This is the fourth in a planned series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.

In my previous post, I talked about how I used sense making as a powerful motivator for the ‘Notice and Wonder’ routine. My next step was to have my pre-service teachers experience ‘Notice and Wonder’ for themselves.

I deliberately chose to start with an everyday, seemingly non-mathematical image. Study the image below. What do you notice? What do you wonder?

I displayed the image and asked the two magic questions. There was silence. Inwardly I was thinking ‘Oh, crap — this is going to be disastrous.’. I think it was at this stage that I reminded them that non-mathematical ‘notice and wonderings’ are as important as mathematical ones. After another quiet moment the buzz started. I quickly walked to the back of the room and made myself invisible. Once the intensity of the discussion subsided, each group shared some of their ideas. I wish I could remember them all, as there were as many non-mathematical ideas as mathematical ones, but their wonderings were questions like:

• Why does a shot cost the same as a small?
• Do you need to buy a coffee to get a free babycino for kids?
• Where was the photo taken? (If you are Australian, you might recognise some clues. But there are also red herrings such as the units, which are in ounces.)
• How can you get a shot of tea?
• Which size is best value for money?
• Do they really mean that skim milk is free? Or just doesn’t attract an extra charge?
• How many people buy the large size when they really want a medium because it’s ‘just an extra 50 cents’?
• Why do we measure coffee in ounces in Australia?
• How did they decide the pricing structure?
• Are the diagrams to scale?
• Why does anyone buy coffee at a petrol station? (Yes — this is where I took this photo, while filling up my car.)

We then reflected on what had happened. I supplemented their ideas with those from Max Ray-Riek’s fabulous book, Powerful Problem Solving. Chapter 4 is dedicated to ‘Noticing and Wondering’ and can downloaded as a sample chapter from Heinemann here. Here is an extract:

These activities are designed to support students to:

• connect their own thinking to the math they are about to do
• attend to details within math problems
• feel safe (there are no right answers or more important things to notice)
• slow down and think about the problem before starting to calculate
• record information that may be useful later
• generate engaging math questions that they are interested in solving
• identify what is confusing or unclear in the problem
• conjecture about possible paths for solving the problem
• find as much math as they can in a scenario, not just the path to the answer.

#### The #math1070 photo challenge

Next I shared how people were sharing images on Twitter. Some of my favourites are at the end of this post. Then I shared examples from the 2016 Maths Photo Challenge. Part of their weekly task is to take two photos from the world around them, and describe any mathematical ideas that they see. I look forward to seeing their ideas, and perhaps sharing them with you soon!