I’ve been wanting to write another blog post for a while now. Throughout this year, I have been inspired by my students to get typing and start sharing my thoughts online. It’s been great, but now that they’ve finished up, I feel the need to have a look back to see what the year entailed. Here’s what I’ve learnt as a teacher in 2016:

Maths is more than just a subject

Ask ten different people what they think of maths and there will be at least one person (if not many) who will openly tell you how they’re either not a maths person or don’t like it. Our students hear these comments from a very young age, sometimes, before even learning any maths at all. This has a number of negative effects on the learner, two of which I think can determine their likelihood of being successful in maths.

With confidence, students are more willing to extend their understanding and fail. Failure provides an opportunity to learn. For students learning maths, it’s whether or not they choose to learn from their mistake that will impact their growth as a learner. Confidence to take a risk and continue persevering, knowing that it’s just another learning opportunity, is what all successful learners need at some point in time.

My dad was great with arithmetic and numbers in general, and so was my older brother. I was told frequently that this was the reason I too was good at maths. While I was in school, I could possibly admit that performing to people’s expectations of my mathematical ability was what motivated me to continue achieving high results in maths, but it didn’t help me as a learner. I relied on memorisation and focused on developing a procedural understanding to give the illusion that I was fluent in the specific topic that we were studying.

What do you notice? What do you wonder?

In May, Amie Albrecht (@nomad_penguin) presented two workshops titled Promoting Productive Mathematical Discussions and Building (and Rejecting!) Mathematical Intuition. She had just been to the 2016 NCTM Conference in San Francisco. Apart from the brilliant souvenirs from the conference, she also brought some great insights and resources from educators around the world. A big takeaway for me from her sessions was the process of Noticing and Wondering. For me, this changed the way that I perceived the learning experience in mathematics. Through noticing, students are encouraged to talk about their initial thoughts, highlight aspects that they find interesting, and listen to their peers’ input. From wondering, students inquire and construct a purpose that drives the learning process. It was a nice format that made the learning in the classroom more real. Real in more ways than the context of the problem. In my classroom, the impact on the learning environment was phenomenal – students’ natural curiosities were recognised and given a voice, which amplified the level of inquiry and the pursuit for learning.

Wonder in Mathematics

180 Opportunities to Notice & Provide Massive Space to Notice

The Learning process isn’t easy.
We need to talk more about it

Why I ask my students how they “feel” when they’re working through a problem.

Firstly, I think that teachers actually need to talk a whole lot less than they think they do. When we do talk, however, we need to be discussing how the students are feeling while they’re learning. For many students, they actually don’t feel comfortable with the learning process (especially in maths) because appearing to be developing an understanding can imply inadequacy. One of Jo Boaler’s (@joboaler) Setting up Positive Norms in Math Class  captures this perfectly: Maths is about learning not performing. As teachers, we seek validation from our explanation or teaching, which isn’t a bad thing. The bad thing is when we stop seeking validation; when the students appear to be doing the maths and accept it as them understanding the maths. This can also be the preference of the learner in an attempt to feel successful by memorising a sequence of steps or focusing purely on the procedural understanding. Heck, I was that student.

This year, I started talking more about what it feels like while learning something. Some people call this, “being in the learning pit”. I would ask the students questions and say things like:

• Who feels like it makes sense when I go through it on the board, but you get lost when you try it for yourself?
• This won’t make sense to you yet. That’s ok, because you haven’t learnt it yet, but eventually you will.
• (towards the end of a unit) What can you tell me about this, that you previously wouldn’t have been able to before?
• Do you need me to explain this in a different way?
• Learning sometimes feels like we’re wondering through the dark with a weak torch. It’s ok, as you learn more, your torch will shine brighter. (Sorry about the analogy!)
• Knowing things is easy. Learning them isn’t.
• You have to first not know something to learn it. So, if you don’t feel like you understand it, that’s ok. It’s just another opportunity to learn.
• Mistakes are valuable, yes. The lessons you learn from them are far more important.

Struggle needs to be productive

This is probably one of the most important things that I have learnt about this year.

If you haven’t watched or read the Lord of The Rings trilogy, stop reading this post now. Find a way to get your hands on a copy and watch it. For those of us who have watched it before (probably at least twice because it’s worth it) imagine what Frodo and the fellowship’s story would have been like if they could just catch an Uber to Mordor and back? John Orr (@MrOrr_geek) describes this much more eloquently in his post about the Hero’s Journey and Productive Struggle. I often find myself talking with my students about the whole learning process, just as Tolkein did with the Lord of The Rings. Jon Orr talks about how the tension is built up over time and how this translates to a learner’s experience in a maths lesson.

