Games for teaching and learning Mathematics

We all know children love playing games, but how can we turn this love of games into rich mathematical learning experiences? What are the qualities of a good maths game, and should we be incorporating games into regular lessons and homework rather than a Friday afternoon filler activity?

Why use games in the mathematics classroom? First and foremost, they’re fun! Of course, that alone isn’t a good enough reason to use them. However, when children talk about fun and school, they often perceive fun lessons to be those where they felt challenged and learnt something new. In my research on student engagement, many students talked about fun maths lessons they had experienced, and these are some of their quotes:

“Maths is kind of fun when you get to play some maths games” (Year 6 )

“…if you sit on the carpet and the teacher goes on and on about what we’re learning it gets boring and you get restless so that’s why I like doing fun games.” (Year 6)

“Ms. C was a great maths teacher cause she kept giving us different kinds of g

games that we didn’t do before that’s about maths. But now it’s kind of boring because all we have to do is maths tests, maths stuff, nothing fun about it.”(Year 7)

“I loved maths in primary. I remember how we always had these games and we would rotate.” (Year 8)

I like the iPad games because they are really fun and they make me improve on my maths and I like the maths games that tells you when you are wrong or you are right because if you get it wrong you can improve on that”(Year 4).

A good game provides engagement at cognitive, affective, and operative levels. That is, there must be challenge embedded with the game – if it’s too easy, children will get bored and no learning will occur. The game must be enjoyable to play, and it must promote interaction and dialogue. There are many maths games on the market that are basically drill and practice with the intention of building fluency with number facts. There are also an infinite number of traditional non-maths based games that have a range of mathematical skills and processes embedded within them. The best ones, however, are those that promote the Australian Curriculum: Mathematics proficiencies: Problem Solving, Understanding, Reasoning and Fluency. Take for example, the board game Mabble (photographed). The game requires an understanding of place value and computation, but also requires the players to engage in problem solving and reasoning, while building fluency and demonstrating understanding. Mabble is self-differentiating; meaning anyone of any ability can play successfully. It is also easy to assess students’ work with Mabble as they have to record their work and their scores.

Is it enough to simply allow children to play the games? Definitely not! This is where I get serious. If children play a maths game at school or at home without reflection afterwards, then chances are they have wasted an opportunity for learning. It’s important that children consider the mathematics involved in the game, the challenges that were faced and the strategies that were included. Often we don’t know if children learned anything while playing a game unless we ask some very strategic reflection questions which can be answered verbally or recorded in written form. Here are some examples of good reflection prompts, organised into the cognitive, affective and operative domains of engagement.

Cognitive:

  • Write a memo to someone about the most important mathematics you learned while playing the game.
  • What was the tricky part about the game?
  • What maths strategies did you use to help you play the game?
  • Write two things that were difficult in this game.
  • Can you connect the maths you used in this game to something you already know?
  • Where would this knowledge be useful?

Affective:

  • What were the fun bits in your learning when you played the game?
  • Why do you think the fun bits were fun?
  • How did you feel playing the game with your group?
  • Survey the members of your group about how they felt during the game and align them with your own.

Operative:

  • What were your strengths when playing this game?
  • What is the most valuable advice you could give students who are going to play this game in the future?
  • How could we change this game next time we do this?
  • What would you do differently in your next game given the knowledge you have gained from this game?
  • What did you find out about your problem solving skills and strategies during this game?

And finally, here is a list of some of my favourite games that promote both mathematical processes and content:

The following are some iPad apps that are mathematics based games:

  • 2048
  • Threes
  • Tangram
  • Maya Numbers
  • Banana Hunt
  • Concentration

Of course, there are many more great games for mathematics teaching and learning. The important thing is that we encourage children to engage with them in a meaningful way and provide opportunities for them to reflect on the mathematics and learning involved. If we can do this, games can become part of our everyday routines and even homework tasks, rather than those Friday afternoon time fillers!

 

Woolworths and Dominoes (Part 2): Even more mathematical opportunities for parents and teachers!

