Tag Archives: Mathematics education

Programming & planning dilemmas in primary mathematics

Often when I work with teachers I am asked for advice regarding the design of a scope and sequence for mathematics. The programming and planning of mathematics seems to cause much concern, and often the reason is that there is no ‘magic fix’ or one-size-fits-all solution.

Traditionally, schools have planned their mathematics teaching using a topic-by-topic or strand-by-strand approach. Sometimes there is a formula for teaching the Number and Algebra strand for a certain number of days per week, with the other days dedicated to the remaining syllabus strands. Often, the strands are split into single, stand-alone topics. Unfortunately, there are issues with this approach. Teaching individual topics in mathematics hinders students in gaining a deep understanding of mathematics and the connections that exist between and amongst the strands. Teaching in this way can promote a traditional, rote learning approach where the opportunities for mathematical thinking are limited. Our curriculum places the proficiencies (Working Mathematically in New South Wales) at the forefront of teaching and learning mathematics – teaching topics in isolation does not promote the proficiencies.

So what’s the solution? Consider planning and programming using a ‘big idea’ approach. What’s a big idea? Big ideas are hard to define and different people have differing ideas on what the big ideas in mathematics actually are. However, all the definitions in literature have one thing in common – they all refer to big ideas as the key to making connections between mathematical content and mathematical actions, and they all link mathematical concepts. Take, for example, the big idea of equivalence. This relates to number and numeration, measurement, number theory and fractions, and algebraic expressions and equations. Connections can be made across the strands and these links should be made explicit to students.

Charles (2005) presents a total of 21 big ideas across the mathematics curriculum, however he states that these are not fixed – they can be adapted. He also states that a big ideas approach has implications for curriculum and assessment and professional development – teachers need to develop their pedagogical content knowledge to ensure they have a deep understanding of the connections within the curriculum if they are to teach mathematics successfully.

Of course, there are challenges to teaching using a big ideas approach. Teachers often feel under pressure to address all curriculum outcomes, and often this is the reason that the topic-by-topic approach is adopted. Using a big ideas approach can feel messy – it is not linear and in some ways feels as though it is conflicting with the organisation of our curriculum. However, we must remember that although our curriculum is separated into strands and sub-strands, this is simply an organisational tool and does not mean that mathematics should be taught in this same way.

My advice would be to take our curriculum, pull it apart and try seeing it differently – what areas of the curriculum have obvious links? How can you link aspects of measurement to the number strand? Where does measurement and geometry link? And how can you use the statistic and probability strand to teach number concepts? Making connections will make your teaching easier in the long run, and more importantly, will result in deeper learning and deeper engagement with mathematics.

 

Randall, C. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. NCSM Journal, 7(3), 9-24.

 

Thanks for the iPads, but what are we supposed to do with them?

This blog was originally posted back in November 2012, on the UWS 21st Century Learning site. It was written when iPads began to appear in schools. We’ve come a long way since then in terms of the increasing popularity of iPads and other tablet devices. However, I wonder how much has changed in relation to the way they are being used to teach and learn primary mathematics? I thought it would be interesting to revisit this post, so I have adapted it slightly to contextualise it into 2015.

The fast pace of technology development has seen a rapid uptake in mobile technologies such as the iPad computer tablet. Although not originally intended for use within educational settings when introduced in 2010, the iPad has fast become the ‘must have’ item in today’s classrooms.

One result of this is that teachers are often expected to integrate iPads or similar technologies into teaching and learning without the support of appropriate professional development, particularly in relation to using the technology to enhance teaching, learning and student engagement. While some claim iPads and other similar mobile devices have the potential to revolutionise classrooms (Banister, 2010; Ireland & Woollerton, 2010; Kukulska-Hulme, 2009), there is still little research informing teachers exactly how the iPads can be integrated to enhance learning and teaching, and whether their use will have a long-term positive impact on student learning outcomes.

