From Policy Fidelity to Schema Evolution: A Culturally Mediated Assessment Framework for Optimizing Early Numeracy in West Nusa Tenggara

The Great Map of How Teachers Think

Imagine you are trying to teach a whole city how to bake the perfect cake. The government hands out a strict, national recipe book that says, "Use exactly 200 grams of flour and bake for 30 minutes." But in some neighborhoods, people have been baking with special local spices and clay ovens for generations. If you force them to use the national recipe exactly as written, the cakes might taste bland or even burn. This is the problem of "policy fidelity": making sure everyone follows the rules exactly. But what if the real magic isn't in following the rules, but in how the bakers think about the ingredients?

This paper dives into the world of education, specifically how young children learn numbers (early numeracy). It focuses on a concept called "schema," which is just a fancy word for the mental map or "folder" in a teacher's brain where they keep their ideas about what counts as "knowing math." Usually, researchers just check if teachers are using the right worksheets (policy fidelity). But this study suggests that instead of just checking the worksheet, we should look at how the teacher's mental map changes when they see kids using local tools, like traditional games or rhymes. The paper uses a mix of observing classrooms, interviewing teachers, and a special computer math trick called "Bayesian learning" (which is like a GPS that constantly updates your route based on new traffic) to see how these mental maps evolve. Why does this matter? Because if we can help teachers update their mental maps to include local culture, we might finally make math feel real and exciting for kids everywhere, rather than just a boring test they have to pass.

From Rigid Checklists to Living Maps

The authors of this paper, based in West Nusa Tenggara, Indonesia, noticed a big gap. The national government wants all children to learn math the same way, using standard tests. But in places like West Nusa Tenggara, children are growing up with rich local cultures, like the Sasak and Samawa people, who have their own ways of counting, measuring, and playing. Often, teachers are stuck in the middle. They see a child using a traditional bamboo stick to measure or singing a local rhyme to count, but the official checklist doesn't have a box for that. So, the teacher might ignore it, thinking it's just "play," not "math."

This paper proposes a new way to fix this. Instead of just training teachers to follow the national checklist, the authors created a system to watch how teachers' minds change over time. They call this "Schema Evolution." Think of a teacher's brain as a web of connected ideas. One idea is "counting," another is "number sense." In a rigid system, these connections are fixed. But in this new system, the connections are like rubber bands that stretch and shrink based on what the teacher sees in the classroom.

How the Experiment Worked

The researchers followed 42 teachers across three different districts (Central Lombok, West Lombok, and North Lombok) for 24 weeks. They didn't just give the teachers a test; they set up a multi-step process that felt more like a detective story than a classroom study.

First, they mapped out what the teachers thought before anything changed. They asked teachers to sort math concepts and saw that teachers in different areas had very different mental maps. For example, teachers in North Lombok already had strong connections between "oral traditions" (like rhymes) and "patterning," while teachers in Central Lombok were more focused on "formal counting" and "number recognition."

Then, the real magic happened. The researchers watched the teachers in their classrooms for six months. Every time a teacher saw a child use a local tool (like a bamboo stick or a traditional game) to do math, they recorded it. But they didn't stop there. After each observation, they showed the teachers a video of what happened and asked, "What were you thinking when you saw that?" This is called a "stimulated recall interview."

Here is where the computer math comes in. The researchers used a "Bayesian update algorithm." Imagine a teacher's mental map is a digital drawing where lines connect different ideas. Every time a teacher sees a child using a local tool and realizes, "Hey, that's actually math!", the computer strengthens the line connecting that tool to the math concept. If a teacher sees a worksheet and realizes it doesn't capture the child's skill, the line connecting the worksheet to the skill might get weaker. This happens over and over, 20 times for each teacher, slowly reshaping their entire mental map.

What They Found: The Maps Changed

The results were fascinating. Over the 24 weeks, the teachers' mental maps shifted dramatically.

• The Shift: At the start, teachers were mostly focused on "National Standard Concepts" (like worksheets), which made up about 62% of their focus. By the end, this dropped to 38%. Meanwhile, "Indigenous Cultural Concepts" (like local rhymes and games) jumped from 18% to 34%.
• The Ranking: Before the study, the link between "local counting methods" and "number sense" was ranked 14th in importance. By the end, it jumped to 3rd! Conversely, "worksheet completion" dropped from being the 2nd most important thing to being much lower.
• The Differences: The change wasn't the same everywhere. In North Lombok, where oral traditions are huge, the teachers' maps changed the fastest. They only needed 2 rounds of discussion to agree on a new way to assess kids. In Central Lombok, where teachers were used to strict national training, it took 4 rounds of discussion to get everyone on the same page. This suggests that you can't just roll out a new policy everywhere at once; you have to move at the speed of the local culture.

The New "Living" Policy

The paper argues that we need to stop treating policy like a stone tablet that never changes. Instead, they propose a "layered policy architecture." Imagine a policy as a tree. The roots are the core principles (like "math should be fun" or "math should be fair"), which never change. But the branches and leaves are the specific rules and checklists. These can grow and change depending on the local soil.

Using the data from the teachers' changing mental maps, the researchers helped the teachers and local officials design new "rubrics" (checklists) that actually work for their specific communities. For example, instead of just checking if a child can count on a worksheet, the new rubric might check if a child can keep a rhythm in a traditional song while counting.

The authors are careful to say this isn't a magic wand that fixes everything instantly. They found that the process works best when it respects the local culture and when the teachers are part of the design. They also added "fairness constraints" to their computer model to make sure that minority groups or specific types of teachers didn't get ignored in the final map.

The Big Takeaway

This paper suggests that the best way to improve math education isn't to force teachers to follow a rigid national rulebook. Instead, it's to help teachers update their own mental maps so they can see the math in their children's local lives. By using a mix of careful observation, honest conversations, and a little bit of computer math, they showed that teachers can evolve from just "following rules" to "creating culture." The result is a system where a child's ability to count using a traditional rhyme is seen as just as smart as counting on a worksheet. It's a move from a system that demands compliance to one that celebrates understanding.