Using Maths Games During Remote Learning

It was late March when schools in my local area (Melbourne, Australia) began to shut down.  Facing a rising number of COVID-19 cases, our government decided to declare an early start to the school holidays.  When term two began in the middle of April, schools were essentially closed to the general population of students, with only the children of essential workers allowed to attend.  This meant that teachers in Melbourne joined many of their colleagues from across the globe in making the rapid switch to remote learning.     

For me personally, this swift adoption of virtually delivered learning was somewhat problematic.  As an education consultant who does most of his work in classrooms, many of my immediate bookings were either cancelled or postponed.  With some extra time on my hands, I began thinking about what I could contribute to help the teachers and students that I usually work with.  It was around this time that I noticed photos popping up in the social media accounts of well-meaning parents sitting their kids down in front of maths textbooks, with accompanying posts about the beginning of their remote learning journeys.   

When I am working with educators, one of the messages that I consistently preach is the need to make mathematics education engaging.  And perhaps the best way to create disengaged maths students is to have them work their way through a textbook.  Mathematics is about understanding the world around us.  Therefore, students learn maths best by interacting with their environment, not by having a textbook placed in front of them. 

To my mind, one of the most productive activities that students could undertake during remote learning was playing maths games.  Asking students to spend time at home playing games has several benefits- some more obvious than others.  Firstly, games are perhaps the most engaging way for students to develop fluency with basic facts.  In order for students to become fluent, they need to devote substantial time to practise whatever skill they are working on.  For example, if a student is trying to learn all of the multiplication facts up to 10 x 10, they are much more likely to enjoy playing Target Os and Xs (see below) than they are working on a worksheet filled with column upon column of multiplication equations.  The more enjoyable the activity or task assigned, the more likely that students will spend longer periods of time engaged with it. 

There are many reasons why games have been identified as having a positive impact on student engagement, including the fact that in most cases they incorporate a social element, as you are playing against another person.  Therefore, informal conversations, jokes, trash talk, and words of encouragement may all feature while you are playing, helping to create a relaxed mood amongst students.  Many students also enjoy the competitive element of games, happily losing themselves in strategic thinking, while still constantly practising whatever maths skills are incorporated into the design of the game. 

Another important benefit of maths games is that they are interactive, allowing one player to observe the other’s thinking while playing (Bay-Williams & Kling, 2014).  It is amazing to see the profound impact that this can have on the ability of students to learn and apply computation strategies.  Throughout my teaching career, I have seen many students rapidly learn new strategies through the simple act of playing a well-designed game with a more proficient peer. 

One final benefit of using maths games during remote learning is that they can help illustrate the value of a strategies approach to learning basic facts, as opposed to a more traditional, memorisation approach.  There have been many studies conducted over the years that compare the two approaches and all of them have found that teaching students to think flexibly via a strategies-based approach leads to better student outcomes (Bay-Williams & Kling, 2019).  Despite this evidence, many families understandably favour the memorisation approach, as it matches the way they learnt when they were at school and is thus the method they are most comfortable with.  Playing maths games can help parents/families see first-hand that there is more than one way to work out the solution to an equation.  And hopefully, through watching the way their children approach questions, there can be a growing appreciation for the strategies approach to learning basic facts. 

I decided that one simple step I could take was to film some of my favourite maths games that I had learnt over the years and share them via my YouTube channel.  When it came to selecting which games to use, I tried to follow three basic principles.  Firstly, each game needed to have a simple structure.  I knew that a large proportion of my audience was going to be students and parents, rather than teachers.  Therefore, the games needed to have simple structures and rules, as anything that was overly complicated was likely to not be played.  Secondly, the materials needed to play each game had to be objects that were likely to be found in most households.  In instances where this was not the case, for example games that needed ten-sided dice, we were always sure to suggest an alternative that families could use.  And finally, each game had to allow for easy differentiation (Russo, Russo & Bragg, 2018).  I wanted teachers to be able to suggest the same game to their entire class.  However, for this to occur, I needed to make sure there were opportunities provided to easily modify the games, either to raise or lower the level of challenge. 

