Category Archives: Problem Solving

Promoting Creative and Critical thinking in Mathematics and Numeracy

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies: Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities, one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson. Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys  are also very worthwhile tasks. For good mathematical problems go to the nrich website. For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300  (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Screen Shot 2017-06-25 at 5.40.37 pm

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

  • higher-level thinking within authentic and meaningful contexts;
  • complex problem solving;
  • open-ended responses; and
  • substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

 

 

 

 

When a Maths Curse is a Good Curse!

In one of my previous posts I wrote about the use of children’s literature to encourage rich mathematical investigations and improve student engagement with mathematics. One of my favourite books, Math Curse by John Szieska and Lane Smith, is described in the blog post as a great way to engage reluctant learners. Even better, Math Curse encourages children (and their teachers) to see the mathematics that is embedded in every aspect of our lives. In this post I am going to share some student work from a Grade 3 classroom. In this classroom, the teacher read the book to the students before challenging them create their own class maths curse. The children took their own photographs, and working in small groups, they came up with a range of mathematical problems and investigations, which they then gave to other groups to solve.

Here are some of the photos with their accompanying questions:

Beyblades:

  1. If one of the beyblades spins for 2 minutes and 31 seconds and the other one spins for 1 minute and 39 seconds what is the difference between the two times?
  2. If one of the beyblades spins for 1 minute and 1 second and another spins for 78 seconds, which beyblade spun for the longest and by how long?

Hair:

  1. If there are 31 people in the class (10 boys and 21 girls) and all of them have hair that is 30cm long. Half of the boys cut 10cm off their hair, the other half cut 20cm off their hair. How long is the classes hair now altogether? How long was it before? How much hair has been cut altogether?
  2. Check your friend’s hair. Estimate how long it is when it is out, how long it is when it is in a ponytail, and how long it is when it is in a braid. List some different ways you could check if your estimate is accurate? What are the potential problems with your methods?
  3. I’m 9 years old. I had really long hair for 6 years, then I cut it. How long did I have short hair for?
  4. I have 5 friends that are girls and 2 friends that are boys. All 5 girls have hair length of 50cm. The boys both have different lengths of hair. The 1st boy has 30cm of hair, the second has 25cm of hair. What is the difference between the 1st boy and the girls and the 2nd boy and the girls?

Birthday Balloons:

  1. Write down the dates of important celebrations. If you add all the dates together, what is the value of their numbers?
  2. How many days are there in 6 years?
  3. If everyone’s birthday occurred every three years (starting the year you are born) what years would your birthday fall on?
  4. If Lisa and Jane went on a holiday every 2 months, how many holidays could they take in a year?
  5. If you could rearrange the seasons, what months would you choose to be Spring? Why?
  6. What is the most popular letter in the days of the months?
  7. Why do you think there are 4 seasons in a year?

From Problem Solving to Problem Posing

What is the purpose of getting students to write mathematical problems? First of all, the problems give us good insight into whether students recognise mathematical situations, and whether they understand where, how, and what mathematics is applied in day to day situations. An added bonus is that the students are highly engaged because they have ownership of the mathematics they are generating, the topics they choose are of interest to them, and stereotypical perceptions of school mathematics are disrupted.

Student Reflection

The students who wrote the examples above completed a structured written reflection following the sequence of designing and solving each others’ maths curses. Here are some of reflection prompts and a sample of responses:

What did you enjoy about today’s learning?

“working with my team”
“working at the problems for a long time and then finally getting them after a long, hard discussion”

“solving questions that my friends wrote”

“I felt challenged and I learnt more about what maths is”

“working with my group, choosing our own questions and learning something new”

“I liked the chess card the best because we had to solve it together and use problem solving”

“having a go at tricky questions even if i got them wrong”

Did you learn anything new?

“how to work things out in different ways”

“working in groups helps you learn more skills”

“not every question uses just one skill like addition, division, multiplication or subtraction”

“when I am challenged I learn more”

“Maths is not always easy”

“how to work together”

“Everyone in the group has different responses so we needed proof to figure out the right one”

What surprised you about this task?

“It surprised me how hard my own questions were”

“I didn’t know that we could come up with so many interesting questions”
“I got a shock! We had to research to solve some problems, Adam even taught me how to add a different way”

“I got some questions wrong “

“It was hard but if we put our brains into gear we could figure it out”

“I was able to play while doing maths” 

Using activities such as this provides multiple benefits for students. Contextualising the mathematics using students’ interests highlights the relevance of the curriculum, improves student engagement, and makes mathematics meaningful, fun and engaging!

