Category Archives: mathematics

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!

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.

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 (http://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.

“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.