Experience First, Formalize Later: A Research-Based Framework for High-Impact Math Instruction
- Math Medic
- 4 hours ago
- 7 min read
Mathematics has long been identified as a crucial subject for developing analytical thinking and problem-solving skills. Many important fields in science, technology, engineering, and economics rely on workers’ abilities to interpret data, notice patterns, and reason abstractly and quantitatively. Yet large-scale international assessments, such as the OECD’s PISA report, indicate that mathematics remains one of the most challenging and disliked subjects for students worldwide.
A core issue is that math classrooms do not reflect the activities of real mathematics: conjecturing, questioning, justifying, noticing, generalizing, visualizing, and critiquing.
Too many students are memorizing and mimicking, following step-by-step procedures rather than thinking deeply (Liljedahl, 2021; Civil, 2002).
Systemic barriers resulting in inequitable access to resources and emotional factors such as math anxiety further exacerbate the challenges of learning mathematics (Schoenfeld, 2022).
Research points to an alternate approach, where mathematics instruction is grounded in inquiry, and students participate through investigation and problem-solving, leading to deeper learning (Hiebert & Grouws, 2007). One obstacle to this pedagogical approach, however, is that many educators lack the training or resources to implement it well, leading them to rely on traditional instructional techniques with which they are more familiar (Mangarin & Caballes, 2024).
The Experience First, Formalize Later (EFFL) model is Math Medic’s teacher-created, classroom-tested teaching approach that has students working in small groups on an accessible and carefully sequenced activity designed to build conceptual understanding.
This is followed by a whole class debrief where student work is discussed and formalized and the teacher connects students’ thinking with the key ideas, definitions, and formulas associated with the lesson.
The EFFL model represents a significant departure from the institutionalized practices of typical math instruction.Traditional lecture-based models place the teacher at the front of the room giving the vocabulary, formulas, and procedures needed in the lesson and then asking students to work individually on practice problems (Bature & Atweh, 2016). The EFFL model flips this by asking students to generate the ideas and to construct understanding collaboratively. The teacher supports the students in this endeavor by asking assessing and advancing questions, facilitating discourse so that students' thinking becomes visible and usable to other students, and connecting students’ ideas (Smith & Stein, 2011). In the debrief, the teacher makes explicit the mathematics embedded in a student's response and formalizes what students have already discovered for themselves.
An Experience First, Formalize Later lesson has four main components:
Activity
Students work collaboratively in small groups to reason about and discuss a carefully sequenced set of questions that builds toward the main idea of the lesson. The Activities are designed to be intuitive and to foster students' sense making.
Debrief
Students share their thinking in a whole group discussion and the teacher formalizes the learning by adding vocabulary, notation, and formulas to the margins of the Activity. Student responses are carefully selected and sequenced to build towards the main idea of the lesson.
QuickNotes
The teacher records a summary of the big ideas of the lesson that relate to the learning targets. Rather than presenting this information at the on-set of the lesson, this portion of the lesson consolidates what students have already discovered in the Activity.
Check Your Understanding
Students work on a few problems in new contexts to check and apply their understanding of the lesson.
The clear structure of each lesson not only helps students acclimate to the work of doing mathematics but also facilitates the teacher’s implementation of the lesson. The EFFL model is exploratory but still purposeful, structured but not scripted, and works across all math courses, ranging from Algebra 1 all the way to AP Calculus. Because teacher implementation is crucial to the use of any instructional resource, we’ve considered teachers at all levels of experience when designing our lessons and support materials.
Why does EFFL work? A Look at the Research
Students begin exploring concepts using the natural language available to them. The margin notes attach academic language at the precise point in the lesson when the naming is necessary, rather than pre-teaching the vocabulary (Dougherty et al., 2021).
The original work completed by students in small groups can be seen as their “rough draft” thinking and the red margin notes as the formalized version of those ideas. When teachers facilitate discussion around and add on to student ideas in the debrief, the goal is not to “fix” student thinking, but to add layers of formality and update the draft. Students in an EFFL classroom are always open to revising their ideas (Jansen, 2020).
Students use their intuition and existing knowledge to work through the task and must problem solve to answer questions that are unfamiliar to them. The process of problem solving leads to the construction of new understanding (Schoen, 2003; Cai & Lester, 2010).
Students do mathematics, rather than watching, memorizing, or learning facts about mathematics (NCTM, 2014).
Tasks have easy entry points and evolving complexity. They are designed in a low-floor, high-ceiling format (Liljedahl, 2021).
