Collaborative Inquiry for Learning

Mathematics for Teaching

CIL-M

Co-Teaching in a Public Research Lesson – Process and

Sample Key Learning DSBN and NCDSB

Collaborative Planning for the 3 part lesson

Key concepts come from an analysis of the curriculum expectations continuum

A developmental sequence comes from an analysis of the continuum of expectations across the grades, the Guide to Effective Instruction in Mathematics and other research including landscapes, learning trajectories, etc.

Focusing on the Big Idea Knowing what math vocabulary to draw out and

emphasize also comes from the curriculum expectations Choosing a problem DO the problem several ways connect strategies

and/or solutions to continuums, landscapes, or trajectories

Before (Activation)

Consistencies (Fundamentals) Careful thought to the big idea in the lesson mathematics is

connected to the During part of the lesson Students DO mathematics Develop a context that is both engaging and grounds the

mathematics Relatively short (5 to 10 mins) Activating student schema Use of mathematical terms (used in a previous lesson)

Not an abstract review of the previous lesson - it builds on students’ knowledge

Record students’ thinking on the far left side of the board

During (Working on it) Pose and post the problem (working left to right) Ask, “What is the important information we will need to

solve this problem?” Record and post students’ thinking Students work to solve the problem Teachers only ask questions to provoke students’ thinking The purpose is to get the students’ existing thinking onto

the paper the solution is evidence of mathematical understanding at that point in time

Within the struggle is the new learning

The 3-part After AFTER – Teaching and Learning Focused on knitting ideas together from one solution to another

towards the learning goal Mathematical annotations - either on the board or on the student

solutions to make explicit mathematical ideas, strategies, and tools and to show relationships between the solutions

AFTER – Consolidation - Highlights / Summary Focuses on the learning in the learning goal or Big Idea

AFTER – Practice Students solve a similar problem or problems independently using the

work of the class that is still posted to guide their thinking

Teacher Moderation of Classroom WorkAssessment for Learning

Three key questions for analysis:

What mathematics are evident in students’ communication (oral, written, modeled)?

What mathematical language should we use to articulate the mathematics we see and hear from students? (e.g., mathematical actions, concepts, strategies, tools)

What mathematical connections can be discerned between students’ different solutions?

Analyse student work by: identifying the mathematics

evident in student solutions to a lesson problem

discern the mathematical connections between the solutions,

Purpose - Plan next steps instruction

during the lesson for the next day lesson

(activation)

After – Teaching and Learning

Make the mathematical thinking in the room visible:

To get students thinking about their thinking (metacognition) as they solve problems

To make connections among the many different ways to solve a problem AND to the mathematical Big Idea through discussion and annotation

To allow young mathematicians to describe and defend their mathematical ideas and conjectures

After - Consolidation Highlights/ Summary

Preparing revisit: Key concepts that came from an analysis of the

curriculum expectations A developmental sequence that came from an analysis of

the continuum of expectations across the grades and the Guide to Effective Instruction in Mathematics and other research including landscapes, learning trajectories, etc.

The math vocabulary to draw out and emphasize that also came from the curriculum expectations

In the classroom: Coordinate discussion based on student work Record the highlights / summary based on student work

in light of the Big Idea of the lesson

After – Practice

Pose one or two questions that are closely related to the problem just solved

Students may use the work of the lesson as an anchor chart:

mathematical annotations - either on the board or on the student solutions to make explicit mathematical ideas, strategies, and tools and to show relationships between the solutions

Co-Teaching Roles

Classroom teacher 2 co-teachers – may ask questions of

students but may not tell how The other teachers are student

observers • These teachers may not interact with the

students but feed to the co-teachers information about the way students are learning.

The Heart of our Learning Even though it’s messy, we are engaging students

deeply in mathematics We are not telling them or showing them

mathematics but are engaging them in mathematics; in other words…

We are not transmitting our mathematics to students but rather establishing conditions through which students construct their own mathematics

Research shows that students do better in all measures if they are actively engaged in mathematical problem solving

Let’s look at the math!We focus on what

the students DO KNOW, not on what the students don’t know or what we think they should know

As you walk around, you see these solutions. What is your response to the students in the class?

Students are playing a dice game. The red die is the numerator and the white die is the denominator. Plot the dice rolls on the number line to decide who got the highest fraction.

Student Work

What do the students know?

What do we learn about the mathematics by looking at what students do know?

Math We Noticed The numbers were often ordered from smaller quantity of

pieces to larger number of pieces The whole number intervals were generally evenly

spaced Students seemed to use the number line 0-6 for some

fractions and ‘saw’ the number line as 0-1 for others the half-way point on the number line was correctly

placed and was labeled as either ½ or 3 The 1/3 point was correctly placed between 0 and 1/2 The fractional intervals on the number line varied Many students reflected an understanding of the

relationship of the improper fraction to a mixed number

Learning Goals for Next Steps and Further Classroom Discussion

Intervals on the number line should be equidistant

Placing of whole number quantities Counting fraction manipulatives beyond

1 and representing of fractional quantities on the number line

Developing benchmark of 1/3

Things we still wonder…

How many fractions is too many to place on a number line at the beginning? later on?

How much work should students have with the 0-1 number line before moving to a ‘0 – more than 1’ number line?

Are some fractions nicer stepping stones for developing understanding than others?

What is the impact of using area, set and linear models for fractions on this work with a number line?

Sustainability

How has this informed OUR practice? How will this influence our board

improvement plans? How will this help improve student

learning in mathematics in our boards? How will we use our learnings to build

teacher capacity in mathematics