All posts by cattard2017

Tips for Teachers: Setting up Your Students for Mathematical Success

Engaging Maths

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.

  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…

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The 12 Days of Christmas: A Maths Task!

As you may have already realised, I am a great believer in providing a context for the learning of mathematics. At this time of year it is especially challenging to keep students engaged in learning, so you may be interested in this set of activities based on The Twelve Days of Christmas. The activities can be adapted to suit students from any grade and is only limited to the imagination. The following suggestions would be appropriate for upper primary/lower secondary students and can either be used as investigation starters or activity task cards. Enjoy!

Resources Required:

  • A copy of the lyrics of “The Twelve Days of Christmas”
  • The book “The Twelve Days of Christmas” (There are several versions available)
  • Price Lists
  • Other resources as required, eg. shopping catalogues
  • Possible Investigations starters/Task cards

Teaching/Learning Activities:

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

Examples of possible questions are:

  1. What is the total number of gifts given?
  2. Is there an easy way to work this out?
  3. What is the total cost of the gifts?
  4. If the department store was holding a pre-Christmas sale and offered a 15% discount for all purchases, what would the new cost be? (This does not include performers, maids, lords etc)
  5. What if the discount offered only applied to live animals?
  6. The maids give a 10% when booked for more than two consecutive days (based on a 7.5 hour working day). What would their new fee be?
  7. The musicians charge a 10% Goods and Service tax and this must be added to the total cost.
  8. How many people arrive at the true love’s house on the twelfth day?
  9. What would it cost to feed all the people and the animals? (Internet would come in handy here!)
  10. Use some Christmas shopping catalogues to replace the gifts with something more appropriate for a modern true love and calculate the cost.
  11. Re-write the lyrics to fit your new list of gifts.
  12. What if there were 15/20/100 days of Christmas? How many gifts would there be? Can you find a pattern to help you work this out?

 

GIFT PRICE LIST

 

GIFT

Cost

$

 

A partridge in a pear tree

 

150.00

 

One turtle dove

 

75.00

 

One French hen

 

7.50

 

One calling bird

 

65.00

 

A simple gold ring

 

180.00

 

One goose a-laying

 

32.50

 

One swimming swan

 

300.00

 

A milkmaid hired for 1 hour

 

12.50

 

One lady dancing

 

375.00

 

One Lord a-leapin’

 

385.00

 

One Piper piping

 

225.00

 

One Drummer drumming

 

145.50

 

Using Children’s Literature for Mathematical Investigations

This week in Australia is The Children’s Book Council of Australia Book Week. Why not celebrate by using children’s literature to teach mathematics?

Engaging Maths

How often do you progress from problem solving to investigation-based work in your mathematics classroom? Have you ever considered using children’s literature in your mathematics lessons to provide an interesting and creative context for mathematical exploration?

The idea of investigation is fundamental both to the study of mathematics itself and also to an understanding of the ways in which mathematics can be used to extend knowledge and to solve problems in very many fields” (Cockroft, 1981, p.250). 

Mathematical investigations move beyond problem solving, yet are not ‘project work’ They are inquiry based and support a constructivist approach to learning in which learners actively construct their own knowledge through reflection on physical and mental actions. During investigation-based work, learning is placed within a purposeful context that requires students to engage in mathematics by combining content knowledge with higher order thinking skills and creativity. Investigations provide insights into the work…

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Engaging children with mathematics: Are you an engaged teacher?

“The first job of a teacher is to make the student fall in love with the subject. That doesn’t have to be done by waving your arms and prancing around the classroom; there’s all sorts of ways to go at it, but no matter what, you are a symbol of the subject in the students’ minds” (Teller, 2016).

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.

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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.

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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

Improving primary mathematics: The challenge of curriculum

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

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

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

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

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

 

Using mathematical inquiry to make mathematics meaningful

At the moment I’m involved in a project with the Sydney Metro (Transport for NSW), currently the largest infrastructure project in Australia. When complete, the Sydney Metro project is going to change the way many Sydney residents work, live and socialise. My involvement with this project has required me to design, deliver and research the effectiveness of a professional learning program. In this program, teachers from all stages of schooling and a range of curriculum areas learn about using inquiry based learning and then design, implement and evaluate units of work that use the Sydney Metro project as the stimulus for inquiry.  So what’s that got to do with engaging maths? My work in this project has confirmed what I’ve always believed – contextualising learning makes mathematics (and other disciplines, of course) more meaningful, purposeful and relevant for students. It shifts the traditional approach of ‘just in case’ learning to ‘just in time learning’.

Using contexts from student’s lives, such as Sydney Metro, makes mathematics come alive. For example, some of the students participating in the inquiry based units are looking at the social implications of having a rail station constructed in their community where there previously wasn’t one. This inquiry provides a purpose for designing survey questions, collecting, representing and analysing data that has meaning and purpose. Others are looking at the engineering aspects of the project relating to the tunnelling that is currently underway. Some are working on design aspects relating to the trains themselves or the stations and some are looking at mapping – planning future metro lines, or timing (the system won’t have a timetable).

The possibilities are endless, but for these units of work (or indeed, any inquiry based unit of work) to be successful, the teachers planning them have to consider carefully the potential directions that students will take their inquiry if the units are to be true inquiry based learning that is driven by students’ interests. This requires a strong knowledge of curriculum and a willingness to hand over some control of the learning to the students. It may even involve the introduction of content beyond the students’ current grade.
Another consideration when planning inquiry units is the inclusion of other aspects of our curriculum, beyond content. For example, in mathematics we have the Proficiencies (Working Mathematically in NSW) that represent the processes of mathematics. It’s impossible to conduct inquiry based learning without these processes and inquiry learning is a perfect opportunity to develop, refine and show evidence of these processes. Then we have the General Capabilities. Again, inquiry based learning provides an opportunity to access mathematics while accessing these capabilities, enhancing the relevance of the learning.

Where do you find resources for inquiry? Take a look around at what is happening in your community, in the media, or simply the things that your students are interested in. Consider how those things could spark curiosity in your students (or how you could promote that curiosity within  your students). Model how to ask good questions (students need to know how to do this – it doesn’t always come naturally). Be prepared for it to get messy, search for resources that the students might need or help them find resources. Be prepared to teach a range of mathematical concepts as the need arises.

I’ll you with an example of a resource that I believe would be a great stimulus for inquiry – take a look, at let me know what you think! Every Drop Counts

What does maths engagement look like?

BTP Blog

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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