1. MED 6312 Content 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 and procedural 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 Mathematics Particular 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 Mathematics Content 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;build new mathematical solve problems that arise in mathematics and in other contexts;solve problems that arise apply and adapt a variety of appropriate strategies to solve problems;apply and adapt monitor and reflect on the process of mathematical problem solving.monitor and reflect

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;recognize reasoning and proof make and investigate mathematical conjectures;make and investigate develop and evaluate mathematical arguments and proofs;develop and evaluate select and use various types of reasoning and methods of proof.select and use

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;organize and consolidate communicate their mathematical thinking coherently and clearly to peers, teachers, and others;communicate their mathematical thinking analyze and evaluate the mathematical thinking and strategies of others;analyze and evaluate use the language of mathematics to express mathematical ideas precisely.use the language of mathematics

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;recognize and use connections understand how mathematical ideas interconnect and build on one another to produce a coherent whole;understand how mathematical ideas interconnect recognize and apply mathematics in contexts outside of mathematics.recognize and apply 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;create and use representations select, apply, and translate among mathematical representations to solve problems;select, apply, and translate use representations to model and interpret physical, social, and mathematical phenomena.use representations to model

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 1.A profound, flexible, and adaptive knowledge of mathematics. 2.Persistence 3.Positive attitude 4.Readiness for change 5.Reflective disposition