Discussing the Big Ideas

1. Discussing the Big Ideas Kindergarten to Grade 3

2. Getting to 20 • You are on a number line. You can jump however you want as long as you always take the same size jump. • How can you land on 20?

3. How Would You Use Student Responses to That Task? • What’s different about …’s way of counting than ….’s way? • Do you have to start at 1 to get to 20? • Anna jumped to 2, then 4, then 6,… . Ryan jumped to 4, then 8, then 12,… Who took more jumps? Why? • Lee started at 6. Could he get to 20? • Which way of counting to 20 was easiest?

4. Explain Your Thinking Number Riddle • I am a number between 15 and 40. • I am the sum of the numbers on 3 coins. • What number could I be?

5. What I will try to avoid

6. Agenda • WE WILL • Start discussions around what big ideas are • Share the brilliant teaching happening in our classrooms • Look at what is ‘needed’ in math classrooms • Discuss expectations of learners • WE WILL NOT • Have time to get into specific big ideas in the math strands • If Time Permits • Take a look at the mathematical processes as BIG IDEAS behind TEACHINGthe outcomes • All In Hopes That • We Create a PLC that will meet throughout the year

7. What are Big Ideas? • These are ideas that underpin a great number of problems, concepts, or ideas that we want students to learn. • A big idea is NOT a topic like fractions, but might be an idea like a fraction only makes sense if you know the whole of which it is a fraction. • Some people use language like “key concepts” or “enduring understandings”.

8. Making Big Ideas Explicit • We cannot assume that students will see the big ideas if we do not bring them to the student’s attention. • Many we do not know what the big ideas in a lesson are, even if they know the lesson goal. This is the WHY ARE WE TEACHING THIS?

9. What Teaching to Big Ideas Means • Provide an Internal Map of the subject • Interconnections, importance • Helping Interpret Curriculum • Assess the attention or emphasis required to achieve outcome • Avoids looking at outcomes as a checklist of concepts or skills to accomplish

10. What Teaching to Big Ideas Means • Basis to look critically at lessons and activities • Big Ideas are not meant to be implicit, rather EXPLICIT • Instruction & Assessment are planned around big ideas

11. So What do I Mean? i.e. Estimate & measure length, height and distance using non-standard units • What does this outcome mean to you? What are the big ideas? • Working with nonstandard units helps describe the same object with diff measurements • Any measurement can be determined in a variety of ways • The choice of units affects the numerical value of the measurement

12. Big Ideas of Math vs Big Ideas of Math Learner • Table Talk: As a group you will have approximately 10 minutes to answer the following: What skills do students need to have in order to be successful in math?

13. Debrief: Key Words

14. So if we believe what we say… • How do we grow, assess and evaluate _____? • What does _____ look like? • What does _____ sound like? • Do we have rubrics? Checklists or assessment tools we can share? Collaboratively create?

15. Break?

16. Placemat Activity • Create a placemat with as many sides as you have members • Each participant will record three different numbers that describe them in some way • GO!

17. Placemat Activity • As a group, choose one number from each section so that the chosen numbers have something in common (mathematical or contextual) • Choose a speaker who will share what your middle number are and how you came up with your middle category

18. What direction? The driving force behind learning can be boiled down to ten common “needs”. These student needs are identified by Sfard (2003) and must be present if effective learning is to take place.

19. Need for Meaning • Learners look for order, logic, causal dependencies behind things, events, and experiences. • Because the need for meaning motivates us to learn, instruction that focuses on it is more effective than instruction that ignores this need. • Abbott and Ryan (1999) describe learning as an active process in which an individual assimilates new facts and experiences into a pre-existing web of knowledge and understanding.

20. Need for Structure • The need for structure follows from the need for meaning. Meaning involves relations among concepts, not just concepts as such. Understanding of these concepts requires the ability to see structure from the relations among concepts. • If understanding means seeing structure, then the connections between concepts already learned and new concepts being introduced must be an integral part of the curriculum and instruction. • Such connections must include not only real-world applications and relevance, but also assistance in building mathematical abstractions, so students can see how the results can be transferred from one context to another (Wu, 1997).

21. Need for Repetitive Action • A person who has created meaning and structure for a mathematical concept is aware of a repetitive, constant structure of certain actions. He is able to think and speak about the process as an object whose inner structures do not have to be remembered each time one deals with it. • A reasonable level of mastery of basic skills is an important element in constructing mathematics knowledge (Fuson & Briars, 1990; Fuson & Kwon, 1992; Hiebert & Wearne, 1996; Siegler, 2003; Stevenson & Stigler, 1992).

22. Need for Difficulty • True learning implies coping with difficulties. • The goal of learning is to advance a student from abilities he now possesses to those he has not yet developed. The best way to accomplish this is to present the student with tasks beyond his present developmental level but within his zone of proximal development (ZPD). Tasks must be demanding, but still within reach of the student.

23. Need for Relevance and Significance • Significance is the ability to understand and appreciate the place and importance of what is to be learned • Significance means linking new knowledge to existing knowledge, so again stresses the importance of helping students build connections. • Recent research has shown that people tend to do significantly better in applying mathematics to real-life problems than in attempting to deal with the same mathematical content in the context of typical school problems (Nunes, Schlieman, & Carraher, 1993; Saxe, 1991; Schlieman & Carraher, 1996).

