Quiz 1 math prek-2, High School Math Education: Key Concepts, Practices, and Strategies

Ignatian Pedagogical Paradigm

A model of learning that emphasizes context, experience, reflection, action, and evaluation as interconnected steps in the learning process.

Constructivism

A theory of learning suggesting that learners actively construct knowledge through experiences and interactions, building on what they already know.

Number Talk

A daily classroom routine where students mentally solve math problems, share answers, and explain their reasoning strategies to deepen understanding.

Worthwhile Task

A math task that is cognitively demanding, offers multiple entry/exit points, and is rooted in relevant and well-designed contexts.

Cardinality

The understanding that the last number name used when counting a set of objects represents the total quantity of the set.

Relational Understanding

Knowing both what to do and why; deeper comprehension of mathematical concepts.

Instrumental Understanding

Knowing how to apply a rule or procedure without necessarily understanding why it works.

Teaching Through Problem Solving

An approach where students learn new mathematical ideas while solving problems, often organized in a three-phase lesson (before, during, after).

Traditional Approach

A teaching style structured as 'I do, we do, you do,' where the teacher models, guides, then students practice independently.

PA Core Standards

Pennsylvania's state academic standards outlining what students should know and be able to do in each subject at each grade level.

CCSSM (Standards for Mathematical Practice)

Eight key practices in the Common Core State Standards for Mathematics, including reasoning, modeling, precision, and structure recognition.

Five elements of the Ignatian Pedagogical Paradigm

Context, Experience, Reflection, Action, Evaluation.

Constructivism view of learning

Learners build knowledge actively through prior experiences and interactions.

Key features of a Number Talk

Mental problem solving, sharing answers, discussing reasoning and strategies.

Features of a worthwhile math task

High cognitive demand, multiple entry points, real-life relevance, well-designed context.

Traditional Teaching

'I do, we do, you do'; emphasizes procedures.

CCSSM Standards for Mathematical Practice

Make sense of problems; Reason abstractly/quantitatively; Construct arguments; Model; Use tools strategically; Attend to precision; Look for structure; Look for repeated reasoning.

Example of Relational Understanding

Knowing why multiplication works as repeated addition.

Worthwhile Math Task for 2nd Graders

Example: 'If there are 12 cookies and 4 kids, how many cookies does each get?'; meets criteria because it's real-life, has multiple entry points, and requires reasoning.

Comparison of Teaching Approaches

Traditional focuses on procedures and direct instruction; Problem solving focuses on exploration, reasoning, and sense-making.

PA Core Standards (Math)

Define grade-level expectations for math knowledge & skills in PA.

CCSSM Practices (8)

Make sense of problems & persevere.

Understanding Mathematics

'Understanding' = proficiency strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition.

Teaching for Understanding

Relational vs. Instrumental Understanding.

Number Talks

Routine: Present problem → Mental solve → Share answers → Discuss reasoning. Builds fluency, flexibility, and reasoning.

Teaching Approaches

Traditional: 'I do, we do, you do.' Teacher models first.

Worthwhile Tasks

High cognitive demand.

Levels of Cognitive Demand

Range from memorization → procedures → procedures with connections → doing mathematics (highest).

Sample Tasks

Snack Task: Do we have enough snacks for everyone?

Key Terms Recap

Ignatian Pedagogical Paradigm: Context, Experience, Reflection, Action, Evaluation.

What is conceptual understanding in mathematics?

Comprehension of mathematical concepts, operations, and relations.

Define procedural fluency.

Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

What does strategic competence refer to?

The ability to formulate, represent, and solve mathematical problems.

What is adaptive reasoning?

Capacity for logical thought, reflection, explanation, and justification.

What is a productive disposition in mathematics?

A habitual inclination to see mathematics as sensible, useful, and worthwhile.

What does CCSS-M stand for?

Common Core State Standards for Mathematics, providing critical areas for grades K-8.

What is the first mathematical practice outlined in CCSS-M?

Make sense of problems and persevere in solving them.

How should children reason abstractively and quantitatively?

By reasoning with quantities and their relationships, representing and manipulating situations symbolically.

What is the importance of constructing viable arguments in mathematics?

Children use mathematical reasoning to justify ideas and solutions, examining and making sense of arguments.

What does it mean to model with mathematics?

Using mathematics to describe, explain, and solve real-world problems.

Why is attending to precision important in mathematics?

Children learn to be explicit about their reasoning and clarify details.

