Imagine this: A student volunteers to answer a math question in an elementary school classroom. The teacher knows from having previously worked with the student that, although he can easily follow the algorithmic steps of the math problem, he has difficulty with his reasoning and his ability to make sense of the steps he is taking.
As he struggles to answer the question, two other students begin to whisper questions in his ear. Her questions help create a situation that allows her to reflect on the connections between the algorithmic steps she knows to solve the math problem and deepen her reasoning about why those steps work. Suddenly, after thinking about her classmates’ questions, she smiles and proudly leads the class through her correct reasoning for the math problem.
In this situation, the teacher has made an effort to get to know each and every student, the mathematical knowledge they bring to the lesson, and how that knowledge can be used to advance learning of more complex mathematics. The teacher has also set up a classroom that nurtures curiosity and questioning that leads to learning.
As hopeful as this scenario sounds, math is a struggle for many students. Nationwide, average math achievement remains low with clear and persistent disparities across racial and ethnic groups. According to the National Assessment of Educational Progress in 2022, students in grades four and eight had the further decline in mathematics since 1990. Additionally, data shows that only 35 percent of fourth grade students were proficient in mathematics, falling to 26 percent proficient at the eighth grade level. As a result, the achievement gap between white students and black and Hispanic students has widened.
Many efforts to improve student math achievement focus largely on adapting grade level lessons to an entire class of students. Students are expected to learn mathematics by participating in the lesson activity; however, this approach ignores differences in how students use their own knowledge in each lesson to advance their learning.
To foster success in mathematics, we must consider what students already know as a way to build on what they don’t already know.
Changing mathematical thinking
Learning mathematics is a cognitive process based on the student’s experience. The change from not knowing to learning a mathematical concept, also known as reorganizationoccurs when a student uses their existing ideas and understanding as a way to develop more advanced ideas.
The reorganization occurs through two related mental processes that the psychologist Jean Piaget called assimilation and accommodation. Assimilation is how we, including students, see the world with the current knowledge that we have. Accommodation is how we learn and change our lens to reorganize what we know into more advanced thinking. Facilitation of learning by a teacher can go a long way in helping a student move from assimilating to adapting a new mathematical concept.
To promote reorganization, create a more student-centered classroom, and change students’ mathematical thinking, instruction must include both a second-order model and consideration of social and cultural contexts.
second order model
TO second order model (SOM) is a teacher’s recognition of his students’ mathematical conceptions and the differences between the teacher’s mathematical thinking and the student’s mathematical thinking, the end product being assimilation. By inferring and understanding the different conceptions that students have, teachers can meet specific learning needsassess progress toward the intended math goal and adjust instruction as necessary to advance student understandings.
As a facilitator of mathematical learning, one needs to develop a clear distinction between “my students reason the same way I do, so I can teach them how I understand it” and a SOM that instead says, “my students have different conceptions than I do.” ”. do, so I need to consider your understanding to guide my instruction.” As teachers develop a SOM, they become more aware of student math facts and their classroom can become more student-centered. A master operating with a SOM You can choose the most appropriate activities and tools to advance student learning from an asset-based perspective, taking students from what they know to what they can easily learn next.
Social and Cultural Contexts
As psychologist Lev Vygotsky has shared, learning mathematics is also social and cultural. Social interaction within the context of a classroom serves as a way for students to build understanding through increased awareness of multiple cultural perspectives and meanings negotiated through interaction. Specifically, social interaction is an important component for the development of a mathematical concept and can help the process of cognitive reorganization of the student by providing situations that lead to questions, interruptions and reflections.
To support a student in realigning their existing understanding to more advanced concepts, social interactions should include teacher facilitation, which is specifically designed to support learning by enabling students to use existing understanding of mathematics as a way to engage in higher-level thought processes. reasoning and problem solving in more advanced mathematics.
Recognize what students already bring to learning
As former teachers and educational leaders, we have a duty to provide students with opportunities to advance their mathematical reasoning in student-centered classrooms. To provide such opportunities for student-centered mathematics classrooms, it is important to understand how learning occurs, recognize students’ assets and existing understanding, and create awareness of the differences between teachers’ mathematical thinking and mathematical thinking. student mathematics.
As we think about the future of math classrooms, we continue to explore how academic standards and cultural, social, and emotional development intersect to support math learning. We look forward to a future that recognizes students’ existing mathematical knowledge as a starting point for thinking about new ideas and concepts.