Learning happens by interacting with others and talking through ideas (Olanrewaju, 2019 & Gerlach, 1994). Studies show that collaborative learning increases students’ interest in mathematics as well as their achievement (Lawrence, 2004).
Giving students opportunities to explain and clarify their thinking in small groups before sharing in a whole-class discussion has been shown to be particularly effective for supporting English language learners (Malloy et al., 2009).
The EFFL model shifts mathematical authority from the teacher to the student. Students are responsible for the intellectual heavy lifting and are seen as the primary resources for learning. Students are asked to be producers rather than consumers of mathematical ideas (Dunleavy, 2015, Gresalfi & Cobb, 2006).
The activity and debrief are designed to foster equitable, equal status interactions, positioning all students as mathematically competent with ideas worth attending to. In this way, students form new ideas about their identities as mathematics learners (Cohen & Lotan, 2014; Horn, 2014; Bartell et al., 2017).
Traditional lecture-based instruction that focuses on memorization and repeated exercises has not been shown to produce meaningful learning or develop deep understanding of concepts (Boaler, 2016). When students passively obtain information, they are less likely to remember it and less likely to be able to apply it to new situations (Mangarin & Caballes, 2024).
Students in active learning environments learn more, but feel like they learn less. Conversely, students in passive learning environments report higher feelings of confidence in their understanding but demonstrate weaker performance on assessments (Deslauriers et al., 2019).
There is a critical difference between learning and performance. Conditions that result in quick, short-term gains in scores (performance) often don’t promote retention, whereas desirable difficulty and productive struggle are necessary for meaningful, lasting learning (Bjork & Bjork, 2011).
Evidence of Impact: What Schools Are Saying
Math Medic improves classroom engagement and discourse. Educators describe Math Medic as a program that gets students talking, thinking, and learning from each other. At Milton High School in Vermont, teachers saw students engaging in “on-topic and engaged discourse” as they wrestled with problems together, reinforcing retention and understanding. At Desert Hot Springs High School in California, teachers noted that lessons are “scaffolded, context-based, and accessible to all students from the very first question,” leading to richer class discussions, stronger mathematical vocabulary, and more equitable access to rigorous math. Across districts, schools point to the EFFL model as the engine driving this engagement: students explore and make sense of mathematics before formalizing their learning, which builds ownership and confidence.
Math Medic strengthens retention and long-term understanding. Teachers repeatedly emphasize that Math Medic helps students hold on to what they’ve learned long after the initial lesson. For example, educators in Vermont note that because students spend time exploring problems and discussing their thinking with peers, “retention is being affected” in ways traditional lecture-based methods cannot match. At Atlantic City High School, leaders report that the EFFL approach directly supports “long-term student understanding and retention of key concepts.” In California, department leaders observed that Math Medic lessons are scaffolded and context-based, which results in students “develop(ing) understanding before formalizing ideas” - an approach that deepens memory and application. And at Grafton High School, teachers saw students not only thrive in AP-level courses but also transition more smoothly between Algebra 2 and Precalculus because their learning carried forward. Schools are clear: Math Medic doesn’t just produce short-term gains, it builds lasting mathematical understanding.
Math Medic measurably improves student learning outcomes. Teachers consistently highlight significant gains in student achievement when Math Medic is implemented with fidelity. For example, Milton High School reported double-digit percentage increases in standardized test proficiency for multiple years in a row, including over 30% growth in a single year for 9th graders. As one teacher explained, “The curricular resources we’ve used (Math Medic) would be the only variable that has changed since our proficiency scores have seen their meteoric rise.” Similarly, at Grafton High School in Wisconsin, the adoption of Math Medic in Algebra 2 Honors and AP Precalculus resulted in a 90% AP Precalculus pass rate in its very first year. Schools attribute these gains not only to rigorous content, but to Math Medic’s structure that balances exploration, conceptual understanding, and practice.
Math Medic brings joy and motivation back to math learning. Beyond measurable gains, teachers emphasize the joy their students feel when learning math through Math Medic. Students describe lessons as engaging and relatable, with investigations that “have a clever hook or engaging scenario that students can relate to.” Teachers at one school report that
students “instantly embraced the EFFL model” and now look forward to exploring new ideas,
often leaving class having already completed their practice work through lively collaboration. One department chair in Wisconsin reflected, “My students are engaged, talking about math, and really enjoying being in my classroom.” This combination of rigor and enjoyment not only raises achievement but also reshapes students’ attitudes toward mathematics in lasting ways (Martin, 2009).
References
