How the “Area of a Triangle” changed the way I taught Maths

Thinking back to my schooling experience Primary School through to High School I was always “good at maths” as I was told by peers, teachers and family.  I was quick with number facts, could recall any times table up to my 15x facts (division included) and would rarely struggle with number races and punching numbers into formulas. I tell you this not to brag but to help set the story.

I now look back and wonder “was I just quick?”, “was fluency my strength?”. I remember being able to problem solve well but probably wasn’t that strong with  understanding or reasoning of concepts.

We all learnt that the area of a triangle is “1/2 b x h” or some variation of that.  Give me a triangle with dimensions of the base and height written on there and I could punch the numbers in and work the answer quickly (see below)

area of a triangle(Image:

I was good mechanically with it, but would struggle with a triangle that didn’t didn’t look this straight forward.

When I began as a teacher, I taught students “the formula is 1/2 b x h”… so “here’s some problems”.. and wonder why kids would struggle. There were a cohort that just got it and would punch in the numbers and fly through 20 questions but fall down when given questions requiring some reasoning and problem solving.

What happened next was the question that changed the way I taught maths….

Student: “Mr Millar why is the area 1/2 base times height?”

Me: “Great question, let me check” (in my head I’m thinking “Bugger how do I answer this question”)

I actually had no explanation as to why this was the case, as I didn’t have a deep understanding of the concept.

I began thinking… “How can I provide kids with an opportunity develop conceptual understanding before drilling formulas and pages of problems into them?”

The next lesson I took a completely different approach by providing kids will rectangular pieces of paper and creating a triangle from that (see example below). I then posed the question “What is the connection between the original rectangle and the triangle you created from it?” Something interesting happened…we weren’t practicing problems, formulas weren’t being used, numbers didn’t matter. We did go through half a ream of coloured paper though!


Kids were trying all different types of triangles and combinations.  There was a buzz in the room not normally present in a maths lesson.  They we coming up with their own formulas and variations of the formula… “Times the base and height together and half that answer” was one I recalled. They were all able to explain why the formula was what it was.  They had a better conceptual understanding.

Fast forward to 2011, when I started as a Numeracy Coach and I came across Peter Sullivan’s “6 Principles for Effective Teaching of Mathematics” (More can information can be found at this link in Section 5). This reinforced my thinking…

Principle 6: Promoting Fluency and Transfer.  Sullivan discusses “With mechanical practice, students have limited capacity to adapt the learnt skill to other situations. With automatic practice, built on understanding, students can be procedurally  fluent while at the same time having conceptual understanding”.

I have also heard this referred to as “Experience before Instruction”, giving kids experiences to develop conceptual understanding before getting into the mechanics.

This has now filtered to my current practice and planning. I’ll be teaching multiplication facts to my 4/5’s over the coming weeks. Rather than drilling the mechanical practice, I will be starting with creating arrays using manipulatives and making connections between these and the facts to give kids a deeper understanding.

Are we providing kids with the opportunities to develop conceptual understanding and reasoning skills before moving to fluency?

PS. The “Teaching and Learning in South Australia” YouTube channel has a fantastic clip that reiterates these messages and is well worth the 4 minutes of viewing.


Posted in Uncategorized | 1 Comment

A day with Dan Meyer

You know when someone recommends a movie, you watch it and then walk out thinking there’s 90 minutes of my life I will never get back? Definitely not the case in Adelaide when Dan Meyer presented in Adelaide on the 4th of August 2014.

I had seen the TED Talk, read the blogs, taught the 3 Acts, spread the good word to anyone who was interested…let’s just say I was primed for the day and Dan Meyer did not disappoint.

Intellectual need in the maths classroom

Dan was straight into the main objective for the day:


The question had me hooked from the start as this has always been an interest of mine, how can I engage students in mathematics whilst providing a level of challenge or ‘intellectual stretch’? I began thinking…What are some strategies, principles or processes I can put into practice? What are some examples of how not to do this? How can 3 Acts do this? How reflective of my current practice is this? My questions were answered plus more.IMG_0002

Dan provided some examples of where we go wrong when trying to engage students in maths and our textbooks can be the biggest culprit.