Promote Struggle – A Hero’s Journey in Math Class

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As a teacher, this analogy really helped me understand what my students were experiencing throughout the learning process and allowed me to acknowledge that, just like when Frodo was navigated through Middle Earth, EVERYTHING was new to them. It’s, at times, confusing and challenging, but rewarding. Dropping the ring into the depths of Mordor was much more than saving Middle Earth. It was an achievement. A culmination of a long, arduous journey with multiple smaller challenges. Sometimes, because we generally understand the content we teach pretty damn well, we fail to see and acknowledge the smaller challenges that our students face along the learning journey. Not only does this promote the concept of transmitting knowledge to students without challenge, but it also assumes that the path to understanding a concept is smooth and straight-forward. As teachers, we take on different supporting and guiding roles to help our students and work hard to maintain an appropriate level of challenge for them to continue growing and building their understanding. At times, we’re rolling off one-liners to help our students persevere with their learning, taking on a mentoring role like Gandalf. Other times, we might be doing some of the legwork for the students to give them scaffolding to engage with the learning at their level of readiness, like Samwise Gamgee.

The important lesson here is that learning is tough, challenging, and a struggle, but our role, as teachers, is to help our students to appreciate the struggle and challenge themselves to continue learning and growing.

“Real World” doesn’t necessarily = Real Learning

A brilliant explanation of why real world doesn’t necessarily mean real learning, which I am continuously suggesting to people to watch, is Dan Meyer’s talk at this year’s NCTM Annual Conference: Beyond Relevance & Real World. Rather than tell you to go to the link and watch it, I’ve embedded it below:

The big message that I was able to take from this talk was that the learning needs to be purposeful and the teaching intentional. The teaching needs to be catered to the learners’ needs in every aspect. One important one that I think makes a difference in students’ engagement and attitude towards learning is the development of the need to learn. I would consider this as catering for the learners’ needs, as real-world learning is generally on a needs basis. That is, the learning is sought after, and valued, when there is a need to learn it. Jon Orr links this to Productive Struggle exceptionally well:

Mix it up

Just like any other teacher, when I see a great way of presenting a problem, I have a go at creating my own and, soon enough, it’s used in every single lesson. I can’t help myself when using formats like Which One Doesn’t Belong or Open Middle, but it’s critically important that I don’t get too carried away. Just like a delicious meal, students can get tired of seeing the same type of question or problem over and over again. It’s important to keep it fresh and mix it up. You’ll get a more positive response and a wider breadth of understanding developed amongst your students through doing so.

Here’s some new things I’ve tried this year and had a go at trying for myself:

• Open Middle
• Which One Doesn’t Belong
• Reversing the Question
• Would You Rather
• Agree or Disagree
• Estimation 180

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Desmos

In Australia, we have a range of Professional Standards that, as teachers, we are expected to meet through our teaching practice. One particular focus area that I never felt like I ever authentically addressed was: Use effective teaching strategies to integrate ICT into learning and teaching programs to make selected content relevant and meaningful. This isn’t just using computers every now and then, but actually using technology to make selected content relevant and meaningful. Using Desmos classroom activities has done this for me (here are some that I’ve made).

The Cartesian Plane has been a MASSIVE hurdle for many of the students that I have taught. Often, students will avoid graphing concepts and focus on procedures to help them develop quite a closed understanding of the maths being learnt. Tools like Desmos can help students leap over the hurdles with greater confidence and understanding. It’s not the same as other graphing tools, like Geogebra, for a number of reasons:

1. Greater margin for error
   To operate Desmos, students don’t need to bang their heads against the wall to try to understand and can make errors while still being able to graph something. Desmos will even tell you what you might have done wrong. They’ve got tutorials if you still can’t figure it out.
2. Sensational Activities (made by teachers)
   The Desmos team push out fantastic activities and bundle them with a huge amount of support. You can even create your own to customise a learning activity to suit your students’ prior understanding and learning intentions.
3. The Dashboard
   See which screen your students are looking at on their device, pace the whole class to view the same screen, see individual responses or overlay everyone’s sketch, or pause the class to start a classroom discussion. You can do so much that you just couldn’t do with any other tool. Did I mention it’s free?

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We’re all in this together

As I’ve said numerous times through most of my posts this year, the Global Math Department (Maths Twitter BlogoSphere #MTBoS) has provided me support, inspiration, and resources. Without my Twitter account, I doubt that I would have taught my students in the way that I did and enjoyed teaching nearly as much.

Here’s a menu of resources that I share with participants of the workshops that I run, which is my go-to when planning lessons and activities.