My last blog about the marketing promotion being run by Woolworths and Disney Pixar attracted so much interest that I thought I would look deeper into the mathematical potential of the whole campaign. Somehow, the incentive of receiving a domino for every $20 spent seems to be very appealing to consumers, young and old. What is it about these little plastic objects that is so attractive? Perhaps the appealing aspect of the dominoes is the fact that children can actually play with these, as opposed to collections of character cards that are usually given away in such promotions.

So why are the dominoes appealing to teachers like me? My research on student engagement with mathematics has shown that when children have an interest in something, they are more likely to want to learn. They also like to use concrete materials to help them learn – things they can see, touch and manipulate (as opposed to the traditional maths worksheets and textbooks). In the case of Woolworths and dominoes, this is a perfect opportunity for parents and teachers alike to seize this amazing opportunity, take advantage of the hype and do some really good, interesting mathematics!

During the week, as I watched the statistics on my blog increase, I thought I would explore the Woolworths web site and dig around a little. I didn’t find too much of interest, although they have made an effort to publish some very basic educational ideas relating to the dominoes. What I did find, however, was that people are actually selling dominoes on eBay! You can buy whole sets (of characters), individual dominoes of specific characters (some up to $3 each), or unopened dominoes. At this point my head started to hurt…..so many mathematical possibilities! Imagine children investigating the cost of dominoes (in shopping dollars), compared to the apparent worth of dominoes as advertised on eBay. All week I have had fantastic (well, I think they’re fantastic) ideas popping into my head, and these are a few that you might want to try out at home (if you are a parent), or at school, if you are a teacher. I will begin my list with simple tasks for younger children, and finish it with more complex tasks for older children:

  • How many dominoes do you think you could hold in one hand? Try it and see if you were right or wrong. How close were you? What if you could use two hands? How many dominoes can you hold? Is this the same as an adult?
  • How many dominoes have a one dot? Two dot? Three dot pattern?
  • If I lay my dominoes flat, end to end (the short end), how long will my line be? How many dominoes will I need if I wanted to make a flat line that is as long as my foot? My leg? My arm? My body?
  • Keep your character doubles, and use pairs of doubles to play a game of memory.
  • Using the picture side of the dominoes (the characters are numbered), order the dominoes from 1 to 44.
  • Are you missing any dominoes? What numbers are missing and how do you know?
  • Using the picture side of the dominoes, imagine that the number of the character is equivalent to its worth. That is, character number 1 is worth $1, character number 2 is worth $2, etc. What would be the value of your collection? If you had every domino from number 1 to number 44, what is it worth?
  • If I lined up my dominoes so they were standing (like in the photo), what would be the best distance apart (if they’re too close together, you might knock them down accidently).
  • How many (standing) dominoes would you need to make a line of 1 metre? Imagine you needed to make a domino line for one kilometre – can you use the number of dominoes you have to work out how many dominoes you would need? How much would you have to spend at Woolworths to have enough dominoes?
  • How long would it take to knock down a one metre line of standing up dominoes? Who can make the longest line?
  • I received 18 dominoes with my shopping this week. How much did I spend?
  • Do you think the Woolworths marketing campaign has been successful? Design a set of survey questions and conduct some research at your school. Analyse your data and prepare a report that you could send to the Chief Executive Officer of Woolworths.

Of course, there are many more ideas – perhaps there will be a Part 3 blog post over the Easter weekend. Oh, and by the way, Woolworths are giving away ‘double’ dominoes at the moment – this opens up another world of mathematical opportunity!

Woolworths and Dominoes: How parents can turn marketing ploys into mathematical learning opportunities

At the moment across Australia Woolworths stores are giving away one ‘free’ Disney Pixar Domino with every $20 spent. This is not the first of this kind of promotion, however this particular one took my interest because of my love of dominoes. Last weekend I couldn’t wait to do my weekly shopping at Woolworths so I could check out this domino deal. The child in me wanted to collect as many dominoes as I could, while the adult in me realised what a clever marketing ploy this was. Surprise, surprise, I had the option of purchasing a whole range of accessories to go with my dominoes – a total of 35 different items (which I resisted).

