Summer Institute Thunder Bay

1. Summer InstituteThunder Bay MaryLou Kestell John Rodger Wendy Telford Plenary 1

2. What’s Important About the Math we Teach?A Focus on Big Ideas Marian Small MaryLou Kestell www.onetwoinfinity.camkestell@sympatico.ca

3. Minds-On • The 3rd term in a linear growing pattern is negative. • The 30th term is 20. What might the 20th term be?

4. Minds-On • Could the 20th term be either positive or negative? Why is that?

5. Characteristics of Minds-On How does this minds-on engage students? How is it open?

6. Characteristics of Minds-On What do you think the important math underlying idea is?

7. Characteristics of Minds-On How would this question prompt students to deal with that underlying idea?

8. Might lead to students being able to respond to… • What makes a pattern linear is… • There are a lot of linear patterns that include the same term because…

9. Or… • If the 100th term of a linear pattern is relatively small, then…

10. What are the Big Ideas? Randall Charles: A Big Idea is a statement of an idea that is central to the learning of mathematics.

11. What are Big Ideas? Marian Small: A Big Idea is one that connects numerous mathematical understandings into a coherent whole.

12. A Big Idea • is NOT a topic name nor an overall expectation. • is a statement (sentence) that a student could walk away with that makes a fundamental mathematical connection. • provides a lens in which to embed new learning.

13. Big Idea #1 Algebraic reasoning is a process of describing and analyzing (e.g. predicting) generalized mathematical relationships and change using words and symbols.

14. Mathematical Processes Representing Reflecting Reasoning and Proving Connecting Selecting Tools and Computational Strategies Reasoning and Proving Communicating Problem Solving Reflecting Communicating Representing Selecting Tools and Computational Strategies Connecting Problem Solving

15. Notice the processes Big Idea #1 Algebraic reasoning is a process of describing and analyzing (e.g. predicting) generalized mathematical relationships and change using words and symbols. • communicating • reasoning • connecting Reasoning and Proving Communicating Connecting

16. Big Idea #2 Comparing mathematical relationships helps us see that there are classes of relationships and provides insight into each member of the class. Which processes do you see embedded in this 2nd big idea?

17. Big Idea #3 Different representations of relationships (e.g. numeric, graphic, geometric, algebraic, verbal, concrete/pictorial) highlight different characteristics or behaviours, and can serve different purposes. Which processes do you see embedded in this 3rd big idea?

18. Big Idea #4 Limited information about a mathematical relationship can sometimes, but not always, allow us to predict other information about that relationship. Which processes do you see embedded in this 4th big idea?

19. Getting a feel for the big ideas Two sets of questions will be circulated which are designed to bring out the big ideas.

20. Getting a feel for the big ideas Choose one of those sets of questions. Match each question to the big idea it is most likely to elicit.

21. Some questions about your task Which big idea did you find easiest to match first? Which did you find hardest to match first?

22. Some questions about your task Which of the questions did you like best? Why?

23. Some questions about your task How do the questions that matched Big Idea #1 show the notion of generalization?

24. Some questions about your task How do the questions that matched Big Idea #1 show the notion of describing or analyzing relationships or change?

25. Some questions about your task How could the question that matched Big Idea #2 broaden a student’s notion of what a “class” of relationships might be?

26. Some questions about your task How could the questions that matched Big Idea #3 broaden a student’s sense of what different representations mean and/or what their purpose is?

27. Some questions about your task Can you think of other examples that you’ve used in the past (with or without realizing it) to make students see Big Ideas #4 that from limited information you can get more?

28. You just experienced a parallel task We will talk more about these, but these two very related tasks were adjusted to meet your needs but treated together in our consolidation.

29. Why use big ideas? • to build connections students need in order to learn both through grades and within grades • to prioritize instructional goals

30. Sharing big ideas with students When students articulate big ideas, it becomes easier for them to make connections to prior knowledge.

31. Building lesson goals You can use a big idea to hone in on an appropriate lesson goal.

32. For example… Consider the expectation: “Solve first degree equations with non-fractional coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).”

33. What is my lesson goal I am going to propose that it is not “students will use a balance to solve a linear equation”, but…

34. What is my lesson goal maybe: “recognizing that solving an equation means determining an equivalent equation where the unknown value is more obvious.”

35. What I mean These equations are equivalent: x = 4 2x – 7 = 1 3x + 7 = x + 15

36. What I mean… It certainly is easier to see the value of the unknown in one of these equations. These equations are equivalent: x = 4 2x – 7 = 1 3x + 7 = x + 15

37. What does this mean for consolidating the lesson? I need to ask a question or two that gets RIGHT to my goal.

38. Consolidation option Agree or disagree? The equation 5x – 4 = 17 + 3x is really the equation x = 10.5 in disguise.

39. Consolidation option Which equation would you find easier to solve? Why? 5x – 4 = 17 + 3x x = 10.5

40. Consolidation option Why might someone say that solving an equation is about finding what easier equation is being disguised?

41. One more example The curriculum expectation reads: construct tables of values and graphs using a variety of tools to represent linear relations derived from descriptions of realistic situations

42. Possible goal Students will see that it is useful to write the table of values where the independent variable increases in a consistent way, but that’s not required for all tables of values.

43. Related consolidation question Here are two tables of values. Determine whether or not they represent linear relationships. Which table makes it easier to tell?

45. and… Could you use the other table too, if you wanted to?

46. Or… My goal could have been, instead, to ask students to consider how a graphical representation of a relationship described verbally gives other insights into the relationship.

47. That consolidating could have been … What characteristics of the relationship did the graph make easy to see that were not so obvious before?

48. Consolidate today’s work Think/pair/share: What is the difference between an expectation and a big idea? OR What’s so big about big ideas?

49. In preparation for our next time together: Could you think about what you consider important issues in developing lesson goals?

50. The Important Book We would like to introduce you to Margaret Wise Brown’s, The Important Book.