: investigating links to teachers’ knowledge and conceptions. Journal for Research in Mathematics Education, 45(5), 636–674.

    Article  Google Scholar 

  • Woolley, M. E., Strutchens, M. E., Gilbert, M. C., & Martin, W. G. (2010). Mathematics success of black middle school students: direct and indirect effects of teacher expectations and reform practices. Negro Educational Review, 61, 41–59.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Education, Monash University, Wellington Rd, Clayton, VIC, 3800, Australia

    James Russo & Sarah Hopkins

Authors
  1. James Russo
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  2. Sarah Hopkins
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Correspondence to James Russo.

Appendix: Example of challenging tasks used in the current study

Appendix: Example of challenging tasks used in the current study

For further information about the tasks, and in particular, to view the associated enabling and extending prompts, please refer to the published versions of these tasks (see Russo 2015, 2016c).

About our room and the people in it

Context

Students were instructed that all of the maths that would be done during today’s lesson related to things in the classroom and specifically the people in the room.

Challenging task 1: how many feet?

Without leaving your seat, or talking to anyone, can you work out how many feet are in the room right now? How did you count them? Can you count them a different way? What do you think is the easiest way to count them?

Challenging task 2: how many fingers?

Without leaving your seat, or talking to anyone, can you work out how many fingers are in the room right now? How did you count them? Can you count them a different way? What do you think is the easiest way to count them?

Fiona the frog goes hopping

Context

Below is the context for the tasks which was shared with students:

Fiona the frog loved to hop. In fact, she loved to hop so much that, instead of swimming, she even hopped through the water. But as much as Fiona loved hopping, she need to rest once in a while. She knew she could rest on land or on a lily pad.

Today, we are going to go hopping with Fiona the frog. We are going to pretend that all the numbers ending with zero are lily pads, which means that Fiona can take a little rest if she lands on one of these numbers.

Challenging task 1: hopping forwards from zero

Fiona the frog needs to get from one side of the lake (0) to the other (100). The only way she can do so is by landing on lily pads (numbers which end with a zero, 10 to 90). She must land on at least five lily pads, or she won’t make it.

Once she starts on her journey, Fiona always covers the same distance with each hop. You have to decide how far Fiona should go with each hop. Should she hop in 1s, 2s or 3s to get to the other side of the lake, as quickly and safely as she can? How did you decide?

Challenging task 2: hopping forwards from seven

Fiona the frog needs to get from one side of the lake to the other (100). This time a big gust of wind picked her up and dumped her on number 7, so she gets a bit of a head start. Also, Fiona has been exercising a lot and does not need to take as many rests anymore. She now only has to land on three lily pads to arrive safely.

Once she starts on her journey, Fiona always covers the same distance with each hop. You have to decide how far Fiona should go with each hop. Should she hop in 1s, 2s, or 3s to get to the other side of the lake, as quickly and safely as she can? How did you decide?

Challenging task 3: hopping backwards from 100

Now Fiona has to get back over to the other side of the lake. So this time, she needs to start at 100 and end at 0, so you will have to count backwards! Again, Fiona is fitter now because of all her exercising, so she only needs to land on three lily pads to make it.

Again, once she starts on her journey, Fiona always covers the same distance with each hop. You have to decide how far Fiona should go with each hop. Should she hop back in 1s, 2s or 3s to get back over to the other side of the lake, as quickly and safely as she can? How did you decide?

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Russo, J., Hopkins, S. Student reflections on learning with challenging tasks: ‘I think the worksheets were just for practice, and the challenges were for maths’. Math Ed Res J 29, 283–311 (2017). https://doi.org/10.1007/s13394-017-0197-3

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