NCTM Institute on Reasoning and Sense Making - Activities That Promote Reasoning and Sense Making in Algebra

aThe last breakout session I attended at NCTM's Institute on Reasoning and Sense Making was "Activities That Promote Reasoning and Sense Making in Algebra." This session was led by two math teachers and a principal (and former math teacher) at Adlai E. Stevenson HS in Lincolnshire, IL. John Carter is the principal, Gwen Zimmermann is the department head (of a math department with 43 teachers! - they have 4200 students in their school), and Darshan Jain just finished his first year teaching.

Gwen started off by referencing the 5 Practices book by Peg Smith from my morning keynote. She did briefly go over them as well. Like with the morning breakout session I attended with Bill Thill, I was very actively engaged in the tasks they presented us. Here are my notes of some key points they made:

Learning is a social construction – a lot of learning happens when we talk about the mathematics.

Gwen talked about a “typical” lesson having three phases -
-Launch phase
-Explore phase
-Discuss and summarize
The challenge is to maintain high level of student reasoning throughout task.

U.S. teachers take high cognitive level tasks and reduce to teaching from a procedural perspective (Hiebert research). We tend to tell kids how to do it and/or give too much scaffolding.

We did three activities -
1st Activity: Knots activity (Key Press Discovering Algebra)

Take a piece of rope. Begin tying non-overlapping knots. What might be some questions related to an algebra curriculum that could be asked?

Learning Goal: Approximate and interpret rate of change from graphical and numerical data.

2nd activity: Tile problem **when I have access to the presentation slides, I will link to them so you can see the activity better.
There was a picture of 4 stages of a rectangle constructed with square tiles and an empty space in the middle. The first one was 3 across by 4 wide, with a 1 across by 2 wide empy space in the middle. The second one was 4 by 5 with a 2 by 3 empty space. The third was 5 by 6 with a 3 by 4 empty space, and the fourth was 6 by 7 with a 4 by 5 empty space. We were to answer the following:
How many tiles are needed for a model at stage 5? Stage 11?
Explain how you determined the number need for stage 11.
Determine an expression for the number of tiles in a model of any stage, n.
Make explicit the connection(s) between your expression and the physical model.

This was the first in a series of three tasks that are presented to a beginning algebra class. The second figure is a cross (start with 5 squares, then I think the next was 9, etc.) and the third figure is three-dimensional and is found here (it is the second lesson in the pdf, starting on page 10).

3rd Activity: Rational Functions and Representations
-used with Pre-Calculus students
Students work collaboratively in groups of 3-4.

We had an envelope with 6 different limits, 6 different tables (graphing calculator generated), and 6 different descriptors of what is happening at those specific x-values. We also had a graph to check against. Our tasks were to:
Match the given mini-tables to their corresponding limit representation, then check your paining against the graph provided.
Write the graphical significance of the values shown in the tables and limit representations.
Write the equation of a rational function that may produce the table and graph provided.

At the end of this task, they shared with us that they use this mnemonic device to represent the various representation:
VeGAN TRw (vegan tomorrow)
Verbal description
Graphical representation
Algebraic representation
Numerical representation

Technology as a tool
Real world context

Here are some thoughts they shared with us in closing:
Not all activities that we are going to do are going to be that wide – it’s the questions we ask students that make activities involve reasoning and sense making.
Reasoning and Sense Making in Algebra
• Clear learning goal
• Tasks that supports reasoning and sense making
• Intentional, thoughtful planning and implementation of instructional moves (“planned improvisation”) that actively engage all students in productive mathematical discussions

Like my earlier breakout session, I did not have a lot of notes to share. We spent most of the time doing and discussing the tasks at hand. I found these two sessions very valuable because by the time I was done with them, I felt a little better as far as what I should be pursuing in my planning to promote reasoning and sense making in my classes. Although not many people used the terms "Rich Tasks" or "Rich Problems" often (Bill Thill being the exception), in all reality, those are the types of problems and tasks we should be doing on a much more regular basis.

Am I at a point today to totally do rich problems/tasks in my classroom daily? No. Does it make sense to do these types of tasks everyday. I don't think so. I think there still has to be a place in the classroom for instruction. However, the rich tasks can promote an introduction to a topic and lead students to come up with some conclusions on their own before you tie them together better. I think Henri Picciotto was getting at that in his session and I may have that somewhere in my notes for it - but I am just now making that connection.

I think once I get home and have a little more time to digest this all, I may do one more post on the institute. But I can say that it was worthwhile, I had a lot of good sessions, and I am taking away many things from it.