My final dilemma was who was going to be my opponent in these videos.  I initially asked my eldest son Nash to film some with me, as he was already familiar with many of the games from playing them together over the years.  He quickly showed that his place in the project was invaluable, as he was a living, breathing student who could provide the audience with excellent examples of the types of questions that can be asked and the different strategies that might be used when playing the various games.  Nash proved to be very good at thinking aloud, making it easy for the audience to “get inside his head” and understand what he was doing.  In fact, based on the feedback we have received from many teachers and parents, the authentic interactions that take place between Nash and me have at times proven to be more valuable than the games themselves. 

My goals when we started were modest.  We planned to film and upload a new game each day.  I thought that this bank of games would be useful to many of the schools where I was working and hoped that they would share them with their families.  However, what started as a local project quickly spread well beyond the suburbs of Melbourne.  We have had picked up more than 3,000 followers and subscribers, teachers, students and parents from 39 different countries, all the way from Argentina to Zimbabwe.

One of my favourite games that we filmed is called Target Os and Xs (Downton, 2019).  The rules are listed below or you can use this link (https://www.youtube.com/watch?v=lwIfKu_fdqE&t=18s) to watch our video for this particular game.

Equipment

*Paper

*Pens

*Two 10-sided dice

*Deck of Cards 

Rules

1. Set-up a regulation Os and Xs board, with nine empty boxes (i.e. a 3 x 3 grid)

2. Use two ten-sided dice (or a deck of cards) to generate nine 2-digit numbers and write each number in one of the empty boxes on the board

3.  Each player is dealt six cards

4. Use two or more cards and any operation to create equation that equals one of the numbers on the board- if you can do this, you fill in that place with a O or X

5. Replace any used cards with new ones from the deck, so that each player has six cards in front of them at the start of each turn

6.  If you are unable to make a target number on any turn, you can pass.  This allows you to exchange as many cards from your hand as you want for new ones

7.  The first player to get three Os or three Xs in a row is the winner

 I love this game for a number of reasons.  Firstly, the target numbers are generated by rolling two dice, therefore they are random numbers.  This means that players who can think flexibly are rewarded in this game, rather than players who have memorised multiplication facts.  For example, 76 cannot be created by simply adding or multiplying two cards together.  However, it can easily be created by using three cards and a bit of flexible thinking (e.g. 7 x 10 + 6 or 8 x 10 – 4).  If you don’t have a ten card, players need to consider if they can make a ten (e.g. 7 + 3 or 5 x 2).  Again, this requires number sense and cannot be rote learned. 

The other reason that this game is one of my favourites is its dynamic nature.  On every turn, things are changing.  You are asked to work with different cards and the target number/s on the board that you are aiming to make will also change, depending on the move made by your opponent.  This adds an extra level of challenge to the game, which only makes it more engaging to play. 

Life appears to be slowly returning to some form of normality in Melbourne.  With schools reopening, it is back to the classroom for Nash and back to work for me.  Poison numbers was the 39th game that we filmed and it was also the final of our daily postings.  We have since created a new page on our website (www.lovemaths.me/games) where these games will be housed.  They are organised into categories (e.g. lower/upper primary, number/operations/other), in order to make it easier for people to find what they are looking for.  However, Nash’s enthusiasm for playing and filming the games has remained, and so we are planning to add to this resource over time. 

My hope is that what started as a resource designed to help with remote learning can continued to be used in classrooms and in the homes of students.  Hopefully, these games can play a role in building stronger home-school connections in the area of mathematics learning well beyond whenever this current pandemic comes to an end.  If you visit the site and play some of our games, we would love to hear from you.

References

Bay-Williams, J.M., & Kling, G. (2014). Enriching addition and subtraction fact mastery through games. Teaching Children Mathematics, 21 (4), 238-247. 

Bay-Williams, J.M., & Kling, G. (2019). Math Fact Fluency, National Council of Teachers of Mathematics. 

Downton, A. (2019). It’s More Than a Game. Prime Number, 34 (3), 38-39.

