Usha led us through an exploration of a visual pattern, building off of the work we’ve done at the last two meetings. She used problem posing to enable us to have greater ownership on the problem and to widen options to explore.
We also stepped back a few times throughout the activity to focus on the teaching.
The Problem-Posing Steps of Usha’s Activity
Grouping to Consider (and Why)
Usha wrote the following statement on the board:
“I feel like I understand the big ideas of 9th grade algebra”
Then, she asked everyone to write their names underneath on a spectrum in terms of our individual level of agreement with the statement.
Usha asked, “Why did I ask you to write your name on the spectrum?”
|This lead to some discussion of questions like “what does it mean to say you understandsomething?” and “what does it mean for something to be a big idea?”.
There was also some discussion as to whether this statement was talking about the actual math content or the specific standards required in the 9th grade.
To explore the concept of “Big Ideas”, Usha recommended taking a look at “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics” by Randall Charles.
|Questions of intent are great in both professional development and classroom discussions. Usha asked us, as teachers, to reflect on a teaching choice she made to engage us in a teaching discussion. It conveys the idea that as teachers we should be intentional with the steps of our lesson and sometimes even about the exact wording we use to pose specific questions.
Someone said the statement can serve as a kind of quick assessment of students.
Usha raised the important question of how we pair students in our math classes. She pointed out that when it comes to putting strong students together with slower students or keeping students of similar abilities together, it depends entirely on the kind of question/activity. A high/low level pairing might work well for certain problems (if we want teaching to happen between students), but for others, especially when students are beginning to formulate an approach to a non-routine problem, it can be frustrating for both students (and we might want students to work from their own understanding and not be pulled into a stronger student’s orbit). The weaker student often loses out in the activity.
We want to pair students so that they can support each other, but we also need to make sure both students have the opportunity/time/space to grapple with the problem in their own way.
Problem Posing: Posing Questions
Usha gave out a Visual Growth Pattern and two questions:
What do you see?
Pose some questions.
|This method came from a book Usha recommended called, “The Art of Problem-Posing“. The problem came from “Algebra and Algebraic Thinking in School Mathematics (Vol. 70). NCTM (2008).
After a few minutes of walking around and looking over our shoulders to see where we were all at, Usha encouraged us to start on the second prompt.
Problem Posing: Sharing Questions
Next, Usha asked everyone to “Share some of your questions with a partner and then write more questions (either together or individually)
|After a few more minutes of walking around, and listening to our sharing, perhaps noting the group slowing down in writing new questions, Usha suggested we try to think big, offering a few model questions (“What might pile 1 look like?” and “Would pile 10 have an even number or odd number of squares?“). This gave several groups, my own included, a second wind in writing more questions.|
Problem Posing: Partners selecting a problem to explore
Next, Usha said, “Together with your partner, decide which questions you’d like to share with the group and write them on these large strips of paper”
She collected the strips as they are written and tape them up on the wall.
You can hang the questions as they come in, or you can try to group them in categories if you want.
|During our meeting, Usha wasn’t grouping the questions in any particular way, but she said that you could.
What are some different ways to group/categorize the problems posed?
Here are a few examples of questions posed:
Here’s are all the Questions from Problem-Posing Activity we wrote on strips.
Problem Posing: Thinking about the process
Next, Usha asked:
“What did you think of this activity?”
|Usha asked this to a group of teachers, but this questions works just as well with students, to help them reflect on the openness of the activity.
We can get conditioned to approach a picture like this in a particular way and jump immediately to the problem of looking for the nth figure (# of squares, for example). Posing it in this way can help break us out of that habit.
The way the problem was presented really opens up the problem
Great extension problems are build by the group – this can really help students who are getting the traditional types of questions quickly because they have a lot more to do
There are all kinds of points of entry
This is a great extension to the last few meetings where we’ve been looking at visual patterns
The questions makes you think of more and more questions
Students have more ownership on how to work on the problem
Maybe some students would shut down in response to this level of openness – We need students to struggle a bit, but if that struggle becomes unproductive, Usha suggested we can always talk to students individually and explain, “Here’s what I mean…” and perhaps offer some model questions.
Problem Posing: Working independently and working with a partner
Next, Usha gave everyone a chance to look over all the questions that were hanging on the wall.
Then she said, “Choose a problem/pick a question, with your partner, and then work on the question alone.”
|This set of instructions gives problem-solvers time to try some things on their own, but with the expectation that they are going to have to explain their ideas and talk about what they did with a partner.
Problem Posing: Brief sharing
Next, she gave the pairs time to talk about what they each did.
“Get together and share with your partner” (Pair Share)
Finally, she said:
“Let’s come together to share”(Whole Class Discussion)
|Usha raised the teaching question of thinking about when we want students to start sharing (how long we want to give them to work on their own). What does sharing mean in this context? Would we consider two students to be sharing ideas if one student has a great idea and is explaining to another student who hasn’t been able to start thinking through the problem yet? How we can insure different student voices are heard? Both learners need to be in a place where they are ready to contribute ideas and hear the ideas of a partner.
While we were sharing with our partners, Usha visited each group to get a sense of both the work each person had done and the conversation of the pair. This allowed her to guide the whole-class discussion that followed.
|In terms of sharing the pair work with the whole group, Usha offered several suggestions:
Have student pairs create posters – there is power in the visual
You can have one partner stand by poster and field questions while the other walks around to look at the other posters, and then they can switch.