24. Need for Social Interaction • There is an inherent social nature to learning and making meaning. • The most obvious forms are student-teacher or student-student exchanges, but even interaction with a textbook is a form of social interaction (Sfard, 2003). • Cooperative learning is another form of learning interaction that does not have the teacher in the central role. It has become popular because of research findings suggesting positive effects on student achievement of collective effort (O'Connor, 1998; Siegler, 2003; Webb, 1991; Webb & Farivar, 1994)

25. Need for Symbolic-Verbal Interaction • Interaction in learning means communication, and communication means using both language (speech) and symbols (written language as well as special mathematics symbols) to convey thoughts. • If mathematics learning is to take place in an interactive setting, students must be encouraged to “talk” mathematics. Research shows that classroom discussion provides many learning opportunities (Ball, 1991;Cobb, Wood, & Yackel, 1991, 1993; Lampert, 1990; Schoenfeld, 1996). • The role of the teacher is particularly important. For the discourse to be effective, students must be taught to communicate mathematically.

26. Need for Well-Defined Discourse • Refers to all communication practices of the classroom - both written and verbal.

27. Need for Belonging • Learning by participation requires one to be a part of a learning community. • Implications for mathematics education: Students need to feel respected and free to speak their mind in the classroom. • However, the extent to which students value belonging to a mathematical community is influenced by the value given to mathematics by the wider community culture (Comiti & Ball, 1996).

28. Need for Balance • To meet learners’ many varied needs, the pedagogy must be varied and rich in possibilities. • It does not imply that old and new are mutually exclusive. For instance, the profound constructivist views of the learner building his or her own knowledge sometimes become trivialized into “teaching by telling”. • Balance individual and cooperative learning opportunities along with problem solving and basic skills acquisition.

29. The reality is that there must be a bit of everything in the classroom: problem solving as well as skills practice, teamwork as well as individual learning and teacher exposition, real-life problems as well as abstract problems, learning by talking as well a silent learning (Sfard, 2003).

30. So what are we really trying to achieve? Our goal should be to have every student become mathematically literate!

31. Table Talk • How has what we talked about today resonate with you, your experience, and your students? • Be prepared to share some of your Big Ideas

32. In Summary • Math really is not a set of tiny little pieces. It is a connected whole. • It is our job to help our students see those connections. • We have to focus in, therefore, on the big ideas, but we also have to ask open and directed questions and be explicit.

33. Next Steps • Please complete the evalution found at: http://cesdsummerinstitute.wikispaces.com/ • Tackle Big Ideas Strand-by-Strand this year?

34. It’s the HOW we teach Mathematical processes

35. What are the Mathematical Processes?

36. Process Breakdown

37. Process Breakdown

38. Process Breakdown

39. Process Breakdown

40. Mathematical Processes

41. Mathematical Processes & Curriculum • As a table group…. • Choose an outcome from an agreed grade level • On the chart paper come up with an activity or activities that would support each process of your outcome. • Be prepared to share with the large group.

42. The Seven Math Processes permeate teaching and learning Communication Connections Mental Math and Estimation Problem Solving Reasoning Technology Visualization ‘filters for understanding’

43. The Seven Math Processes permeate teaching and learning Communication Connections Mental Math and Estimation Problem Solving Reasoning Technology Visualization FOR EXAMPLE ... ‘filters for understanding’

44. Mental Math and Estimation Use mental math to find these differences. As you solve each question, keep track of the processes you are using. 35 – 16 = 92 – 56 = 1001 – 692=

45. Mental Math & Estimation Grade 3: Number Describe and apply mental mathematics strategies for subtracting two 2-digit numerals, such as: • taking the subtrahend to the nearest multiple of ten and then compensating • thinking of addition • using doubles. [C, ME, PS, R, V] Grade 4: Number Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3 and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction. [C, CN, ME, PS, R]

46. Mental Math and Estimation • Types of Calculations Used in Everyday Life • 200 volunteers recorded all computation over a 24-hour period • 84.6% involved some form of mental math • Only 11.1% involved a written component • Almost 60% of all calculations required only an estimate rather than an exact answer • What mathematics do adults really do in everyday life? • - Northcote, M., & McIntosh, M. (1999)

47. Mental Math and Estimation Mental mathematics is the cornerstone of all estimation processes Draw a number line like this one. Place the fractions 3/8 and 4/7 on the number line. Explain to a partner how you decided where to place each fraction. 0 1

48. Benchmarks and Referents Benchmark: something (for example a number) that serves as a reference to which something else (another number) may be compared. Glossary: Alberta Online Guide • Place given numerals on a number line with benchmarks 0, 5, 10 and 20 • Order a given set of decimals by placing them on a number line that contains benchmarks, 0.0, 0.5, 1.0. • Using 0, ½ , 1 to compare and order fractions • Estimate the quotient of two given positive fractions and compare the estimate to whole number benchmarks • Estimate the square root of a given number that is not a perfect square using the roots of perfect squares as benchmarks.

49. Benchmarks and Referents Referent: a personal item that is used to estimate. • Known quantities: five-frame ten-frame • Using 10 and 100 as a referent for estimating quantities • Real-life referents for measurement units: cm, m, mm, g, kg, mL, L, cm2, m2, cm3, m3, minute, hour • 1 mm is about the thickness of a dime • 1 L is like the small milk container • 50 g is the mass of a chocolate bar To estimate the length of my eraser, I use my referent for a cm, the width of my baby finger, and mentally iterate it.

50. Mental Math and Estimation “ME” In Your Curriculum… Kindergarten – 2 (Number) Grade One – 8 (Number) Grade Two – 8 (Number/S&S) Grade Three – 12 (Number/S&S) Grade Four – 6 (N/S&S)