Australia’s Declining Maths Results: Who’s Responsible?

Once again, mathematics education is in the spotlight. The most recent TIMMS  and PISA results highlight a decline in Australia’s mathematics achievement when compared to other countries, which will no doubt perpetuate the typical knee jerk reactions of panic and blame. So, what are we doing about this decline? Who’s responsible? Typically, the first to get the blame for anything related to a decline in mathematics are teachers, because they work at the coal face, they spend significant amounts of time with students, and they’re an easy target. But shouldn’t we, as a society that considers it acceptable to proudly claim “I’m not good at maths” (Attard, 2013), take some portion of the blame?

Numeracy and Mathematics education is everyone’s business

As a society, we all need to take some responsibility for the decline in mathematics achievement and more importantly, we all need to collaborate on a plan to change the decline into an incline. From my perspective, there are three groups of stakeholders who need to work together: the general community, the policy makers and school systems that influence and implement the policies, and the teachers.

Let’s start with the general community. It seems everybody’s an expert when it comes to mathematics education because we all experienced schooling in some form. Many say: “I survived rote learning – it didn’t hurt me”. The world has changed, access to information and technology has improved dramatically, and the traditional ‘chalk and talk’ practices are no longer appropriate in today’s classrooms. Many hold a limited view of school mathematics as drill and practice of number facts and computation. Although it’s important that children build fluency, it’s simply not enough. We must promote problem solving and critical thinking within relevant contexts – making the purpose of learning mathematics visible to students. It is, after all, problem solving that forms the core of NAPLAN, TIMSS and PISA tests.

The community pressure for teachers to use text books and teach using outdated methods, along with a crowded curriculum and an implied requirement for teachers to ‘tick curriculum boxes’ causes significant tensions for teachers, particularly in the primary school where they are required to be experts at every subject. If we consider the limited number of hours allocated to mathematics education in teacher education degrees compared with the expectations that all primary teachers suddenly become experts on graduation, then we should understand that teachers need continued support beyond their tertiary education to develop their skills. In addition, rather than focusing on students’ learning, the crowded curriculum  leads them to focus on getting through the curriculum (http://v7-5.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=2#page=1) and this often leads to a ‘back to basics’ approach of text books, work sheets and lots of testing that does not create students who can problem solve, problem pose and problem find.

This is where the policy makers and school systems must come into play by providing support for high quality and sustained professional learning and encouraging primary teachers to gain expertise as specialist mathematics teachers. We already have a strong curriculum that promotes problem solving and critical thinking both through the Proficiencies and through the General Capabilities. The General Capabilities provide teachers with the opportunity to embed mathematics in contextual, relevant and purposeful mathematics. However, teachers need to be supported by all stakeholders, the community and the policy makers, to use these tools and focus less on the teaching of mathematics as a series of isolated topics that make little sense to students.

What can we do?

There are no easy solutions, but one thing is clear. We need to disrupt the stereotypical perceptions of what school mathematics is and how it should be taught. We need to support our teachers and work with them rather than against them. Let’s band together and make some changes that will ultimately benefit the most important stakeholders of all, the children of Australia.

 

 

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.

 

Christmas Maths: Open ended investigations for Grades 4-6

In this final Christmas themed post, I am including a range of open-ended investigations that are suitable for upper primary and lower secondary students (from the book Engaging Maths: Everyday Investigations Years 3 to 6). You will notice that some of the investigations extend beyond the mathematics curriculum and integrate quite easily into other key learning areas. This is intentional. If we want to engage students in mathematics, then making it contextual often requires it to either be embedded within another subject area or at least have some connections to other areas. Another consideration is the General Capabilities of the Australian Curriculum: Mathematics. When we incorporate contextual mathematics and investigation-based tasks, we are more likely to include the General Capabilities and this is evidenced in the activities below.

Short activities:

  1. If you have a Christmas tree in your house or school, how tall is it? Can you reach the top of the tree by reaching up? How much taller than you is the Christmas tree? What fraction of the height of the tree is your height?
  2. Draw a picture of a Christmas tree. Use your drawing as a plan to show where you will place the decorations.
  3. Tie a piece of tinsel to the very top of the Christmas Tree. Wind the tinsel around the tree until you reach the lowest branch. What is the length of the tinsel?
  4. If the individual lights of a string of Christmas lights are 30 cm apart, how many lights would you need so decorate the perimeter of the classroom?
  5. How would you work out how much wrapping paper needed to wrap 10 presents that were each the size of a shoe box? Record all of your working out. What mathematics did you use?

Investigations:

  1. Plan a Christmas party for some of your friends. Show all the mathematics that you need to use for your planning.
  2. Many families start to budget for Christmas presents several months before Christmas day. Design a budget for the Christmas presents that you would like to give to your family members, relatives and friends. Perhaps you might like to include your teachers.
  3. Survey the other students in your class using the question, “Do you have a Christmas tree in your home?” “Is it a real tree or an artificial tree?” “Which type of tree do you prefer and why?” Present the data that you have collected and present a report to your class.

Extension Activities:

  1. Investigate and research the tradition of decorating a tree for Christmas. Answer questions such as “When did the tradition start?”
  2. Plan menus for the meals for family for Christmas Day and Boxing Day and include a budget.
  3. Make a list of the things you would like for Christmas. Sort your items into needs and wants. How would your list compare to the list of a child in a different country? Investigate.

I hope you have enjoyed this series of posts that have included many rich activities to keep students engaged with mathematics until the very last day of the school year. If you do implement any of the tasks, I would love to hear from you and see your students’ work samples!

More Christmas Maths: Open-ended tasks and Investigations for the Early Years Classroom

The use of open-ended tasks and mathematical investigations provides opportunities for students to demonstrate their abilities in a creative, non-threatening and meaningful way while promoting high levels of engagement and providing rich assessment data. Although the end of the year is near, the use of Christmas as a context for meaningful mathematics is an opportunity that is too good to miss. Providing students with a context that is exciting and relevant will ensure they maintain their engagement with mathematics until the end of the school year.

This week I am sharing a set of tasks that are taken from a book written by John Pattison and myself: Engaging Maths: Everyday Investigations for Early Years (2014). The tasks are separated into short activities, investigations, and extension activities. The short activities are intended as a warm up for the more complex investigations.

Short activities:

  1. This year how many days holiday will you have before Christmas Day? How many days will there be between the beginning of the school holidays and the last day of the year?
  2. Do you have a Christmas tree? How tall is the tree? Can you touch the top of the tree if you stand on tip toe? Is the tree taller than your dad or mum? How many lights are there on the tree?
  3. Does your family put presents under the Christmas tree? How many presents did each member of your family receive? How many of the presents were yours?
  4. How much tinsel would you need to decorate the Christmas tree?
  5. Your grandmothers, grandfathers, uncles, aunts and cousins are coming to your house for Christmas. If each person has a Santa bag full of presents under the Christmas tree, how many bags would there be?
  6. If each person is given a knife, fork and spoon with which to eat their Christmas dinner, how many pieces of cutlery would you need altogether?

Investigations:

  1. Make a list of the ten presents you would like Santa Claus to bring you for Christmas. Put the presents in order starting with one (1) for your first choice. Write a letter to Santa giving reasons for your choice of presents.
  2. Use store catalogues to help you to find the cost of your list of presents. Santa has said that he can only supply one hundred dollars worth of presents. Which presents will he choose to give you?
  3. Make a list of all the food items that Mum and Dad have to buy for the Christmas dinner. How many shopping bags will they need to take to the shops to carry all the food?

Extension Activities:

  1. Christmas Day always takes place on the 25th of December. Christmas Eve is the day before Christmas Day and Boxing Day is the day after Christmas Day. In 2013 Christmas Day was a Wednesday. What day was Christmas Eve and what day was Boxing Day in 2013? On which days of the week will Christmas Eve, Christmas Day and Boxing Day take place in the next five years? What did you discover?
  2. Christmas celebrations are very different in other countries. Use the Internet and the books in your library to investigate how people in other countries celebrate Christmas. Share the information you discovered with your classmates and teacher.
  3. There are many books with stories about Christmas in Australia. Find some of these books in the school library or on the Internet. Read your favourite story to the rest of your class.

Mathematics and Christmas: Keeping Students Engaged!

It’s that time of year again! Last year I wrote a series of blog posts to encourage teachers to continue to provide rich teaching and learning activities until the very end of the school year. I thought it would be a good idea to repost these activities for those who might want a reminder of some engaging Christmas themed mathematical explorations.

As the end of the school year approaches, report writing is almost complete and Christmas is on the horizon. It’s easy to lose focus, get distracted, and keep students occupied with ‘busy work’. However, it’s critical that we don’t waste a minute of students’ learning time, particularly when we know that over the long Christmas break some students may regress in relation to mathematical fluency and understanding.