What is constructivism in learning?

Connecting existing ideas to new information through assimilation and accommodation.

Define assimilation in the context of learning.

New concepts fit prior knowledge, expanding on it.

What is accommodation in learning?

New concepts do not fit existing ideas, creating cognitive conflict.

What does sociocultural theory emphasize in learning?

Active engagement and assistance from those with more knowledge.

What is mathematical understanding?

The ability to justify why a mathematical claim or answer is true.

What are the five dimensions of a productive classroom?

Mathematics, cognitive demand, access to content, agency, and use of assessment.

What is the 'I-we-you' approach in teaching?

Aids problem solving by transitioning from teacher-led to student-led learning.

What are worthwhile tasks in mathematics?

Tasks without prescribed rules, open-ended, and promote procedural fluency and conceptual understanding.

What is the difference between high and low cognitive demand tasks?

High demand tasks challenge students and connect procedures to concepts, while low demand tasks are routine and straightforward.

What are entry points in mathematical tasks?

Challenging approaches that accommodate diverse learners and reduce anxiety.

What is the role of mathematical discussions in learning?

Facilitates understanding through anticipating, monitoring, and connecting ideas.

What is revoicing in a classroom discussion?

Restating a student's statement to clarify or emphasize their idea.

How can mistakes be viewed in a learning environment?

As learning opportunities that contribute to understanding.

What are the three dimensions of lesson plans?

Before: prior knowledge and expectations; During: support and engagement; After: discussions and summarizing.

What are the five strands of mathematical proficiency?

Conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.

What does CCSS-M stand for?

Common Core State Standards for Mathematics.

What is conceptual understanding in mathematics?

Comprehension of mathematical concepts, operations, and relations.

Define procedural fluency.

Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

What is strategic competence?

The ability to formulate, represent, and solve mathematical problems.

What does adaptive reasoning refer to in mathematics?

The capacity for logical thought, reflection, explanation, and justification.

What is a productive disposition in mathematics?

A habitual inclination to see mathematics as sensible, useful, and worthwhile.

What is the role of mathematical practices in education?

To help children analyze problems, use tools, and monitor their strategies to find solutions.

What is constructivism in learning?

The theory that people connect existing ideas to new information through assimilation and accommodation.

What is the difference between assimilation and accommodation?

Assimilation is when new concepts fit prior knowledge, while accommodation occurs when new concepts create cognitive conflict.

What is the sociocultural theory of learning?

The idea that learners are actively engaged and can be assisted by working with more knowledgeable peers.

What are the five dimensions of a productive classroom?

The mathematics, cognitive demand, access to content, agency/authority/identity, and use of assessment.

What does the 'I-we-you' approach in teaching involve?

A method that aids problem-solving by gradually shifting responsibility from teacher to student.

What is the significance of high cognitive demand tasks?

They challenge students and connect procedures to underlying concepts, promoting deep engagement.

What are worthwhile tasks in mathematics?

Tasks that have no prescribed rules, are open-ended, and incorporate both procedural fluency and conceptual understanding.

What is meant by entry points in mathematical tasks?

Challenging tasks that can be approached in various ways, accommodating diverse learners.

What is the purpose of mathematical discussions in the classroom?

To facilitate productive discussions, connect strategies, and enhance understanding.

What are some strategies to increase learning potential in mathematics?

Allowing multiple approaches, making tasks exploratory, postponing solution methods, and increasing entry points.

What is the role of mistakes in learning mathematics?

Mistakes are seen as learning opportunities that can lead to deeper understanding.

What are the three dimensions of lesson plans?

Before (prior knowledge and clear expectations), during (support and worthwhile extensions), and after (active listening and summarizing).

What is revoicing in mathematical discussions?

Restating a student's statement to clarify or emphasize their point.

What does waiting mean in the context of questioning?

Allowing quiet thinking time for students to formulate their responses.

How can teachers support productive struggle in learning mathematics?

By helping students see mistakes as learning opportunities and allowing time for struggle.

What is the importance of agency, authority, and identity in the classroom?

It encourages student-to-student engagement and collaborative learning.

What is the goal of mathematical understanding?

To justify why a mathematical claim or answer is true and to understand the underlying rules.

What is the significance of using appropriate tools strategically?

Children should become familiar with various problem-solving tools and know which are most appropriate for different tasks.

What does it mean to attend to precision in mathematics?

Children learn to be explicit about their reasoning and clarify details in their work.