The dominoes have the usual dot arrangement on one side, while on the other side, they have a picture of a Disney Pixar character. The interesting thing is that although there are 28 dominoes in a complete set (double six dot), there are a total of 44 characters to collect….anyone who knows me will know that this doesn’t sit well with me! How can this work? My guess is that young children collecting the dominoes will probably be more focused on the character side rather than the domino side, which is a real shame.

When I got home from shopping last week I sat down to open the individually wrapped dominoes to find out which ones I had collected (I was still wondering how this would work with 44 characters). Out of the 11 dominoes I had collected, three had the same domino arrangement (5 and 3), yet each had a different character on the back. Now, hopefully you’ll know where this is leading….I started to think about the mathematical potential of these dominoes and how perhaps families could take advantage of this marketing ploy to encourage children to ‘play’ with mathematics.

So, what kind of cool ‘mathy’ things could you do with the dominoes? Here are a few things that have popped into my mind, but I’m sure there are many more:

  • Order the dominoes from the smallest total to the highest total (or back to front)
  • How many combinations of dominoes add to….(pick a total – being aware that some totals have more combinations than others)
  • How many dominoes will you get if you spend……
  • Are there any strategies you can use to increase the number of dominoes you get?
  • Play a game of dominoes (great for young children practicing matching and subitising)
  • How many different ways can you make a total of 15 out of two dominoes? What about a total of 20?
  • Estimate how long it will take to collect a full set of 28 dominoes
  • Turn your dominoes face down (picture face up), choose three dominoes and arrange them so that when you add them, the total is as close to 100 as possible. How close did you get?
  • Make the longest possible domino train that adds up to a total of 15/20/25
  • Predict which dominoes you will get the next time you go shopping. How did you make your prediction?
  • Work out which dominoes you need to have a complete set
  • Sort your dominoes (any way you like) and ask someone if your family to work out how you sorted them

Those are just a few suggestions – there are many mathematically based ideas for dominoes. If you are a teacher and looking for ideas, Paul Swan has a great book called Domino Deductions, and there are also some ideas in my book Engaging Maths: Exploring Number. Don’t forget there are also mathematical opportunities relating to the other side of the dominoes as well.

If you are a parent, I encourage you to take this great opportunity and make the most if it – store promotions are meant to encourage spending, but if you are going to spend the money anyway, you might as well make the experience educational!

 

These are a few of my favourite things: Essential materials for every maths classroom

What concrete materials do you have in your mathematics cupboard and why bother investing in concrete resources? Concrete materials provide opportunities for children to construct rich understandings of mathematical concepts. In addition, providing opportunities for children to physically engage with materials is much more meaningful than working with drawn or even digital representations. For example, if you are teaching students concepts relating to 3-dimensional space, it makes sense that it is better for children to be able to manipulate objects in order to explore their properties and relate their learning to real-life. Concrete materials also promote the use of mathematical language, reasoning, and problem solving.

I often get asked about the essential resources required for primary mathematics classrooms. There are quite a few, but if you have a limited budget or space, there are a few resources that are what I would consider to be essential, regardless of the year level that you are teaching. My advice would be to invest in materials that are flexible and able to be used in a variety of ways, perhaps in conjunction with other materials. Also consider collecting things that are not necessarily intended as educational resources but may have some mathematical value, such as collections of things (keys, lids, plastic containers, etc.) for activities that require sorting and classifying. Here is a list of basics that can be purchased from educational resources suppliers (some of the items can also be sources at normal retail and/or discount stores):

  • Counters
  • Dice (as well as the standard six sided dice, you could purchase many other variations including blank dice)
  • Calculators (yes, these are great, even in the early years. Think about using them to investigate numbers rather than simply computational devices)
  • Base 10 material (be careful how you ‘name’ these – using terms like ones, tens, hundreds and thousands limits their use. It is best to use the terms minis, longs, flats and blocks so they can be used flexibly to teach a range of whole number and measurement concepts)
  • Pattern blocks (great for more than just exploring 2D shape – these can be used to teach fractions, place value, area, perimeter etc.)
  • Dominoes (one of my truly favourite things!)
  • Playing cards
  • Unifix blocks

Of course, any resource is only as good as the teacher using it and the way it is integrated into teaching and learning. Prior to using any concrete material, think about the purpose of the lesson and the mathematical concepts being taught. Also consider how you can make the most out of the resources – how will you differentiate the task, and how will you capture evidence of learning? This is where technology can play a useful role and allow teachers and students to capture evidence when working with concrete materials. Technology can also be used alongside concrete materials. For example, work with pattern blocks can be recorded using the Pattern Block App on an iPad. Or students could integrate their use of concrete materials with a verbal reflection or explanation using the Explain Everything app.

