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.

“If you like the teacher, you’ll ‘get’ maths more”: Students talk about good mathematics teachers

This post was originally published in 2010 on the UWS 21st Century Learning Blog. The discussion relates to my PhD research on the influences on student engagement in maths during the middle years of school and findings have subsequently been published in academic journals (see, for example, Attard 2011, 2012). I thought it would be interesting to revisit since things don’t seem to have changed much in relation to the issue of students ‘turning off’ maths.

Many students during the middle years of schooling (Year 5 to Year 8 in New South Wales) are experiencing emotional, social, physical, and cognitive changes that must be dealt with in the mathematics classroom. Mathematics curriculum and instruction must address the particular needs of these students because so many jobs and indeed the demands of everyday living now and in the future, require complex mathematical thinking. Over the last 20 years research has overwhelming documented an increasingly smaller percentage of students pursuing the study of mathematics at upper secondary level and beyond. The choice not to pursue mathematics has been seriously influenced by students’ attitudes towards and performance in mathematics, in turn deeply shaped by school mathematical experiences and the teaching they experienced in school (Nardi & Steward, 2003).

So, what makes a good mathematics teacher? There are several frameworks that address ‘good’ teaching including the Quality Teaching Framework (NSW Department of Education and Training, 2003) and the Standards for Excellence in Teaching Mathematics in Australian schools (Australian Association of Mathematics Teachers [AAMT], 2006). But how do the frameworks compare to what students think about the qualities of a good mathematics teacher? My PhD thesis was a longitudinal study on engagement in middle years mathematics and early in the study I asked a group of 20 Year 6 students at a Western Sydney school to name the qualities that make a ‘good’ mathematics teacher. The students perceived a good maths teacher to be someone who:

  • is passionate about teaching mathematics;
  • responds to students’ individual needs;
  • gives clear explanations;
  • uses scaffolding rather than providing answers;
  • encourages positive attitudes towards mathematics; and
  • shows an awareness of each students’ prior knowledge.

The study followed the same group of students through their transition to high school, and into Year 8. During their time in secondary school, the students’ experiences included a wide range of practices and teachers, and significant exposure to technology within the mathematics classroom (a one-to-one laptop program). Despite being exposed to an integrated curriculum and a school that was purpose built to cater for ‘next-practice’ learning and teaching, it was the teachers and the relationships that were developed within the classroom that had the most significant impact on student engagement in mathematics. It appeared that the introduction of technology during Year 7 had removed many of the opportunities for student/teacher and student/student interaction that are such an integral aspect of learning mathematics. During their time in Year 7 the students experienced lowered engagement as a result.

Two years after the study began, when the students were in Year 8, their secondary school underwent some significant changes in terms of its curriculum delivery (no longer integrated) and the use of technology in the mathematics classrooms. There was significantly less reliance technology and a much heavier emphasis on direct instruction. The students began to build relationships with their teachers and in turn, this saw their engagement in mathematics begin to build. The students spoke about how they now felt their teachers ‘cared’ about them and ‘knew’ them. This comment from one of the students indicates the importance of positive student/teacher relationships: “if you like the teacher, you’ll get maths more. You’ll know what’s going on more.”

Although some of the pedagogies these students experienced during the study were not necessarily considered ‘best practice’, it appears the students were able to overcome this where it was difficult for them to overcome the lack of positive interactions with some of their mathematics teachers. It is proposed that regardless of the school context, students in the middle years have a need for positive teacher-student and student-student relationships as a foundation for engagement in mathematics. This relationship is built on an understanding of students and their learning needs. Unless such a relationship exists, other pedagogical practices including the use of technology may not sustain engagement in mathematics during the middle years.

Attard, C. (2011). “My favourite subject is maths. For some reason no-one really agrees with me”: Student perspectives of mathematics teaching and learning in the upper primary classroom. Mathematics Education Research Journal, 23(3), 363-377.

Attard, C. (2012). The influence of pedagogy on student engagement with mathematics during the middle years of schooling. In A. L. White & U. H. Cheah (Eds.), Transforming School Mathematics Education in the 21st Century (pp. 140-157). Penang: SEAMEO RECSAM.

Association of Mathematics Teachers [AAMT]. (2006). Standards of Excellence in Teaching Mathematics in Australian Schools. Adelaide: Australian Association of Mathematics Teachers.

Nardi, E., & Steward, S. (2003). Is mathematics T.I.R.E.D? A profile of quiet disaffection in the secondary mathematics classroom. British Educational Research Journal, 29(3), 345-367

NSW Department of Education and Training. (2003). Quality Teaching in NSW Public Schools. Sydney: Professional Support and Curriculum Directorate.

Mathematics, technology, and 21st Century learners: How much technology is too much?

