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Geometric Reasoning involves using abstract thinking to define, analyze, and make arguments about shapes and spatial relationships. Students must also be able to formalize these arguments in written proofs. Students' geometric knowledge provides concrete representations and models for abstract math concepts, which can serve as an entry point to higher-order math thinking skills.
Children begin with intuitive geometric knowledge (e.g., about shape and symmetry) which builds through exposure to events in the world. In school, this informal knowledge is built upon and expanded into more formal understanding of geometric concepts including:
Symmetry: Reflection across a line (e.g., folding) within a shape;
Congruence: The relationship between shapes through rotation, reflection, or translations; and
Similarity: The relationship between shapes whose sides are proportional and angles are equal.
Geometric Reasoning is scaffolded by Spatial Skills, which allow students to understand two and three-dimensional shapes and space, and by communicating and making arguments about these concepts through different channels, including sensorimotor and embodied activity.
Providing math tasks with high cognitive demand conveys high expectations for all students by challenging them to engage in higher-order thinking.
Students activate more cognitive processes by exploring and representing their understandings in visual form.
Continual use of foundational skills with different problems reinforces a conceptual understanding of math skills.
Daily review strengthens previous learning and can lead to fluent recall.
Thinking of and about patterns encourages learners to look for and understand the rules and relationships that are critical components of mathematical reasoning.
Discussing strategies for solving mathematics problems after initially letting students attempt to problem solve on their own helps them understand how to organize their Algebraic Thinking and intentionally tackle problems.
Analyzing incorrect worked examples is especially beneficial for helping students develop a conceptual understanding of mathematical processes.
Adding motions to complement learning activates more cognitive processes for recall and understanding.
In guided inquiry, teachers help students use their own language for constructing knowledge by active listening and questioning.
Spending time with new content helps move concepts and ideas into Long-term Memory.
Math centers with math games, manipulatives, and activities support learner interests and promote the development of more complex math skills and social interactions.
Math games allow students to practice many math skills in a fun, applied context.
Rhyming, alliteration, and other sound devices reinforce math skills development by activating the mental processes that promote memory.
By talking through their thinking at each step of a process, teachers can model what learning looks like.
Visualizing how ideas fit together helps students construct meaning and strengthens recall.
Providing physical and virtual representations of numbers and math concepts helps activate mental processes.
Connecting information to music and dance moves enhances Short-term and Long-term Memory by drawing on auditory processes and the cognitive benefits of physical activity.
When teachers connect math to the students' world, students see how math is relevant and applicable to their daily lives.
Math games and manipulatives for vision differences support math development for learners with visual needs.
Children's literature can be a welcoming way to help students learn math vocabulary and concepts.
When students create their own number and word problems, they connect math concepts to their background knowledge and lived experiences.
When students explain their thinking process aloud, they recognize the strategies they use and solidify their understanding.
Analyzing and discussing solved problems helps students develop a deeper understanding of abstract mathematical processes.
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