Patterns
Are my students noticing or identifying patterns? Are they keen to play with numbers, pulling them apart, combining or reassembling them in a bid to find patterns?
Do they decompose numbers into their place value parts (e.g. into hundreds, tens and ones) to find patterns? Are they combining place value parts and noticing patterns? The 9s (as in 9, 18, 27…) are worthwhile exploring here. Do they notice when patterns break down and are they interested in exploring why?
Are they excited to talk about or describe patterns? Students are, in my experience, often fascinated or intrigued by patterns especially if their teacher regularly explores patterns with them through counting routines. The ‘Count Around‘ routine developed by Pam Harris and Kim Montague is a favourite of mine especially as it takes us beyond skip counting.
Relationships
Are my students noticing or using number relationships such as Partners to 10, 20, 100 and 1000? The thinking routine, ‘I Have, You Need’ by Pam Harris is useful here. The teacher signals the target number, eg. 10. Then says, ‘I have …’ with students responding, ‘you need…’ to get to 10.
The focus here is two-fold: the answer and how the students arrived at their answer (their strategy). As students gain familiarity with their Partners to 10, can they use these facts to help with Partners to 20, or 100. For example if I know 1 + 9 = 10, what about 11 + ? = 20, or 11 + ? = 100 (can I use the 1 + 9 fact here?)
There is an opportunity here to ‘nudge’ students to use relationships. For example in solving 11 + ? = 20: “Did anyone use their Partners to 10 or Rainbow Facts to help solve 11 + ? = 20?”
For further examples of nudging students to use relationships, see Kim Montague and Pam Harris using a Problem String.
Strategies
What mental strategies are my students using? Are they efficient strategies? Perhaps they need to be nudged into using more sophisticated strategies? Maybe they’re overusing particular strategies and need to be exposed to, or supported to use, other strategies
Representations
How are my students representing their maths thinking or maths strategies? Are their representations clearly laid out – in terms of placement and spacing on the page or whiteboard? Could an outsider without any understanding of what we have been learning, get the gist of what the student is representing?
Do they show evidence of understanding proportions such as in their placement of numbers on a number line? Or, in the size of their proportions when using an area model to represent their thinking in multiplication? Pam Harris often makes the case that we (teachers) need to model proportionality when we represent students thinking on the whiteboard. We also need to draw attention to the way we are laying out their thinking giving reasons for our choices in layout.
Reasoning
Are my students willing and keen to explain how they arrived at their answer or conjecture, or why they made particular choices in a problem-solving task?
In my experience, there’s some common foundations for students reluctant or unwilling to reason. They are often good at memorisation and timed tests, and have a strong preference for standard algorithms over mental strategising. They often respond, “that’s what I got”, “that’s the answer”, or “I just knew it.”
Jo Boaler in her book ‘Mathish‘ and Pam Harris and Kim Montague on the ‘Math is FigureOutAble‘ website and podcast discuss the costs of privileging standard algorithms over mental strategising. They also present alternatives to build more sophisticated mathematical reasoning skills.
An Interest in Mistakes
How do my students respond to mistakes? As a teacher, my role is to model enthusiasm for learning especially when mistakes are made. These are opportunities to learn if we’re willing to persist and unpack where and why the mistake occurred. Also, if we’re keen to identify important take-aways.
In my classrooms I’ve noticed these common responses to mistakes: Students seeing mistakes as catastrophic and further evidence that they “can’t do maths”. Students claiming, “it’s too hard. I just can’t do it.” It’s like a form of learnt helplessness.
Here I tend to draw on mindset teaching practices to challenge their responses, pointing to other occasions where they have persisted and unpacked their mistakes, and learnt what to do from thereon. Also, highlighting what happens in our brains when we seek to unpack and learn from mistakes.
I have used Jo Boaler’s Mindset videos on YouCubed with students in years 4 to 6 and found that they respond positively to them, and apply messages from them in our maths classes. Carol Dweck’s foundational texts on Mindset which inform the practices on YouCubed have also been beneficial to me as a teacher.