Russo, J., Russo, T., Bragg, L. (2018). Five principles of educationally rich mathematical games.  Australian Primary Mathematics Classroom 23 (3), 30-34.

(This article originally appeared in the Journal of the Association of Teachers of Mathematics (273), September 2020) 

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)

Maths and Literature: The Narrative-First Approach

By Michael Minas

At the beginning of 2018 I moved into the year 5/6 team, which meant I was about to start working with a new group of teachers.  As we approached the first day back, I knew that I wanted problem solving to be a focus of our planning and I began to think about the best way to get the rest of the senior team on board with this plan.  Luckily, I had a fresh and exciting idea to bring to our team’s first planning session.

At the 2017 MAV Conference, I was lucky enough to attend a professional development workshop on the “Narrative-First Approach”.  At this session, James and Toby Russo outlined a new way to use picture story books as a launch pad for engaging mathematics lessons.  Rather than looking for books with obvious links to particular mathematical concepts (e.g. using “The Doorbell Rang” by Pat Hutchins when exploring division), the Russos were advocating an entirely different approach.  They suggested that teachers first select an engaging text and then use the key components of the narrative to develop an appropriate mathematical problem. 

Planning

Feeling confident in the knowledge that my colleagues were open to exploring new ideas, I suggested that we try this approach to plan a problem solving unit for term one.  We agreed that each teacher would bring a picture story book along to our team’s first planning session, with the only selection criteria being that it was a great book. 

Planning day got off to a good start, as everyone arrived with a picture story book in hand.  We first had a team discussion on how to plan a problem solving lesson using the Narrative-First Approach, using an example that the Russos had shared at the MAV conference (“Where the Wild Things Are” by Maurice Sendak”).  We then worked in smaller groups of two or three teachers to plan problem solving tasks linked to our selected texts.   

I worked with an experienced teacher named Kim and we first planned around the text I had brought- “Have You Seen Elephant?” by David Barrow.  This book is about a young child playing hide and seek with an elephant.  On each page, the elephant is attempting to hide in a different location but, despite his best efforts, he can be easily seen due to his enormous size.  However, for some unexplained reason, (SPOILER ALERT) the boy in the story can never find the elephant and he eventually gives up looking. 

We had little trouble coming up with a problem solving activity that we felt had a strong connection to the central theme of the book.  We gave each student one of the illustrations from the book (showing the elephant standing next to the boy, his pet dog and a fence) and asked them to use the information contained in the picture to estimate the elephant’s height.  For an enabling prompt, we asked them to work out their own height and prompted them to use this information to assist with the initial problem.  And the extending prompt asked them to find some places around the school where the elephant would be able to effectively hide, using the estimated height they had just calculated. 

However, while “Have You Seen Elephant?” was fairly easy to plan for, we had significant difficulties when it came to our second text.  Kim had selected “The Red Tree” by Shaun Tan, a book about a lonely red-haired girl who is experiencing depression.  We spent a reasonable amount of time looking at the text from a variety of angles but, despite our best efforts, we found it very difficult to design a problem solving task with authentic links to the story.  In the end, we decided to abandon this text and went with another option.   Despite our difficulties with this particular text, the rest of the planning session was a success and at the end of the day, the team had a problem solving unit that featured seven good lessons, all linked to an engaging picture story book. 

Teaching

When it came time to use these problems with our students, the team all followed a similar lesson structure.  First we read the books to our classes without mentioning the accompanying problem solving task.  This was to ensure that the texts were being honoured for the great books that they all were. 

After reading and discussing the texts with the students, we introduced the problem solving task.  This part of the lesson was very similar to what our students were already familiar with.  After launching the task, the class was asked to work in complete silence for five minutes, to give each student an opportunity to get started on the task without their thinking being influenced by their peers.  At the conclusion of the silent time, the students were free to work independently or collaboratively.  Enabling prompts were left in a familiar spot in the room, where students could access them as required.  And extending prompts were handed out by the teacher to students who successfully completed the initial task. 

Reflections

At the end of the unit, students were given the opportunity to write reflections on it.  Below is a sample of their thoughts.

