All posts by cattard2017

What does maths engagement look like?

May 22, 2018 cattard2017 Leave a comment

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.

Catherine Attard

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…

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Education, mathematics, Professional development, teachers

Teachers and Mathematics: Making the most of professional development

April 10, 2018 cattard2017 1 Comment

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.

Active Participation

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.

Reflection

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.

Education, Home Schooling, mathematics, Mathematics lessons, Professional development, teachers, Uncategorized

Tips for beginning primary teachers: What’s in your maths toolbox?

March 2, 2018 cattard2017 Leave a comment

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!

Another ‘must have’ for beginning teachers is a bank of good quality resource books. Don’t fall into the trap of purchasing Black Line Masters or books full of worksheets to photocopy. You don’t want your students to be disengaged! Books such as my Engaging Maths series (http://engagingmaths.co/teaching-resources/books/ ), or any of Paul Swan’s books or resources (http://www.drpaulswan.com.au/resources/) are a great place to start. Explore some of the excellent free resources available online such as http://nrich.maths.org/teacher-primary and http://illuminations.nctm.org/, but do be aware that some resources produced outside of Australia will need to adapted for the Australian Curriculum: Mathematics.

In my 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.

Education, Home Schooling, mathematics, Mathematics lessons, parents, teachers, technology

More tips for teachers: Essential materials for every mathematics classroom

February 18, 2018 cattard2017 Leave a comment

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):

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 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!)
Playing cards
Unifix blocks
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.

Assessment, Education, Home Schooling, mathematics, Mathematics lessons, Problem Solving, Professional development, teachers

Tips for Teachers: Critical ingredients for a successful mathematics lesson

February 8, 2018 cattard2017 1 Comment

What are the ingredients for an effective mathematics lesson? Teachers are continually faced with a range of advice or ideas to improve their mathematics lessons and often this just creates confusion. It’s a little bit like being a cook. New recipes appear online and in cookbooks on bookstore shelves, but often they’re just adaptations of classic recipes that have been around before, their foundation ingredients are tried and tested, and often evidence based. There are always the staple ingredients and methods that are required for the meal to be successful.

The following is a list of what I consider to be important ingredients when planning and teaching an effective mathematics lesson. The list (or recipe) is split into two parts: lesson planning and lesson structure.

Lesson planning:

Be clear about your goal. What exactly do you want your students to learn in this lesson? How are you going to integrate mathematical content with mathematical processes? (The proficiencies or Working Mathematically components) Will you consider the General Capabilities in your planning?
Know the mathematics. If you don’t have a deep understanding of the mathematics or how students learn that aspect of mathematics, how can you teach it effectively? Where does the mathematics link across the various strands within the mathematics curriculum?
Choose good resources. Whether they are digital or concrete materials, make sure they are the right ones for the job. Are they going to enhance students’ learning, or will they cause confusion? Be very critical about the resources you use, and don’t use them just because you have them available to you!
Select appropriate and purposeful tasks. Is it better to have one or two rich tasks or problems, or pages of worksheets that involve lots of repetition? Hopefully you’ve selected the first option – it is better to have fewer, high quality tasks rather than the traditional worksheet or text book page. You also need to select tasks that are going to promote lots of thinking and discussion.
Less is more. We often overestimate what students will be able to do in the length one lesson. We need to make sure students have time to think, so don’t cram in too many activities.
You don’t have to start and finish a task in one lesson. Don’t feel that every lesson needs to be self-contained. Children (and adults) often need time to work on complex problems and tasks – asking students to begin and end a task within a short period of time often doesn’t give them time to become deeply engaged in the mathematics. Mathematics is not a race!

Lesson Structure:

Begin with a hook. How are you going to engage your students to ensure their brains are switched on and ready to think mathematically from the start of each lesson? There are lots of ways to get students hooked into the lesson, and it’s a good idea to change the type of hook you use to avoid boredom. Things like mathematically interesting photographs, YouTube clips, problems, newspaper articles or even a strategy such as number busting are all good strategies.
Introduction: Make links to prior learning. Ensure you make some links to mathematics content or processes from prior learning – this will make the lesson more meaningful for students and will reassure anxious students. Use this time to find out what students recall about the particular topic – avoid being the focus of attention and share the lesson with students. Talk about why the topic of the lesson is important – where else does it link within the curriculum, and beyond, into real life?
Make your intentions clear. Let students know what they’re doing why they’re doing it. How and where is knowing this mathematics going to help them?
Body: This is a good time for some collaboration, problem solving and mathematical investigation. It’s a time to get students to apply what they know, and make links to prior learning and across the mathematics curriculum. This is also a time to be providing differentiation to ensure all student needs are addressed.
Closure: This is probably the most important time in any mathematics lesson. You must always include reflection. This provides an opportunity for students to think deeply about what they have learned, to make connections, and to pose questions. It’s also a powerful way for you, the teacher, to collect important evidence of learning. Reflection can be individual, in groups, and can be oral or written. It doesn’t matter, as long as it happens every single lesson.

There are many variables to the ingredients for a good mathematics lesson, but most importantly, know what and how you are teaching, provide opportunities for all students to achieve success, and be enthusiastic and passionate about mathematics!

Assessment, Education, Home Schooling, mathematics, Mathematics lessons, parents, Problem Solving, Professional development, teachers

Beach Towels and Pencil Cases: Interesting, Inquiry-based Mathematical Investigations

January 25, 2018 cattard2017 Leave a comment

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!

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:

https://honey.nine.com.au/2018/01/19/14/35/pencil-case-missing-letter

Short activities:

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?

Investigations:

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.

Extension Activities:

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?

Education, mathematics, Mathematics lessons, Professional development, teachers

Tips for Teachers: Setting up Your Students for Mathematical Success

January 22, 2018 cattard2017 2 Comments

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 this year to make your work more engaging for both you and your students. The following are some ideas to consider.

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.

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)

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.

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.

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.

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?

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.