Teller (2016), makes a powerful point about teaching and engagement, and how important it is that we, as teachers, portray positive attitudes towards our subject and towards teaching it. Do you consider yourself an engaged teacher? Are your students deeply engaged with mathematics, and how do you know? In education we talk about student engagement every day, but what do we actually mean when we use the term ‘engagement’? When does real engagement occur, and how do we, as teachers, influence that engagement? In this post, I will define the construct of engagement and pose some questions that will prompt you to reflect on how your teaching practices and the way you interpret the curriculum, influences your own engagement with the teaching of mathematics and, as a result, the engagement of your students.

**Student Engagement: On Task vs. In Task**

In education, engagement is a term used to describe students’ levels of involvement with teaching and learning. Engagement can be defined as a multidimensional construct, consisting of operative, cognitive, and affective domains. Operative engagement encompasses the idea of active participation and involvement in academic and social activities, and is considered crucial for the achievement of positive academic outcomes. Affective engagement includes students’ reactions to school, teachers, peers and academics, influencing willingness to become involved in school work. Cognitive engagement involves the idea of investment, recognition of the value of learning and a willingness to go beyond the minimum requirements

It’s easy to fall into the trap of thinking that students are engaged when they appear to be busy working and are on task. True engagement is much deeper – it is ‘in task’ behaviour, where all three dimensions of engagement; cognitive, operative, and affective, come together (see figure 1). This leads to students valuing and enjoying school mathematics and seeing connections between the mathematics they do at school and the mathematics they use in their lives outside school. Put simply, engagement occurs when students are thinking hard, working hard, and feeling good about learning mathematics.

There are a range of influences on student engagement. Family, peers, and societal stereotypes have some degree of influence. Curriculum and school culture also play a role. Arguably, it is teachers who have a powerful influence on students’ engagement with mathematics (Anthony & Walshaw, 2009; Hattie, 2003). Classroom pedagogy, the actions involved in teaching, is one aspect of a broader perspective of the knowledge a teacher requires in order to be effective. The knowledge of what to teach, how to teach it and how students learn is referred to as pedagogical content knowledge (PCK). The construct of PCK was originally introduced by Schulman (1986), and substantial research building on this work has seen a strong focus on PCK in terms of mathematics teaching and learning (Delaney, Ball, Hill, Schilling, & Zopf, 2008; Hill, Ball, & Schilling, 2008; Neubrand, Seago, Agudelo-Valderrama, DeBlois, & Leikin, 2009). Although this research provides insight into the complex knowledge required to effectively teach mathematics, little attention is paid to how teachers themselves are engaged with teachers.

**Engaged Teachers = Engaged Students**

It makes sense that teachers need to be engaged with the act of teaching in order to effectively engage their students. If we take the definition of student engagement and translate it to a teaching perspective, perhaps it would look something like Figure 2, where teachers are fully invested in teaching mathematics, work collaboratively with colleagues to design meaningful and relevant tasks, go beyond the minimum requirements of delivering curriculum, and genuinely enjoy teaching mathematics in a way that makes a difference to students. In other words, thinking hard, working hard, and feeling good about teaching mathematics.

**Are you an engaged teacher?**

Teaching is a complex practice with many challenges. Teaching mathematics has the additional challenge of breaking down many stereotypical beliefs about mathematics as being difficult and only for ‘smart’ people, mathematics viewed as black and white/right or wrong, and mathematics as a simply focused on arithmetic, to name a few. However, there are elements of our day to day work that we can actively engage with to disrupt those stereotypes, make teaching more enjoyable, and promote deeper student engagement. The following section provides some thoughts and questions for reflection.

*Curriculum*

How do you interpret the curriculum? Do you view it has a series of isolated topics to be taught/learned in a particular order, or do you see it has a collection of big ideas with conceptual relationships within and amongst the strands? How do you incorporate the General Capabilities and Cross-curriculum priorities in your teaching? Do you make the Working Mathematically components a central part of your teaching?

*Planning*

How do you plan for the teaching of mathematics? Does your school have a scope and sequence document that allows you to cater to emerging student needs? Does the scope and sequence document acknowledge the big ideas of mathematics or does it unintentionally steer teachers into treating topics/concepts in isolation?

*Assessment*

How often do you assess? Are you students suffering from assessment fatigue and anxiety? Do you offer a range of assessment tasks beyond the traditional pen and paper test? Do your questions/tasks provide opportunities for students to apply the Working Mathematically components?