“I think the book problem solving was really fun so I think we should do more of it.  Some of the problems were slightly easier but some of them were challenging.”

Sasha

“I loved term1 problem solving because each session we read a really good book and then did a problem based on the book we read.  I much preferred the picture story book problem solving to usual problem solving.”

Jordan

“Problem solving was quite challenging but I am proud of my work.  Most weeks I would get into the ‘zone of confusion’ but most weeks I find a way out of it.  I enjoyed the picture story problem solving because I liked how all of the problems were connected to the book.”

Emma

“I liked the approach of the picture stories because we got to read great books.  I also think the problems were too much of a mixture of levels because some were too easy or too hard.”

Asher

The above comments are a fairly good representation of the entire sample of reflections.  The vast majority of students reported that they really enjoyed working on problems with links to books which they had just read.  However, a reasonable proportion also reported that they felt that some of the problems we planned were either too difficult or not challenging enough. 

The difficulty in pitching the problems at an appropriate level seems to be related to the fact it was the first time using this approach for the teachers who planned this unit, myself included.  Some of the teachers addressed this issue in reflections they wrote for this article.

“I found it quite difficult to get my head around at first, particularly during the planning phase. After some practice with colleagues and developing the prompts independently, I started to feel more comfortable when planning for the narrative first approach. When using this approach in the classroom, I found it extremely engaging for my students and an amazing way to incorporate literature in the maths classroom.”

 “Some texts lend themselves to it better than others, but once you've done one or two lesson plans, you can see possibilities all over the place.”

The teachers involved also spoke about how the use of a picture story book helped particular students who usually find problem solving especially difficult.

“I noticed that some of my students who have difficulties applying skills to problem solving found it easier when the problem came from a story, so I think that additional context is really supportive for some students.” 

 “In one particular session, using ‘Have You Seen Elephant?’ by David Barrow, a student who at times found problem solving quite challenging, flourished as he used life knowledge to help solve the problem.  When confronted with the problem, many students looked to the height of the boy to help solve the problem, this student used his knowledge about his backyard at home. As he had understanding of the fence height, he was able to transfer this to understand the scale.”

 Finally, teachers also commented on the positive effect that this approach had on the books that their children selected to read.

“I also noticed an effect the other way, that using picture story books in maths lent those books a kind of gravitas.  Some students who normally would dismiss them as 'too easy' or have that snobbishness about them actually began to ask to read those books during independent reading time. Almost like it legitimised picture story books because maths is important and weighty, so now the books are too.”

In summary, I thought this unit was really successful and I would encourage other teachers to try this approach with their own classes.  More information on the “Narrative-First Approach” can be found at bit.ly/narrativefirst

(This article originally appeared in Prime Number- Term 2, 2019.)

Mathematics in Popular Culture

By Michael Minas

At the start of the 2018 school year, I realised that I had a major problem on my hands.  One of the first things I did with my new class of year 5/6 students was a getting-to-know-you activity, where they were asked to respond to a range of questions.  To my surprise, when the students were asked “What is one thing you don’t like about school?”, five of the 24 responses were primarily related to mathematics. 

The troubling realisation that almost a quarter of my new class seemed to be disengaged from my favourite subject was later confirmed when I asked the students to complete an attitude to maths survey.  After analysing the results of this survey, I discovered that 21% of my class said that they did not like studying mathematics at school and 25% had a negative self-perception about their ability to do well in the subject.  When the results were split along gender lines, it was even more alarming.  The female students in my class were twice as likely to provide negative responses as their male counterparts when asked whether they enjoyed maths.    

Considering the important role that student mindset plays in influencing learning outcomes, it was clear that these results were of great concern.  I immediately began contemplating what steps I could take to improve my students’ opinion of both the study of numeracy, as well as their view of themselves as capable mathematicians. 

From my own experience, as both a classroom teacher and numeracy coach, I knew that there were two groups of people who were likely to be heavily influencing my students’ view of mathematics- their immediate family members and their teachers.  Fortunately, I had significant control over one of these factors and was also able to somewhat influence the other, via events such as information evenings and parent/teacher meetings. 