Textbooks putting photos on the front of ‘fun’ contexts will translate into kids enjoying maths? No!

How about putting grizzly bears and elephants into texts? Probably not, if anything it makes a mockery of the concepts being taught.

What about if we put a concept into something “real world”? Doesn’t work as well as we think!

How bout if I change the context to something else will that make it more interesting? Perhaps but just because the context might be suited, the content around the concept is the issue.

Just because the textbook provides images, context and scaffolds doesn’t necessarily equate to engagement.

I also found the things we say interesting and cringe-worthy (particularly as I have used and witnessed them more than I’d like to admit!)

  • “maths is everywhere” to entice students – terrifying rather than motivating
  • Teacher: “this example shows how an interpreter uses statistics, one day you might use this if you become an interpreter” – Student: “I might never be that…”

Our answers are insufficient!

Developing the Question

Dan spoke of deconstructing the experience. Looking at what the teacher moves are and how they are different from a similar kind of problem in a textbook.

IMG_0009It was a rewarding experience to not only experience Dan himself facilitate a 3-Act lesson (super stairs) but also have the opportunity to look closely at the pedagogy and thinking behind the Acts of the lesson, why each part was critical in its own way. How the opening Act is fragile and the way you handle student questions can make or break future lessons when asking for student questions.

I  found myself moving from the “that looks like a good lesson I’m going to try that with my students next week” to “ok this lesson looks good but what are the elements/aspects/teacher actions involved that make it this way?” The latter I believe makes for more sustainable improvement in the teaching and learning experience over time.

If there was one thing that I took away more than others from the day was the reflection of the ‘how’ and ‘why’ more so than the task itself.

More on Developing the question can be found here.

Teaching in 3-Acts

The thinking and background behind 3-Act lessons was shared brilliantly. Dan made connections between some of the biggest blockbuster movies of all time and how they are created in 3 acts and how these stories can provide a framework for mathematical tasks. Greater detail on this can be found on Dan’s blog here.

The connection between stories and the 3-acts were made transparently, no words, no talking, you can clearly see what is happening and it has you curious “what’s going to happen next?”. It’s not often a text book will do that and made me wonder with my favourite maths tasks how I could strike that curiosity in a better way for students.

The first act (developing the question/setting the scene) for me personally is the one I generally do not do as successfully at the 2nd and 3rd acts. This is a fragile part of the lesson, the way you ask for guesses, handle students questioning, how simply the scene is set can be the make or break not just for that lesson in particular but the success of future experiences like this.

Dan modeled the lesson well but it was how we were able to reflect on the teacher actions that hit the home run for me. We were challenged to look at the pedagogical actions more than the surface of “the teacher showed a video”. For me this is what provides the most sustainable future for improvement in pedagogy. It wasn’t so much the task itself but the actions taken by the teacher that stuck in my mind.

Creating intellectual need in maths

IMG_0023I think the most interesting part here was not necessarily the activities (found here under Intellectual need activities) but once again the “why” behind it. Each of these left my mind “perturbed” or “rippled” something on reflection I haven’t done well in the past for my students. They would enter my class with a “still” mind but I begin to question whether they left with that “ripple” to create the intellectual need.

In summary


I left the day on a high! Not only because I had an opportunity to talk with and hear from someone I look up to in my career but because my thinking had been “rippled” further than I originally thought it would. It has given me another aspect I can use for improving pedagogy in mathematics, new ideas I can share with colleagues and another way of making over maths.

If this was a movie I just saw, would I recommended it wholeheartedly?……….. You betcha!!

Posted in August 2014 | Tagged , , , , , , | 2 Comments

Numeracy Zombies…. Bringing them back to life


There are some dead statements that keep coming back” (@pkcc1 on Twitter)

These are the numeracy zombies, statements which many will argue should be dead and buried but keep coming back to life and can be extremely difficult for many to kill dispel. We see the same numeracy zombies coming back to life time and time again! Many of these are myths developed from personal experiences of ourselves, our parents and friends both as students and teachers. And the scary thing is they keep coming back.

