Reflective Questions on Elementary Mathematics Instruction

1. MED 6312Content Instruction in the Elementary School: Mathematics Session 1

2. 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?

3. 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.

4. The changing landscape of elementary mathematics teaching and learning • A distinction between conceptual knowledge andprocedural knowledge • Mathematical 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.

5. 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

6. The changing landscape of elementary mathematics teaching and learning • Communication in a mathematics classroom • Students learn best in a community • Expressive and receptive communication • Make 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

7. The changing landscape of elementary mathematics teaching and learning • Reasoning • Inductive • Deductive • Geometric • Symbolic • Visual • Proportional • Numeric

8. The changing landscape of elementary mathematics teaching and learning • Patterns and change • Repetition of an event or sequence of events • Pattern searching • Predict change • Mathematical connections • Understanding how things are connected and related to each other

9. The changing landscape of elementary mathematics teaching and learning • Connecting concrete and abstract • Apply mathematics in real settings • Situated mathematics • Connecting mathematics and other school subjects • Connecting mathematics and real life

10. 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 Mandates • NAEP • TIMSS

11. 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.

12. NCTM Standards of MathematicsContent and processes that students should learn • Number and Operations • Algebra • Geometry • Measurement • Data Analysis and Probability • www.nctm.org CONTENT

13. PROCESSES • Problem Solving • Reasoning and Proof • Communications • Connections • Representation

14. 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.

15. Problem Solving • What does a problem solving-based classroom . . . • Look like? • Sound like? • Act like? • What is the difference between exercises and problems?

16. 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

17. 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

18. 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.

19. 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?”

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

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

26. What does it mean to learn mathematics?

27. 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

28. 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

30. 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.

31. Becoming a Teacher of Mathematics • A profound, flexible, and adaptive knowledge of mathematics. • Persistence • Positive attitude • Readiness for change • Reflective disposition