Plenary 1

What’s important about the Math we Teach? A Focus on Big Ideas Marian Small

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

Minds-On 4 Term Number 123…20…30 Term Value NE GA TIV E ??20

Minds-On Could the 20 th 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 was the important underlying idea? 7

Characteristics of Minds-On Would OR how would this question force students to deal with that underlying idea? 8

Would the student be able to respond to… What makes a pattern linear is… There are a lot of linear patterns that include the same term because… 9

Would the student be able to respond to… If the 100 th term of a linear pattern is relatively small, then….. 10

Teacher struggles My experience is that setting lesson goals beyond reciting an expectation or simply using a topic name is a struggle for teachers. 11

For example… Instead of reciting this curriculum expectation as a goal: solve problems involving percents expressed to.. whole- number percents greater than 100%... 12

For example… it could be: If one number is less than 100% of another, the second number is more than 100% of the first. OR 13

For example… it could be: If a percent is greater than 100%, its decimal equivalent is greater than 1. OR 14

For example… it could be: The same strategies are used to solve problems involving percents greater than 100% as problems involving percents less than 100%. OR…. 15

Why you want to do this… If you decide on the goal, you are more likely to know what questions to ask, what activity to use,…. 16

Make it yours Even if you get a lesson from a valued resource, you have to make your OWN decision about what to pull out of that lesson. 17

For example… Let’s look at this lesson from Grade 7 TIPS. 18

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Maybe… Complete this: The way someone figures out terms 3, 4, 5, and 6 might be very different from the way that person figures out term 100 because…. 25

Big Ideas Randy Charles: A Big Idea is a statement of an idea that is central to the learning of mathematics, 26

Big Ideas Marian Small: ….one that links numerous mathematical understandings into a coherent whole. 27

Big Ideas It is not a topic name nor is it an overall expectation. It is a statement (sentence) that a student could walk away with that makes a fundamental mathematical connection. 28

Big Ideas for CAMPPP #1 Algebraic reasoning is a process of describing and analyzing (e.g. predicting) generalized mathematical relationships and change using words and symbols. 29

It’s not… It is possible that, on first blush, this may sound like a definition, but it’s not. It provides a new lens in which to embed the learning. 30

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

Notice the processes Notice the embedded processes in the 1 st big idea- communication, reasoning, connecting 32

Big Ideas for CAMPPP #2 Different representations of relationships (e.g. numeric, graphic, geometric, algebraic, verbal,concrete/pictorial) highlight different characteristics or behaviours, and can serve different purposes. 33

Which processes do you see embedded in this big idea? 34

Big Ideas for CAMPPP #3 Comparing mathematical relationships helps us see that there are classes of relationships and provides insight into each member of the class. 35

Big Ideas for CAMPPP #4 Limited information about a mathematical relationship can sometimes, but not always, allow us to predict other information about that relationship. 36

Getting a feel for the big ideas Two sets of questions will be circulated which are designed to bring out the big ideas. 37

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. 38

Some questions about your task Which big idea did you find easiest to match first? Which did you find hardest to match first? 39

Some questions about your task Which of the questions did you like best? Why? 40

Some questions about your task How do the questions that matched Big Idea 1 show the notion of generalization? 41

Some questions about your task How do the questions that matched Big Idea 1 show the notion of describing or analyzing relationships or change? 42

Some questions about your task How could the questions that matched Big Idea 2 broaden a student’s sense of what different representations mean and/or what their purpose is? 43

Some questions about your task How could the question that matched Big Idea 3 broaden a student’s notion of what a “class” of relationships might be? 44

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 that from limited information you can get more? 45

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. 46

Why use big ideas? to build connections students need in order to learn both through grades and within grades to prioritize instructional goals 47

Why use big ideas? It helps for students to know what the big ideas are so that the connections to prior knowledge they are making are more explicit. 48

Building lesson goals You can use a big idea to hone in on an appropriate lesson goal. 49

For example… Consider the expectation: Solve first degree equations with non- fractional coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1) 50

What is my lesson goal I am going to propose that it is not that “students will use a balance to solve a linear equation”, but… 51

What is my lesson goal maybe: recognizing that solving an equation means determining an equivalent equation where the unknown value is more obvious. 52

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

What I mean These equations are equivalent: X = 4 2x – 7 = 1 3x + 7 = x But it’s sure easier to see the unknown value in one of them.

What does this mean for consolidating the lesson? I need to ask a question or two that gets RIGHT to my goal. 55

What does this mean for consolidating the lesson? Agree or disagree: The equation 5x – 4 = x is really the equation x = 10.5 in disguise. 56

What does this mean for consolidating the lesson? Which equation would you find easier to solve? Why? 5x – 4 = x x =

What does this mean for consolidating the lesson? Why might someone say that solving an equation is about finding what easier equation is being disguised? 58

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 59

My goal today might be… for students to 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 a table of values. 60

So I could ask… Here are two tables of values. You want to find out if they represent linear relationships. Which table makes it easier to tell? 61

62 xy xy

and… Could you use the other table too, if you wanted to? 63

Or.. My goal could have been, instead, to ask students to consider how a graphical representation of that same relationship gives other insights into it. 64

Consolidate 65 Think/pair/share: What is the difference between an expectation and a big idea? OR What’s so big about big ideas?

The Important Book We would like to introduce you to Margaret Wise Brown’s The Important Book. 66

A Sample page 67

We will 68 use this book throughout the week as a way for you to consolidate what you explore in our CAMPPP.