In December 2017, I attended the Maths Association of Victoria conference at La Trobe University, Melbourne, which provided an opportunity to learn more about engagement in maths. The topic of student engagement in mathematics (or lack of) is a cause for concern among maths educators. A lack of maths engagement can “limit one’s capacity to understand life experiences through a mathematical perspective” (Attard, 2012a, p. 9).
I gained inspiration particularly from sessions led by Catherine Attard (an Australian researcher and maths educator), and Dan Finkel (a researcher and maths educator from Seattle, USA). In this blog post, I share with you some key insights and tasks from those sessions.
Attard immediately captivated us, as we got busy (and totally engaged!) folding a paper circle in many ways. The task involved us in rich discussion of the mathematical language associated with geometry and the attributes of 2D and 3D…
Over the past week I have been involved in a number of professional development events for primary and secondary teachers of mathematics. This included presentations at a primary and middle years conference and a number of sessions involving the development of teachers as action researchers. This weekend I will be travelling to the US to attend the NCTM Annual Meeting and Exposition in Washington DC and will be presenting a session there. All of these engagements with teachers reminded me of a post I published last year about what teachers do with the information they gain from attending professional development, particularly when it happens away from school. The following are some thoughts I wrote about last year – a timely reminder for those teachers who are taking time away from their students or in their personal time to deepen their knowledge about mathematics teaching and learning.
How do you make the most of professional development?
Too often teachers attend PD sessions, get enthusiastic, try a few new things, but quickly get bogged down in the day-to-day challenges of life in a busy school and the demands of administration and curriculum authorities. How can you translate the underlying philosophy being promoted in the professional development sessions into sustainable change that can be shared amongst colleagues to improve and transform mathematics teaching and learning?
PD is expensive, and it’s important that opportunities aren’t wasted. I’ve been talking and writing a lot recently about promoting critical thinking in the mathematics classroom. It’s equally as important for teachers to engage critically with professional development. The following list contains a few thoughts that might help teachers get the most out of PD opportunities.
Choose the right PD
Do a little research on the person presenting the PD. What are their credentials? Are they a self-proclaimed expert or do they have an established reputation? A simple Google search should reveal some insights, and, if the presenter is an academic, you could search Google Scholar for some of their academic publications. Spending time researching the presenter’s background can save you from attending a PD session that may not be right for you, and can provide some good research background should you choose to go ahead with the session. You also need to consider what you want out of a PD session. If you want a ‘bag of tricks’ in the form of a handful of ready to go activities, then you probably shouldn’t be wasting your school’s money. Rather, think about PD that is going to cause you to think deeply about your practice, and have a long-term effect on students’ educational outcomes.
Does the presenter understand the school context and curriculum in your state/country?
When you attend PD, you expect that the presenter is aware of the school/state/country context, and more importantly, the curriculum. This assists you, the teacher, in applying the learning to your practice, and also makes the content of the PD more relevant to you and your students.
Understand the structure of the PD session
Before you commit to attending a PD session, ensure you understand what is going to happen in that session. Nobody likes sitting down and being lectured to for hours on end, nor do you want to listen to a presenter talk about themselves for an entire day! Look for presentations that are interactive and allow participants to apply theory to practical activities. If we are going to ask our students to do something differently, we need to experience it ourselves first. It’s also a better way of retaining information.
When you’re at the PD session, don’t be afraid to ask questions. It’s also important to think critically about the information you are receiving. Presenters are usually very happy to answer questions that spark discussion – this often results in deeper learning, and better value for your school’s money! If the presenter doesn’t welcome questions, this is a sign that they may not have expert knowledge. During the PD session it’s important that you participate in any activities – there’s usually a good reason a presenter has asked you to engage in a task. Active participation gives insight into the student experience and possible challenges, and it’s a great way to make links between theory and practice.
Use the session as a networking opportunity
Often one of the most valuable aspects of professional development sessions is the opportunity to connect with teachers from other schools. It’s a great opportunity to discuss practice, students and school procedures. Networks developed at PD sessions can be maintained easily using tools such as LinkedIn, Twitter, and Facebook.
