How does it feel different to experience a problem string as a teacher learner versus planning to lead a problem string as the teacher facilitator? How might we, as teacher leaders and instructional specialists, support teachers in this work of preparing to lead problem strings with their students?
A few years ago I was introduced to Pam Harris’ work with problem strings in secondary mathematics. After experiencing these powerful instructional formats as a learner in one of Pam’s workshops, I was hooked. I loved how Pam’s carefully crafted problem strings helped me develop strategies and make sense of mathematical relationships I hadn’t known existed. I wanted to replicate these same learning experiences for students and teachers in my care.
But how is the experience I had—specifically as a learner in a problem string—different than leading a problem string myself?
I was blessed to learn from Pam Harris & Kim Montague during a two-day Building Powerful Numeracy: Proportionality workshop that was hosted by the Texas Regional Collaboratives in Austin, Texas in May 2016. This workshop leveraged two of Pam’s resources (Building Powerful Numeracy for Middle and High School Students and Lessons & Activities for Building Powerful Numeracy) to support student (and teacher!) sense-making about proportionality and proportional reasoning.
Early in the workshop, Pam led us through a problem string that included the following problems:
This list of problems does not represent the learning experience, though! For example, Pam didn’t start by asking us “What is 2 x 47?” This list of problems contains some of the information from the string, but definitely not the whole picture.
Here is a picture of the notes I took as a learner during this problem string. What do these notes add to the story of what this problem string learning experience might have looked like?
While these notes capture many of the ideas that surfaced in the problem string, they still do not tell the whole picture. If I looked back on these notes months later, would I know what to ask in order to orchestrate this problem string myself? What would I draw out from learners? What would I minimize? What would the story of the problem string be?
This summer, I had an opportunity to lead a similar Building Powerful Numeracy: Proportionality workshop for teacher learners. I found myself revisiting my notes from the summer before and wondering Where do I start? How can I best recreate these problem string experiences now as a facilitator?
In other words…how do we go from a list of problems to a plan of action?
One great first step is to read through and consider a vignette of the problem string in action…if one exists! Pam’s books (linked above) include many create examples of classroom vignettes that illustrate a rich picture of what these lesson structures look like in action. You can also find vignettes for certain problem strings on Pam’s website (see a variety of problem string posts here). Also, if you’re looking for great problem strings (and accompanying vignettes!) for high school algebra courses, be sure to check out Pam’s two newest books Algebra Problem Strings and Advanced Algebra Problem Strings:
I started by reading the following two pages from Pam’s Building Powerful Numeracy for Middle and High School Students book. Take a few minutes and read the vignette for yourself. (Note that this particular vignette deals with boxes of 27 widgets instead of packages of 47 M&Ms. Slight difference. Incredibly similar structures throughout.)
This vignette gives us a mental picture of what this problem string might look like.
But, even still, how do we take this mental picture and make it happen for learners in our care? Do we replicate the questions from the transcript line-by-line? Is this a script that we should follow? If not, then what might our next planning step look like?
Enter Kim’s Page. Pam & Kim first shared this tool as part of one of their Focus on Algebra: Exponential Functions workshops a few years ago. Kim had inadvertently designed this instructional planning tool when she was trying to calibrate a problem string to better draw out certain noticings and connections from a class of students. A “Kim’s Page” is a one page overview of the string that includes pacing notes, key questions, ideas to draw out, etc., all at-a-glance—an instructional cheat sheet (if you will) for the problem string.
After reading the vignette referenced above, I created the following “Kim’s Page” to prepare for my facilitation of this string:
Let me highlight a few of my notations on this page and what they meant to me as the facilitator.
I wanted to remember the exact wording I would use to introduce the context of the string and then ask the first problem, so I wrote this information down.
I purposefully wanted to use a horizontal table format to track the group’s work throughout the string, so I noted that on my planning page, as well.
I noted specific strategies that I would be looking for and then drawing out into the whole-group discussion. Here I was looking for people who multiplied by 8 by doubling, then doubling, then doubling again.