A colleague of mine created a Haiku Deck (seen here) about Digital Learning Zombies which had me thinking “What are the Numeracy Zombies we encounter in our schools and communities?” As a teacher, coach and community member I have heard many (and possibly been guilty of resurrecting some). My initial thoughts on some zombies and my argument to them are as follows:

Numeracy isn’t about having hands-on fun and playing games

Not all students are engaged by one single way of learning, nor will one way help develop understanding.

There is only one way or one formula to solve problems

There are multiple ways of solving problems and a variety of tools. How often have you been to a restaurant to split the bill and used long division to work out your share? Not often I would imagine. Some might use the calculator on their phone, others in their heads, some might do a quick estimate and others may divide the total bill by the amount of people eating. There are a variety of ways to work out a solution and people will use their own learning style to do so.

Keep giving more drill and skill questions until children get it!

Developing a stronger understanding of concepts is much more effective than memorising skills. For example memorising the formula for area of a triangle 1/2 b x h is not as important as knowing that a triangle’s area is half the area of the quadrilateral it fits into. I would argue that a concept will stick longer when you have developed a strong understanding of it rather than just regurgitating more of the same.

Problem solving is for those that have mastered the concepts

Problem solving is a proficiency all students needs to access to. How can all students be good at problem solving if they haven’t had an opportunity to do so? Would you prevent a baby from learning to walk because they haven’t mastered crawling?

Not everyone is born to do maths

Maths needs to be nurtured with a supportive learning environment that promotes risk taking and creativity, one that focuses on problem solving.

We are progressing into a world that requires more than the basic recall of numeracy concepts and regurgitating the formula to solve a problem. The way the 21st Century is moving we need to be better equipped with creativity, problem solving and critical thinking. If we are to move forward in promoting more numerate attitudes then we need to start putting these zombies to rest for good.

Posted in July 2014 | Tagged , , , , , | 2 Comments

How the Zen Master helped me be a better Maths coach


Who is the Zen Master and what does he have to do with maths coaching? Phil Jackson, one of the most successful coaches in NBA history with an impressive resume! Does he teach maths? No! But there are some strong links to why his success as a basketball coach can translate into maths coaching.

If you haven’t read my bio, I have previously been a maths coach in schools and continue to work with some amazing maths coaches today. I am also a huge Chicago Bulls fan and remember the 90’s and them winning 6 championships like it was yesterday. Yes they had Scottie Pippen, Dennis Rodman and some guy called Michael Jordan, but it was Phil Jackson and his coaching that played an integral role in their success.

Phil is renowned for coaching a variety of personalities, egos, experience and abilities and coaching them towards a common goal. It’s not only is non-one size fits all approach but his way of connecting with his players in a more spiritual sense.

On reflecting on my craft as a coach I felt I was able to do a ‘good’ job. I was able to move some teachers along and sometimes not much at all (it’s hard to stay undefeated for a whole season!). Let’s be real I was no Phil Jackson, but after reading his book ’11 rings’ I found some insights that translated well into my coaching of teachers. I had a few all-stars, rookies and developing talents and used some of Phil’s philosophies to work with the varied roster.

It was the parallels between Phil’s coaching philosophies and that of mine as a maths coach that really changed the way I worked with teachers (and students for that matter).

How the Zen Master helped my coaching:

Connect with every player as an individual –
One size doesn’t fit all. What works for one won’t work for all. Phil doesn’t treat everyone the same. He works to connect with each player on a much deeper level than most coaches do. I began starting to look at what made teachers tick rather than what I thought was best for them. Some were motivated by data, others by hands-on activities, some had a real passion for reflecting on and improving their practice. I made it paramount to tap into each teacher as an individual and use their interests as a means to work on their weaknesses.

Let each player discover their own destiny

Jackson says “My approach was always to relate to each player as a whole person, not just a cog in the basketball machine. That meant pushing him to discover what distinct qualities he could bring to the game beyond taking shots and making passes. How much courage did he have? Or resilience? What about character under fire? Many players I’ve coached didn’t look special on paper, but in the process of creating a role for themselves they grew into formidable champions.”

Similar to the previous philosophy I would also find ways to push teachers to discover talents they previously thought they didn’t have. I think through knowing when to push and back off helped. The article “Are you coaching heavy or light?” also contributed to this. At times I would back off and let them step to the edge and only put the safety net out if absolutely necessary (something we could do more for our students too!).