On a recent visit to a shopping centre in Sydney, I noticed a new children’s playground had been installed. On closer inspection I was amazed to find a cubby house structure that had a number of iPads built into it. There was also a phone charging station built less than a metre off the ground, for users of the playground to access. The playground had obviously been designed for very young children. So what’s the problem? Shouldn’t playgrounds be meant to promote physical activity? What messages are the designers of this playground sending to children and their parents? Does technology have to pervade every aspect of our lives? What damage is this doing to children’s social and physical skills?

While considering the implications of this technology-enhanced playground, I began to reflect on the ways we use technology in the classroom. Is there such as thing as having too much technology? I am a strong supporter of using technology to enhance teaching and learning, and I know there are a multitude of benefits for students and teachers, particularly in relation to the use of mobile technologies (Attard 2014, 2013). However, there are issues and tensions. How do we, as educators, balance the use of technology with what we already know works well? For example, in any good mathematics classroom, students would be manipulating concrete materials to assist in building understandings of important mathematical concepts. Children are engaged in hands-on mathematical investigations and problem solving, arguing, reasoning and communicating through the language of mathematics. Can technology replace the kinesthetic and social aspects of good mathematics lessons? How do we find the right balance? Do students actually want more technology in the classroom, or do they prefer a more hands-on and social approach?

Often we use technology in the classroom to bridge the ‘digital divide’ between students’ home lives and school. We know this generation have access to technology outside the school, and we often assume that students are more engaged when we incorporate digital technologies into teaching and learning. In the The App Generation, Gardner and Davis (2013) discuss how our current generation relies on technology in almost every aspect of their lives. They make some important points that can translate to how we view the use of the technology in the classroom, “Apps can make you lazy, discourage the development of new skills, limit you to mimicry or tiny trivial tweaks or tweets – or they can open up whole new worlds for imagining, creating, producing, remixing, even forging new identities and enabling rich forms of intimacy” (p. 33).

Gardner and Davis argue that young people are so immersed in apps, they often view their world as a string of apps. If the use of apps allows us to pursue new possibilities, we are ‘app-enabled’. Conversely, if the use and reliance on apps restrict and determine procedures, choices and goals, the users become ‘app-dependent’ (2013). If we view this argument through the lens of mathematics classrooms, the use of apps could potentially restrict the learning of mathematics and limit teaching practices, or they could provide opportunities for creative pedagogy and for students to engage in higher order skills and problem solving.

So how do educators strike the right balance when it comes to technology? I often promote the use of the SAMR model (Puentedura, 2006) as a good place to start when planning to use technology. The SAMR model (Puentedura, 2006) represents a series of levels of “incremental technology integration within learning environments” (van Oostveen, Muirhead, & Goodman, 2011, p. 82). However, the model is not without limitations. Although it describes four clear levels of technology integration, I believe there should be another level, ‘distraction’, to describe the use of technology that detracts from learning. I also think the model is limited in that it assumes that integration at the lower levels, substitution and augmentation, cannot enhance students’ engagement. What is important is the way the technology is embedded in teaching and learning. Any tool is only as good as the person using it, and if we use the wrong tool, we minimise learning opportunities.

Is there such a thing as having too much technology? Although our students’ futures will be filled with technologies we haven’t yet imagined, I believe we still need to give careful consideration to how, what, when and why we use technology, particularly in the mathematics classroom. If students develop misconceptions around important mathematical concepts, we risk disengagement, the development of negative attitudes and students turning away from further study of mathematics in the later years of schooling and beyond. As for the technology-enhanced playground, there is a time and a place for learning with technology. I would rather see young children running around, playing and laughing with each other rather than sitting down and interacting with an iPad!

References:

Attard C, 2014, iPads in the primary mathematics classroom: exploring the experiences of four teachers in Empowering the Future Generation Through Mathematics Education, White, Allan L., Tahir, Suhaidah binti, Cheah, Ui Hock, Malaysia, pp 369-384. Penang: SEMEO RECSAM.

Attard, C. (2013). Introducing iPads into Primary Mathematics Pedagogies: An Exploration of Two Teachers’ Experiences. Paper presented at the Mathematics education: Yesterday, today and tomorrow (Proceedings of the 36th Annual conference of the Mathematics Education Research Group of Australasia), Melbourne.

Gardner, H, & Davis, K. (2013). The app generation. New Haven: Yale University Press.

Puentedura, R. (2006). SAMR.   Retrieved July 16, 2013, from www.hippasus.com

van Oostveen, R, Muirhead, William, & Goodman, William M. (2011). Tablet PCs and reconceptualizing learning with technology: a case study in higher education. Interactive Technology and Smart Education, 8(2), 78-93. doi: http://dx.doi.org/10.1108/17415651111141803