1. Differentiating Mathematics Instruction
Session 2:
Focusing on Important Mathematics: Big Ideas
<Welcome the participants. Informing them that today’s topic is one of the 8 criteria for mathematical literacy in the board’s student success action plan.>
Adapted from Dr. Marian Small’s presentation August, 2008

2. but also the big picture
Thinking about the pieces …
but also the big picture
Today’s session is about focusing on the big ideas of the courses you teach - this means considering the pieces and how they connect.

3. Goals for Session 2
Develop an understanding of the notion of instructional trajectories
Become more knowledgeable about big ideas in mathematics
Make connections between big ideas in Ontario mathematics courses and instructional trajectories
Appreciate the value of big ideas for planning and teaching
Specifically we’ll work through these three areas.

4. Differentiated Instruction Synopsis
Stand Up/Hand Up/ Pair Up
Discuss the synopsis you chose to read about.
Highlight something you found interesting, new, not sure of, or…
Find a new sharing partner and repeat the activity.
<Read the instructions, then ask participants to stand, raise a hand in the air, pair with someone whose hand is also in the air to do the activity.>

5. Integer Addition and Subtraction
Individually write down the big ideas on stick-on notes.
In your groups create a concept map, linking your ideas.
Post your concept maps.
Do a ‘Gallery Walk’ to review others’ work.
Share your observations in a group discussion.
<Read the instructions. >
Initiate a discussion with the question: What observations did you make as you created your concept maps and observed those of others?
This provides a chance to see whether we look at curriculum holistically or in discrete bits and whether you think the written curriculum does or does not need adjusting to meet the breadth of students you have.
Creating a concept map and doing a ‘Gallery Walk’ are literacy strategies that could be used for diagnostic assessment with your students. Concept Maps could be created using the ministry licensed software- SmartIdeas®. However for the purposes of this activity posting your concept maps on chart paper allows for a Gallery Walk and possible discussions providing diagnostic assessment.>

6. Big Ideas - Planning Instructional Sequences
Instructional trajectories/learning landscapes/knowledge packages
A description, usually visual, of the development that helps you see where students come from and where they go to
There are a variety of terms to describe a set of mathematics knowledge (big ideas or concepts, strategies, models of representation) that are organized like a “concept map” to show mathematical connections among learning goals and to map out possible learning paths, sometimes referred to as instructional trajectories.
A learning trajectory is usually presented visually to describe thet prior knowledge a student should have for a particular concept and where they can go to next in their learning.
Big ideas should be statements that are important principles for students to come to. They are not definitions, but the essence of some larger teaching piece.

7. Landscape of Learning Big ideas or key ideas (ovals)
Strategies (rectangles)
Models or representations (triangles)
Cathy Fosnot uses the term ‘learning landscape’ to describe sets of mathematical knowledge. For the topic of fractions, decimals and percent, she identifies the big ideas, the possible instructional strategies and models that could be used to represent thinking. Notice the inter-connectively of these ideas and it’s lack of linearity.
Fosnot, 2002

8. Mathematics Knowledge Package
Liping Ma calls these planning sequences: ‘knowledge packages.’
Ma, 1999

9. Integer +/- trajectory
<Distribute copies of BLM 2.1>
Marion Small created this sample instructional trajectory and considers it to be a work in progress that will undergo changes.
Notice it’s much more like a landscape than a trajectory something we might think of as a more linear path. There are many paths and their connections.
The whole crux starts with identifying what negative ‘x’ means - that it’s the opposite of ‘x’. There are different ways of identifying what an opposite means. Take negative three: What makes it the opposite of 3? You could say it’s on the other side of zero, or if you use colour tiles, you could think of it as the other colour to positive three. It depends on your model.
There are different ways of modelling integers which leads to modeling sums of integers (in the top right hand side of the chart). If a student is having difficulty with modelling sums of integers that tells you the student needs to go back to the idea of modeling integers.
In order to recognize the zero principle it’s important to think of addition as combining.
Let’s examine subtraction (in the bottom half). Notice the two ways of thinking about subtraction. One is as ‘take-away’ or separating and the other is ‘missing addend.’ Some models have you think one way and others another way. Other aspects could be included, e.g. thinking of subtraction as adding the opposite.
Thus, a trajectory has many small pieces that need to be considered and it identifies their connections. This is a teacher’s planning tool for instruction and assessment.
M Small

10. Big Ideas Plan instructional path
Identify connections and learning pathways
Big ideas should be used to plan your instructional path of student learning. It should identify connections between concepts and likely learning pathways so that students can make connections between what they know and new knowledge.
Where would you look for these big ideas? It would be natural to look at the expectations in the curriculum document. However, some expectations are broader and more encompassing than others, but not obvious from the curriculum. The curriculum document also does not suggest the order of presentation.

11. Specific Expectations
Construct tables of values, graphs and equations using a variety of tools to represent linear relations derived from descriptions of realistic situations
Describe a situation that would explain the events illustrated by a given graph of a relationship between two variables or…
Examining two specific expectations for this strand, once again informs the teacher what they WHO? are supposed to do but not what it’s for or what it’s about.
Dr. Small doesn’t suggest these are bad; their purpose is to identify what a teacher should be doing and a student should be learning, but she doesn’t consider these expectations as big ideas, but rather as topics to be covered.

12. Examples
With certain relations, all you need to know are two pieces of data and you can describe the whole relation.
With certain kinds of relations, a specific increase in one variable always results in a specific increase in the other.
Big ideas for students should be a sentence they can articulate that describes what they have learned. For a teacher it’s the end point or goal for student learning.
Dr. Small’s examples of big ideas for linear relations include:
With certain relations, all you need to know are two pieces of data and you can describe the whole relation.
This is true of all linear relations. If you know 2 points on the line then you can define the specific line.
That’s what makes linear relations different from other relations..
Another example would be-
With certain kinds of relations, a specific increase in one variable always results in a specific increase in the other.

13. What are the big ideas you teach?
Move to a table with teachers who teach one of the same courses you do.
What is a big idea in that course?
What is important to teach, but not a big idea?
Create an instructional trajectory for your Big Idea.
<Provide curriculum documents for groups to use as reference and chart paper.
Read the instructions.
Facilitate teachers to work together in groups to identify one big idea in a grade or course they teach. Ask them to create an instructional trajectory. Acknowledge there may be things that they teach that are important, but are not big ideas.>

14. Share Your Thinking Post your instructional trajectories.
Do a ‘Gallery Walk’ to review others’ work.
What observations did you make?
<Invite participants to post their trajectories, do another Gallery Walk and share some of their conversations and ideas either during the creation of their trajectories or during their walk.>

15. Exit Card Consider and write a response to:
Something you learned that would be useful
A question you still have or something you have doubts about
< Ask participants to consider the two statements and to submit their written responses.>

16. Home Activity In your reflection journal, write about:
something you learned that you think might be useful in your teaching.
something with which you disagree with or have doubts about.
why considering Big Ideas helps in differentiating learning.
<Assign their home activity and remind participants of the date and location for the next session.>