So what do we do when we are given a set of iPads and told to use them in our classrooms? Early during the iPad ‘revolution’ I conducted two research projects investigating how iPads were being used to teach and learn mathematics in primary classrooms. These projects gave me the opportunity to observe a variety of pedagogies and make some interesting observations regarding practical issues relating to the management of iPads.

In each of the projects, teachers had been provided with iPads for their classrooms with little or no professional development that related to integration into teaching and learning practices. The teachers involved experienced a ‘trial and error’ process of using different strategies to integrate the iPads into their mathematics lessons, a task they found harder to do than with other subject areas. The iPads were used in a wide variety of ways that appeared to have differing levels of success. The success of each lesson was determined by the observed reaction to and the engagement of the students with the set tasks and the teacher’s reflection following the lesson.

Several lessons that incorporated iPads utilised a small group approach where students worked either independently or in small groups of two to three students on an application that was based upon the drill and practice of a mathematical skill. The challenge with this approach was that it was difficult for the teacher to know whether the students were on task, if there were any difficulties, and whether the chosen application was appropriate in terms of the level of cognitive challenge. Often when this pedagogy was implemented it was done so without student reflection at the conclusion of the lesson. Without discussion of the mathematics involved in the task, students did not have the opportunity to acknowledge any learning that occurred.

The pedagogies that appeared most effective were those that were based on using the technologies to solve problems in real-world contexts. When used this way, the iPads were used as tools to assist in achieving a set goal, rather than as a game. An example of one of these lessons was in Year 5, when students were asked to plan a hypothetical outing to the city to watch a movie. The children were able to use several applications on their iPads ranging from public transport timetables to cinema session time applications to plan their day out. The lesson resulted in rich mathematical conversations and problem solving, and high levels of engagement due to the real-life context within which the mathematics was embedded.

The integration of interactive whiteboards with iPads was also a common element in the observed lessons, illustrating how such technologies can enhance teaching as well as learning. In several instances teachers projected the iPads onto interactive whiteboards to demonstrate the tasks set for the students. In other examples, it was the students’ work on the iPads that was projected for the purpose of class discussions and constructive feedback.

The variety of ways in which the technologies were used demonstrated their flexibility when compared to traditional laptop or desktop computers. All of the teachers involved in both projects found it challenging to integrate the technologies into mathematics in contrast with other subject areas such as literacy.

This challenge led to the teachers expressing a need for professional development in relation to integrating the iPads into existing pedagogical practices and a desire to have a platform from which ideas can be shared amongst peers. The incorporation of the iPads led to the teachers becoming more creative in their lesson planning and as a result, tasks became more student-centred and allowed time for students to investigate and explore mathematics promoting mathematical thinking and problem solving.

Overall, the use of iPads appeared to have a positive impact on the practices of the teachers and the engagement of the students participating in the projects. Benefits of the iPads included the flexibility in how and where they could be used, the instant feedback for students and the ability for students to make mistakes and correct them, alleviating the fear of failure and promoting student confidence.

The disadvantages of the iPads were mostly management issues relating to the sourcing and uploading of appropriate applications, the difficulties associated with record-keeping and supervision of students while using the iPads and the number of iPads available for use. The interactive nature of the technologies was engaging for the students at an operative level. However, when the tasks in which they were embedded did not include appropriate cognitive challenge, students were less engaged and became distracted by the technologies.

The incorporation of iPads in the two projects emphasised their potential to increase student engagement and the importance of providing professional learning experiences for teachers that go beyond learning how to operate the technologies. Rather, continued and sustained development of teachers’ technological pedagogical content knowledge (TPACK) (Mishra & Koehler, 2006) that builds on their understanding of mathematics content, ways in which students learn, the misconceptions that occur, and ways in which technology can enhance teaching and learning is required.

 References:     Banister, S. (2010). Integrating the iPod Touch in K-12 education: Visions and vices. Computers in Schools, 27(2), 121-131.    Ireland, G. V., & Woollerton, M. (2010). The impact of the iPad and iPhone on education. Journal of Bunkyo Gakuin University Department of Foreign Languages and Bunkyo Gakuin College(10), 31-48.     Kukulska-Hulme, A. (2009). Will mobile learning change language learning? ReCALL, 21(2), 157-165.     Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108(6), 1017-1054.

