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Toggle navigation Learner Variability Navigator. Algebraic Thinking Add to Workspace. Factor Connections Hover to see how factors connect to Algebraic Thinking. How Algebraic Thinking connects to Explore related strategies related to all selected factors. Main Ideas Algebraic Thinking allows students to move away from thinking and working with particular numbers and measures to understanding and reasoning with generalized relationships among them.
Algebraic Thinking practices occur in these mathematical domains: Equivalence, expressions, equations and inequalities: Includes developing an understanding of the equal sign as expressing a relationship between equivalent quantities, representing and reasoning with expressions that include unknown quantities, and reasoning with and describing relationships among quantities that may or may not be equivalent; Generalizing and reasoning with arithmetic relationships: Includes reasoning about the structure of arithmetic expressions and relationships including core properties of number and Operations; Functional thinking: Includes representing and reasoning with generalized relationships between co-varying quantities using verbal, symbolic, graphical, and tabular using tables representations; and Proportional Reasoning: Includes reasoning abstractly about the relationship between two generalized quantities.
Learn More View Measures and References. Research-Based Strategies For This Factor Cognitively Demanding Tasks Providing math tasks with high cognitive demand conveys high expectations for all students by challenging them to engage in higher-order thinking. Concrete-Representational-Abstract CRA CRA is a sequential instructional approach during which students move from working with concrete materials to creating representational drawings to using abstract symbols.
Cumulative Review Continual use of foundational skills with different problems reinforces a conceptual understanding of math skills. Daily Review Daily review strengthens previous learning and can lead to fluent recall.
Direct Instruction: Patterning Thinking of and about patterns encourages learners to look for and understand the rules and relationships that are critical components of mathematical reasoning. Direct Instruction: Problem Structures Teaching students to recognize common problem structures helps them transfer solution methods from familiar to unfamiliar problems.
Direct Instruction: Problem-solving Strategies Discussing strategies for solving mathematics problems after initially letting students attempt to problem solve on their own helps them understand how to organize their mathematical thinking and intentionally tackle problems. Error Analysis Analyzing incorrect worked examples is especially beneficial for helping students develop a conceptual understanding of mathematical processes.
Explaining Their Thinking When students explain their thinking process aloud with guidance in response to questions or prompts, they recognize the strategies they use and solidify their understanding.
Gestures Adding motions to complement learning activates more cognitive processes for recall and understanding. Guided Inquiry In guided inquiry, teachers help students use their own language for constructing knowledge by active listening and questioning. Guided Practice Spending time with new content helps move concepts and ideas into Long-term Memory.
Math Centers Math centers support learner interests and promote the development of more complex math skills and social interactions. Math Games Math games allow students to practice many math skills in a fun, applied context. Math Songs Rhyming, alliteration, and other sound devices reinforce math skills development by activating the mental processes that promote memory. Model Assignment or Skill By talking through their thinking at each step of a process, teachers can model what learning looks like.
Multiple Representations: Graphic Organizer Visualizing how ideas fit together helps students construct meaning and strengthens recall. Multiple Representations: Manipulatives Providing physical and virtual representations of numbers and math concepts helps activate mental processes. Peer Teaching Having students teach their knowledge, skills, and understanding to their classmates strengthens learning.
Rich Resources: Children's Literature Children's literature can be a welcoming way to help students learn math vocabulary and concepts. Student-generated Problems When students create their own number and word problems, they connect math concepts to their background knowledge and lived experiences.
Worked Examples Analyzing and discussing solved problems helps students develop a deeper understanding of abstract mathematical processes. Click algebraic thinking to see the collection of articles and lessons in this blog about this topic. You may also want to check on these book for other lessons.. View All Posts. I admire the valuable information you offer in your articles. NZCER I find the description clear, concise, can easily be committed to memory and can form part of teachers everyday discussion with a little effort.
Patterns about what? Author: View All Posts. Post navigation Previous Post Levels of understanding of function in equation form. When higher-level courses regularly incorporate opportunities to build on students' algebraic understanding, students are far more likely to succeed than if the courses present just one mathematical perspective.
The development of algebraic thinking is a process, not an event. It is something that can be part of a positive, motivating, enriching school mathematics experience. Watch for opportunities to develop your own understanding of this important topic throughout the year, including as part of conferences, journals, publications, and the NCTM Web site. For this month's questions, consider the following: How can we build algebraic thinking into the pre-K—12 curriculum at all levels? How should secondary school mathematics be organized to capitalize on the inclusion of algebraic thinking throughout the elementary and middle grades?
What can NCTM do to support teachers in fostering the development of algebraic thinking? Toggle navigation. Log out. Log in. View Cart. Join Now. NCTM Store. Toggle navigation MENU. Log In Not a member?
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