MED 6312Content Instruction in the

Elementary School: Mathematics

Session 1

Reflective Questions

• What does it mean to do mathematics?

• What kind of language do you use when doing math?

• What kind of classroom environment needs to be developed?

What is Mathematics?

Mathematics is a study of patterns and relationships.

Mathematics is a way of thinking.

Mathematics is an art, characterized by order and internal consistency.

Mathematics is a language that uses carefully defined terms and symbols.

Mathematics is a tool.

The changing landscape of elementary mathematics teaching and learning

• A distinction between conceptual knowledge and procedural knowledgeMathematical procedures (algorithms)

enable you to find answers to problems according to set rules

Conceptual understanding enables you to find answers to problems in a variety of ways because you understand the underlying concepts of the problem.

The changing landscape of elementary mathematics teaching and learning

• Problem-focused teaching Instead of teaching mathematics as if it were

some naked, abstract, symbol manipulation, we are teaching mathematics as it occurs in our lives.

• Ask students to solve problems in a way that makes sense to them

• Then build new knowledge on their previous understandings

The changing landscape of elementary mathematics teaching and learning

• Communication in a mathematics classroomStudents learn best in a communityExpressive and receptive communicationMake and test mathematical conjectures

• Conjecture: an idea proposed as a possible explanation or generalization that seems to be true but which should be tested further.

Metacognition

The changing landscape of elementary mathematics teaching and learning

• ReasoningInductiveDeductiveGeometricSymbolicVisualProportionalNumeric

The changing landscape of elementary mathematics teaching and learning

• Patterns and changeRepetition of an event or sequence of

eventsPattern searchingPredict change

• Mathematical connectionsUnderstanding how things are

connected and related to each other

The changing landscape of elementary mathematics teaching and learning

• Connecting concrete and abstractApply mathematics in real settingsSituated mathematics

• Connecting mathematics and other school subjects

• Connecting mathematics and real life

What determines the math we teach?• National Curriculum Standards

National Council of Teachers of Mathematics (NCTM)

• New Common Core

• State Core Curriculum

• District Standards

• Federal and State MandatesNAEPTIMSS

NCTM Principles of MathematicsParticular features of high-quality mathematics education

• Equity – Excellence in mathematics education requires equity—high expectations and

strong support for all students.• Curriculum

– A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.

• Teaching – Effective mathematics teaching requires understanding what students know and

need to learn and then challenging and supporting them to learn it well.• Learning

– Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

• Assessment – Assessment should support the learning of important mathematics and furnish

useful information to both teachers and students.• Technology

– Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.

NCTM Standards of MathematicsContent and processes that students should learn

• Number and Operations

• Algebra

• Geometry

• Measurement

• Data Analysis and Probability

• www.nctm.org

CONTENT

PROCESSES

• Problem Solving

• Reasoning and Proof

• Communications

• Connections

• Representation

Problem Solving

• Instructional programs from prekindergarten through grade 12 should enable all students to—

• build new mathematical knowledge through problem solving;

• solve problems that arise in mathematics and in other contexts;

• apply and adapt a variety of appropriate strategies to solve problems;

• monitor and reflect on the process of mathematical problem solving.

Problem Solving

• What does a problem solving-based classroom . . . Look like? Sound like? Act like?

• What is the difference between exercises and problems?

What kinds of things do you need to consider as a teacher as you orchestrate problem solving activities?

• Build a positive classroom atmosphere where students feel secure to express their developing ideas.– Knowledge - give experiences that are successfully reachable.– Beliefs and affect – success, everyone can do it– Control – students learn to monitor their own thinking

• Time – give time to think

• Planning – coordinate when students can do practice exercises, write in journals, try challenging problems.

• Resources – have many additional resources available including real life materials

• Use Technology

• Classroom management – train them from the beginning of the year to work in groups

• Pose problems effectively

Problem Solving Strategies

• Act it out

• Draw a picture

• Use simpler numbers

• Look for a pattern

• Make a table

• Make an organized list

• Look at all the possibilities

• Guess, check, and improve

• Work backward

• Write an equation

Reasoning and Proof

• Instructional programs from prekindergarten through grade 12 should enable all students to—

• recognize reasoning and proof as fundamental aspects of mathematics;

• make and investigate mathematical conjectures;

• develop and evaluate mathematical arguments and proofs;

• select and use various types of reasoning and methods of proof.

Reasoning and Proof

• What is reasoning? Is it separate, the same, or overlapping problem

solving? What reason, what proof do you have that

something is true?

• Reasoning builds thinking through making connections and generalizations

• Example: What is the definition of “quarter?”

Communication

• Instructional programs from prekindergarten through grade 12 should enable all students to—

• organize and consolidate their mathematical thinking through communication;

• communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

• analyze and evaluate the mathematical thinking and strategies of others;

• use the language of mathematics to express mathematical ideas precisely.

Communication

• How can we communicate mathematically? Writing Conversation Prepared presentations Graphs Pictures Symbolic representations

• Communication helps students identify, clarify, organize, articulate, and extend thinking.

• Writing helps review, reiterate, consolidate thinking.

Connections

• Instructional programs from prekindergarten through grade 12 should enable all students to—

• recognize and use connections among mathematical ideas;

• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;

• recognize and apply mathematics in contexts outside of mathematics.

Connections

• Connections within mathematical ideas Example: what does division mean?

• what it means for whole numbers, decimals, fractions, integers, etc.

• Connections between math symbols and concepts How can the area of a circle be measure in square

units?

• Connections between math and the real world Make a scale model of the classroom and arrange

desks.

Representation

• Instructional programs from prekindergarten through grade 12 should enable all students to—

• create and use representations to organize, record, and communicate mathematical ideas;

• select, apply, and translate among mathematical representations to solve problems;

• use representations to model and interpret physical, social, and mathematical phenomena.

Representation

• Different representations for an idea can lead us to different ways of understanding and using that idea.

What does it mean to learn mathematics?

Sociocultural Theory - Vygotsky

• Mental processes exist between and among people in social settings

• From social settings, the learner moves ideas into his or her own psychological realm

• Information is internalized is it is within the learner's ZPD

• Semiotic mediation -- the way information is internalized -- through social interaction and interaction with diagrams, pictures, and actions

Implications

• Build new knowledge on prior knowledge

• Provide opportunities to talk about mathematics

• Build in opportunities for reflective thought

• Encourage multiple approaches

• Treat errors as opportunities for learning

• Scaffold new content

• Honor diversity

A note about manipulatives . . . .

Use them to help students understand and explore mathematical concepts and relationships, to see patterns. Don't prescribe their use or fail to help students connect the dots.

Becoming a Teacher of Mathematics

1. A profound, flexible, and adaptive knowledge of mathematics.

2. Persistence

3. Positive attitude

4. Readiness for change

5. Reflective disposition