Before you leave your PD session, pause and consider what you have learned (a good presenter will actually give you opportunity to reflect). Think about how you might apply what you have learned (not just the activities, but the educational philosophy underpinning them) to your classroom, and don’t limit yourself to just replicating the activities. What are the underlying messages? How can you use those messages to adapt your practice? What will be different in the way that you plan and implement lessons? It doesn’t have to be a big change. Often subtle differences have huge effects.
Sustainability: Sharing the Learning
Finally, it’s important to share the learning. It’s difficult to sustain any kind of change that will have ongoing benefit for students if it’s not supported by others in your school. This may not be easy, but small changes are better than no changes. Sometimes it’s a good idea to try out new things in your own class first, then use evidence of your success to convince others.
When it comes to PD, one of the most important things to remember is the reason we do what we do. We want our students to be the best they can, and when it comes to mathematics, we want to give them confidence, skill, passion and excitement that will ensure they continue to study and use mathematics beyond their school education.
If you’re an early career teacher, chances are you spend lots of your spare time looking for good maths resources. Some of you may have your own class, while others are beginning their careers as a relief teacher, having to move from one class to another, and often between different schools. 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 of diverse abilities?
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!
In my 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. Whether you are lucky enough to have your own class or have to begin your career as a relief teacher moving from class to class, 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.
What hands-on materials and resources do you have in your mathematics classroom? Concrete materials, coupled with good teaching practice and strong teacher content knowledge, provide opportunities for learners to construct rich understandings of mathematical concepts. In addition, allowing opportunities for children to physically engage with materials can be much more meaningful than working only with visual or even digital representations, particularly when learners are still in the concrete phase of their learning about specific concepts. For example, if you’re teaching concepts relating to 3-dimensional space, it makes sense that it is better for children to be able to manipulate real objects in order to explore their properties and relate their learning to real-life, as opposed to exploring objects through graphical representations only. Concrete materials also promote the use of mathematical language, reasoning, and problem solving.
I’m often asked about the essential resources required for primary mathematics classrooms. There are quite a few, but if you have a limited budget or storage space, there are some 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):
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 as , 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)
Measurement materials (you’ll need a range of things to cover all aspects of measurement, eg. scales, tape measures, rulers, )
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!)
Paper shapes (circles, squares, etc.) to promote a range of concepts including fractions, shape, and measurement
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 or resource, consider the purpose of the lesson and the mathematical concepts being covered. Also consider how you can make the most out of those 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 do some reading. 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 listed above, your students will become much more engaged with mathematics and will develop deeper conceptual understandings.
And one last thing…students are never too old or too smart to benefit from hands-on materials so never keep them locked away in a cupboard or storeroom (the materials, not the students)! Students should feel they can use concrete materials when and if they need them. After all, we want our students to be critical, creative mathematicians, and hands-on materials assist learning, and promote flexibility in thinking and important problem solving skills.
In several of my previous posts I discussed the importance of promoting critical thinking in mathematics teaching and learning. I’ve also discussed at length various ways to contextualise mathematics to provide opportunities for students to apply prior learning, build on concepts, and recognise the relevance of mathematics in our world. In addition, investigations provide excellent assessment material – usually when we assess in mathematics we ask for specific answers. In investigations, students can show us a range of mathematics, often beyond our expectations. They are also a great way to integrate other subjects areas such as literacy and science.
In this blog post I am going to share some ideas for open ended and inquiry-based mathematical tasks based on two items that most students would be familiar with – beach towels and pencil cases!
Let’s start with pencil cases. It’s the start of the 2018 school year next week and many children begin each school year with brand new stationery, in brand new pencil cases. Even if they’re not brand new, most children have a pencil case. I came across an interesting article relating to pencil cases a few days ago, and I think this could be used to spark interest and curiosity. The article can be found here:
Who has the heaviest pencil case? Compare the mass of your pencil case with the pencil cases of your group members. Who has the lightest? Estimate the mass, then use scales to test your estimations. How close were the estimations?
Estimate, then calculate the surface area of your pencil case. What units are the most appropriate to use? Explain how you measured the surface area.
Faber Castell is a famous brand of pencils. Investigate the history of Faber Castell and illustrate this on a timeline.
