As a teacher in inclusive settings, number strings were a critical part of my daily mathematical work. Routines are highly effective in inclusive classrooms, particularly routines like number strings which *externalize complex cognitive processes*. By that delicious turn of phrase, I mean that a number string is not the kind of routine that teaches low-level thinking like memorization. Instead, it allows kids to participate in the strategic thinking of other kids, giving them access to complex processes that too often only go on in individual minds. Students with learning disabilities, for example, can often benefit from explicit instruction in strategic thinking. Daily number strings immerse students in strategic thinking, and it is all the more powerful because it is the strategies of other students, rather than the teacher, that are valued and explored. The routine is also a powerful way to build students’ cognitive flexibility, a goal that can be important for a variety of disabilities including autism. But how to maximize the participation of all students in the number strings routine? To jumpstart your own thinking, here are 30 ways to increase student participation in number strings.

- Anchor number strings in context. Many of the most complex number strings are already set within contexts, such as using a brownie pan as a model for multiplying fractions. Some students will need context for many other number strings as well. Contexts can relate back to the investigation they began with, or use another familiar context.
- Never underestimate the importance of models. Carefully plan your representation using models, so that you can be prepared to represent thinking using the model that best supports the strategy you are developing. Models such as the rekenrek, the array, the open number line and the ratio table are critical scaffolds for children’s thinking, and are particularly important when working with kids who need more support in mathematics.
- Explicitly represent language that helps students be precise mathematically. For example in multiplication, it often helps students to see and hear the phrase “groups of.” In early fraction work, it helps kids to see and hear the words for fractions before they see the notation, so saying and writing “one fourth” can be helpful.
- Write down student thinking word for word as they describe particularly important strategies or the beginnings of conjectures. Write down the student’s name.
- Make strategies explicit by creating strategy posters that students can use as resources.
- Explicitly ask students to try new strategies; “Can you try Alisha’s strategy?” Some students might need to hear that they are not being asked to always use Alisha’s strategy, but are just trying it in this number string. This can help students push themselves to be more flexible in their strategy use.
- Many teachers have students use their thumbs to signal that they are ready during a number string. Thumbs are less tiring than hands to hold up for a long period of time, allowing the teacher to increase wait time for students. In addition, you can use the thumbs as an assessment tool to gauge understanding.
- Use turn and talks strategically and often. Most children need to talk through complex ideas in order to understand their own thinking. When most of the students thumbs are not up, and you have waited a minute or so, try a turn and talk.
- Teach children how to engage in productive turn and talks through interactive modeling (responsiveclassroom.com).
- Make sure that you adhere to time limits during strings. Strings as a daily practice should be 10 – 15 minutes, extending to 20 minutes only when all kids are engaged and there is a juicy question on the table. If you have to end, you can give the last problem as a quick written assessment that you can collect and use to prepare for the next day’s string.
- Strings can regularly end with written assessment. You can give a problem that would be next, or a final challenge problem that students should solve individually. You can differentiate this by writing two problems at different levels of complexity that use the strategies in the string for students to choose from. Let students pick which one to tackle.
- Move around the room during turn and talks to gather information about where the kids are. This will help you call on a greater range of students.
- Equalize participation. Have a colleague keep track of whom you call on so that you can equalize participation.
- Support participation by previewing content. Certain learners can benefit from seeing the first two problems ahead of time and having time to work out a solution. This is particularly useful if you have one or two students who struggle to engage in the first two problems.
- Sentence frames/stems can assist with dialogue, and are particularly effective when they are tailored to the mathematical content of the number string.
- Have clear structures and expectations for behavior around the number string.
- Use a notebook or a separate whiteboard for enthusiastic sharers to write down their ideas that they can show the teacher, either during or after the number string.
- Use divergent strings. When teaching a string and you feel that a chunk of the group is ready to move on to a challenge problem and another group is not, give two problems. Ask students to chose the problem that makes sense for them. Share strategies for both problems to end the number string. This is a useful strategy for the final challenge problem as well. I often challenge students to use a particular strategy in the final challenge problem. By providing two challenge problems, you can make sure that all learners get a chance to try the strategy on a problem with no helpers.
- If you end the string with multiple challenge problems, put each problem up in a different spot around the room. Allow students time to think, and then dismiss them to the space in the room where their choice of problem was listed. Assign one child at each spot to write out their strategy on the poster for the other students in their small group.
- Attend to multiple ways of communication. Learners communicate complex ideas through gestures in mathematics— from highly proficient adults as well as English Language Learners and D/deaf learners (Shein, 2012; Goldin-Meadow et al., 2013; Wittman, Flood & Black, 2013). Look for and highlight gestures that communicate mathematical understanding, such as indicating number space on a number line, constant difference with hands marking number space that shifts placement, but stays the same size, or compensation which children often communicate with a gesture of exchange. Allow children to use their linguistic strengths in your classroom, encouraging the creative mixtures of different languages, gesture, and movement in mathematics. If talking/signing is thinking in development, then such flexibility is necessary to allow kids to truly think through complex ideas.
- Another way that kids communicate abstract ideas is through using manipulatives to communicate abstract ideas. Children who have difficulty with certain aspects of communication may have difficulty explaining compensation in addition, for example, but the same child may be able to explain it with connecting cubes (Foote & Lambert, 2011). Give these moments status in your classroom, and you will encourage all students to develop their mathematical thinking.
- Allow for flexibility in how children sit in a number string. Some kids may need movement breaks, to sit at a desk, etc.
- Allow for flexibility in how children participate in the number string. For children with communication devices, you can preview critical questions for them, so that they can prepare their questions or responses either digitally or scribed.
- Use thumbs to gauge student understanding. If the string seems too difficult for the majority of the class, change it. Or end it, and revise it for the next day. Experienced teachers of number strings make constant revisions to their problems on the fly.
- Number strings are a great routine for the whole class, and for small groups. Whole group number strings allow the class to build a community of respectful strategic thinkers and allow all learners access to rigorous content. Small group work allows students to focus on kinds of computational problems they need additional work with. Small groups should always be flexible and shifting based on assessment and/or student choice.
- Ask students to write a number string as an assessment of their understanding of the routine and of a particular strategy.
- Think carefully about whether or not students should have access to paper or whiteboards during the number strings. Such tools support some kids, while other students use them to perform procedures and avoid thinking with models. A UDL (Universal Design for Learning) approach would recommend offering the tools to all learners, and then having conversations with individual learners if they are not using the tool in a useful way. Such a conversation helps students develop metacognition around their engagement in strings, and offering choice puts responsibility for learning on the child.
- If you have two teachers, offer two strings. Write out two strings on separate chart paper and ask students if they would like to do a number string like this one, or like this one. They can then chose the number string that makes sense for them.
- While number strings are great to develop computation at grade-level, they are very effective at developing computational fluency for topics at earlier grade levels. Kids often know what operations they would like to get stronger at. During a day in which kids are playing math games, offer a choice of attending a Multiplication Clinic or a Subtraction Clinic, stressing that a clinic is used in sports as a way to get better through focused practice.
- For all these methods to be successful, teachers need to develop students’ understanding that all learners have different strengths and challenges in math, and that the way to get better is to engage deeply with what you need to work on, rather than comparing yourself to others. Try a mindset intervention (Dweck, 2008), or use the Multiple Abilities Treatment from Complex Instruction (Cohen & Lotan, 1999).

Any more ideas?