Flipping our classrooms liberates significant time for richer meaning-making activities and higher-order thinking lesson segments.

Extra work from video creation was not enough to dissuade me. I have coached a number of my colleagues at my school in flipped learning and instructional video creation; however, none of them are Maths teachers – Ken is lone wolf. I create all of my own videos currently, and have no ‘buddy’ such that I can do a part create part curate mode. Finding videos that are just right for my classes’ needs often takes more time than making them myself. So, I’m lightboarding and screencasting pretty hard at this point. Flipping the lot is still worth it because it’s not about the videos. Confused?

It’s not really about the videos, this flipping practice. I’m dedicated to it not because I’m lazy (it is actually more work), nor because there is anything particularly wrong with my lecturing skills (they’ve worked at least satisfactorily for over 20 years), It’s not because of my enthusiasm for ‘edutech’ either. We flip our classrooms because it employs technology to efficiently package and present the lecture part of a lesson (Bloom’s lower two levels) and sends that to the out-of-class space, which liberates significant time for richer meaning-making activities and higher-order thinking lesson segments in the classroom. The removal of the classroom lecture also allows more interaction between teacher and students, making it possible for a teacher to interact with every student meaningfully, every lesson, leading to improved working relationships. In the study of mathematics at high school, Problem Solving Teaching is one important example of a strategy for which we need to liberate supplementary time.

I have observed, unsurprisingly, that there are substantial issues, to do with many students’ dispositions and skills in solving problems. I have thought that it would be great to be able to use the freed-up time due to flipping videos to improve on the status quo. Problem solving is an immensely important element of a mathematics education, and in the twenty-first century, it is cited increasingly as a vital skill.

In my opinion, when BYOx (Bring Your Own Device) practice works well in schools, benefits such as increases in engagement, flexibility and opportunities for personalised instruction lead to more effective pedagogy and learning, and improved dispositions to learning the subject Mathematics. The BYOx allows students to access learning content in a myriad of contexts and locations. The BYOx also allows increased facility for student collaboration and sharing, and creative educational pursuits. Cloud-integrated applications such as OneNote allow for such advances inside and outside the classroom. I feel glad that mine is a BYOx school. I feel part of the Twenty-first Century, and less part of the old industrial model of schooling.

**Problem Solving Teaching – A background narrative**My focus on student problem solving first started in a resolute manner in 2006 when I first became a Mathematics HoD. I was initially concerned about senior student problem solving performance on exams. Since then my interest has deepened, and I am active in teaching problem solving at all high school year levels, as an element of both Mathematics, the subject, and numeracy, the cross-curricula ability. One of the key junctures was in 2013, when every faculty in my school had to action a mandate to implement Reciprocal Teaching authentically in their subject areas. As far as I was aware, no maths educator had tried this. This mandate required me to find a way to blend Problem-Solving Teaching (PST) – was not going to dump this! – with Reciprocal Teaching (RT) – what the English teachers did. Whilst I value RT and had used RT when teaching English myself, I was at a loss to find how in the world to make these two great strategies work as one machine. So, I used a truly unique plan one Sunday morning. Google it!

My search of “Problem Solving Teaching + Reciprocal Teaching” yielded an immediate bulls-eye. Implausibly, the first result listed was an article about RT in Maths, written by Yvonne Reilly, Jodie Parsons and Elizabeth Bortolot of Sunshine College, Victoria. Later, a team from my school comprising of the principal, several deputies and some HoDs took part on an expedition to some of the best schools we could find in Victoria. Sunshine College was of these. We saw the aforementioned authors in action in their classrooms. This school seemed to have every excuse not to be producing great outcomes, with many different languages spoken in students’ homes, and an ICSEA index of well below 1000. However, the opposite was true. What the staff were doing there was amazingly successful – especially with regard to problem solving in Maths classes.

Reciprocal Teaching in Hattie’s Visible Learning, has a high effect size of d = 0.74. It is a renowned strategy for reading comprehension, and it is a collaborative group process.

- Predicting
- Clarifying
- Questioning
- Summarising

However, the key change Reilly et al made, and thus, my way of meshing PST and RT, involved replacing Questioning, which usually has little relevance in a high school mathematical problem-solving context. Questioning in RT is where students textually analyse, and interrogate an extended text and explore deeper meanings. The vital change was to use Solving instead:

- Predicting
- Clarifying
- Solving
- Summarising

This key change, plus refining the nature of the other three steps to suit a high school Maths context was what we needed. Now for the job of making it our own.

As a fan of PST (d = 0.61), and in particular the heuristic work of George Pólya, I set to work on modifying the ideas from Sunshine College for our needs. Over four years and a few iterations in collaboration with staff (past and present) from my current school, we have arrived at the following process (see Figure 1) which we call Reciprocal Maths Teaching (RMT), or just Reciprocal Maths. RMT dovetails Pólya’s PST steps with a reading comprehension strategy, in a collaborative context. Pólya’s steps are inherent in RMT.

Pólya’s steps:

- Understand the problem
- Devise a plan.
- Carry out the plan.
- Look back.

