Using Enabling Prompts to Effectively Support Teaching with Challenging Tasks
/Introduction and Background Literature
Challenging tasks are mathematical tasks designed to encourage students to “connect different aspects of mathematics together, to devise solution strategies for themselves and to explore more than one pathway to solutions” (Sullivan et al. 2013, p. 618). In addition, work on such tasks typically requires students to record, explain and justify their mathematical thinking, and, importantly, allows students opportunities to takes risks and to struggle with the mathematics (Sullivan & Mornane, 2014). This struggle has been described as ensuring students have opportunities to be in the “Zone of Confusion” (Sullivan, Borcek, Walker, & Rennie, 2016, p. 168).
It has been argued that all students should be provided with meaningful opportunities to work on such tasks (Clarke, Roche, Cheeseman, & Sullivan, 2014). In order to ensure that such tasks are accessible to (almost) all students, they are often structured such that they inherently have a “low- floor, high-ceiling” (Bobis et al., 2018, p. 501). To further support differentiated instruction, challenging tasks are typically developed to include enabling and extending prompts. Enabling prompts can involve “reducing the number of steps, simplifying the complexity of the numbers or varying the forms of representation” (Sullivan et al., 2015, p. 126). Research suggests that teachers view enabling prompts as effective for assisting students to think about the core mathematical ideas, for providing an entry point into the main task, and for ensuring an appropriate fit between student understanding and the level of challenge in the task (Cheeseman, Dowton, & Livy, 2017). By contrast, extending prompts “are posed to extend the thinking of students who complete the learning task quickly” and communicates to students the expectation that finishing the task does not mean they stop “thinking and learning” (Sullivan et al., 2015, p. 126).
Research has found that teaching with challenging mathematical tasks is effective for improving student performance (Russo & Hopkins, 2019b), however this approach can be pedagogically demanding for teachers (Stein, Engle, Smith, & Hughes, 2008). In part, this demand is a consequence of a teacher’s uncertainty around how to structure the learning session, including how to use enabling and extending prompts to support the learning of all students, and how much to allow students to struggle (Russo et al., 2019).
The purpose of this paper is to present two case studies exploring how primary school teachers have managed to overcome some of the obstacles they faced when incorporating challenging tasks, with some outside professional learning support. The case studies focus on the teachers’ and students’ use of enabling prompts in the classroom. At the outset, I need to note that I was in a dual role of providing these teachers with professional support (in my capacity as a Numeracy coach), whilst also documenting and analysing their experiences to share with other educators. Pseudonyms have been used to protect the anonymity of the students and teachers included.
Case Studies
Case Study 1: Charlotte
Charlotte was an experienced teacher, working in a Year 5 classroom. She previously incorporated lots of problem solving into her classroom program but felt that it had “dropped off” over recent years. After observing a modelled lesson with her class, Charlotte decided that she wanted challenging problem solving tasks to be the focus of our time together.
Charlotte initially expressed one major concern about using this approach:, that some students would not know what to do and thus be unlikely to experience success. She felt that these students would be reluctant to independently access enabling prompts when required, due to not wanting to signal to their peers that they needed any additional help. She also suspected that these same students would copy from others rather than truly collaborate and thus get little out of the sessions. We spoke about possible steps we could take to support these students and I encouraged Charlotte to make them a focus of her observations when lessons were being modelled, so she could see what they were actually doing when they were stuck or in the “Zone of Confusion”.
When the identified students were given opportunities to make genuine decisions in the course of the modelled lessons, we found time and again that they independently took a variety of steps to assist themselves, enabling them to successfully engage with the task at hand. The majority of students were quite happy to independently access the available enabling prompts, which were always placed in the same spot in the room (in this case, on the teacher’s chair). And this differentiation strategy proved very capable of generating its own momentum, as the more people who used enabling prompts led to a greater proportion of students willing to walk up to the teacher’s chair and grab one for themselves. A culture of students being able to select the right level of support appropriate for them without having to worry about judgement from peers was established in the space of three to four weeks, which meant there was little to no stigma attached when students accessed the enabling prompts. This case study provides a concrete example of how consistent routines coupled with the expectation that students are resourceful and resilient can lead to students accessing enabling prompts autonomously as needed (Russo et al., 2019).
Another feature of the challenging tasks lesson structure which supported the students that Charlotte had been concerned about was the freedom given to students to collaborate. This particular class was used to sitting in assigned seats, thus the ability to move about the room and choose who you would like to work with was something of a novelty. Rather than students copying from more confident peers, as Charlotte had initially feared, we instead found that the students did an excellent job of finding people who were working at a similar pace to them and that genuine collaborations were sprouting up all around the room. And Moreover, because so many of the students were engaged in the problems they were working on, there was very little classroom management required, with nearly all the students remaining on task for the entire session.