So how can you keep mathematics engaging until the last day of school? Consider the elements required for sustained engagement to occur. Three factors are critical: cognitive, operative, and affective engagement. In terms of mathematics, true, sustained engagement occurs when students are procedurally engaged and interacting with the mathematics and with each other; when there is an element of cognitive challenge within the task; and when they understand that learning mathematics is worthwhile, valuable, and useful both within and beyond the classroom. It is easy to mistakenly think that students are engaged when they appear to be busy working, or ‘on task’. True engagement is much deeper than ‘on task’ behaviour, rather, it can be viewed as ‘in task’ behaviour, where all three elements; cognitive, operative and affective, come together. This leads to students valuing and enjoying school mathematics and seeing connections between the maths they do at school and the maths they use in their lives outside school (for more information see my FRAMEWORK FOR ENGAGEMENT WITH MATHEMATICS).

As this time of year it is easy to design mathematics tasks that promote high engagement and have the potential to stimulate learning. The following is a set of problem solving activities based on the famous Christmas Carol, The Twelve Days of Christmas. The activities are suitable for children in the middle years (grades 5 to 8), however can be easily adapted to suit younger or older learners.

Resources Required:

A copy of the lyrics of “The Twelve Days of Christmas”

The book “The Twelve Days of Christmas” (There are several versions available)

Price Lists

Other resources as required, eg. shopping catalogues

Possible Investigations starters/Task cards

Teaching/Learning Activities:

  • Read the book/lyrics or listen to the song “The Twelve Days of Christmas”
  • Discuss how the gift giver has to increase the number of gifts to his true love each day.
  • Provide students with a price list (this can be adapted according to the ability of the group)
  • At this point students can be asked to investigate the cost of the gifts, turning the activity into an open-ended investigation, or, specific questions can be posed to the students.

Examples of possible problems to explore are:

  1. What is the total number of gifts given?
  2. Is there an easy way to work this out?
  3. What is the total cost of the gifts?
  4. If the department store was holding a pre-Christmas sale and offered a 15% discount for all purchases, what would the new cost be? (This does not include performers, maids, lords etc)
  5. What if the discount offered only applied to live animals?
  6. The maids give a 10% when booked for more than two consecutive days. What would their new fee be?
  7. The musicians charge a 10% Goods and Service tax and this must be added to the total cost.
  8. How many people arrive at the true love’s house on the twelfth day?
  9. What would it cost to feed all the people and the animals? (Internet would come in handy here!)
  10. Use some Christmas shopping catalogues to replace the gifts with something more appropriate for a modern true love and calculate the cost.
  11. Re-write the lyrics to fit your new list of gifts.

The full list of activities and hypothetical price list are available on PDF here: The Twelve Days of Christmas

Other engaging end-of-year mathematics tasks are:

All of the above activities have the potential to promote high levels of engagement at a time of year when it is difficult for students and teachers to remain focussed. However, it is important to remember that any activity is only as good as the teacher implementing it. To enhance students’ engagement and learning, ensure there is regular student reflection, and rich discussion about the mathematics embedded with the tasks.

Primary Mathematics: Making the Most of Technology to Assess Student Learning

As the school year rapidly draws to a close, many teachers are beginning the task of reporting student achievement. For some, there may be a scramble to collect assessment data, and often, due to a sense of panic, teachers revert to pen and paper testing to gain a snapshot of their students’ ability measured against syllabus outcomes…one of the main reasons students develop a dislike of mathematics in the first place. The purpose of this blog post is to ask you to consider using alternative assessment evidence, and in particular, consider taking advantage of some of the educational software tools you may already be using in your classroom.

Regardless of what technological devices you use, if you do use technology in your mathematics lessons, chances are you already have some good assessment data that you can use in your reporting. Take, for example, the use of apps on an iPad or other mobile device. If your students are engaging in different apps to either build on their mathematical fluency (typically game-type apps) or to express mathematical reasoning and communication (with apps such as Explain Everything, Educreations or ShowMe), then it’s rather easy to collect evidence of learning. Some apps offer the affordance of being able to save student progress, and others simply require students to take a screen shot of their results.

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Educreations allows you to save files that record audio and written mathematics, allowing assessment of content and process outcomes.

I recently conducted a research evaluation of the Matific suite of resources (access the research report here). One of Matific’s affordances is that it allows teachers to track student progress.