Figure 1: Reciprocal Maths Teaching (RMT)

By explicitly teaching students these RMT steps, as well as the “toolkit” of mathematical strategies in Solve, students gradually build an arsenal of tactics to tackle problems they may encounter in future. Our Maths faculty has spent up on physical resources to facilitate the implementation of RMT. Students from Years Seven to Nine have a trolley in their Maths classroom with laser-etched, ceramic-coated, tablet-sized whiteboards. One on side is a blank template version of Figure 1, where students may write their responses with whiteboard markers, and the other side is blank for rough working, of a temporary nature. We found from Sunshine College that mini-whiteboards give some students more courage to take a risk and have a go at the problem. We supply the stationery too. Additionally, we have a truly gigantic outdoor RMT template (as per Figure 1) with three RMT template full-size whiteboards.

**Flipped RMT in a BYOx Classroom – Observations Thus Far**

When RMT in a BYOx setting is working well, we see students solve a problem collaboratively in a team. The team then ‘publishes’ its full solution as a video or a PowerPoint, or in text using a writeable interface such as a touch screen and stylus, or a Wacom tablet connected to a laptop. Typically, this work is kept on the Collaboration Space section of our OneNote Class Notebook. The class analyses these problems and their solutions.

Problem solving is messy. Problem Solving Teaching can be likewise. Just like when a teacher trains a new class in how to watch flipped videos, take Cornell Notes, and use the classroom space responsibly, Flipped RMT is a lot of work early in the year. Certainly, by flipping classes, I have significantly more contact time available than traditional non-flipped Maths classes to train students, however.

Group norms comprise an essential set of skills and protocols that students must be trained in. Some students take over, others try to avoid the tasks at hand, some disrupt other groups because they don’t like the group I’ve put them in, and some initially refuse, as they still haven’t decided that problem solving is worth the effort. However, collaborative skills are essential twenty-first century skills. Students learn by hearing their peers explain mathematical concepts or strategies, and deep learning occurs when students teach other students.

Some students often like to immediately go to the Solve step (I’m looking at you boys!). After all, it’s the answer that matters – nothing else, right? My students sometimes come from households where parents can be heard to say (e.g. at PT interviews) that “It’s maths. It’s only right or wrong…”. Whether we flip or not, teachers must be patient and never give in to students who baulk at the other key stages Predict, Clarify and Reflect.

A strategy I have adapted for RMT from what English teachers often do when running RT activities is to provide students with one fixed role e.g. the Clarifier, as opposed to the whole group all working the same step in a synchronous way. Thus, students have a carefully defined job and a set responsibility to the team. Of course, the others need to pay attention and ask questions as students perform their role in the solving of the problem. This strategy mitigates the above issues, and I am keen to keep employing and developing it. I am using a rotation schedule to make sure students are gaining essential experience in all roles of the problem solver: Predictor, Clarifier, Solver and Reflector.

The amount active learning has been great to watch during RMT sessions. It is also often the reason why some students initially resist. Groups are typically small (I like four), and provided the ‘right’ questions are provided, students wrestle with collective reasoning, apply their mathematical concepts and strategies in partnerships, and justify the reasonableness (or not!) of their results. Higher order thinking is required, as are initiative and insight. Sometimes, we can get students to re-watch an Individual Space video if no students in the group can recall the maths underlying the problem. This is sometimes, my job as a facilitator – to intervene and diagnose “just in time”, what the blockers might be. Other times, I like to just fire questions at student to prompt them, in little ways. The thinking needs to come from the students, wherever possible. As a result, I get to speak to lots of my students and have substantive conversations about the maths during RMT. It certainly does not do working relationships any harm.

**Flipped RMT – what next? **I am confident in the validity and effectiveness of this strategy. However, it is crucial that I do some formal diagnostic work to more scientifically measure the effect of RMT. This will be a future focus. Additionally, I’m still not quite satisfied with some of the finer elements in our RMT process. For example, I want to “beef up” Predict, as students demonstrate some uncertainty around how to respond to this phase. We need to provide training that is more meaningful to students to help them understand and value this part of the process. I’d also like to put in an iterative loop between Solve and Reflect.

Additionally, I have just begun to incorporate elements Newman’s Error Analysis (Newman’s Prompts) and Dweck’s Growth Mindset strategies in our RMT process. Again, a flipped classroom provides more opportunities for the application of these approaches.

No doubt, what I find from trying these new elements will be interesting.

I will be presenting a session in the FREE Seminar program at the National Education Summit Brisbane on Saturday 5 June 2021 at the Brisbane Convention & Exhibition Centre - come along, check out all of the resources, samples and sessions on offer - all for free.

**References**

Garrett, M. (2016). The Problem with Problem Solving. URL: https://calculate.org.au/2016/04/13/problem-with-problem-solving/ [Retrieved 04/01/2019].

Hattie, J. (2012). Visible Learning for Teachers: maximizing impact on learning. London: Routledge.

Palincsar, A. S. (1984). Reciprocal teaching of comprehension-fostering and comprehension-monitoring activities. Cognition and Instruction, p. 59.

Polya, G. (1945). How To Solve It. New Jersey: Princeton University Press.

Queensland Curriculum and Assessment Authority. (2017). 21st century skills for senior curriculum: a position paper.

Queensland Curriculum and Assessment Authority. (2017). Specialist Mathematics 2019 v1.0: General Senior Syllabus. Brisbane: QCAA.

Yvonne Reilly, e. a. (2009). Reciprocal Teaching in Mathematics. MAV Conference 2009 (p. 8). Melbourne: The Mathematical Associatin of Vicotria.

Photo by j.mt_photography from Pexels