Underlying Theory:
This tool was developed to capture components of mathematical instruction that exist in the interaction between students, teachers, and content. We leverage the instructional triangle (e.g., Hawkins, 2002) (or didactic triad) as means for conceptualizing these interactions. We leverage an expanded version of the triangle that includes attention to the way that teachers influence student interactions with content (Lampert, 2001) and how students interact with each other (Cohen et al., 2003).
As is the case with any instrument, we make no claims that we are capturing all components of high quality mathematics instruction, rather we are foregrounded elements of the student-teacher-content relationships. In particularly, we focus on the elements that can be in support of a classroom culture where students and teachers engage in mathematical argumentation (justifying and generalizing).
Tool components:
Student Habits of Mind
Student Habits of Mind capture productive ways that students can engage with mathematics including reasoning with representations, making connections, noticing and reasoning form regularity, patterns and structure, engaging in metacognition and reflecting, analyzing mistakes and stuck points, and making meaning of tasks and terms. These habits are productive on their own but can be particularly useful towards the goals of the capstone habits: justifying and generalizing.
Student Habits of Interaction
Student Habits of Interaction capture productive ways students can interact with other actors in the classroom. These include activities private reasoning time, explaining mathematical reasoning, revoicing and recapping each other’s ideas, asking genuine questions, exploring multiple pathways, comparing logic and ideas, and critiquing and debating. These activities, when paired with Habits of Mind can lead to fruitful engagement and discussion around mathematics.
Teaching Routines
Teaching routines refer to the various structures, repeated actions, a teacher takes to provide opportunities for students to engage in meaningful mathematics. While a classroom may have many types of routines for many purposes, this set of a routine types in the Math Habits Tool specifically focus on routines that can promote student engagement in meaningful mathematics. They are many ways analogous to other constructs in the literature such as teaching practices. The teaching routine types are broad categories of routines that will often include a set of particular teaching moves (see next section), and vary in form in different classrooms. However, they share overarching goals including: Structuring mathematically worthwhile student talk in pairs or small groups, engaging students in making meaning of tasks, contexts, and/or language, conferring to understand the student thinking of groups or individuals as they engage in mathematical tasks, engaging students in public records of each other’s ideas, selecting and sequencing student ideas, and orchestrating productive mathematical discussion with multiple students.
Catalytic Teaching Habits
The catalytic teaching habits are then individual teaching moves that can serve to engage students productively with the mathematics in the classroom. The term catalytic reflects the potential for such moves to catalyze student engagement in habits of mind and interaction. These moves have purposes such as providing opportunity for students to access mathematics, prompting students to engage deeply in mathematics, researching student’s mathematical thinking, publicizing student reasoning, and engaging students meaningfully in each other’s ideas.
Overall Lesson Codes
In addition to the specific activities that are monitored over time, we also include categories of overall lesson codes. More details on these categories can be found in the training.
- Overall Teacher Activity
- Overall Student Activity
- Lesson Cohesion
- Records of Essential Mathematics
