Situation 1: Senior Class
The class is working on a division problem. One student proudly proclaims, “I’m finished.”
I ask them to talk me through their answer – they have used long division to solve the problem.
They look at me decidedly non-plussed:
“That’s the answer.”
“I know, I’m just asking you to talk me through how you solved it.”
“There it is,” they say, pointing to their representation of long division.
“Tell me what you were thinking here.”
They just stare at me.
Try something else Greg!
“Ok, let’s see if you can you solve this division in another way.”
“Why would I do that? That’s the answer.”
Try as much as I can, employing my best rhetoric about mental strategising, they won’t budge.
Answer gained = task done.
Situation 2: Junior Class
We’re mathematising, solving an addition problem using mental strategies.
One student looks around, sees that they’ve finished first. Cue: big grin on their face.
They start devising additional problems with larger numbers, and solves them the same way. Standard Algorithm or Column Method. Beaming, they rush to show me their work.
I see you’ve used the Column Method to solve those addition problems. Now use a mental strategy to solve the first one.
They’re not happy.
It’s great that you know how to do the Column Method. What do I say about the Column Method?
We can use it to check our answers.
Yeah. We start with mental strategies then you can use the Column Method to check your answer. What are some addition mental strategies we use?
Situation 3: Junior Class
They race across the classroom as they see me coming.
Mr. Curran look at what I’ve done!
They proudly hold up a page of 5-6 digit addition problems they’ve generated and then solved using the standard algorithm.
It’s like finding worksheets chock full of standard algorithms near the photocopier. Or, checking in on a relief teacher and finding a whiteboard full of standard algorithms to greet the students as they arrive.
I observe this student closely whenever we’re focusing on mental strategies.
Their first move is generally towards the standard algorithm.
I catch them: “remember, mental strategies first then Column Method to check.”
They sigh, appearing to find using mental strategies challenging.
Interestingly, they often tune out when we share or talk through our mental strategies. They also frequently make computation errors with their standard algorithms particularly as the size of the numbers they’re adding increases.
Yet in talking to them, across the year, it’s apparent that they think:
- correctly solving tasks with the standard algorithms = high level maths skills and intelligence
- they’re a level beyond their peers in getting the answer first.
Beyond Answer Getting
If we value the development of mathematical reasoning, especially as we seek to move students beyond counting and additive reasoning towards multiplicative and proportional reasoning, then building students’ repertoires of mental strategies (across operations) along with their capacities to represent such strategies is critical.
This approach doesn’t preclude standard algorithms. They can, as I emphasise with my students, be used to check answers. They have a place but they’re not given priority.
Jo Boaler in her text, Mathish (2024), summarises the research relating to the privileging of standard algorithms over mental strategising or number sense as she refers to it. It can lead to students not wanting to make sense of their answers, unpacking them, and explaining the maths moves they’ve made. This lack of interest in conceptual thinking is linked to the “rule-following mode” associated with the too early teaching of standard algorithms “according to Boaler.
Boaler (2024) argues that teaching standard algorithms too early can lead to students having difficulties in developing a range of mental strategies. In my experience, this is linked to their belief in the superiority of standard algorithms. Subsequently they think other approaches aren’t worth devoting time to and they either tune out or focus intermittently, leading to knowledge and skill gaps.
Pam Harris and Kim Montague from ‘Maths is FigureOutAble‘ advocate for sense-making, conceptual understanding and the development of mathematical thinking and reasoning. Their mathematising teaching approaches focus on using relationships and connections (“doing maths”) particularly through the development of mental strategies and models (for representing mental strategies).
I have implemented many of teaching approaches advocated by Harris and Montague in my primary/elementary classrooms. Consequently, my students have become much more positive about maths. They have developed a wider range of mental strategies for all operations along with an interest in talking about them. They have learnt to clearly represent their thinking or strategies in different ways. And since they know patterns, connections and relationships between numbers are critical in maths, they are often on high alert for them.
We love challenges and persist when we’re finding it difficult to see a way through. The time spent on this (along with the reflection), is just as important, sometimes more important than answer getting – especially when we can apply our learnings in future lessons. And the moments when a mistake is made but it’s not the end of the world as we know it but the start of an investigation is magic:
Mr Curran we found where we made the mistake.
Yay! Where? Tell me about it.
What can we learn from this for the future?
In these moments I know we’re really getting somewhere, somewhere well beyond rote memorisation, rule following, and answer getting.
Check Out
Mathish: Finding Creativity, Diversity, and Meaning in Mathematics (book and website) – Jo Boaler (2024)
Math is FigureOutAble (website and podcast) – Pam Harris