‘When will I ever use this?’ How to build bridges and make maths meaningful

One of the most common questions children ask in relation to mathematics is ‘When will I ever use this?’ Often they don’t realise that we use mathematics in almost every aspect of our lives, from the minute we wake up each morning and estimate whether we should push the snooze button, to working out how many minutes or hours there are until we get to finish school or work for the day. The perception that mathematics has little or no relevance to their lives beyond the classroom is one of the reasons children begin to disengage from mathematics during the primary years. In order to bridge the gap between children’s lives and the mathematics classroom I firmly believe that primary teachers should take every opportunity to make mathematics meaningful by using the real world, whether through the use of objects, photographs or physically taking children into the world beyond the classroom and engaging them in rich, worthwhile activities.

So how can you do this? If you are new to teaching with contextual mathematics, I would suggest that you begin by designing a mathematics trail at your school or somewhere out in the community – it could even take place at the local shopping centre. Find points of interest that have mathematical potential, photograph them and then plan a set of activities. For example, if you have a giant chessboard in the school playground, you might pose the following questions:

  • Estimate the following and explain your thinking: The area of the chessboard, the perimeter of the chessboard, and the area of each tile
  • Use words to describe the position of the chessboard without coordinates and in relation to its surroundings.
  • Locate the chessboard on a map of the school grounds. What are the coordinates?
  • Investigate the total number of squares (of any size) in the chessboard.
  • Design a new maths game that can be played on the chessboard and write a set of instructions for another group to follow.

You will notice that the questions above are quite open-ended. This will allow for all students to achieve some success and provides an important opportunity for children to show what they can or cannot do. Open-ended questions are more engaging for students and often require them to think harder and more creatively about the mathematics they are engaging in.

Another idea for contextualising mathematics is to use objects or photographs of real life objects, items or events. It could be something as simple as a school lunchbox, with questions such as the following:

  • Explore the ways sandwiches are cut. What different shapes can you see? Can you draw them?
  • Before recess, compare the mass of your lunchbox with five other lunchboxes. Can you order the lunchboxes from lightest to heaviest?
  • List the types of food in the lunch boxes today. Can you sort them into different categories? What categories do you have? Is there another way to sort them?
  • Conduct a survey to find out the most popular recess or lunch food in your class. Do you think this is a healthy food?
  • How many Unifix cubes do you think would fit in your empty lunchbox? Write down your estimate and then test it out. Was your estimate close? Find someone with a different size or shape lunch box and repeat the activity.
  • Use a special bin to collect rubbish from your lunch boxes. How much rubbish did you collect?
  • Sort out the lunch box rubbish and organise it into a graph. What information does your graph give you?

Another idea is to collect interesting photographs from around the world. I took the photograph above in Essaouiera, Morocco yesterday (I’m here to attend a conference). The photograph could be a stimulus for mathematical discussion and investigation, and there are several interesting mathematical questions you could pose:

  • Can you work out the number of hats in the photograph without actually counting them one by one? How? Is there another way?
  • The hats at the top of the photograph are called a ‘fez’ or ‘tarboosh’. Investigate their history and construct a timeline.
  • If each fez cost 80 Moroccan Dirham, how much would each one cost in Australian currency? Would the entire contents of the shop be worth more than $200?

A great free resource (and one of my favourites) that often has fantastic mathematical potential is the website, Daily Overview (http://www.dailyoverview.nyc/). Each day Daily Overview post a different aerial photograph from somewhere in the world. The photograph is accompanied by background information that could also be explored within a mathematics lesson.

There are many ways to bridge the gap between school mathematics and children’s lives. If we can promote the relevance of mathematics to children while at primary school, then we have a much better chance of sustaining their engagement through the secondary years, when mathematics becomes more abstract. We want children to continue the study of mathematics beyond the compulsory years and this is more likely to happen when they no longer ask ‘When am I every going to use this?’.