*Tasks*

What gets you excited about teaching mathematics? Do you implement the types of tasks that you would get you engaged as a mathematician? Do your tasks have relevance and purpose? Do you include variety and choice within your task design? Do you take into account the interests of your students when you plan tasks? Do you incorporate student reflection into your tasks?

*Grouping*

How do you group your students? There are many arguments that support mixed ability grouping, yet there are also times when ability grouping is required. Is the way you group your students giving them unintended messages about ability and limiting their potential?

*Technology*

How do you use digital technology to enhance teaching and learning in your classroom? Do you take advantage of emerging technologies and applications? Do you use digital technology in ways that require students to create rather than simply consume?

*Professional Learning*

How do you incorporate professional learning into your role as an educator? Do you actively pursue professional learning opportunities, and do you apply what you have learned to your practice? Do you share what you have learned with your colleagues, promoting a community of practice within your teaching context?

There are many other aspects of teaching mathematics that influence our engagement as teachers, and of course, the engagement of our students. Many factors, such as other non-academic school-related responsibilities, are bound to have some influence over our engagement with teaching. However, every now and then it is useful to stop and reflect on how our levels of engagement, our enthusiasm and passion for the teaching of mathematics, can make a difference to the engagement, and ultimately the academic outcomes, of our students.

**References:**

Anthony, G., & Walshaw, M. (2009). *Effective pedagogy in mathematics* (Vol. 19). Belley, France.

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.

Delaney, S., Ball, D. L., Hill, H. C., Schilling, S. G., & Zopf, D. (2008). “Mathematical knowledge for teaching”: Adapting U.S. measures for use in Ireland. *Journal for Mathematics Teacher Education, 11*(3), 171-197.

Hattie, J. (2003). *Teachers make a difference: What is the research evidence?* Paper presented at the Building Teacher Quality: The ACER Annual Conference, Melbourne, Australia.

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualising and measuring teachers’ topic-specific knowledge of students. *Journal for Research in Mathematics Education, 39*(4), 372-400.

Neubrand, M., Seago, N., Agudelo-Valderrama, C., DeBlois, L., & Leikin, R. (2009). The balance of teacher knowledge: Mathematics and pedagogy. In T. Wood (Ed.), *The professional education and development of teachers of mathematics: The 15th ICMI study* (pp. 211-225). New York: Springer.

Teller, R. (2016) Teaching: Just like performing magic. Retrieved from http://www.theatlantic.com/education/archive/2016/01/what-classrooms-can-learn-from-magic/425100/?utm_source=SFTwitter

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Thousands of children in Australian schools have recently sat the national literacy and numeracy test (NAPLAN), and many teachers have been busy administering a whole range of assessments because it’s report writing season and boxes need to be ticked. My question is, how often do we ask ourselves why we’re assessing? What are we doing with the results apart from using them for reporting purposes? I’ve spent quite a bit of time in schools lately, and after talking to lots teachers and seeing a range of mathematics assessment tasks and work samples, I’ve begun to reflect on some of the things we could do better.

“Effective pedagogy requires effective assessment, assessment that provides the critical links between what is valued as learning, ways of learning, ways of identifying need and improvement, and perhaps most significantly, ways of bridging school and other communities of practice” (Wyatt-Smith, Cumming, Elkins, & Colbert, 2010, p. 320)

It’s through our assessment we communicate most clearly to students those learning outcomes we value, yet it’s often held that no subject is as associated with its form of assessment as is mathematics (Clarke, 2003). Assessment practices in mathematics often consist of formal methods such as tests and examinations (Wiliam, 2007), and it’s believed that such strategies need as much consideration for renewal as does content and classroom pedagogy. Although lots of progress has been made in terms of improving mathematics teaching and learning and curriculum, many such improvements have failed due to a mismatch between assessment practices and pedagogy (Bernstein, 1996; Pegg, 2003). It’s been suggested that in mathematics, there should not be more assessment, but more appropriate assessment strategies implemented to inform learning and teaching as well as report on progress and achievement (Australian Association of Mathematics Teachers, 2008; Clarke, 2003). And this is the point I want to highlight – *assessment to inform teaching*. Regardless of the type of assessments we use, are we using assessment data in the right way?

What do you do with your assessment work samples? Do you simply use the scores to determine how students are grouped, or what aspects of a topic you need to cover? How often do we, as teachers, take the time to analyse the work samples in order to identify specific misconceptions? Imagine a scenario where students are grouped according to assessment scores. Each of those groups are then exposed to pedagogies intended to address the ‘level’ of the group. What if, within each group, there were a range of misconceptions? And what about the top groups? What if work samples that resulted in accurate answers exposed misconceptions despite being correct?

