What is the difference between a Problem Talk and a Problem String?

Problem (Number) Talks and Problem Strings are both powerful instructional routines for the mathematics classroom. They are mini-lessons intended to work hand in hand with rich tasks, math congresses, and other instructional routines. 

Unfortunately, the two routines are often mixed up and meshed into one activity. In that combined form, the result is not as effective. I worry that if we don't get this right, teachers will try the combined form, not get good enough results, and stop doing everything that looks like them. I really don't want that to happen. So, let's get clear about the similarities and differences; it's all about purpose.

The purpose for a Problem String is to construct mathematical relationship and connections.

What is the purpose of Problem Strings?

The purpose for a Problem String is to construct mathematical relationship and connections. Because the problems are sequenced and the teachers represent student thinking in a visual way, students have the opportunity to take up the thinking. In a Problem String, participants solve the problem the best they can and then students see what others were thinking. But then, they solve the next problem. The next problems are sequenced in such a way as to make patterns jump out at students and as students pick up on those patterns so they can use them in the subsequent problems. As they continue to use the new relationships, they cement the connections.

What are the characteristics of Problem Strings?

• are series of related problems. These problems are in a certain order, purposefully designed to promote patterns to become apparent. It's not just one problem, it's a sequence of problems. The teacher gives a problem, students solve it, students share their strategies, the teacher models as they share, and then the teacher gives the next problem.
• are focused toward constructing a particular model, big idea, or strategy. Problem Strings are more focused than a Problem (Number) Talk or a rich task. In those tasks, there are lots of things being developed, but in a Problem String there is a tighter emphasis on one goal. 
• systematically nudge toward more efficient and sophisticated strategies. Problem Strings help students realize that there are better or more clever ways to solve problems. Because they are focused toward constructing one strategy, students get better and better at that strategy. Then the next Problem String is designed to work on a different strategy. As you progress through series of Problem Strings, students get more and more efficient and sophisticated. 
• Of the time that you do Problem Strings and Talks, about 80% of that time should be doing Strings. Strings is where the constructing happens so they need to be more often.

 What are the purposes of Problem Talks?

The purposes of a Problem Talk include:

• proving there is more than one way to solve a problem. This is helpful for a short period of time to help students believe that they do not have to remember the one way that their teacher taught them. Once students know this, we do not need this purpose any more. Unfortunately, this is the purpose I see being enacted the most as I travel across the country. Teachers launch a problem, have students share their thinking, and teachers rejoice that students can solve problems more than one way. And teachers mistakenly think that somehow students are supposed to start to pick up others' strategies, but this doesn't happen often enough. Students need more support to construct other strategies - and that support is best in Problem Strings!
• poking around for formative assessment. This is a great use of Problem Talks, but we don't need it very often. Teachers could throw out a problem at the beginning of a unit to see what students do with it. They can use this information to guide instruction. Then, after students have begun to develop a couple of strategies, it might be time to do another Problem Talk to see if they are using the strategies or if there's more work to be done. This is also a fairly short lived purpose. We only need to do this kind of poking around every so often.
• compare already constructed strategies for efficiency, cleverness, sophistication (where are they on the progression of strategies). This is the most important and most often used purpose. Once students have constructed several strategies, teachers can throw out a problem or two to see if students are using the strategies that are best suited to each problem. 
• Of the time that you do Problem Strings and Talks, about 20% of that time should be doing Talks. Strings is where the constructing happens so they need to be more often.

What are the characteristics of Problem Talks?

The characteristics of Problem Talks include:

• fewer problems, often just one problem
• this is where you compare already constructed strategies. For these numbers, for this structure, which is clever, elegant, efficient? The teacher throws out a question, students solve it the best way they can, and the teacher represents several strategies on the board. Students compare and connect strategies. They seek to understand each others' thinking.
• In the case of a rich problem (where many strategies work well), like 15 x 18, teachers would hope to see all of the strategies being used across the classroom. If not, if students are all using the same strategy - this is good information for the teacher. Perhaps the teacher was too direct with that strategy. Perhaps more work needs to be done with the other strategies. For a problem like 37 x 99, we would hope students would gravitate toward the over strategy, finding 100 37s and subtracting the extra 37. If students are using other strategies, that gives the teacher the important information that they need to work more on the over strategy. 

What do Problem Strings and Problem Talks have in common?

They are both instructional routines. The beauty of instructional routines is that they can be routine. When students are clear on their roles and what it happening in a routine, then they can put all of their mental energy into solving the problems, not figuring out what they are supposed to be doing. Both Strings and Talks:

• are mini-lessons, taking between 10 to 20 minutes. The graphic lists 5-15 minutes but I have since changed this because they actually take longer. The first few times you do one, it may take you longer because you are establishing the routine. It will also take longer the first time you do a Problem String of its type. 
• are routines in which the teacher purposefully chooses who shares. This means that it is not a "free for all". The teacher carefully considers which strategies will help move the mathematics of the class forward. One might be a mistake. One might be a slightly confusing explanation that will allow the class to parse out and make sense of the strategy. One might be one of the least sophisticated strategies in the room, so that it can be connected to a strategy that is slightly more sophisticated. Rarely is the least sophisticated or the most sophisticated shared-- because with both of these, the mathematics doesn't tend to progress for students. 
• are routines in which the teacher models student thinking using a purposefully chosen model(s). The teacher chooses the model to best help others take up the thinking, compare the different strategies, or to engender disequilibrium that needs begs to be resolved. 
• are routines in which the students and teacher realize that all strategies are not equal. If it feels like every time you do a Talk or String that you are just celebrating everyone's strategy, you are missing the power of these routines. Instead, use these routines to help students construct more sophisticated reasoning.

Look here for more information about Problem Strings.