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The Matific website allows teachers to view assessment data in a number of ways

Interestingly, out of the 16 teachers involved in the study, only nine teachers used the ability to track student achievement and even fewer considered using it as assessment data. However, those who did use this affordance, considered it a valuable tool that allowed them to differentiate future tasks, tailoring the learning for individual student needs:

It was perfect in a sense that we made it a point that we started at the middle and we went down for those who needed extra support, which was fabulous because they were still doing it visually, they were doing the exact same thing, and then we also gave the option that they could go up if they felt confident enough but at the same time visually, it was exactly the same for those kids that don’t want to be different, that maybe do need that little bit of extra support (Year 6 teacher). 

Data from students’ interactions with educational apps such as Matific, game apps and productivity apps can provide valuable formative and summative assessment data that can remove the anxiety associated with formal pen and paper testing, particularly during the primary years when it’s critical that we foster high levels of student engagement. Consider the apps you currently use – how can you collect evidence and use it to your advantage and the students’ advantage…and also save you time? Isn’t it better to spend class time on learning rather than testing?

Improving Primary Mathematics: The Challenge of Curriculum

Arguably one of the biggest challenges for most primary teachers is the struggle to address the many components of the mathematics curriculum within the confines of a daily timetable. How many times have you felt there just isn’t enough time to teach every outcome and every ‘dot point’ in the entire mathematics curriculum for your grade in one year? It is my belief that one of the biggest issues in mathematics teaching at the moment stems from misconceptions about what and how we’re supposed to be teaching, regardless of which curriculum or syllabus you are following.  The way we, as teachers, perceive the content and intent of our curriculum influences whether students engage and achieve success in mathematics. The way we experienced the curriculum when we were at school also influences how mathematics is taught in our own classrooms.

This struggle arises partially from the common perception that every outcome (in NSW) or Content Descriptor (from the Australian Curriculum) must be addressed as an individual topic, often because of the way the syllabus/curriculum is organised (this is not a criticism – the content has to be organised in a logical manner). This often results in mathematical concepts being taught in an isolated manner, without any real context for students. A result of this is a negative impact on student engagement. Students fail to see how the mathematics relates to their real lives and how it is applied to various situations. They also fail to see the connections amongst and within the mathematical concepts.

Imagine if you could forget everything you remember about teaching and learning mathematics from when you were at school. Now think about the three content strands in our curriculum: Number and Algebra, Measurement and Geometry, and Statistics and Probability. Where are the connections within and amongst these strands? If you could, how would you draw a graphical representation of all the connections and relationships? Would your drawing look like a tangled web, or would it look like a set of rows and columns? I’m hoping it would like more like a tangled web! Try this exercise – take one strand, list the content of that strand, and then list how that content applies to the other two strands. If you can see these connections, now consider why we often don’t teach that way. How can you teach mathematics in a different way that will allow students to access rich mathematical relationships rather than topics in isolation? How can we make mathematics learning more meaningful for our students so that maths makes sense?

This leads me to my second point and what I believe is happening in many classrooms as a result of misunderstanding the intention of the mathematics curriculum. If students are experiencing difficulties or need more time to understand basic concepts, you don’t have to cover every aspect of the syllabus. It is our responsibility as teachers to ensure we lay strong foundations before continuing to build – we all know mathematics is hierarchical – if the foundations are weak, the building will collapse. If students don’t understand basic concepts such as place value, it doesn’t make sense to just place the ‘strugglers’ in the ‘bottom’ group and move on to the next topic.

We need to trust in our professional judgement and we need to understand that it’s perfectly okay to take the time and ensure ALL learners understand what they need to before moving on to more complex and abstract mathematics. It most definitely means more work for the teacher, and it also means that those in positions of leadership need to trust in the professional judgement of their teachers. Most importantly, it means that we are truly addressing the needs of the learners in front of us – the most important stakeholders in education.

 

Using Contexts to Make Mathematics 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 all mathematics teachers should take every opportunity to make mathematics meaningful by using the real world where appropriate, whether through the use of objects, photographs or physically taking children into the world beyond the classroom and engaging them in rich, worthwhile activities. This blog post was originally posted in 2015 and I thought the messages here would be a timely reminder, given that I have continue to receive invitations to assist teachers and schools in engaging their students with mathematics.

So how can you make mathematics more meaningful? 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 recently in Oslo, Norway. What sorts of questions could you ask students to explore relating to the interesting shapes you see in the bridge and the building? Here’s another interesting photograph from Morocco.

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There are several interesting mathematical questions you could pose relating to the phtograph:

  • 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?’.