When students transition from one level of schooling to another, it’s not uncommon to hear teachers complaining about the broad range of abilities, and more specifically, those students who appear not to have achieved the most basic skills. How have these students managed to get to kindergarten/Year 3/Year 6/high school/university without knowing how to……? Mathematics content is hierarchical – when students miss out on learning concepts in the early years, the gaps in knowledge continue to widen as they progress through school. Whether caused by inattention, absence from school, or any other reason, students find it hard to catch up when they’re missing pieces of the mathematical jigsaw puzzle. It’s like building a house on faulty foundations.

So how can we fix this? A teacher recently told me that she didn’t have time to analyse the responses in an assessment task. Isn’t this our job? How can we manage workloads so that teachers have the time to really think about where students are going wrong, and how can teachers access professional learning to assist them in being able to identify and address students’ misconceptions?

I think one way we can address this situation is to think carefully about the design and the quantity of assessment tasks. Administer *fewer, better quality tasks that are designed to assess both the content and the processes of mathematics*. That is, tasks that require students to show their working, explain their thinking, and produce an answer. The more they show, the more we see. Another strategy to assist teachers is to provide time for teachers to look at assessment samples and analyse them collaboratively, discussing the identified misconceptions and planning strategically to address them.

The knowledge that teachers need to effectively teach mathematics is special. We need to know more about mathematics than the average person – we need to understand where, why and when our students are likely to go wrong, so we can either avoid misconceptions occurring, or address them when they do. This specialist knowledge comes from continued professional learning and collaboration with peers. Don’t just rely on the curriculum documents – we need to look beyond this to ensure we have that specialist knowledge.

This post posed more questions than answers in relation to assessment in the mathematics classroom. Hopefully it will spark some conversation and thinking about what we are doing with the assessment work samples we gather, regardless of why type of assessments they are. If we don’t try and change the way we use assessment, we’ll always have those students who will struggle with mathematics, and while there will always be a range of achievement levels in every group of students, that doesn’t mean we shouldn’t keep trying to close those gaps!

References:

Australian Association of Mathematics Teachers. (2008). The practice of assessing mathematics learning. Adelaide, SA: AAMT Inc.

Bernstein, B. (1996). *Pedagogy, symbolic control and identity: Theory, research, critique*. London: Taylor and Francis.

Clarke, D. (2003, 4-5 December). *Challenging and engaging students in worthwhile mathematics in the middle years.* Paper presented at the Mathematics Association of Victoria Annual Conference: Making Mathematicians, Melbourne.

Pegg, J. (2003). Assessment in mathematics. In J. M. Royer (Ed.), *Mathematical cognition* (pp. 227-260). Greenwich, CT: Information Age Publishing.

Wiliam, D. (2007). Keeping learning on track: Classroom assessment and the regulation of learning. In F. K. J. Lester (Ed.), *Second handbook of mathematics teaching and learning* (pp. 1053-1098). Greenwich, CT: Information Age Publishing.

Wyatt-Smith, C. M., Cumming, J., Elkins, J., & Colbert, P. (2010). Redesigning assessment. In D. Pendergast & N. Bahr (Eds.), *Teaching middle years: Rethinking curriculum, pedagogy and assessment* (2nd ed., pp. 319-379). Crows Nest, NSW: Allen & Unwin.

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Here are some of the photos with their accompanying questions:

Beyblades:

*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?**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:

*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?**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?**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?**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:

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

**D id 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!

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“Effective pedagogy requires effective assessment, assessment that provides the critical links between what is valued as learning, ways of learning, ways of identifying need and improvement, and perhaps most significantly, ways of bridging school and other communities of practice” (Wyatt-Smith, Cumming, Elkins, & Colbert, 2010, p. 320)

It’s through our assessment we communicate most clearly to students those learning outcomes we value, yet it’s often held that no subject is as associated with its form of assessment as is mathematics (Clarke, 2003). Assessment practices in mathematics often consist of formal methods such as tests and examinations (Wiliam, 2007), and it’s believed that such strategies need as much consideration for renewal as does content and classroom pedagogy. Although lots of progress has been made in terms of improving mathematics teaching and learning and curriculum, many such improvements have failed due to a mismatch between assessment practices and pedagogy (Bernstein, 1996; Pegg, 2003). It’s been suggested that in mathematics, there should not be more assessment, but more appropriate assessment strategies implemented to inform learning and teaching as well as report on progress and achievement (Australian Association of Mathematics Teachers, 2008; Clarke, 2003). And this is one of the points I want to highlight – *assessment to inform teaching*. Regardless of the type of assessments we use, are we using assessment data in the right way?