According to the Faber Castell website, it takes one ‘pinus caribaea’ tree 14 years to be ready to be used to manufacture pencils. Each tree can produce 2500 pencils. If one tree was allocated to each school, how many pencils do you think each child in your school might receive? How did you work this out?
If each of the 2,500 pencils were sold for $1.50, how much do you think the entire tree be worth in pencil sales?
At the beginning of each school year many children get brand new pens and pencils to take to school. Investigate how much it would cost to buy your stationary. Which shop offers the best value for money?
Some pencil cases like the one in the photo and in the Missing Letter article have small clear plastic pockets to put your name in. If a pencil case has only eight pockets, is this enough for your name? Investigate the length of names in your class. What would be the average length name in your class? What else could you explore about names?
The pencil case in the picture came with some pre-printed letters for the clear pockets. There are more of some letters than others. Investigate the most common letter occurring in students’ Christian names. Do you think it would be the same in all countries?
Design and make a pencil case to suit your individual stationery needs. Write about the mathematics you use to do this.
Design a new and improved pencil and explain the changes you have made.
Design, justify, and create a marketing campaign for a new, ‘miracle’ pen.
Research and discuss the following statement: “To save the environment, wooden pencils will no longer be manufactured”.
Promoting Curiosity and Wonder
Mathematical investigations should promote curiosity and wonder. The pencil case questions and investigations are open, yet provide some structure and support. They give enough detail to communicate the type of mathematics required to complete the task or investigation. Students should eventually be able to feel confident enough to come up with their own questions and follow their own path in terms of the mathematics they access and apply, just like mathematicians do.
Round Beach Towels?
In the last year or two a new beach towel has emerged onto the beach towel scene. It’s round. Now this idea immediately caused some concern for my mathematical brain. I had questions.
Is there more fabric in a round beach towel than a regular, rectangular beach towel?
Is there more fringe, and wouldn’t this make the towel more expensive?
How does one fold a round beach towel?
Could you wrap a round beach towel around you the way you wrap a rectangular beach towel?
How much more area on the beach gets taken up by people spreading round beach towels?
Does this mean less people get to lay on the sand?
Could you design a round beach towel that has a tessellating pattern?
All of the questions above can be explored using a range of mathematics…I wonder how many more questions your students could come up with?
Christmas is over, the novelty of new toys has worn off, and the holiday chorus of “I’m bored” is echoing in households everywhere. What can you do to stop the boredom?
First, it’s important to understand why children feel bored. According to the literature, boredom signals a person’s need for physical or mental activity to keep him or her occupied and to vent energy, just like needing food to satisfy feelings of hunger. Apart from the constant nagging, having bored children can lead to negative behaviour, and that’s the last thing parents want during the long summer holidays.
What’s wrong with kids being bored?
Apart from maintaining the peace, there are other important reasons to keep your children from being bored. The long summer holiday period can result in what’s called learning loss. That is, children who aren’t kept mentally and physically active during the long school holidays can lose some of the skills they learned during the school year.
The phenomenon of learning loss is well documented in research, and studies have shown that often schools have to spend several weeks bringing students back to their pre-holiday learning levels.
At best, children learn little or nothing during the summer holidays, and at worst, they can lose weeks of learning. The greatest losses occurring in the area of mathematics, and then spelling. The children most at risk of learning loss are those from low-income families, because of the differences in holiday experiences and activities of children from high-income families.
Regardless of income, there are many ways to stop children being bored, maintain their learning and keep them busy and happy during the holidays.
Activities at home
Home is where you can get back to basics with your children.
Encourage your child to read a book.Research has proven that reading for pleasure improves reading attainment and writing as well as general knowledge, and community participation. Reading also provides insight into human nature and decision-making. If you don’t have books at home your child is interested in, take a trip to the local library and let them choose.
Play games with your children. If you want to help maintain your child’s mathematical learning and keep them having fun, there are lots of simple games you can play. All you need is a deck of cards, a set of dominoes or some traditional board games such as Monopoly, Guess Who or Yahtzee.