However, there is another factor that exerts great influence over students’ perceptions and opinions of mathematics that teachers have little to no influence over- how the subject is presented in the media.  Over the past few years, I have frequently found myself noticing the negative way that maths is portrayed in children’s books, movies and television shows.  When a writer wants to signify that school is boring, the class in question will undoubtedly be studying mathematics.  Or if a director wants the audience to know that a character is awkward or a bit of a social outcast, he (or theoretically she, but as we are about to discover, it is almost always a he) is likely to be interested in maths. 

These representations made me both angry and upset.  Here I was, trying to show my students that maths was a beautiful, exciting, practical subject, one that they could use to help them understand our complex world and yet this message was being drowned out by authors, directors and publishers.  I decided that it was time to take some action.

Media portrayal of mathematics in “the Big Bang Theory”

At the beginning of term one, I set-up a class investigation where students were asked to be on the lookout for examples of mathematics from books, television shows and movies.  Every time a student came across an example, we made note of it on a large class table that was displayed in the room.  If the example was found in a book, the student would read the relevant passage of text to the rest of the class.  If the example came from a movie or a television show, I attempted to find the clip in question so it could be shared (see Figure 1).  After sharing each example, we added to our class table, recording a summary of how the maths was being used and whether we believed it was a positive or negative representation of the field of mathematics.

While I was planning this investigation, I received Jennifer Hall's article summarising her research into Maths and Mathematicians in Children's Media (Hall, 2018), as part of our research-practice collaboration (a current editorial focus of Prime Number).  After reading Jennifer's work, I became more aware of the issues surrounding gender and mathematics in the media, particularly the lack of positive portrayals of girls and women engaging with mathematics.  This prompted me to add an additional column to the table that my class was going to use to record their findings, one that simply made note of the gender of character/s who are using mathematics. 

At the start of the investigation, the students found the process of locating examples of maths being used in the media quite challenging.  This meant that there were large time lags between the discussions we were having after sharing examples, thus making it difficult for the investigation to gather any momentum.  After a few weeks, I began to supplement the examples the students were providing with some of my own and this seemed to be an effective way to get the class to engage with the investigation. 

By the end of the term, students were coming to school with examples to share on a regular basis.This meant that when the investigation concluded, we had almost 30 examples on our class poster.We then used these to look for trends or patterns that the students thought were interesting.

our class table recording media portrayals of mathematics

When I planned this investigation, I had two very simple aims.  I wanted to develop a greater awareness of how mathematics is depicted in popular culture and also improve the critical thinking skills of my students.  Initially, I thought that the negative manner in which mathematics is portrayed in the media would be the focus of the project.  However, as the term progressed, the students became much more concerned about the under representation of female characters using mathematics in a confident, capable manner and how this would impact on the mindsets of other children, especially girls. 

At the end of the investigation, I asked the class to write reflections, including what they would like to see changed about the manner in which mathematics is portrayed in the media.  Here is a selection of their thoughts:

“I noticed there were more negative views on maths.  This would tell primary students that maths isn’t a good subject, only increasing the level of dislike towards maths.”

-Sasha

“I would demand an increase in the number of female mathematicians in the media because a majority of the time, males are performing complicated mathematical equations whilst females struggle to perform basic mathematics.  This sends an internal message to girls, putting their mindset on a downward slope, when the reality is that we all have the same mathematical capability.”

-Doug

“I think some of the TV shows/movies/books could be quite offensive.  Like on ‘Mean Girls’, the contestants said, ‘Choose the girl,’ like they thought girls are incapable.”

-Matthew

“As a girl, it made me think that girls cannot do math and are not good at it but I love maths and boys are not any better at it, just because they are boys.  Anyone can be good at maths if they try hard.”

-Tayla

“I would change how maths is presented because the media are saying how maths is bad.  It influences kids to think maths is a menace and to have a bad attitude to it, which causes noise, disruption and bad behaviour.”