Setting up Your Students for Mathematical Success : Tips for Teachers

Many children begin the new school year with feelings of fear and anxiety. Will they like their new teacher or teachers? Will the work be difficult? What will the homework be like? As you prepare programming and planning for a new teaching year and new students, give some thought to the strategies and activities you and your students can do in the first few weeks of term to ensure everyone gets the most out of their mathematics lessons for the entire school year. Think about what you can do differently in 2017 to make your work more engaging for both you and your students. The following are some ideas to consider.

  1. Be a positive mathematical role model

I’m sure this won’t come as a surprise, but there are teachers in our schools who actually don’t like maths and don’t like teaching it. Why is this a problem? Student know! This knowledge perpetuates the common misconception that it’s okay to dislike mathematics, and worse still, it’s okay to be considered ‘bad’ at maths.  Unless the teacher is an award-winning actor or actress, it’s really difficult to hide how you feel about a subject – it’s obvious in body language, tone of voice and of course, the way you teach the subject and the resources you use. If you know someone like this, suggest they seek some support from a colleague or colleagues. Often the reason a person dislikes mathematics is related to a lack of confidence.

  1. Get to know your students as learners of mathematics

The foundation of student engagement requires an understanding of students as learners, in other words, the development of positive pedagogical relationships (Attard, 2014). Positive relationships require teachers to understand how their students learn, and where and when they need assistance. It’s also important to provide opportunities for ongoing interactions between you and your students as well as amongst your students.

Another way to get to know your students as learners is to use existing data. For example, if your school takes part in external testing such as PAT, you can use this data as a guide. However, keep in mind that things change quickly when children are young – what they knew or understood three months ago may be very different after a long summer holiday.

A great activity to do in the very first few maths classes of the year is to ask your students to write or create a ‘Maths Autobiography’. If required, provide the students with some sentence starters such as “I think maths is…” “The thing I like best about maths is…” “The thing or things that worry me about maths is…” They could do this in different formats:

  • In a maths journal
  • Making a video
  • Using drawings (great for young children – a drawing can provide lots of information)
  1. Start off on a positive note

Have some fun with your maths lessons. I would strongly recommend that you don’t start the year with a maths test! If you want to do some early assessment, consider using open-ended tasks or some rich mathematical investigations. Often these types of assessments will provide much deeper insights into the abilities of your students. You can even use some maths games (either concrete or digital) to assess the abilities of your students.

A great maths activity for the first lesson of the year is getting-to-know-you-mathematically, where students use a pattern block and then need to go on a hunt to find other students who have specific mathematical attributes. Encourage your students to find someone different for every attribute on the list, and change the list to suit the age and ability of your students. For example, in the younger years you could use illustrations and not words. In the older years, you could make the mathematics more abstract.

  1. Take a fresh look at the curriculum

Even if you’ve been teaching for many years, it’s always good to take a fresh new look at the curriculum at the start of each year. Consider how the Proficiencies or Working Mathematically processes can be the foundation of the content that you’re teaching. For example, how can you make problem solving a central part of your lessons?
Take a close look at the General Capabilities. They provide a perfect foundation for contextual, relevant tasks that allow you to teach mathematics and integrate with other content areas.

  1. Consider the resources you use: Get rid of the worksheets!

Think about using a range of resources in your mathematics teaching. Regardless of their age or ability, children benefit from using concrete manipulatives. Have materials available for students to use when and if they need them. This includes calculators in early primary classrooms, where students can explore patterns in numbers, place value and lots of other powerful concepts using calculators.

Children’s literature is also a great resource. A wonderful book to start off the year is Math Curse by Jon Scieska and Lane Smith. Read the book to your students either in one sitting or bit by bit. There are lots of lesson ideas within the pages. Ask your students to write their own maths curse. It’s a great way to illustrate that mathematics underpins everything we do! It’s also a great way to gain insight into how your students view mathematics and what they understand about mathematics.

  1. How will you use technology in the classroom?

If you don’t already integrate technology into your mathematics lessons, then it’s time to start. Not only is it a curriculum requirement, it is part of students’ everyday lives – we need to make efforts to link students’ lives to what happens in the classroom and one way to do that is by using technology. Whether it’s websites, apps, YouTube videos, screencasting, just make sure that you have a clear purpose for using the technology. What mathematics will your students be learning or practicing, and how will you assess their learning?

  1. Reach out to parents

As challenging as it may be, it’s vital that parents play an active role in your students’ mathematical education. They too may suffer from anxiety around mathematics so it’s helpful to invite them into the classroom or hold mathematics workshops where parents can experience contemporary teaching practices that their students are experiencing at school. Most importantly, you need to communicate to parents that they must try really hard to be positive about mathematics!

These are just a few tips to begin the year with…my next blog post will discuss lesson structure. In the meantime, enjoy the beginning of the school year and:

Be engaged in your teaching.

Engaged teachers = engaged students.

 

 

Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34(3), 1-14.

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.

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.

OLYMPUS DIGITAL CAMERA

 

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?’.