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Today, I want to talk to you about one of the most powerful and cross-cutting strategies we use at Innovamat: the known facts–derived facts strategy. We work on it from first grade all the way through the final years of middle school, and it connects with fundamental principles of cognitive psychology about how we learn, remember, and transfer knowledge to new situations.
This strategy encourages students to rely on facts they already know to derive new ones, instead of calculating from scratch each time. For example, if they know that 40 + 20 = 60, they can easily deduce that 35 + 25 will also be 60. In other words, it’s not about recalculating — it’s about reasoning from what they already know.
Beyond calculation: a way of thinking mathematically
The known facts–derived facts strategy is one that permeates the entire teaching and learning process. It’s a way of training mathematical thinking — such as deduction, the ability to make connections, and the development of number sense — from early childhood through secondary school.
In classrooms that use Innovamat materials, the known facts–derived facts strategy comes to life through rich mathematical conversations, where students explore relationships between numbers, generalize patterns, and learn to think flexibly.
From counting forward and backward in early childhood to using doubles, halves, or equivalences in primary and secondary school, known facts–derived facts supports their mathematical growth every step of the way.
👉 If you’d like to see how this develops step by step in the classroom, Cecilia Calvo, the conceptual designer of the materials, explains it in detail in this blog article.
From the product team, our challenge is to bring that experience to different formats and environments — finding the best way for students to systematically practice this strategy until they achieve fluency.
To give you some context, here are examples from student workbooks at different grade levels, where they practice the known facts–derived facts strategy.
And after working systematically on paper, it’s time to move into the digital environment with Atlas. Below, I’ll explain how we’ve developed the Atlas applets to meaningfully practice the known facts–derived facts strategy.
From paper to applet: the first version
When we designed our first applet on known facts–derived facts, we started from what seemed like a brilliant idea: letting students write their own known fact and then generating new derived operations from it.
We wanted to encourage autonomy — for each child to start from a point they recognized as their own. But when we observed its use in the classroom, we realized that something wasn’t working as we had expected.
What we learned from observing in the classroom
Watching students interact with the applet revealed three important design flaws:
1. Operations that were too simple. Because students could choose freely, many started with trivial examples (like 10 + 1). The derived operations ended up being just as basic (11 + 1, 9 + 1), and they weren’t prompted to apply the derivation strategy. As a result, they simply calculated directly — without engaging in real reflection.
2. Lack of focus on change.The design didn’t guide students to notice what had changed between operations. The operations seemed independent, and students had no incentive to derive.
3. Visual disconnection from the notebook.The digital aesthetic was too far removed from the paper material, making it difficult totransferbetween contexts. Students did not recognize that it was the same strategy.
Redesign to guide reasoning
With these learnings, we redesigned the applet from scratch in the new version, the algorithm directly proposes a carefully selected initial operation, in which deriving is more efficient than recalculating.
The student begins by solving for this basic fact, and from there generates derivatives that involve small modifications. If they make a mistake, the interface displays a visual aid that highlight swhat has changed between the initial operation and the derivative. If the error persists, a second aid explains how that change affects the outcome.
Thus, the applet guides the step-by-step reasoning, promoting a deep understanding of strategy.
What we gained… and what we lost
This new version significantly improved classroom practice: we got more students to intentionally apply derivation and understand numerical relationships.
However, there is an important nuance. Since the system proposes the initial operation, we partially lose the opportunity for the student to search for his known fact on his own, a key step in cognitive autonomy. It’s a balance we continue to explore: how guide without limiting the student’s initiative.
A network of connections that makes sense of mathematics
Known facts – derived facts – apply to a multitude of situations and are not limited to addition or subtraction. They extend to times tables (if 6 × 4 = 24, then 6 × 8 will be double: 48), at percentages, to the decimals or even to algebraic reasoning.
Apply this strategy in different contexts creates deep connections between concepts, encouraging students to build a coherent and meaningful knowledge network.
Iterative learning: for us too
At Innovamat we believe that designing education is also learning. The story of this applet is an example of how we iterate on our own ideas, observe in the classroom, and improve based on real-world evidence.
Just as we encourage students to derive new facts from those they already know, we derive learning from every experience, on paper and digitally. Because learning—in the classroom or in product design—is always a process of connection, reflection, and continuous improvement.
References
Baroody, A. J. (2016). The development of arithmetic fluency. Cognition and Instruction, 34(2), 93-96.
Carpenter, T. P., & Fennema, E. (2014). Children’s Mathematics: Cognitively Guided Instruction (2nd ed.).