It is important to note that both of the above processes (use of enabling prompts and student collaboration) required some outside support from the teacher in order to achieve the level of success described. There were a few students who were more reluctant than others when it came to accessing the enabling prompt. In these cases, neither Charlotte nor myself ever brought the prompt over to the child’s desk. Instead, we verbally suggested that this might be a good next step and also highlighted some of other people in the room who were using enabling prompts, especially if this list included some students with high levels of social capital. This was usually enough to give the child the necessary push.
In terms of supporting genuine collaboration, a brief mention was always made of what is classified as working together on a task and how this differs from copying. During the summary stage of the lesson, we would often select groups of students who had been collaborating and ask them to reflect on how they ensured that everyone understood what the group was doing. These discussions served as a good model for others in the class to follow in future sessions.
At the end of our time together, Charlotte agreed to model a lesson for me to observe and give her feedback on. This lesson showcased how well her students had adapted to tackling challenging problem solving tasks. As Charlotte was introducing the task, one of her students put his hand up to confirm that there would be an enabling prompt available, an event which I believe is noteworthy for a number of reasons. Firstly, it demonstrates that the students now viewed enabling prompts as a crucial element to any problem solving lesson. And secondly, the willingness to ask this question in front of the entire class also reflects that this student did not see using enabling prompts as something to be embarrassed about, which was consistent with our observations of the class from the previous few weeks.
When the time came for Charlotte to rove around the room to see which students were having difficulty getting started, she was pleasantly surprised to find that there was just one student who was yet to make a start on the task. A number of other students had accessed the available enabling prompt and were using this to help them, while others were working collaboratively. When Charlotte spoke with the child having difficulty, she simply asked him if he had seen the enabling prompt yet and then reminded him that this might be a good first step. This entire interaction took less than 20 seconds and was enough to get this child started without any further intervention.
Case Study 2: Tia
Tia was another experienced teacher who I had the opportunity to work with across an entire term. At the time, she was teaching a Year 1 class. However, unlike Charlotte, Tia was initially less enthusiastic about making challenging problem solving tasks the focus of our work together. After watching a modelled lesson, she expressed doubts that this approach was going to offer anything significantly different to her students, when compared with her current practice. This was due to the fact that when I prepared for the initial modelled lesson, I did so with no prior knowledge of her students and their capabilities. Thus, the problem I selected was not challenging enough for this particular cohort of Year 1s, which meant that Tia did not get the chance to observe how her students reacted when given the opportunity to genuinely struggle with a task.
During our first debriefing session, I asked for a second opportunity to model a challenging problem solving lesson, hoping for a better outcome if I planned something that was pitched at a higher level. This second lesson was more successful, with a much greater proportion of students spending time in the “Zone of Confusion” and thus engaged in genuine problem solving. This lesson gave Tia the opportunity to witness her students being challenged in a manner that was different to her regular classroom program and afterwards, she was very keen to learn more about this approach. This example illustrates how the process of observing lessons taught by someone with expertise in teaching with challenging tasks can persuade a generalist teacher that such approaches have utility (Clarke, Cheeseman, Roche, & van der Schans, 2014; Russo & Hopkins, 2019a).
Over the next few weeks, I continued to model lessons for Tia using challenging tasks, while she also took the opportunity to independently trial this approach with her class on the days I was not there. Some of my observations during this time were very similar to what took place in Charlotte’s Year 5 classroom. Tia had a small collection of students who found this approach particularly challenging, as the lack of teacher modelling meant that they had to figure out what to do for themselves. This was clearly something that they were not used to doing. As with Charlotte’s class, we ensured that the enabling prompts were always placed in the exact same spot in the room, to make it easy for the students to locate them. We also consistently asked students the same question whenever they were having trouble getting started- “Have you got the enabling prompt yet?”
As the weeks progressed, the students became more familiar with the structure and features of this type of lesson and we found that the vast majority of the class were accessing enabling prompts when required. The students in Tia’s class made a smoother adjustment to the introduction of enabling prompts than the older students in Charlotte’s room. One possible reason for the Year 1 students’ greater willingness to use enabling prompts is that they seemed less concerned about the perceptions of their peers. The younger students displayed fewer outward signs that they believed there was a stigma attached to using enabling prompts (e.g. looking around at what their friends were doing before leaving their seats to access the enabling prompt). This was consistent with what I have observed in other settings over time, with older students generally being less willing to use enabling prompts when this approach in initially introduced.
During the debriefing sessions, Tia and I spent considerable time discussing the process for creating prompts. Due to the fact that some of the students in Tia’s class were still learning to read, the need for well-crafted enabling prompts was of particular importance. We found that the most beneficial modification to make was to vary the form of representation by including additional visual cues. In some cases, re-presenting the exact same task with a supporting diagram showing what the problem was asking was all that was required to help students get started. Visual cues were also a feature of the enabling prompts that we used in Charlotte’s class but they were used less frequently with the older students. Again, the power of using visual cues to support students to access enabling prompts has been documented in the literature (Russo et al., 2019).
