Tag Archives: middle years

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.

 

More tips for teachers: Essential materials for every mathematics classroom

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.

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

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

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Short activities:

  1. 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?
  2. 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.
  3. Faber Castell is a famous brand of pencils. Investigate the history of Faber Castell and illustrate this on a timeline.
  4. 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?
  5. 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:

  1. 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?
  2. 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?
  3. 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?
  4. Design and make a pencil case to suit your individual stationery needs. Write about the mathematics you use to do this.

Extension Activities:

  1. Design a new and improved pencil and explain the changes you have made.
  2. Design, justify, and create a marketing campaign for a new, ‘miracle’ pen.
  3. 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?IMG_4837

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?

Critical Thinking, Mathematics, and McDonald’s

You might be wondering what McDonald’s has to do with mathematics and critical thinking. Recently I found a copy of the original McDonald’s price list dating back to the 1940s when McDonald’s was owned by the original founders, Dick and Mac McDonald. Since that time, the fast food franchise has become a global fast food brand recognised by most. It is because of this recognition that the 1940s menu makes a perfect stimulus for mathematical investigation and critical thinking. The links between mathematics and children’s lives are not always obvious for students, so opportunities such as this are important to ensure our students understand how mathematics can help to make important decisions that affect our finances, health and general well-being. Although you might consider rejecting this idea so as not to promote a fast food culture, consider this an opportunity for students to think critically about food choices.

The Maths and McDonald’s graphic below contains some suggestions for mathematical investigations and would best be suited to students in upper primary or lower secondary classrooms. However, they can be adapted quite easily for younger students.

Maths & McDonald_s (3)

Below, the prompts are listed in a table that details some of the potential mathematical content that students would be expected to apply, and the processes they would use in the application of the mathematics. Although not included in the table, the tasks also address several of the General Capabilities from the Australian Curriculum: Mathematics. In addition, the tasks lend themselves well to integration with other curriculum areas.

Investigation

Use Mathematics to:

Mathematical Content

 

 

Processes
(Working Mathematically components/Proficiencies)
Notes

 

 

Investigate how prices have changed over time (comparing similar items) · Addition

· Subtraction

· Fractions (percentages)

· Problem Solving

· Reasoning

· Communicating

· Fluency

· Understanding

· Provide access to Internet where possible to allow students to compare current prices

· Students could access census information to explore changes in cost of living

Explore the popularity of McDonald’s food compared to other fast food options · Statistics · Reasoning

·Communicating

·Fluency

·Understanding

· Students will need to spend time considering appropriate questions to ask

· Encourage students to analyse data and formulate conclusions resulting from the data

Analyse the nutritional value of a McDonald’s meal compared to a typical home cooked meal · Addition· Subtraction

· Multiplication· Division· Fractions

·    Problem Solving·    Reasoning·    Communicating·    Fluency·    Understanding · The beauty of this investigation is that it is personalised. If students are working in groups, they will need to negotiate what a ‘typical’ home cooked meal is.

· Grocery store apps would be handy for this investigation if students have access to mobile devices

· There are multiple ways this task could be completed

Consider the cost of a McDonald’s meal for your family, compared to your favourite home cooked meal · Addition

· Subtraction

· Multiplication

. Division

· Fractions

·    Reasoning·    Communicating·    Fluency·    Understanding · The beauty of this investigation is that it is personalised. If students are working in groups, they will need to negotiate what a ‘typical’ home cooked meal is.

·Grocery store apps would be handy for this investigation if students have access to mobile devices· There are multiple ways this task could be completed

Analyse the financial cost of eating takeaway compared to cooking the same food at home · Addition

· Subtraction

· Multiplication

·Division

.Fractions

·    Problem Solving

·    Reasoning

·    Communicating

·    Fluency

·    Understanding

. The takeaway food considered in this task may not necessarily be McDonald’s.

. It is important to allow students to draw from personal experience to ensure they are engaged with the mathematics and the task.

Using the Investigations in the Classroom

Once you have given students time to look at and discuss the original McDonald’s menu, you can choose to allow students to choose one or more of the investigations to explore. Better still, once they have completed an investigation they may be able to come up with one of their own – this is a great way to promote mathematical curiosity and wonder. Allow students to choose how they present their work, and encourage them to document all of the mathematics they do. It is also critical to build reflection into the investigation, so make sure you have some reflection prompts prepared for either verbal or written reflection.

The Maths and McDonald’s investigation provide opportunities for students to learn and apply mathematics in context. This improves student engagement, allows them to see the relevance of mathematics, promotes critical thinking and provides important and authentic assessment data.

The McDonald’s menu: https://www.thesun.co.uk/fabulous/food/3564107/mcdonalds-original-menu-1940-first-ever/

Beyond Monday’s Maths Class: Making the Most of Teacher PD

Last weekend I travelled interstate to attend a professional development day for teachers of mathematics. It was a good day, with lots of ideas shared and great enthusiasm from the 500+ audience. The presenter was well informed and, in fact, created quite a lot of hype due to her international reputation. Everyone went home happy and the word on Twitter was that Monday’s maths lessons were going to be different. Fantastic! But what about Tuesday’s lesson, and what about next week’s, next month’s, and next year’s lessons? What about the lessons of other teachers in the school?

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.

  1. 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.

  1. Does the presenter understand the Australian school context and curriculum?

When you attend PD, you expect that the presenter is aware of the Australian school context, and more importantly, the Australian 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.

  1. 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.

  1. 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.

  1. 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.

  1. 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.

  1. 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.

Promoting Student Reflection to Improve Mathematics Learning

Critical reflection is a skill that doesn’t come naturally for many students, yet it is one of the most important elements of the learning process. As teachers, not only should we practice what we preach by engaging in critical reflection of our practice, we also need to be modelling critical reflection skills to our students so they know what it looks like, sounds like, and feels like (in fact, a Y chart is a great reflection tool).

How often do you provide opportunities for your students to engage in deep reflection of their learning? Consider Carol Dweck’s research on growth mindset. If we want to convince our students that our brains have the capability of growing from making mistakes and learning from those mistakes, then critical reflection must be part of the learning process and must be included in every mathematics lesson.

What does reflection look like within a mathematics lesson, and when should it happen?Reflection can take many forms, and is often dependent on the age and abilities of your students. For example, young students may not be able to write fluently, so verbal reflection is more appropriate and can save time. Verbal reflections, regardless of the age of the student, can be captured on video and used as evidence of learning. Video reflections can also be used to demonstrate learning during parent/teacher conferences. Another reflection strategy for young students could be through the use of drawings. Older students could keep a mathematics journal, which is a great way of promoting non-threatening, teacher and student dialogue. Reflection can also occur amongst pairs or small groups of students.

How do you promote quality reflection? The use of reflection prompts is important. This has two benefits: first, they focus students’ thinking and encourage depth of reflection; and second, they provide information about student misconceptions that can be used to determine the content of the following lessons. Sometimes teachers fall into the trap of having a set of generic reflection prompts. For example, prompts such as “What did you learn today?”, “What was challenging?” and “What did you do well?” do have some value, however if they are over-used, students will tend to provide generic responses. Consider asking prompts that relate directly to the task or mathematical content.

An example of powerful reflection prompts is the REAL Framework, from Munns and Woodward (2006). Although not specifically written for mathematics, these reflection prompts can be adapted. One great benefit of the prompts is that they fit into the three dimensions of engagement: operative, affective, and cognitive. The following table represents reflection prompts from one of four dimensions identified by Munns and Woodward: conceptual, relational, multidimensional and unidimensional.

Picture1(Munns & Woodward, 2006)

Finally, student reflection can be used to promote and assess the proficiencies (Working Mathematically in NSW) from the Australian Curriculum: Mathematics as well as mathematical concepts. It can be an opportunity for students to communicate mathematically, use reasoning, and show evidence of understanding. It can also help students make generalisations and consider how the mathematics can be applied elsewhere.

How will you incorporate reflection into your mathematics lessons? Reflection can occur at any time throughout the lesson, and can occur more than once per lesson. For example, when students are involved in a task and you notice they are struggling or perhaps not providing appropriate responses, a short, sharp verbal reflection would provide opportunity to change direction and address misconceptions. Reflection at the conclusion of a lesson consolidates learning, and also assists students in recognising the learning that has occurred. They are more likely to remember their learning when they’ve had to articulate it either verbally or in writing.

And to conclude, some reflection prompts for teachers (adapted from the REAL Framework):

  • How have you encouraged your students to think differently about their learning of mathematics?
  • What changes to your pedagogy are you considering to enhance the way you teach mathematics?
  • Explain how your thinking about mathematics teaching and learning is different today from yesterday, and from what it could be tomorrow?

 

References

Munns, G., & Woodward, H. (2006). Student engagement and student self-assessment: the REAL framework. Assessment in Education, 13(2), 193-213.

 

 

 

 

Promoting Creative and Critical thinking in Mathematics and Numeracy

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

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

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

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

How and when do we embed critical and creative thinking?

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

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

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

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

Who should be engaging in critical and creative thinking?

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

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

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

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