The best way to get the most out of concrete materials is to research. There are many high quality resource books and there are also many great websites such as NCTM Illuminations that provide excellent teaching ideas. Once you see the potential of high quality, flexible concrete materials such as those in the list above, you and your students will become much more cognitively, affectively and operatively engaged with mathematics.

“I wonder…..”: Promoting curiosity in the mathematics classroom

I recently came across an article published in the neuroscience journal, Neuron that caught my attention. The article, by Gruber, Gelman and Ranganath (2014), describes a scientific investigation that explored how curiosity influences memory. The authors found a “link between the mechanisms supporting extrinsic reward motivation and intrinsic curiosity and highlight the importance of stimulating curiosity to create more effective learning experiences” (p. 486). In other words, students will learn more about topics they are interested in – something we’ve known along in the education world, but now we have scientific evidence!

Gruber et al. (2014) claim high curiosity results not only in the learning of interesting information but also incidental material. They also discuss how most of the events a person experiences in a day will be forgotten. If we translate this to children and their classroom experiences, can we expect that they won’t remember much of what happens during the average school day? This certainly presents a strong argument against the use of traditional approaches to teaching and learning, particularly the use of textbooks. How can we expect children to get excited and curious about mathematics from a worksheet? We need to ensure we find ways to ‘hook’ students into mathematics and provide opportunities for them to experience the joy of mathematical exploration and discovery.

So what kind of mathematics tasks and activities could be used in the primary classroom to promote curiosity? We know that the teacher is the biggest influence on student engagement with mathematics, and I firmly believe that curiosity is something that must be modeled by the teacher. There are many types of activities that would assist in promoting curiosity amongst students. For example, mathematical magic tricks, or ‘mathemagic’ is a great place to start.

Here’s one (it’s a favourite of mine) that uses three dice:

This trick is based on a simple mathematical fact: Each pair of opposite faces on a six-sided die always adds up to seven. All you need for this trick is three six-sided dice and basic multiplication, addition and subtraction skills! If you’ve got that, you’re ready for the trick.

Instructions:

  • Hand a student the three dice and ask he or she to stack them together so that they form a column
  • Turn your back to the student while he/she silently adds up the numbers on the five hidden dice faces. Tell your student to memorise the sum and keep it a secret.
  • When three dice are stacked together there are five faces that you can’t see: the bottom and top faces of the lowest die, the top and bottom faces of the middle die and the bottom face of the top die. Altogether you get five hidden faces.
  • When your student is ready and has figured out the sum of the numbers on the five hidden faces, you can turn around. Tell him/her that you will use your magical powers to name the sum of the five hidden faces, without looking.
  • Look at the top face of the stacked column, and subtract the number from 21 (For example, if the top number is 3, subtract three from 21) “Abracadabra, the sum is 18!”

When students (and most adults) first see this trick performed, they are amazed. Perform it a couple of times to prove that you are, indeed, magical, before asking them to explore how the trick works. Non-threatening, engaging activities such as this not only spark curiosity, they provide opportunities for mathematical discussion, reasoning, and generalising. An added bonus is that when students ‘get’ the trick, they feel empowered because they can go home and trick their families and friends!

Other activities that promote curiosity include explorations of magic squares, investigating number patterns, which can be as simple as using ten-point circles to explore the patterns with the multiplication tables or simply asking questions that begin with “I wonder …” about some of the day to day contexts that students find themselves in.

There are endless ways that teachers can arouse mathematical curiosity in their students and many resources, educational and otherwise, that could be used. Consider using picture books, non-fiction books such as the Guinness Book of Records, puzzles, video clips, and the list goes on. Anything that gets children interested in mathematics and encourages them to continue with and be successful in the study of mathematics has to be a good thing!