Forget the ring (referring to championship ring) –

Jackson admits that personally, he hates losing, but he knew that it was of greater importance to emphasize “the journey rather than the goal. What matters most is playing the game the right way and having the courage to grow, as human beings as well as basketball players.”

Coaching discussions were a time to reflect about how far we had come along the journey, celebrating the small wins, the small steps along the way. As philosophical as it sounds it was about being in the present and what was happening now. It wasn’t about winning, more about what challenges had been overcome and the improvements made.

The key to success is compassion-

“Now, ‘compassion’ is not a word often bandied about in locker rooms. But I’ve found that a few kind, thoughtful words can have a strong transformative effect on relationships, even with the toughest men in the room.”

Relationships are key to any coaching. This was something I made a conscious effort to develop but most importantly maintain. I would ensure that giving positive feedback about their progress would sometime ease the tension that happens in the classroom, particularly when things don’t go to plan. It was about understanding that “hey it’s not perfect all the time, but what worked well?”.  Many would sell themselves short so I made it my goal to highlight their strengths, new learning and victories with others. This went a long way to developing a relationship built on trust.

Whilst these are just a few of the 11 qualities that resonated with my role, they can be applied to any leadership role in education, business or even in our everyday lives. To me they are about being mindful of what’s happening in the here and now.

Posted in July 2014 | Tagged , , , , , , | 4 Comments

Dieting for maths teaching!

12weekchallenge-programs-title-03“The 12 Week Maths Teaching Transformation” – makeover your maths teaching. Feel great and teach better in just 1 term!

Ok so this is not the type of thing gaining 50,000 likes on Facebook or making hundreds of thousands of dollars, but if it was out there would it gain interest? Maybe. Is a 12 week makeover the answer to better teaching of maths? Probably not, especially with what we know about change in an education setting, along with Carol Dweck’s research into growth and fixed mindset. In my short experience but steep learning curve as a Numeracy Coach along with the experiences of a number of my colleagues, I have noticed some elements of improving maths teaching in relation to losing weight.

I have personally used the services of a nutrition planner and personal trainer and noticed sound results in doing so. Living with someone who is into healthy eating also has its benefits, as you are constantly around good habits. Similarly, having the support of a coach, mentor or trusted colleague to support with maths pedagogy has its positives. Being someone on both sides I have seen (in most cases) significant growth over time to suggest this is a valuable mode of operation.

Throughout the coaching process we look at making small improvements in the hope of producing significant growth over time (slow and steady wins the race). has a 10 Fast Weight Loss tips that while they don’t have a direct link to improving maths pedagogy raise the question, is it about making small changes to pedagogy one step at a time? – swap the cake for a piece of fruit ditch page 29 of the text for an open ended task…Sometimes this is easier said than done.

Professor Peter Sullivan discusses ‘6 Principles for effective teaching of mathematics’ which have provided me as a coach a structure to follow. While they don’t give 10 weight loss tips or a plan to follow for 12 weeks, they do present a set of principles that guide and provide advice for teaching practice… much like visiting the nutrition planner.

We measure results in weight loss by the numbers (kilograms lost, body fat %, cm lost) which is generally a solid indicator that efforts put into place have made a positive impact. However this is not an easy task to measure the growth of a teacher by the numbers. Yes improvement in test scores can be influenced by good pedagogy but I have worked alongside brilliant teachers whose efforts aren’t always reflected in student achievement data.

Maybe we need to look at the before and after photos… what does the learning look like in our classrooms before and after? What are students noticing about the teaching? How do we feel as teachers about our pedagogy before and after some learning? Fitter, lighter, more energetic?

It would be an interesting concept to see how a 12 week maths teaching transformation would go for teachers. Would it be a sustainable strategy? Is there something in our schooling systems that need to change? Is dieting for maths teaching the way to go? These are the questions that continue to perplex me as an educational leader.

If a 12 week transformation fad isn’t the way to go, how do we get sustainable improvement in maths teaching?

Posted in June 2014 | Tagged , , , , | 14 Comments