-Nathan

“If I could change one thing, I would tell the authors, directors, producers and game designers to have more female roles.  Being a female myself, I feel quite upset (disappointed) that people think ‘Oh, girls aren’t good enough to do this, or girls aren’t capable of doing that,’ especially in maths.”

-Hope

I believe this investigation was extremely beneficial, as it helped my students understand how the media can influence not only their view of mathematics and how they see themselves as learners, but also their perceptions of the wider world.  I encourage other teachers to undertake a similar type of investigation with their own classes.  It was easy to set-up and once the investigation was established, it did not require very much class time, especially when completed across an entire term.   I also think this experience is a good example of how teachers engaging with research can shape their attitudes and behaviours, which then impacts the student learning experience.

(This article originally appeared in Prime Number- Term 4, 2018.)

Road Trip Investigation

By Michael Minas

As any classroom teacher can attest to, nothing matches the rewarding feeling you get when you watch students get lost in an activity that they are genuinely engaged in.  And when it comes to maths, children love to immerse themselves in tasks where there is a clear and meaningful connection to the real world that exists outside the four walls of their classroom.

Towards the end of 2017, my family was planning on taking a holiday to Queensland.  As my wife and I sat at our dinner table, debating the merits of driving against the prospect of flying from Melbourne to Brisbane, it occurred to me just how much our conversation involved maths.  When we eventually decided on the road trip option, I seized the opportunity to involve my year 3/4 class in the planning of our upcoming journey.

When I returned to school the following week, I told my students that they would be helping to plan my family’s drive from Melbourne to Brisbane.  The children were placed in pairs and asked to design the best possible road trip, taking into account the cost of petrol and accommodation, the time spent in the car on each day of the journey and the places we were to visit along the way.  They were told that we had to stop and sleep somewhere for a minimum of two nights (thus breaking the trip up into three legs) and a maximum of four nights (or five legs). 

For each leg of the trip they were asked to record:

*Starting and finishing cities/towns.

*Total distance travelled.

*Departure and arrival times, while making allowances to stop for breaks.

*Cost of petrol. 

The students were immensely interested in the challenge in front of them.  One reason for this high level of engagement was the very clear link between the work they were doing and the real adventure about to be undertaken by someone they all knew.  The class was also aware that my two children were aged just three and one at the time of the trip and they used this information to help them plan the journey.  Many pairs made links to their own experiences of driving long distances, to help guide them on what would be the appropriate length of time to spend in the car on each day of the trip.  There were also some students who had younger siblings and this helped them to make judgements about how frequently to stop for breaks.

Once each group had decided on the towns where we were going to stay for overnight visits, they then had to select accommodation.  For this part of the activity, the students used the internet to find hotels, motels and apartments that they thought would be a good fit for my family.  It was quite astounding to see the level of thought and care that they put into this section of the project.  Some groups spent time checking out whether the restaurant at the hotel they had selected catered for vegans, in order to ensure my wife could get a decent meal at the end of a long day of driving, while others were asking me if I thought we would like to go swimming, to work out whether it was worth paying extra for accommodation with a pool.  This part of the activity involved a great amount of discussion and reasoning between the students, as they debated the trade-off between the facilities on offer at the various hotels against the cost of the accommodation.  

Finally, the students were asked to visually represent the journey they had planned on a map of the east coast of Australia and write a persuasive piece, outlining why their particular road trip plan should be the one that my family should follow on our upcoming journey. 

The feedback from the class supported my intuition that this activity was one of the most rewarding tasks we had done during the year.  Below is a selection of their thoughts, taken from some written reflections we did at the end of the activity. 

“I loved the road trip because I was able to choose where my teacher was going to stay and look on great websites such as Trivago and Trip Advisor.” 

Thom

“I liked the road trip project because we got to work on a project that actually amounts to something.  I also liked it because we got to work on it every day.  The other reason why I liked it was because we had independence.” 

Magnus

“I absolutely loved “The Road Trip”, for it gave us a chance to use our planning skills and our maths skills at the same time.  I enjoyed the sessions because me and my partner worked really well.  I had never done something like this before and I utterly loved it.  It was such an original idea, I would love to do it again.” 

