The pathways cornerstone aims to help students develop independence and resilience as the complexity increases. Students will build independence through analysis, logic, and discourse. A student who can analyze and reason well is a more independent and resilient student. Gradually students develop the ability to make choices and justify those choices. As they build a mathematics identity, students become better able to make choices in solving problems, justify those choices, and listen to and evaluate different choices made by their peers.
While solutions to problems are important, so are the processes that lead to the solutions and the reasoning behind the solutions. Students should be able to communicate all of this, but this ability is not quickly developed. Students need extensive experiences in oral and written communication regarding mathematics, and they need constructive, detailed feedback in order to develop these skills. Mathematics is, among other things, a language, and students should be comfortable using the language of mathematics. The goal is not for students to memorize an extensive mathematical vocabulary, but rather to develop ease in precisely discussing mathematics being learned.
Pathways Looks Like in the Classroom
The instructional emphasis at all levels should be on a thorough understanding of the subject matter and the development of logical reasoning. Students should be asked “Why?” frequently enough that they anticipate the question, ask it of themselves, and construct compelling arguments to explain. A classroom full of discourse and interaction that focuses on reasoning is a classroom where independent analytic ability and logic are being developed.
Teachers introduce concepts in a way that connects to students' academic background, life experiences, culture and language. They intentionally bridge from informal contextual descriptions to formal definitions. They also clarify the use of mathematical and statistical terminology and symbols, especially those used in different contexts or different disciplines. Teachers engage students in purposeful sharing of mathematical and statistical ideas, reasoning, and approaches using varied representations.
In classrooms with a developed pathways cornerstone, teachers help students develop independence with productive struggle. They anticipate what students might struggle with during a lesson and are prepared to support them productively through the struggle, providing instruction about the role of productive struggle in learning. Teachers allow time for productive struggle and ask questions that scaffold students’ thinking without interfering with their progress. This necessitates providing students with non-graded opportunities that allow them to learn from mistakes without fear of a failing grade.
Students develop the ability to present and explain ideas, reasoning, and representations to one another in pair, small group, and whole-class discourse using age-appropriate, discipline-specific terminology, language, and symbols. Students seek to understand approaches used by peers through clarifying questions, making sense of, and describing approaches used by others. They listen carefully to and critique the reasoning of peers using examples to support or counterexamples to refute arguments. Likewise, they are able to adjust their own thinking and problem-solving strategies after sharing discourse with peers.
Students make sense of tasks by drawing on and making connections with their prior understanding and ideas. They persevere in solving problems and realize that it is acceptable to say, “I am not sure yet how to proceed here,” while staying engaged in the problem solving process. They understand that productive struggle with math tasks is an important step to new insights.High School Pathways
High school pathways described in the
2+1 course model
are an innovation that high school faculty can use to create equitable opportunities that connect mathematics to student goals and interests as educators plan pathway options to create math pathways options for students.
This includes leaning into new and innovative ways to incorporate instructional best practices, such as
NCTM’s Principles to Action
, to create student-centered instructional experiences that should be a focus as we look to implement the standards this next decade. Resources and courses created today can lay a strong foundation for high school experiences in the future.Resources