This reminds me of Peg (Margaret) Smith & Mary Kay Stein’s work from 5 Practices for Orchestrating Productive Mathematics Discussions. Smith & Stein discuss the importance of: anticipating what strategies students will use; monitoring student work; selecting strategies worth discussing in class; sequencing presentations of student strategies to maximize the learning trajectory; and, then, connecting the strategies to help students understand the mathematics. More on this in a little while…
I noted a specific place in the string where I wanted to pause and have a sidebar conversation with the community of learners. Specifically, after learners explained how they determined how many M&Ms are in 8 packs (by doubling, doubling, and doubling again), I wanted learners to step back and consider:
What problem did we just solve? 8 x 47
How? By doubling three times.
And, then, most importantly: Does this always work?? Why or why not?
Learners worked with their partners for a minute or two to create a convincing argument of whether or not this double-double-double method always works for multiplying by 8.
This sidebar led into some interesting work about the Associative Property of Multiplication and its applicability for reasoning through multiplication and proportionality problems.
Later in the document, I noted a place where I was looking for two specific strategies to emerge: one where someone used what they knew about 8 packs to determine how many M&Ms were in 9 packs; and, one where someone used what they knew about 10 packs to determine how many M&Ms were in 9 packs.
I included a reminder to ask learners to generalize their thinking at the end of the string and put into words and/or symbols how they might determine what 99 times anything is. I also included sample notation that I hoped to draw out for this in the resulting discussion.
This marked the end of the “official” string, but not the end of our time thinking about it as teacher learners! One page from Pam’s work encourages teacher learners to consider the following questions after they (as teachers) have experienced a problem string as learners: What is going on in this string? What model, big idea, and/or strategy is it trying to promote? Here are some of my notes-to-self that I hoped to elicit from participants during the discussion of these questions.
Also, I included an extra, optional question to push us back into the string (if desired) and push our emerging thinking about the realms of usefulness for ratio tables. Have we considered that ratio tables can help us think about problems we usually associate with division?
So, there you have it! There is my Kim’s Page—my cheat sheet; my string-at-a-glance planning/implementation page—for this particular problem string.
So…how did the string actually run in the session?? Exactly like in the vignette in Pam’s book. As in word-for-word exactly? No! As in, the exact same strategies emerged. The exact same big ideas emerged. The exact same story of the problem string emerged.
Did this happen because I scripted the problem string?? No. Rather, I believe this happened because I spent time thinking deeply about:
- what this string was about,
- what strategies I wanted to emerge,
- what ideas I wanted to elevate and/or minimize,
- what mathematical ideas I wanted to connect, and
- what the story of this problem string was.
Having considered this whole, I was able to lay out a planning page that let me know what to look for and when, what to move quickly through and what to pause on, what key questions to ask to push thinking deeper, etc.. The problem string ran the way I intended because of the story of the string that was captured and annotated in my Kim’s Page.
Here’s where Smith & Stein’s 5 Practices come back again. As Smith & Stein explain, thoughtful planning with the 5 Practices sets the stage for student discussions that lead to a specific mathematical goal:
“We think of the five practices as skillful improvisation. The practices that we have identified are meant to make student-centered instruction more manageable by moderating the degree of improvisation required by the teacher during a discussion. Instead of focusing on in-the-moment responses to student contributions, the practices emphasize the importance of planning. Through planning, teachers can anticipate likely student contributions, prepare responses that they might make to them, and make decisions about how to structure students’ presentations to further their mathematical agenda for the lesson.” (2011, p. 7)
While Smith & Stein’s work specifically relates to discussions around larger mathematical tasks, I believe that these same practices can be applied to orchestrating productive problem string discussions as articulated in this post.
What do you think? How does it feel different to experience a problem string as a teacher learner versus planning to lead a problem string as the teacher facilitator? How else might we support teachers in this work of preparing to lead problem strings with their students?
Food for thought for a later post: How might we help teachers envision and plan to implement a problem string when they do not have a vignette to base it in? How can we collectively write the story for a string in order to plan a Kim’s Page that helps us implement the string with purpose?
Harris, P. W. (2011). Building Powerful Numeracy for Middle and High School Students. Portsmouth, NH: Heinemann.
Harris, P. W. (2014). Lessons & Activities for Building Powerful Numeracy). Portsmouth, NH: Heinemann.
Smith, M. S., & Stein, M. K. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: NCTM.