*This week's photos show our Grade 2 shopkeepers; the Grade 4 Spanish students learning about Spanish-speaking countries while enjoying some sunny weather; some of our multilingual students sharing morning greetings in their home languages in the run-up to our International Day on Saturday; a Grade 4 student showing her innovative camera prop for the upcoming Grade 4 Show; and, in the center, Grade 1 students tackling a "low-floor, high-ceiling" mathematics task (see below).
12 Legs, So Many Possibilities: The Power of Open-Ended Math
This week, our Grade 1 classroom was filled with a curious collection of pirates, frogs, sea creatures, and playdough sculptures during the mathematics lesson. The catalyst? A deceptively simple question: "There are 12 legs. How many creatures could there be?"
In a traditional classroom, math is often viewed as a race to a single "right" answer. However, in a PYP Mathematics environment (and as emphasized in a recent staff Professional Learning session) we treat mathematics as so much more: a landscape of Inquiry, Thinking and Communication.
The "Low-Floor, High-Ceiling" Approach
This task is an example of what Stanford Professor of Education Jo Boaler calls a "Low-Floor, High-Ceiling" task. It has a "low floor" because every child can begin by counting out 12 legs, but it has a "high ceiling" because it pushes students to justify their reasoning and explore complex combinations. After all, even in a Grade 1 classroom, some children are ready to consider grouping and repeated addition (the foundations of a conceptual understanding of multiplication and division).
Our students were challenged to consider:
- What is the greatest number of creatures we could see? (Perhaps 12 one-legged pirates?)
- What is the smallest number? (Two six-legged insects?)
- How many different ways can we solve the same problem? [At one point, Roberto exclaimed: "I found more than 10 ways to make 12 legs!"]
- How can we show our thinking?
Low-Floor, High-Ceiling activities also have the potential to open up unexpected lines of inquiry. Oli, for example, was keen to initiate a scientific debate about animal structures during this maths lesson, questioning what is and what isn't a leg: "Why are you counting pinchers on the scorpion as legs, because technically they are its claws, not its legs!"
From Manipulatives to Mindsets
By using Play-Doh, toothpicks, and Playmobil figures, students moved through the essential phases of the PYP math journey: from constructing meaning (handling physical objects) to transferring meaning (explaining their logic). There were no rigid rules; the only requirement was that the total must equal 12.
As students realized that their peers had different answers that were also correct, they practiced vital Approaches to Learning (ATL) skills:
- Thinking Skills: Analysing multiple numerical combinations;
- Communication Skills: Explaining their creative "why" and "how" to others;
- Self-Management: Persisting through trials and errors with their manipulatives.
When Zosia noticed that other students at her table were coming up with different solutions, she stated, "We don't all need to have the same answer because we all have different brains!"
Depth Over Speed: Preparing for the Future
You might wonder: Why not just give the students a worksheet full of addition problems? While rote worksheets can reinforce procedural fluency, they often fail to build conceptual understanding (the "why”). Research shows that the highest-achieving mathematicians aren’t necessarily the ones who calculate the fastest, but the ones who can see numbers flexibly and from multiple angles.
By moving away from repetitive sums, we are preparing students for the higher-level mathematics in their futures, such as algebra, calculus, and data science, where problems don't come in neat rows. In diverse fields that involve mathematical thinking, success depends on adaptive reasoning: the ability to look at a complex set of data and "model" a solution. A student who can creatively solve "12 Legs" is learning to think like a mathematician, an engineer or a coder; they are learning to see patterns and relationships, rather than just following a recipe. We are training problem-solvers, not just procedure-followers.
World Word of the Week
Bricolage (French)
The art of creating or problem-solving by using whatever materials happen to be available. It is the process of being resourceful and finding a new purpose for "found" objects to construct something meaningful.
Context: Our "12 Legs" challenge was a beautiful exercise in bricolage. Our students didn't need a formal worksheet or a calculator; they became "bricoleurs," using playdough, animal figures, and ingenuity to construct their own mathematical truths.
Parent Conversation Starters - Keeping the Learning Going at Home
"If we saw 12 legs at the park today, what is the 'weirdest' combination of animals we might have seen?"
"Why was it more fun to have 'no rules' in math today, and how did you make sure your answer was still right?"
"If we changed the number to 20 legs, what would be the first creature you'd add to your list?"
Home-School Connection: Bringing the Inquiry Home
You don’t need a degree in advanced calculus to foster a mathematical mindset at home. Supporting your child’s mathematical development also doesn't require a reliance on flashcards or speed drills. In fact, Jo Boaler’s research suggests that the best way to foster a love for math is to emphasize depth over speed, moving toward what she calls Low-Floor, High-Ceiling (LFHC) tasks. These are activities where the entry point is simple, but the possibilities for complex thinking are endless.
Here are a few ways to turn household moments into "High-Ceiling" inquiries:
- The "Total Number" Challenge: Next time you are setting the table or putting away laundry, give your child a target number (e.g., 20). Ask: "Using these forks and spoons, how many different ways can we group them to reach 20?" Followed up, perhaps, with, "Have you found ALL the different ways they could be grouped? How do you know?"
- The "Shape Hunt": Ask your child to find three objects in the room that share a common concept (e.g., "They all have at least one right angle" or "They are all symmetrical"). The goal isn't just to find them, but to justify their reasoning.
The Golden Rule: Value the "How" and "Why" over the "What"
When your child gives an answer, resist the urge to simply say, "Correct." Instead, ask:
- "How do you know that’s true?"
- "Is there another way you could have solved that?"
- "What happens if we change the rules?"
By focusing on the ATLs of Communication and critical Thinking, you are helping your child see that math isn't a race to a finish line; it’s a brilliant journey of discovery of how mathematics promises something for everyone. As Emilia said: "I love maths in Grade 1. It's the best thing ever!"
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