If you want to help with spelling, try playing Scrabble or teach your child to do simple crossword puzzles. For those who like a challenge, chess promotes important problem-solving skills.
Get messy with some creative art. If your child is feeling creative and you don’t mind a mess, let him or her paint, build, sculpt, design or invent. Creative art has been found to assist in children’s learning and promote well-being.
Play fun and educational video games. If traditional activities don’t do the trick, there are always the digital alternatives. Research has found playing video games can have cognitive, emotional and social benefits. But it’s important to choose carefully.
Rather than choosing games that promote mindless violence or require little or no thinking, there are many educational games and apps that can help your child continue learning over the holidays such as Minecraft, Pick-a-Path, or MathDoodles. Many good apps are free and even if they’re not designed to be educational, they often involve problem-solving skills important in developing critical and creative thinking.
Take your children to the local park or playground. Recently, the NSW Department of Education announced over this year’s summer holidays it will trial having the playgrounds of 40 schools open to the public. This is aimed at providing access to outdoor spaces for families who don’t have backyards for children to play in.
The benefits of outdoor play during the summer holidays are significant. Science says holidays often result in weight gain among adults and children but there are also social benefits like improved self-confidence to be gained from interacting with other children.
If you want to add educational benefits to outdoor activities, play games that involve keeping score to help children maintain their mathematics skills. Younger children could go on “shape hunts” or “number hunts”, or you could play a game of I Spy to ensure there is mental and physical activity happening.
There are lots of other low-cost activities to support your children’s education. Going shopping can help your kids learn about financial literacy. Going to a museum or going hiking can teach children about history and nature.
Most importantly, all of these activities will keep your kids’ minds and bodies active, keep you sane and stress free, and stop the kids from saying “I’m bored”.
You might be wondering what McDonald’s has to do with mathematics and critical thinking. Recently I found a copy of the original McDonald’s price list dating back to the 1940s when McDonald’s was owned by the original founders, Dick and Mac McDonald. Since that time, the fast food franchise has become a global fast food brand recognised by most. It is because of this recognition that the 1940s menu makes a perfect stimulus for mathematical investigation and critical thinking. The links between mathematics and children’s lives are not always obvious for students, so opportunities such as this are important to ensure our students understand how mathematics can help to make important decisions that affect our finances, health and general well-being. Although you might consider rejecting this idea so as not to promote a fast food culture, consider this an opportunity for students to think critically about food choices.
The Maths and McDonald’s graphic below contains some suggestions for mathematical investigations and would best be suited to students in upper primary or lower secondary classrooms. However, they can be adapted quite easily for younger students.
Below, the prompts are listed in a table that details some of the potential mathematical content that students would be expected to apply, and the processes they would use in the application of the mathematics. Although not included in the table, the tasks also address several of the General Capabilities from the Australian Curriculum: Mathematics. In addition, the tasks lend themselves well to integration with other curriculum areas.
· The beauty of this investigation is that it is personalised. If students are working in groups, they will need to negotiate what a ‘typical’ home cooked meal is.
·Grocery store apps would be handy for this investigation if students have access to mobile devices· There are multiple ways this task could be completed
Analyse the financial cost of eating takeaway compared to cooking the same food at home
· Problem Solving
. The takeaway food considered in this task may not necessarily be McDonald’s.
. It is important to allow students to draw from personal experience to ensure they are engaged with the mathematics and the task.
Using the Investigations in the Classroom
Once you have given students time to look at and discuss the original McDonald’s menu, you can choose to allow students to choose one or more of the investigations to explore. Better still, once they have completed an investigation they may be able to come up with one of their own – this is a great way to promote mathematical curiosity and wonder. Allow students to choose how they present their work, and encourage them to document all of the mathematics they do. It is also critical to build reflection into the investigation, so make sure you have some reflection prompts prepared for either verbal or written reflection.
The Maths and McDonald’s investigation provide opportunities for students to learn and apply mathematics in context. This improves student engagement, allows them to see the relevance of mathematics, promotes critical thinking and provides important and authentic assessment data.
The McDonald’s menu: https://www.thesun.co.uk/fabulous/food/3564107/mcdonalds-original-menu-1940-first-ever/