What do you do with your assessment work samples? Do you simply use the scores to determine how students are grouped, or what aspects of a topic you need to cover? How often do we, as teachers, take the time to analyse the work samples in order to identify specific misconceptions? Imagine a scenario where students are grouped according to assessment scores. Each of those groups are then exposed to pedagogies intended to address the ‘level’ of the group. What if, within each group, there were a range of misconceptions? And what about the top groups? What if work samples that resulted in accurate answers exposed misconceptions despite being correct?

When students transition from one level of schooling to another, it’s not uncommon to hear teachers complaining about the broad range of abilities, and more specifically, those students who appear not to have achieved the most basic skills. How have these students managed to get to kindergarten/Year 3/Year 6/high school/university without knowing how to……? Mathematics content is hierarchical – when students miss out on learning concepts in the early years, the gaps in knowledge continue to widen as they progress through school. Whether caused by inattention, absence from school, or any other reason, students find it hard to catch up when they’re missing pieces of the mathematical jigsaw puzzle. It’s like building a house on faulty foundations.

So how can we fix this? A teacher recently told me that she didn’t have time to analyse the responses in an assessment task. Isn’t this our job? How can we manage workloads so that teachers have the time to really think about where students are going wrong, and how can teachers access professional learning to assist them in being able to identify and address students’ misconceptions?

Another concern is related to the quality of assessment tasks. I have seen many tasks that are poorly worded or poorly set out, or have diagrams that can only lead to confusion or misconceptions. Often tasks test mathematical content but do not provide opportunities for students to express their reasoning. A student can achieve a correct answer while maintaining a misconception – if we don’t ask them about their thinking, are we really assessing their true ability?

I think one way we can address these issues is to think carefully about the design and the quantity of assessment tasks. Administer *fewer, better quality tasks that are designed to assess both the content and the processes of mathematics*. That is, tasks that require students to show their working, explain their thinking, and produce an answer. The more they show, the more we see. Another strategy to assist teachers is to provide time for teachers to look at assessment samples and analyse them collaboratively, discussing the identified misconceptions and planning strategically to address them.

The knowledge that teachers need to effectively teach mathematics is special. We need to know more about mathematics than the average person – we need to understand where, why and when our students are likely to go wrong, so we can either avoid misconceptions occurring, or address them when they do. This specialist knowledge comes from continued professional learning and collaboration with peers. Don’t just rely on the curriculum documents – we need to look beyond this to ensure we have that specialist knowledge.

This post posed more questions than answers in relation to assessment in the mathematics classroom. Hopefully it will spark some conversation and thinking about what we are doing with the assessment work samples we gather, regardless of why type of assessments they are. If we don’t try and change the way we use assessment, we’ll always have those students who will struggle with mathematics, and while there will always be a range of achievement levels in every group of students, that doesn’t mean we shouldn’t keep trying to close those gaps!

References:

Australian Association of Mathematics Teachers. (2008). The practice of assessing mathematics learning. Adelaide, SA: AAMT Inc.

Bernstein, B. (1996). *Pedagogy, symbolic control and identity: Theory, research, critique*. London: Taylor and Francis.

Clarke, D. (2003, 4-5 December). *Challenging and engaging students in worthwhile mathematics in the middle years.* Paper presented at the Mathematics Association of Victoria Annual Conference: Making Mathematicians, Melbourne.

Pegg, J. (2003). Assessment in mathematics. In J. M. Royer (Ed.), *Mathematical cognition* (pp. 227-260). Greenwich, CT: Information Age Publishing.

Wiliam, D. (2007). Keeping learning on track: Classroom assessment and the regulation of learning. In F. K. J. Lester (Ed.), *Second handbook of mathematics teaching and learning* (pp. 1053-1098). Greenwich, CT: Information Age Publishing.

Wyatt-Smith, C. M., Cumming, J., Elkins, J., & Colbert, P. (2010). Redesigning assessment. In D. Pendergast & N. Bahr (Eds.), *Teaching middle years: Rethinking curriculum, pedagogy and assessment* (2nd ed., pp. 319-379). Crows Nest, NSW: Allen & Unwin.

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The following is a list of what I consider to be important ingredients when planning and teaching a successful mathematics lesson. The list (or recipe) is split into two: 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)**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 you are teaching, provide opportunities for all students to achieve success, and be enthusiastic and passionate about mathematics!

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**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:

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.