Students can annotate each others posters with questions and comments on post-its
It is an active way for everyone to engage with different student thinking because you have to try to figure things out for yourself. Having posters can help other students focus on the thinking and allow them to ask targeted questions.
For the sake of time, Usha had each group report out and give a general sense of what they discussed. As we did that, Usha made connections between our work, and asked us to repeat things she thought had been explained too quickly for the group.
Since Usha posed the problem in a way that allowed us to choose what to work on, different pairs were working on different problems.
Below you will find some of the thinking on the following questions:
- Using the picture directly, describe with words two different ways you could determine the number of tiles needed to make the pth tile in the sequence.
- Would pile 5 have an even or odd number of squares? Is there a way to figure out if the number of squares in a given figure number will be even or odd?
- How would you describe what the 19th figure would look like so someone could draw it?
- What can we learn by exploring the negative space? What would the first figure look like?
Using the picture directly, describe with words two different ways you could determine the number of tiles needed to make the pth tile in the sequence.
Mark tried to find the explicit rule without a chart. He looked to see if there was a pattern if you take the top row and using it to fill in the rectangle. He tried that with the three figures given and saw that there was. He noticed that when you fill in the rectangle with the top row, you end up with most of a rectangle that is one more than the figure number in height and two more than the figure number in width. He also noticed the chunk missing each time was one less than the figure number.
Would pile 5 have an even or odd number of squares? Is there a way to figure out if the number of squares in a given figure number will be even or odd?
Tyler, Alison, Ida
Tyler saw that the total squares for the three given figure was alternating odd, even, odd, so he made the generalization that the pattern would continue and the fifth figure would have an even number of squares. And then he drew it just to make sure. Then he started thinking about how he could get students to move from the first question, to the second question which involves any figure. If a student draws the fifth figure to answer the question, how can you get them to pull back and get them to look at the relationship between the figure number and the number of squares (and whether it is an odd or even number of squares). He was thinking of maybe having try to set up a chart to help them make and test a generalization.
Usha asked if there was a way to talk about why it would be even (or odd). Tyler shared an algebraic explanation about why an even figure number will produce an odd number of squares and vice versa. (Using the formula n^2 + 2n + 3) If n is odd, the square of an odd number is always odd. Two times n will always be even and 3 is obviously odd. Then he said – “An odd plus an even plus an odd will always be even.”
Is that true? Is an odd plus an even plus an odd always even? Is that always true? Why?
How would you describe what the 19th figure would look like so someone could draw it?
Avril and Mark
Avril created a chart focusing on the largest length and width of each figure number. She noticed that longest length and width of each figure was always two more than the figure number. From that, she was able to construct the 19th figure, beginning with the bottom row and the “second from the right column”, both of which she knew would have 21 squares. Then she made sure there was one square “missing” from the top left corner and that there were two squares “hanging over the edge” for the top and bottom rows.
Mark started off looking at the three given figures and seeing what kinds of patterns he could find. He broke each figure up into three parts – the top row, the bottom row and the square in the middle. He noticed that the top row is always one more than the figure number and the bottom row is two more than the figure number. He also noticed that the middle was always a square with sides equal to the figure number. From that pattern, he wrote up his 5 steps for drawing the 19th figure.
When they came back together, Avril and Mark’s approach complemented each others’ work, because he could check his instructions against her picture and vice versa.
What can we learn by exploring the negative space? What would the first figure look like?
Solange and Eric
Solange and Eric thought about the question: What can we learn by exploring the negative space as the figure grows? After discussing it for a while, they started to refer to the negative space as dark space or empty space, because the idea that this space would be negative is potentially confusing. They didn’t think of the empty space as representing negative numbers. They were looking at the squares that were missing from a larger square formed by (n+2)^2. If you look at the empty space in each figure, there is a constant of 1 missing square in the top left corner of the square and a changing rectangle on the right side. The empty rectangle could be described as 2n. So a formula for the white squares would be the larger square ((n+2)^2) minus the constant empty square (1) and the growing rectangle (2n).
They also started to look at what the 1st, 0th and -1st piles would look like…
Teaching Questions Raised
Specific to Usha’s activity:
- What happened during each step of Usha’s lesson? What did each one accomplish?
- What are some benefits of presenting the pattern in the open manner that Usha did? What are some strategies to deal with the challenges? How can we support students who might struggle with this kind of openness?
- What is gained by omitting figure 1 from the picture?
General/big questions in math teaching:
- How can we give our students more time and space to engage with each other’s thinking?
- What kinds of activities are appropriate for high/low level student pairings? What kinds of activities are best for students to be grouped with students of a similar level?
- What are different strategies we can use to help our students adjust to the discomfort of non-routine problems and questions, without completely alleviating that discomfort?
- Why is the square of an odd number always odd? Why is the square of an even number always even?
- Why is an odd plus an even plus an odd always even? Why is an even plus an even plus an odd always odd?
Usha handed out a handout with directed questions that could be used with this growth pattern in lieu of having students create their own problems.
In attendance: Tyler, Solange, Avril, Alison, Ida, Usha, Davida, Eric, Mark
Programs represented: BMCC, CUNY Adult Literacy, Brooklyn Public Library, CUNY Collaborative Programs, Fifth Avenue Committee
Location: The Brooklyn Public Library, Pacific Street Branch at 25 4th Avenue
Respectfully submitted by Mark (with suggestions from Usha)