Gruber, M. J., Gelman, B. d., & Ranganath, C. (2014) States of curiosity modulate hippocampus-dependent learning via the dopaminergic circuit. Neuron, 84(2), 486-496.

Are you a beginning teacher? What’s in your maths toolbox?

Very recently one of my children began a career as a primary teacher. Like most early career teachers, she has had to begin working as a casual relief teacher. Fortunately for her, she has a ready supply of resources and mathematics activities (thanks to Mum) for those days when she walks into a classroom and has to deliver a day full of engaging activities. However, many teachers who are starting out have to build their toolbox of resources from nothing. Where do you begin? How can you develop a bank of activities that suits lots of different levels and abilities, and engages children you may never have met before?

One of the first things I would recommend would be to invest in a small range of materials that allow you to implement some simple tasks that could then be expanded into interesting and worthwhile mathematical investigations. For example, if you purchase around ten sets of playing cards (go to a cheap two dollar store), you could learn a few basic games (Snap, Making 10, Playing with Place Value – see my book Engaging Maths: Exploring Number) that could then be differentiated according to the students you are teaching. A simple game of Making 10 could be used from Grade 1 all the way to Grade 6 by simply changing the rules.

Other materials that are a ‘must have’ for beginning teachers are dice and dominoes. There are many simple investigations that could lead from simple explorations with these materials. For example, use the dice to explore probability or play a game of Greedy Pig. Play a traditional game of dominoes before adding a twist to it, or simply ask students to sort the dominoes (students have to select their own criteria for sorting)– an interesting way to gain insight into students’ mathematical thinking and a great opportunity for using mathematical language. Once students have sorted the dominoes conduct an ‘art gallery tour’ and ask other students to see if they can work out how others have sorted out their dominoes. Photograph the sorting and display then on an Interactive Whiteboard for a whole class discussion and reflection…the list goes on!

Another ‘must have’ for beginning teachers is a bank of good quality resource books. Don’t fall into the trap of purchasing Black Line Masters or books full of worksheets to photocopy. You don’t want your students to be disengaged and you want to be called back for more work! Books such as my Engaging Maths series (https://engagingmaths.co/teaching-resources/books/ ), or any of Paul Swan’s books or resources (http://www.drpaulswan.com.au/resources/) are a great place to start. Explore some of the excellent free resources available online such as http://nrich.maths.org/teacher-primary and http://illuminations.nctm.org/, but do be aware that some resources produced outside of Australia will need to adapted for the Australian Curriculum: Mathematics.

In my early research on student engagement, I found that students would remember what they would recall as a ‘good’ mathematics lesson for a very long period of time. In fact, some of the students in my PhD study talked about a ‘good’ mathematics lesson two years after it had taken place. Although you might only be in a classroom for a very short time while you begin your career as a relief teacher, you can make an impact on the students in your care and the way the view mathematics by being prepared with your ‘toolbox’ of engaging and worthwhile activities.

Professional Learning and Primary Mathematics: Engaging teachers to engage students

The issue of student engagement with mathematics is a constant topic of discussion and concern within and beyond the classroom and the school, yet how much attention is given to the engagement of teachers? I am a firm believer that one of the foundational requirements for engaging our students with mathematics is a teacher who is enthusiastic, knowledgeable, confident, and passionate about mathematics teaching and learning – that is, a teacher who is engaged with mathematics. Research has proven that the biggest influence on student engagement with mathematics is the teacher, and the pedagogical relationships and practices that are developed and implemented in day to day teaching (Attard, 2013).