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Late in 2016 there were reports from international testing that Australia continues to slip further behind in mathematics when compared to other countries. So, what can you do about this? Relying on teachers alone won’t fix the problem. There are many things parents can and should do to help their children learn mathematics, particularly before they begin school and during the primary school years. The following is a list of tips for parents that will help them to help their children succeed:

**Be positive about maths!**

May people openly claim they don’t like maths or they’re not good at it, unintentionally conveying the message that this is okay. Unfortunately, this can have a detrimental effect on the children who hear these messages. In my research on student engagement, children whose parents made similar comments often used the same comments as mathematics became more challenging during the high school years. These behaviours can lead to children opting to stop trying and drop out of mathematics as soon as they can, ultimately limiting their life choices.

As a parent, be conscious of displaying positive attitudes towards mathematics, even when it’s challenging. Adopting what is referred to as a ‘growth mindset’ allows children (and parents) to acknowledge that mathematics is challenging, but not impossible. Rather than saying “I can’t do it” or “it’s too hard”, encourage statements such as “I can’t do it yet” or “let’s work on this together”. If you’re struggling with the mathematics yourself, and finding it difficult to support your child, there are options such as free online courses like Jo Boaler’s YouCubed website (www.youcubed.org), apps such as Khan Academy, or you can seek help from their child’s teacher.

If you choose to use a tutor to help your child, make sure it’s a tutor who knows how to teach for understanding, rather than memorisation. Too often tutoring colleges use the traditional teaching method of drill and practice, which won’t help a struggling student to understand important mathematical concepts. Find a tutor who understands the curriculum and can tailor a program to work alongside what your child is learning at school.

**Developing a positive working relationship with teachers**

It’s important for parents to work with their child’s teacher to ensure they are able to support the learning of mathematics. This will help the teacher understand the child’s needs and be better able to support the child in the classroom, while at the same time helping the parents support the child at home. Often schools hold information evenings or maths workshops to help explain current teaching methods with few parents turning up. It’s important to attend these events as they are a good opportunity to learn ways to help children with mathematics at home.

**Know what maths your child is learning**

Mathematics teaching and learning has changed significantly over the last few decades. Unfortunately, many of the older generations still expect children to be learning the same maths in the same way, regardless of how much the world has changed! Access to the mathematics curriculum is free to everyone. Parents have the opportunity to find out what their child should be learning simply by accessing the curriculum online, or talking to their child’s teacher. This can help parents who may have unrealistic expectations of what their child should know and be able to do, and will also help them understand that mathematics is not just about numbers or learning the multiplication tables.

One of the most common complaints when it comes to school mathematics is that children don’t ‘know’ their multiplication tables. Is this important? Yes, it’s still important that children gain fluency when dealing with numbers. However, it’s also important that we don’t just rely on rote learning, or repetition. Children need to understand how the numbers work. In other words, they need to be numerate, and have a flexibility with numbers. Once they understand, then fluency can be built. Using maths games (there are lots of apps that help with this) is a good way of getting children to build up speed with number facts.

**Make maths part of everyday activities**

Bring maths into daily conversations and activities with your child. After all, there’s maths in everything we do. For example, if you’re cooking you might ask your child to help you measure out ingredients. If you’re shopping, you could have a little competition to see who can make the best estimation of the total grocery bill or perhaps ask your child to work out the amount of change (this may be challenging given that we use credit cards most of the time).

If your child likes to play digital games, download some maths apps so they can use their screen time to learn while having fun at the same time. Alternatively, traditional games can provide opportunities to talk about maths and help your child. Games that use dominoes and playing cards are great for young children as are board games such as Snakes and Ladders or Monopoly. Even non-numerical games such as Guess Who have benefits for mathematics because the promote problem solving and strategic thinking, important mathematical skills.

Parents who can work with their child’s teacher, be proactive in their child’s education, and demonstrate positive attitudes towards mathematics can make a *big difference *to their child’s success at school. **It’s an investment worth making.**

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**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.

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Short activities:

- 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?
- Draw a picture of a Christmas tree. Use your drawing as a plan to show where you will place the decorations.
- 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?
- 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?
- 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:

- Plan a Christmas party for some of your friends. Show all the mathematics that you need to use for your planning.
- 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.
- 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:

- Investigate and research the tradition of decorating a tree for Christmas. Answer questions such as “When did the tradition start?”
- Plan menus for the meals for family for Christmas Day and Boxing Day and include a budget.
- 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!

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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:

- 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?
- 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?
- 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?
- How much tinsel would you need to decorate the Christmas tree?
- 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?
- 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:

- 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.
- 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?
- 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:

- Christmas Day always takes place on the 25
^{th}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? - 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.
- 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.

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