Teaching with tablets: Pedagogy driving technology, or technology driving pedagogy?

If you are a teacher, then you have probably experienced the introduction of a new technology into your classroom at some point in time. Whether it was an interactive whiteboard, laptops or tablets, it is likely that you would have felt some pressure to use that technology as much as possible because of the expense involved. Often teachers are expected to incorporate new technologies without the support of appropriate professional development. That is, professional development that not only addresses the technical aspects of the devices, but the pedagogical considerations as well.

My research into the use of iPads in primary classrooms has revealed that many teachers find it a challenge to use technology creatively to teach mathematics when compared to other subject areas. I believe that the way technology is used in mathematics lessons often reflects how the teacher views and understands mathematics and the curriculum. The teachers who see mathematics as a collection of facts and rules to be memorised often rely on a drill and practice approach, and therefore limit the use of technology to applications that support this method. The plethora of drill and practice apps now available on tablets help perpetuate this teaching method. On the other hand, teachers who see mathematics as a collection of big ideas that need to be applied to rich, contextual activities are the ones who use tablets and other technologies in more creative ways, steering away from the mathematics specific applications. Often during the drill and practice approach, the technology becomes the focus of the lesson. However, when rich tasks are involved, the focus remains on the learning and the technology is used as a tool to promote the learning, access and present information.

So how can you make your use of technology more meaningful in mathematics lessons? Frameworks are often helpful in encouraging teachers to reflect on their practices, and one that is a good starting point is the SAMR model of technology integration by Puentedura (2006). The model represents a series of levels of technology integration, beginning at the substitution level, where technology simply acts as a direct substitute for traditional practices, with no improvement. The second level, augmentation, provides some functional improvement – imagine the use of a maths game app that gives instant feedback. The feedback component is the improvement. At the third level, modification, the technology has allowed for significant redesign of existing tasks. The final level, redefinition, allows us to create new tasks that were previously inconceivable.

I believe that we should be pushing ourselves to aim for the redefinition level of SAMR, however, this does not mean that technology should not be used at the lower levels. The most important thing to remember is that you must not let the technology determine the pedagogy – it should be the other way around, where the pedagogy is driving the technology. Another thing to think about is that no framework is perfect. Although the SAMR model is a good starting point, a major flaw is that it assumes that any use of technology is going to enhance teaching and learning. I disagree. I have seen lessons where the technology distracts students, and the focus is no longer on the mathematics: it’s on the technology. Technology driving pedagogy.

Apart from adding a ‘distraction’ level to SAMR, I would also like to suggest that consideration of student engagement sits as a backdrop behind the entire model. I would also want to consider how the proficiencies (Working Mathematically) align with the model. In the graphic below you will see that I have made some additions to SAMR, suggesting that the lower levels of the model align with the proficiency of fluency, and as you progress through the model, more proficiencies are added so that tasks that move beyond drill and practice promote understanding, problem solving and reasoning.

From: Engaging  Maths: iPad activities for teaching and learning, Attard, 2015.
From: Engaging Maths: iPad activities for teaching and learning, Attard, 2015.

This adapted model can be used as a tool to help plan and design tasks and activities that incorporate technology. On the other hand, it might help you make the decision to not use technology! Resist the temptation to use devices simply because you feel you have to – if it doesn’t enhance teaching and learning, don’t use it. If you are going to use those drill and practice type apps, then make sure they are embedded in good teaching – always include rich reflection prompts that provide children with the opportunity to talk about the mathematics involved in the task, the problems and challenges they encountered, and ways they can improve their learning. Remember, don’t let the technology drive the pedagogy – mathematics and learning should always be the focus!

Attard, C. (2015). Engaging maths: iPad activities for teaching and learning. Sydney: Modern Teaching Aids.
Puentedura, R. (2006). SAMR.   Retrieved July 16, 2013, from www.hippasus.com

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.