A regular challenge for me as a pre-service and in-service teacher educator is to re-engage teachers who have ‘switched off’ mathematics, or worse still, never had a passion for teaching mathematics to begin with. Now, more than ever, we need teachers who are highly competent in teaching primary mathematics and numeracy. The recent release of the Teacher Education Ministerial Advisory Group (TMAG) (2014) report, Action Now: Classroom Ready Teachers, included a recommendation that pre-service primary teachers graduate with a subject specialisation prioritising science, mathematics, or a language (Recommendation 18). In the government’s response (Australian Government: Department of Education and Training, 2015), they agree “greater emphasis must be given to core subjects of literacy and numeracy” and will be instructing AITSL to “require universities to make sure that every new primary teacher graduates with a subject specialisation” (p.8). While this is very welcome news, we need to keep in mind that we have a substantial existing teaching workforce, many of whom should consider becoming subject specialists. It is now time for providers of professional development, including tertiary institutions, to provide more opportunities for all teachers, regardless of experience, to improve their knowledge and skills in mathematics teaching and learning, and re-engage with the subject.

So what professional learning can practicing teachers access in order to become ‘specialists’, and what models of professional learning/development are the most effective? Literature on professional learning (PL) describes two common models: the traditional type of activities that involve workshops, seminars and conferences, and reform type activities that incorporate study groups, networking, mentoring and meetings that occur in-situ during the process of classroom instruction or planning time (Lee, 2007). Although it is suggested that the reform types of PL are more likely to make connections to classroom teaching and may be easier to sustain over time, Lee (2007) argues there is a place for traditional PL or a combination of both, which may work well for teachers at various stages in their careers. An integrated approach to PD is supported by the NSW Institute of Teachers (2012).

In anticipation of the TMAG recommendations for subject specialisation, I have been involved in the design and implementation of a new, cutting edge course to be offered by the University of Western Sydney, the Graduate Certificate of Primary Mathematics Education, aimed at producing specialist primary mathematics educators. The fully online course will be available from mid 2015 to pre-service and in-service teachers. Graduates of the course will develop deep mathematics pedagogical content knowledge, a strong understanding of the importance of research-based enquiry to inform teaching and skills in mentoring and coaching other teachers of mathematics. For those teachers who are hesitant to commit to completing a full course of study, the four units of the Graduate Certificate will be broken up into smaller modules that can be completed through the Education Knowledge Network (www.uws.edu.au/ekn) from 2016 as accredited PL through the Board of Studies Teaching and Educational Standards (BOSTES).

In addition to continuing formal studies, I would encourage teachers to join a professional association. In New South Wales, the Mathematical Association of NSW (MANSW) (http://www.mansw.nsw.edu.au) provides many opportunities for the more traditional types of professional learning, casual TeachMeets, as well as networking through the many conferences offered. An additional source of PL provided by professional associations are their journals, which usually offer high quality, research-based teaching ideas. The national association, Australian Association of Mathematics Teachers (AAMT) has a free, high quality resource, Top Drawer Teachers (http://topdrawer.aamt.edu.au), that all teachers have access to, regardless of whether you are a member of the organisation or not. Many more informal avenues for professional learning are also available through social media such as Facebook, Twitter, and Linkedin, as well as blogs such as this (engagingmaths.co).

Given that teachers have so much influence on the engagement of students, it makes sense to assume that when teachers themselves are disengaged and lack confidence or the appropriate pedagogical content knowledge for teaching mathematics, the likelihood of students becoming and remaining engaged is significantly decreased, in turn effecting academic achievement. The opportunities that are now emerging for pre-service and in-service teachers to increase their skills and become specialist mathematics teachers is an important and timely development in teacher education and will hopefully result in improved student engagement and academic achievement.

M

References:

Attard, C. (2013). “If I had to pick any subject, it wouldn’t be maths”: Foundations for engagement with mathematics during the middle years. Mathematics Education Research Journal, 25(4), 569-587.

Australian Government: Department of Education and Training (2015). Teacher education ministerial advisory group. Action now: Classroom ready teachers. Australian Government Response.

Lee, H. (2007). Developing an effective professional development model to enhance teachers’ conceptual understanding and pedagogical strategies in mathematics. Journal of Educational Thought, 41(2), 125.

NSW Institute of Teachers. (2012). Continuing professional development policy – supporting the maintenance of accreditation at proficient teacher/professional competence. . Retrieved from file:///Users/Downloads/Continuing%20Professional%20Development%20Policy.pdf.

Teacher Education Ministerial Advisory Group (2014). Action now: Classroom ready

Teachers.