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Visit highlights
How can we get our students to stop repeating mechanical procedures and start reasoning mathematically?
This was the driving question behind Robert Kaplinsky’s visit to Sant Cugat. An international leader in math education and co-founder of the renowned website Open Middle.
We had the privilege of sharing a day with him with the goal of fostering reflection and improving teaching practice. Kaplinsky, who has been an educator since 2003 and is the author of the book Open Middle Math, showed us how seemingly simple tasks can completely transform classroom dynamics.
A modeling session at La Floresta school
The day kicked off with a live session at La Floresta school with 6th-grade students.
With teachers and members of the didactic team observing, Kaplinsky presented open-ended challenges. The children, working in small groups at vertical whiteboards, had to share strategies and use reasoning to solve multiplication problems.
Watching him teach live allowed us, as educators, to see firsthand how decisions are made in the moment and how to facilitate student reasoning effectively.
Following the class, he held a discussion with the teachers and trainers who had observed the class. Together, they reflected on teaching practices and the mathematical learning processes.
Training at Innovamat: Beyond mechanical practice
After the school visit, the training continued at the Innovamat offices with an intensive session for teachers and trainers from Catalonia.
Kaplinsky introduced activities designed to put us in the students’ shoes. The goal was to make us reflect on why we often end up doing repetitive paper-based practice, even when we know it isn’t the path toward deep understanding.
🎥 You can watch the full training session here:
What is an Open Middle problem?
The training activities centered around the Open Middle concept. These problems feature a “closed beginning” (everyone starts with the same rules) and a “closed end” (there is one correct answer or specific goal), but they have an “open middle.” This means there are multiple pathways and strategies to reach the solution.
Here is a practical example we worked on:
Challenge: Using the digits 1 to 9 at most once, find the combination that gives a product closest to 7000. [ _ ] [ _ ] x [ _ ] [ _ ] = closest to 7000.
Tasks like this are incredibly rich. They force students to analyze patterns and place value of numbers, make decisions based on logic rather than memorized algorithms, and learn from their mistakes. Regarding the latter, Kaplinsky emphasized that if students do not see where they make mistakes and do not adjust their strategy, there is no real learning.
Another challenge that sparked an interesting debate was this one:
Challenge: Using the digits 1 to 9 at most once each, place a digit in each box to create an equation where $x$ has the greatest possible value. [ _ ] [ _ ] + $x$ = [ _ ] [ _ ].
At first glance, it might seem like you just need to put the biggest numbers “wherever,” but we quickly realized its complexity. The key to the task is understanding how to maximize the value of $x$. This becomes clear if we rewrite the equation as a subtraction problem: $x = CD – AB$. Therefore, to make $x$ as large as possible, we need to make the number on the right as large as possible and the number on the left as small as possible.
We also reflected on place value. Tens carry much more weight than ones, so it makes sense to place the largest digits in the tens place of the larger number, and the smallest digits in the tens place of the smaller number.
Thus, we saw that solving the equation requires understanding that for $x$ to be maximized, the result (the difference) must also be maximized.
This is the essence of Kaplinsky’s pedagogy: making the student’s productive struggle the main character. Instead of completing a worksheet with 20 identical equations, the student might only solve 4 or 5, but with clear intent and critical analysis behind every calculation.
The mindset of change: scary, but not dangerous
To close the session, Kaplinsky invited us to reflect on our own comfort zones. To explain why many teachers resist implementing rich or Open Middle activities, he used two examples:
- The danger of mechanical practice: We often think that a quiet class, where students are correctly filling out a worksheet of 20 multiplication problems, is a “good class.” It doesn’t feel scary; it feels under control. Kaplinsky describes this as something that might not feel scary, but is actually very dangerous.
In reality, this type of class creates a false sense of security. Students are executing algorithms and solving operations in silence, often without understanding them. When those same students face a problem where the numbers aren’t arranged exactly the same way, their knowledge crumbles. The real danger is that we are raising students who “calculate like machines” but don’t know how to “think like mathematicians”. - Learning through challenge: In contrast, when a teacher proposes an Open Middle challenge, the atmosphere in the classroom changes. There is noise, debate, doubt, and—most importantly—the teacher cannot predict exactly what every student will say.
This can feel scary, but it is not dangerous. It feels scary because the teacher feels they are losing direct control over the pace of the class. But in exchange, there are far more opportunities to think mathematically and learn through error. This is where, according to Kaplinsky, real growth happens, because the student is forced to make decisions and defend them.
- The danger of mechanical practice: We often think that a quiet class, where students are correctly filling out a worksheet of 20 multiplication problems, is a “good class.” It doesn’t feel scary; it feels under control. Kaplinsky describes this as something that might not feel scary, but is actually very dangerous.
Kaplinsky concluded that many teachers operate by trying to maintain constant control, telling every student exactly which path to take at every moment. But his message was clear: we must have the courage to accept the fear of the unexpected.
Bringing open activities into the classroom isn’t throwing students into the void; it’s giving them the tools to climb the mountain themselves. The satisfaction a student feels when they find a solution on their own after having “struggled” a bit with the problem is, in Robert’s words, the ultimate goal of our work.
Conclusion
Robert Kaplinsky’s visit reinforces Innovamat’s commitment to high-quality mathematical education that prioritizes reasoning over memorization. As he says himself, we must give our students opportunities to struggle productively with problems, developing a flexibility that will serve them well beyond the classroom.
Thanks for making us think, Robert!
About Robert Kaplinsky
Robert Kaplinsky has been an educator since 2003, working as a classroom teacher, math specialist, instructor at the University of California, Los Angeles (UCLA), and presenter at international conferences worldwide. He is the co-founder of the website Open Middle, has published articles in leading outlets such as Edutopia and Education Week, and is the author of Open Middle Math: Problems That Unlock Student Thinking. He also spearheaded the #ObserveMe movement and is the founder and president of Grassroots Workshops. Kaplinsky graduated from UCLA with a degree in Mathematics and Applied Science (Computer Science) and holds a Master’s in Education.
