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Includes root memorization and does not guarantee conceptual depth of understanding. This includes facts which are not transferrable and include definitions, formulae in the forms of symbols, and different names of polygons. EX: Knowing the shape of a graph without necessarily understanding how it is generated

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Is an equation that uses mathematical symbols or variables to show a relationship and is represented by FACTS

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Math Theorems List and Important Class 10 Math Theorems (With Proofs)

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Are statements that have been proven and connect explanations of conceptual understandings, represented by PRINCIPLES

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Maths Theorems List and Important Class 10 Maths Theorems (With Proofs)

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The most important maths theorems are listed here. Also, the important theorems for class 10 maths are given here with proofs. Click now to get the complete list of theorems in mathematics.

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Includes broader concepts that help break down specific micro concepts (EX): the topic of integration is supported by concepts such as limits, summation, etc

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Statements that allow students to make connections between two or more concepts. (Principles are theorems in Mathematics) (EX): Fundamental Theorem of Calculus

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Small operations or actions that are embedded in strategies that when applied can allow the strategy to work (EX): Create a Table of Values, Know how to Plot Points

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Broad techniques that students draw upon. They can be seen as the processes through which students acquire and apply mathematical skills, knowledge, and concepts.

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Fundamental building block of math that is a defined series of skills and strategies that form math techniques when a learner face an unfamiliar situation. 1) Understand the Problem, 2) Devise a Plan, 3) Carry out Plan, 4) Review

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Problem-Solving as Central to Mathematics:

A mathematical problem requires using tools for which no immediate solution is apparent. This challenges students to think critically and develop deeper conceptual understanding beyond procedural fluency.

Mathematics education has evolved to emphasize problem-solving as a means of learning, aligning with educational standards like the Common Core.

Role of the Teacher:

The teacher's role shifts from directing lessons to facilitating problem-solving. They must allow students to grapple with problems, encouraging independent strategy development and evaluation.

Teachers construct problem-solving tasks that are appropriately challenging, promoting student engagement and discourse in the classroom.

Benefits of Problem-Solving:

Problem-solving enhances creative, critical, and logical thinking skills while fostering the ability to structure, process information, and enjoy intellectual challenges.

Students develop deeper mathematical understanding, improving their ability to connect ideas and apply strategies across various contexts.

Collaborative and Social Learning:

Mathematics learning occurs within social contexts where students engage with peers to explore solutions. This interaction helps students feel supported, take risks, and refine their ideas through discussion and feedback.

Vygotsky's theory of social learning plays a central role, suggesting that students co-construct knowledge through shared problem-solving experiences.

Students learn problem-solving through phases: understanding the problem, devising a strategy, solving it, and reflecting on the solution.

This process is iterative, requiring students to move back and forth between phases, thus deepening their mathematical reasoning.

Planning for Mathematical Discourse:

Teachers must plan for rich mathematical discussions, encouraging students to justify their solutions, challenge ideas, and communicate their thinking. This builds a community of learners who can approach problems as mathematicians.

Real-World Connections:

Effective problem-solving tasks relate to real-world situations, encouraging students to apply mathematical thinking in meaningful ways.

By embedding mathematics in relevant contexts, students see the subject's applicability and relevance beyond the classroom.

Conclusion:
Mathematics teaching should prioritize problem-solving, allowing students to explore complex tasks that develop their conceptual understanding and reasoning skills. Teachers play a critical role in designing problem-solving experiences that are accessible yet challenging, fostering collaboration, reflection, and meaningful discourse. By focusing on these practices, students not only solve mathematical problems but also prepare for the intellectual challenges they will encounter in life.

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Mathematics as a Complex Problem-Solving Activity | Generation Ready

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By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: ...

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The ability to make generalizations and provides explanations for arguments. Helps build mathematical memory and allows to find out the "why" and critical thinking skills (EX): proving an identity such as double angle in Trig

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Refers to talking and writing about mathematics

Number Talks - students compare relationships and look at different ways to add, subtract, multiply, divide

Social Constructivist - people construct their own understanding and knowledge through experiences and reflecting on them (emphasizes assisted discovery through teacher-student and student-student interactions)

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4) Making Connections & 5) Creating Representations & 6) Investigating 

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4) Making Connections & 5) Creating Representations & 6) Investigating 

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4) Learners can see the link between symbols and procedure, and between facts and how they relate to one another

5) expressions of math ideas such as graphs, tables, figures, etc (helps learners organize and communicate math ideas and helps interpret phenomena)

6) Explore mathematical situations (Ex) ask questions

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Macro, Meso, and micro concepts in mathematics

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Macro, Meso, and micro concepts in mathematics

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MACRO - give breadth of understanding and possess the greatest transferability across disciplines (EX) Geometry, Algebra, Calculus

MESO - breaks down macro concepts to more focused understanding (Ex): Trigonometry

MICRO - gives us the supportive depth of understanding (EX) Ratios, Angles, Similarity

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Consist of skills, which allow learners to follow a set of rules to a specified outcome

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Marriage of Structure of Knowledge and Structure of Process

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Marriage of Structure of Knowledge and Structure of Process

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Creating synergy between facts, skills , and conceptual understanding is important, Lynn Erickson writes, "to stimulate more sophisticated, complex thinking we need to create a synergy between the simpler and more complex processing centers in the brain. This interactive synergy requires the mind to process information on two cognitive levels, the factual and conceptual. The conceptual mind uses facts as a tool to discern pattern, connections, and deeper understandings" (10).