At the end of our time working together, Tia spoke about the renewed levels of enthusiasm that she now had for teaching numeracy. She said that one of the positive elements of this approach was that it allowed all the students in her class to work on the same problem, while also providing additional support and challenges for those who needed it. Tia could see improved levels of independence and reasoning in her students, which was proving to be beneficial well beyond her classroom mathematics program.
Concluding remarks
The case studies in this article were selected because they highlight the important role that enabling prompts play when teaching with challenging tasks. When using challenging tasks, teachers need to ensure that the problems are pitched at a high level, as the aim is for a large proportion of the class to spend part of the lesson in the “Zone of Confusion” (Sullivan et al., 2016, p. 168). This time struggling provides many benefits, including helping students to develop a deep understanding of the mathematics connected to the task. However, due to the wide range of capabilities in an average Australian classroom, the use of challenging tasks means that it is likely that some students will find it difficult to achieve success if they only have access to the main problem. Therefore, enabling prompts are a crucial element whenever planning this type of lesson (Sullivan et al., 2015).
The case studies presented emphasise some important considerations for teachers to contemplate when using enabling prompts. Firstly, these prompts need to be created with consideration given to the particular class/cohort of students who will be working on the challenging task. Teachers need to think about the likely barriers that these students will face when trying to get started on the task and compose enabling prompts to help them overcome these obstacles. Varying the form of representation is a very useful technique, particularly when working with younger students. The use of visual cues allows students to gain a better understanding of what the task is asking them to do. Importantly, these types of enabling prompts also ensure that students are not being deprived of the opportunity to engage in genuine problem solving, as they still need to plan their own approach to the problem and think creatively about how they might solve it.
Another important point illustrated in these case studies is that the students are the ones who are responsible for deciding when/if they need to access the enabling prompts. Teachers should anticipate which students are likely to need the enabling prompt, as this will help when planning what types of variations to make (Cheeseman et al., 2017). However, enabling prompts are not designed to be given out to a pre-determined group of students, thus forming a sub-group in the class who are working on a simpler version of the task. All students need to be given access to the main task and provided with an opportunity to work on this before making the decision to use the enabling prompt. Teachers can facilitate this process by ensuring that the enabling prompts are placed in a consistent location in the room, thus making them easy for the students to find (Russo et al., 2019). Charlotte and Tia both placed enabling prompts on the teacher’s chair at the front of their classrooms, and this worked well for both groups of students.
Finally, when enabling prompts are initially introduced to a new group of students, teachers should expect that there will be a small proportion of students who are reluctant to use them. This is due to a variety of factors but one of the strongest seems to be concern over how they are perceived by their peers, in relation to their capability as maths students. Developing a classroom culture where students are focussed on their own learning, rather than worrying about what their peers are doing, proved to be helpful in overcoming this issue in both Charlotte and Tia’s classes. It is important that the teacher highlights examples of when enabling prompts have been used successfully as part of the reflection stage of the lesson, as this will help strengthen the class’ understanding of the importance of utilising enabling prompts.
References
Bobis, J., Downton, A., Hughes, S., Livy, S., McCormick, M., Russo, J., & Sullivan, P. (2018). Curriculum documentation and the development of effective sequences of learning experiences. Paper presented at the ICMI Study 24: School mathematics curriculum reforms: Challenges, changes and opportunities. Tsukuba.
Cheeseman, J., Downton, A., & Livy, S. (2017). Investigating Teachers' Perceptions of Enabling and Extending Prompts. Mathematics Education Research Group of Australasia. In S. L. A. Downton, & J. Hall. (Ed.), Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 141-148). Melbourne, Australia: MERGA.
Clarke, D., 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.
Clarke, D., Roche, A., Cheeseman, J., & Sullivan, P. (2014). Encouraging students to persist when working on challenging tasks: Some insights from teachers. Australian Mathematics Teacher, 70(1), 3.
Russo, J., Bobis, J., Downton, A., Hughes, S., Livy, S., McCormick, M., & Sullivan, P. (2019). Teaching with challenging tasks in the first years of school: What are the obstacles and how can teachers overcome them? Australian Primary Mathematics Classroom, 24(1), 11-18.
Russo, J., & Hopkins, S. (2019a). Teachers’ Perceptions of Students When Observing Lessons Involving Challenging Tasks. International Journal of Science and Mathematics Education, 17(4), 759-779.
Russo, J., & Hopkins, S. (2019b). Teaching primary mathematics with challenging tasks: How should lessons be structured? The Journal of Educational Research, 112(1), 98-109.
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.
Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane, A., Roche, A., & Walker, N. (2015). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education, 18(2), 123-140.
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, VIC: MERGA.
Sullivan, P., Borcek, C., Walker, N., & Rennie, M. (2016). Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks. The Journal of Mathematical Behavior, 41, 159-170.
Sullivan, P., & Mornane, A. (2014). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education Research Journal, 26(2), 193-213.
(This article originally appeared in Australian Primary Mathematics Classroom, vol 4, 2019)