Phoebe

When I reflected on why the activity was so popular, it was clear that many of the students loved applying what they had been learning to a meaningful activity with strong links to the outside world.  I believe that opportunities to have students working on similar types of tasks are all around us.  The trick is to recognise when they occur and find ways to use them in the classroom to ignite the interests of your students. 

(This article originally appeared in Prime Number- Term 2, 2018.)

Shark Investigation

By Michael Minas

In term two of 2016, the year 2, 3 and 4 students who I was working with were studying animals.  During this time, they became interested in a number of current events related to this topic.  One of these was the death of a 60 year old diver in Western Australia after a shark attack and the subsequent decision to catch and kill the shark that was responsible.  We watched some news clips about the attack and in one, a police officer reported that the shark in question was estimated to be longer than 5.3 metres.  Given that we were also focussing on length, area and perimeter in our maths classes, it seemed like a good idea to connect the two topics in an attempt to make the learning more meaningful for the students.      

Students were first asked what tool they would use if they wanted to measure the length of a shark.  Numerous suggestions were made, including measuring tapes, rulers and trundle wheels.  The students were then placed in pairs and given a handout with the name, length and picture of a particular species of shark, which they were asked to keep secret from the rest of the class.

Each pair was then instructed to make a model that showed the length of their shark, using scrolls of paper.  They were free to use whatever tool they felt was best suited for measuring their shark accurately.  The lengths of the sharks ranged from 17 centimetres to 12 metres, so a variety of measuring devices were used, with some pairs starting with one tool and then deciding to switch to a more appropriate one midway through the activity.  At the end of the session, the students labelled the scrolls with their own names (e.g. Tom and Magnus’ shark) and they were collected by the teacher.

The next session, each pair was given a scroll that was made by another group, as well as a sheet which showed all of the different types of sharks and their average lengths.  Before doing any measuring, students were asked to estimate the length of their mystery shark and guess what species it might be.  Next, they measured the shark model, using the same range of equipment from the previous day.  Finally, they were asked to record what shark they now thought the model represented, based on their measurement.  Pairs were given a table to record this information and they tried to estimate and measure as many different sharks as they could in the allotted time.

One unintended, positive aspect of the activity was that it emphasised the importance of accuracy when measuring.  Some models were longer or shorter than they should have been, which caused great confusion for pairs who were working with these strips on day two.  This in turn lead to a really good class discussion about how these mistakes occurred and situations where we need to be extremely careful when measuring length.  The students were able to list a variety of examples where inaccurate measurement could lead to problems (e.g. being fitted for a suit or purchasing carpet for their bedroom). 

The level of engagement was very high throughout the two sessions, as the students could see a connection to the real-world and a clear purpose for the activity.  As part of our initial discussion on the moral issues involved in trying to catch and kill sharks, we discovered that only sharks that were longer than 3 metres were eligible to be put down under current West Australian law.  As many students were strongly opposed to this policy, they were extremely interested to discover which sharks could be killed and which breeds were safe. 

At the conclusion of the second session, students were asked to write a reflection on the activity.  Their responses indicated that they found the task interesting because of the clear connection between the maths and a current issue that many of them were passionate about.  A sample of some of the students’ reflections is included below. 

I really enjoyed it because I really like animals and measuring, so combining the 2 was fun.  I also enjoyed that it challenged me because I needed to work on my estimating, so it made me aware of that.  It was hard to make sure that we got it correct and measured exactly the right way. 

Phoebe

I found this activity fun because we got to measure sharks.  It was pretty easy at first, then it got harder because we had to estimate the length.  Me and my partner estimated the length by stepping one step and calling that a metre. 

Lachlan

I liked this activity because we got to learn about different species of sharks.  I thought our first shark was hard because we had to do the biggest shark.  It was almost bigger than the room.

Maddy

This activity was fun because we got to use lots of different equipment, like the metre ruler, measuring tape and also chalk. 

Demi

(This article originally appeared in Prime Number- Term 4, 2016.)