A Problem String Is NOT

I was recently at the Florida leadership conference, FAMS, working with leaders and ran into Howie Hua, and we had a great conversation. Fantastic guy. You can listen to Kim and I discussing it here.

In that conversation he asked me why sometimes in a Problem String I have ‘unrelated’ problems that don’t help with getting ‘the answer’. Great question!

The answer is that Problem Strings are an instructional method, not a strategy to solve a problem. A strategy defines a recognizable approach to solving a specific kind of problem, while Problem Strings help students build the relationships necessary to solve problems using those strategies. But I can see where that could be confusing, since Problem Strings often function by demonstrating how a problem might be solved. Through that lens, those ‘unrelated’ problems Hua wondered about are there to build the relationships needed to solve problems like the ones in the string, not just the last problem in the string.

Here’s an example. To solve the problem 42 - 17, one might use the following problems to help:

42 - 20 = 22

20 - 17 = 3

22 + 3 = 25

Howie thought that when I say Problem String, I meant to give students those problems so that they could walk through them to solve the final problem. 

That’s not the instructional routine called Problem Strings.

Problem Strings are routines that teachers can use to help students look for and find patterns they can use to generalize how to solve problems.

Here is an example of a Problem String. Notice how the problems are meant to build a pattern of relationships, not just the steps to solve the last problem:

26 - 10

26 - 9

55 - 20

55 - 19

42 - 17

To facilitate, a teacher gives each of these problems to students, one at a time, discussing each, and representing student thinking. That representation of student thinking for them by the teacher is critical, as the ability to model thinking is a learned skill. By the end of the string, the teacher can ask, “How might you use a similar relationship to solve 42 - 17?” Since each of the preceding pairs suggests the pattern of subtracting a friendly number that is a bit too big and then adjusting, students can try that strategy for the last problem that is given without a helper.

The board could look like this, making students’ thinking visible, point-at-able, discussable.

So, the problems used in a Problem String can look similar to writing out the equations representing steps taken while solving a problem using a strategy such as Over. But a Problem String is much more than the equations used in it, rather it’s a way of teaching.

As an instructional routine, Problem Strings require a teacher to facilitate it. They require the teacher to pick and choose which strategies from the class to share. Which parts to highlight and which to ignore. All of which is informed by where the class is on the landscape of learning, and the strategy the Problem String is meant to develop.

Without this active facilitation by the teacher, a list of problems to solve, whether they build to something or not, is just that. A list of problems to solve. Problem Strings are so much more.