A Call to Action for AI to Promote Mathematical Reasoning

By: Nicola Hodkowski, Ph.D.

As a former upper elementary math teacher for eight years, I often felt frustrated watching my students struggle with mathematics reasoning. My students lacked the fundamental understanding needed to reason about mathematics and I felt stuck as to how to help them gain that reasoning. 

I implemented many common recommendations—such as extended practice, varied manipulatives, and extra review sessions—but these were merely superficial fixes; I was simply treating symptoms instead of diagnosing the root cause. I was missing the essential link between my teaching goals and the students’ actual mathematical operations, a question that eventually set me on a new path. 

The core of the problem, I realized, was that my teaching wasn’t adapting to how my students actually thought about mathematics. My previous approach, like using manipulatives such as crackers to model arrays, was based on trying to help them see the math as I saw it. Or using my first order model of the mathematics and trying to “transfer that knowledge” to students. But I wasn’t linking the goal for learning to the specific ways different students were operating to solve the problems.

Celebrating and Using Math Knowledge Students Have

My journey changed when I started focusing on my students’ thinking. I realized I needed not only to pay attention to their thinking, but to capture and leverage it in a meaningful way. I began to infer students’ reasoning so I could then tailor my mathematical goals and classroom activities to their available conceptions. 

My shift was recognizing that a new mathematical idea entails a transformation of the conceptions available to students. Instead of just transmitting my knowledge or trying to promote what I understood the math to be (e.g. I do, we do, you do), I became a facilitator helping students transform their existing knowledge to actively construct new ideas. By creating a “second order model” (SOM) of my students’ math reasoning, I was better equipped to facilitate their transformation from what students knew to what they didn’t yet know.

For instance, I was able to distinguish between students who were still operating on single units (“1s”) and those who were starting to count using composite units. This distinct difference in operating became the driving force for my planning and instruction. I started purposely designing activities to advance students from counting 1s to counting composite units, which is a necessary conceptual shift.

The Proof is in the Outcomes

This shift toward inferring student reasoning based on the units they pay attention to and how they operate on those units and using it as a way to teach more advanced knowledge was an enormous but rewarding change, and the results speak for themselves. The quantitative data on the state test scores from 2012 (before full SOM implementation) to 2013 (post) showed a remarkable trend:

• The percentage of my students scoring at Proficient or Advanced increased from 58% to a staggering 85%.
• In comparison, the school’s increase was 30% (from 46% to 60%), and the district’s was a mere 4% (from 56% to 58%).

My students exceeded their comparable counterparts in both the percentage scoring at Proficient or Advanced and the growth from the previous year. In fact, these results suggested the achievement gap was closing for my students who were one or more years behind in grade-level math. By learning to distinguish my students’ ways of mathematical thinking—using the units they paid attention to and how they operated on those units—and basing my instruction on my inferences of student reasoning, I was able to create lessons truly conducive to their learning. 

A Call to Action for AI in Mathematics Education 

In my role as Director of Mathematics Education Research at Digital Promise, I partner with many researchers and developers who are exploring how to create artificial intelligence (AI) enabled technology tools to help increase student reasoning in mathematics. But I often reflect on the fact that dramatic improvement in my classroom—an increase from 58% to 85% proficiency—was made possible not by technology, but by deeply understanding and responding to the nuances of student thinking. 

Currently, AI does not seem equipped to reason about students’ thinking and use that thinking to advance students’ knowledge. For example, existing tools may ask for student reasoning, but often, when the student gets a problem wrong, the existing tools will revert to a “show and tell” of the mathematics. As I learned in my classroom, students do not just see the math and it is not transmitted—rather, they need to connect a new concept to what they already know in ways that are relevant to their existing understanding. I see this is an opportunity for education technology tools.

I urge the developers of AI-driven education tools to prioritize true student-adaptive pedagogy. Don’t stop at simply grading, providing hints, or assigning practice problems based on a linear curriculum. Instead, focus on developing tools that:

1. Infer Student Reasoning: Create algorithms that can distinguish a student’s underlying mental operations (like counting in 1s versus composite units), rather than just identifying a correct or incorrect answer.  
2. Guide Conceptual Transformation: Use that deep inference to propose or generate tasks that intentionally challenge and build upon a student’s current conception, fostering the necessary cognitive transformation to the next level of reasoning.  
3. Support Teacher Development: Design AI as a reflective partner for teachers, providing them with clear, actionable analysis about a student’s cognitive state—the “what” and “why” of their current thinking—to inform lesson planning.  

Let’s leverage AI to move beyond surface-level personalization and toward tools that genuinely support the conception-based, transformative learning that leads to improvement in student math outcomes.

Nicola Hodkowski, Ph.D.

As the Director of mathematics education research at Digital Promise, Dr. Hodkowski specializes in bridging the gap between the learning sciences and the rapidly evolving AI edtech ecosystem. Her work focuses on moving beyond “first-order” AI models—which merely correct student errors—to “second-order” models that allow technology to identify and build upon a student’s unique mathematical reasoning. Nicola helps translate mathematics education research into AI features that engage students in meaningful dialogue about their ideas, ensuring tools don’t just provide steps but actually use how a student thinks as a tool to advance mathematical reasoning. She serves as a strategic partner to innovators and educators, ensuring that AI-powered solutions remain authentic to the discipline of mathematics while staying deeply student-centered.