Toward the goal of achieving master mathematicians...

ACQUIRED KNOWLEDGE & SKILLS: It includes the acquisition of knowledge and skills such as….

• choosing and using tools: Students consider the available tools when solving a mathematical problem. Students are familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insights to be gained and their limitations.

• computational fluency: Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently.

• conceptual understanding: Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful.

• degrees of complexity: Students engage in mathematical practices that move thinking across the levels Webb’s Depth of Knowledge: Recall, Skill/Concept, Strategic Thinking, Extended Thinking.

• 8 standards for mathematical practice for students: The Standards of Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.

  • • 1. Make sense of problems and persevere in solving them.

      2. Reason abstractly and quantitatively.

      3. Construct viable arguments and critique the reasoning of others.

      4. Model with mathematics.

      5. Use appropriate tools strategically.

      6. Attend to precision.

      7. Look for and make use of structure.

      8. Look for and express regularity in repeated reasoning.

• making connections: Students recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; and recognize and apply mathematics in contexts outside of mathematics.6

• problem-solving strategies & skills: Students use many problem-solving strategies intuitively when they solve problems. When they become familiar with a collection of strategies, by seeing them modeled and then practicing them, helps students develop useful tools for tackling problems and strengthens their problem-solving abilities. (Example: look for a pattern, construct a table, organize a list, act it out, draw a picture, use objects, guess and check, work backward, write an equation, make a model, solve a similar problem).

• procedural fluency: Students displaying procedural fluency know procedures and when to use them, and they can perform them flexibly, accurately, and efficiently.

• vocabulary: Students who are given the opportunity to develop their mathematics vocabulary knowledge increases their abstract reasoning ability and move beyond operations to problem-solving. Math vocabulary is inextricably bound to students’ conceptual understanding of mathematics.

INSTRUCTIONAL STRATEGIES & CONTEXTS: Which occurs in whole group instruction, small group instruction and/or working with individual students using strategies like...

• 8 mathematical teaching practices

1. 1. 1. Establish mathematical goals to focus learning.

      2. Implement tasks that promote reasoning and problem solving.

      3. Use and connect mathematical representations.

      4. Facilitate meaningful mathematical discourse.

      5. Pose purposeful questions.

      6. Build procedural fluency from conceptual understanding.

      7. Support productive struggle in learning mathematics.

      8. Elicit and use evidence of student thinking.

• assignments that matter: Utilizes assignments that include purposefully selected non-routine, real-world and abstract mathematical tasks that advance mathematical understanding, reasoning, and application. Quality assignments ensure that students spend classroom time as well as time out of class completing assignments that encourage mathematical reasoning and developing multiple solution paths to complete.

• CRA Method of Instruction (Concrete - Representational - Abstract): Teachers begin with concrete manipulative experiences, transition students to using visual representations (drawings), and finally transition to using abstract mathematical notation.

• gradual release of responsibility: An approach to instruction that moves from teacher modeling to shared responsibility between teacher and students to independent student practice. A structure of “I do”, “We do”, “Few do” and “You do” is commonly used.

• guided practice: Guided practice is interactive instruction between teacher and students. After the teacher introduces new learning, they begin the student practice process by engaging students in a similar task to what they will complete later in the lesson independently. Students and teacher collaboratively complete the task as a model.

• hands-on & manipulatives: Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether specifically made for mathematics (e.g., connecting cubes) or for other purposes (e.g. buttons). These tools can be used to introduce, practice, remediate a concept, and/or solve math problems.

• inquiry-based learning: In inquiry-based learning, teachers use questions, problems and scenarios to help students learn through individual thought and investigation. Instead of simply presenting facts, the teacher encourages students to talk about a problem and draw on their intuition to understand it. Students are allowed to ask their own questions - essentially providing their own inquiry.

• math discourse: Meaningful discourse includes students comparing and contrasting ideas and methods, constructing viable arguments, critiquing each other’s reasoning, and helping each other make sense of mathematics. When focused and properly facilitated, discourse provides a powerful means for increasing understanding about mathematical ideas and concepts.

• mathematical mindset: Mathematical Mindsets are practical strategies and activities to help teachers show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Mathematical Mindsets: explain how the brain process mathematics learning, reveals how to turn mistakes and struggles into valuable learning experiences, provides examples of rich mathematical activities to replace rote learning, explain ways to give students positive math mindset, and gives examples of how assessment and grading policies need to change to support real understanding.

• modeling: The purpose of explicitly modeling of a target mathematics concept/skill provides students a clear and accessible format for initially acquiring an understanding of the mathematics concept/skill. Explicit modeling by the teacher provides students with a clear, accurate, multi-sensory of the skill or concept.

• questioning & feedback: Strategically using questioning techniques to ensure all learners have the opportunity to participate in meaningful thinking and discussion. The teacher asks varying types of questions and give feedback to facilitate conversations that support higher-order thinking. Questions are strategically planned prior to the lesson and developed throughout the teaching process.

• technology: It is essential that teachers and students have regular access to technologies that support and advance mathematical sense-making, reasoning, problem-solving, and communication. Effective teachers optimize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics.