In today's world, students who are truly successful in mathematics can not only solve problems but also explain:
- what each problem is asking
- how they will solve the problem
- why their solution plan makes sense
- and how they know their answer is correct.
In other words, students must be able to think metacognitively about each and every problem and explain their thinking.
Background
The 8 Standards for Mathematical Practice presented by NCTM are the same for all grade levels, which makes it difficult to provide much practical guidance for classroom implementation. Therefore, it provides the standard and its learning progression for every grade from pre-kindergarten to high school to understand how the thinking builds in tandem with students' cognitive development and the demands of mathematical content.
Standard for Mathematical Practice 2: Reason abstractly and quantitatively
Fostering metacognition requires a balance of explicit instruction, teacher modeling, student-centered exploration, and responsive coaching._HELP ME UNDERSTAND**
Explanation of What This Thinking Looks Like
At each grade level, this thinking builds upon previous knowledge, moving from concrete and specific to abstract and symbolic. For example, as students progress from second to third grade, they learn to write numerical expressions that describe the number of tiles in different but related ways.
A Step-By-Step Approach to Teaching Practice 2
- Begin by asking students to reflect on what each number in a fraction represents. This can be done through discussions or group activities.
- Next, ask students to discuss different sample operational strategies for a patterning problem, evaluating which is the most efficient and accurate means of finding a solution.
- Encourage teachers to play an active role in teaching metacognition by providing guidance and support to students as they develop their skills.
A Reflection Guide to Support Students as They "Think About Their Thinking"
As students continue to develop their metacognitive skills, their understanding of their own learning process and articulation will become easier. This will, in turn, help them improve their conceptual understanding, procedural skill, and fluency and apply their mathematical thinking.
Fostering Learnership in Mathematics
Ownership of mathematical learning requires metacognition and learnership. Explicit instruction, modeling, and coaching are essential for helping students develop these habits.
Decontextualizing and Contextualizing
Decontextualizing involves abstracting a given situation and representing it symbolically, while contextualizing involves returning to the original context after solving the problem. Both processes are crucial for mastering mathematical thinking.
Decontextualizing
Decontextualizing means taking a specific problem situation and representing it abstractly, symbolically, and then manipulating these symbols without necessarily attending to their referents and contexts.
Contextualizing
Contextualizing is the movement from general to specific. It involves taking an abstract symbol or an equation and looking for its context, its special case.
Levels of Knowing Mathematics
Learning a concept or procedure through several levels of knowing: intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communication.
Students must be able to reason at both levels, to understand context and abstract relationships, and apply that understanding. By fostering metacognition and learnership, teachers can help students develop the skills necessary for success in mathematics.
Reflection
As a teacher, I understand the importance of fostering metacognition in my students. I plan to incorporate metacognitive strategies into my lesson planning and provide regular opportunities for students to reflect on their learning process. I believe that by doing so, I will be able to support my students in becoming more independent learners and eventually, mathematical experts.