What are instructional routines?
"Instructional routines are specific and repeatable designs for learning that support both the teacher and students in the classroom... enabling all students to engage more fully in learning opportunities while building crucial mathematical thinking habits." Kelemanik, Lucenta, & Creighton (2016)
The Instructional Routines on this page are accessible and challenging to all students, can be approached in multiple ways using different types of sense-making strategies, promote understanding through discussion, make student thinking visible, make math visual and encourage practice of mental math
This growing collection features resources and support for bringing the following instructional routines into your classroom:
(1) Notice/Wonder; (2) Dot Talks (3) Number Talks, (4) Number Strings, (5) Visual Patterns; (6) Estimation 180; (7) Which One Doesn't Belong; (8) Slow Reveal Graphs; (9) Three-Act Math Tasks, (10) SPLAT!; (11) Always, Sometimes, Never, (12) Solve Me Mobiles; (13) Clothesline Math, (14) Fraction Talks; (15) What Math Do You See?; (16) SAME or DIFFERENT?; (17) Would You Rather?; (18) Convince Me That...; (19) Contemplate Then Calculate; (20) Connecting Representations; (21) My Favorite No; (22) Push & Support Cards; (23) Numberless Word Problems
Start problem-solving without the problem. Leave off the question, give students a math situation, graph or image and ask them: "What do you notice? What do you wonder?"
Dot talks are a great way to show how visual math can be and how many different ways there can be to look even at a simple image. They take about 10-12 minutes to do.
In a dot talk, a teacher shows an image of dots for a couple of seconds and then asks students how many they saw. (The important thing is that students are not counting the dots one by one.) Then each student describes their way of seeing the dots. The teacher visually records each student's way of seeing and creates (or asks students to create) an expression representing the student's way of seeing.
For the dot talk below, Ann member Patricia Helmuth, NY recorded different ways of seeing by making multiple copies of the image of the dots.
- When Steve Wyborney does dot talks with his students, he does one as described above. Then he gives students another set of dots, but the second time, each student receives a handout with 18 copies of the same collection of dots. Their task is to notice as many ways of seeing the dots as they can, record it similar to how Patricia did and write an expression for each way of seeing. To learn more, read his post, Provide Massive Space to Notice. Also check out 180 Opportunities to Notice - 10 pages of dot patterns he created that you can download and use with your students.
- Cindy Whitehead created a Desmos activity called Crazy Eights, where students need to match equations to a dot pattern, create an equations that match dots patterns and create dot patterns that match equations.
A Number Talk takes about 10-15 minutes and help students develop their number sense and teach them that they have mathematical ideas worth listening to and that they can make sense of calculations in their own ways. Especially for adult education students, many of whom received a fragile understanding of number and calculations from their prior education. The basic routine is students do a mental calculation without using paper. Students explain their thinking and teachers create visual representations of that thinking.
Here are some resources to start using Number Talks in your class.
Pam Harris facilitates a weekly slow chat on Twitter called #MathStratChat. Her collection of computation problems are thoughtfully designed and make for rich number talks. and have the added bonus of multiple approaches.
MATH TALKS is another thoughtful collection from Fawn Nguyen that includes a few sample student responses for each.
Berkeley Everett created adaptable slides so teachers can make customized number talk images using fruit, nuts, and pastries. Custom Number Talk Images
Kristen Acosta has a collection of Number Talk Images for inspiring great math discussions.
Number Strings/Problem Strings
Number strings, also called problem strings, refer to a related series of mental math problems that are designed to draw out student thinking, help students notice relationships and develop explicit reasoning strategies.
Number Strings is a website for math educators to find number strings, share number strings, get feedback, and interact with other teachers using number strings with learners. It is a growing resource, and currently has addition strings, subtraction strings, multiplication strings, division strings, rational number strings, and algebra strings.
Pam Harris has created an introduction to problem strings that include (1) videos showing the string in action with students, (2) a written description of the mathematics, (3) explanation of the teacher moves, and an annotated transcript of the video. These can all be found at her website: Math Is Figureoutable - Problem Strings
Visual Patterns are a great way to utilize patterns to encourage making use of structure in problem-solving. Students make observations and build on those observations to generalizations.
Visual Patterns is a collection of hundreds of visual patterns gathered by Fawn Nguyen. There are patterns that can be used to explore linear function, quadratic functions, cubic functions,
For some ideas about how to use visual patterns with adult students, including handouts and a lesson plan, read Developing Algebraic Reasoning Through Visual Patterns
Mathigon has a visual pattern tool that can help make students' thinking visible to classmates and teachers.
Here are the first 3 steps of a visual pattern.
What comes next?
Estimation 180 is a collection of engaging photos and short videos developed by Andrew Stadel. The visuals are used as estimation challenges (like the one to the right) that help students develop their number sense through reasoning, multiple estimation strategies, explaining their ideas, connecting to different math content, and reflecting on the process.
To learn more about using Estimation 180 with adult education students:
Order the glasses from least to greatest
Which One Doesn't Belong?
With this instructional routine, teachers present four of something (like the 4 numbers to the right) and ask students to come up with reasons why each doesn't belong.
The website Which One Doesn't Belong? (WODB) is a collection of Numbers, Shapes, and Graphs to explore, curated by Mary Bourassa, inspired by the book Which One Doesn't Belong by Christopher Danielson. Start there or create your own!
You can find ideas on how to use WODB with adult students: Learning Through Classification
Use this WODB Template in Desmos to create your own interactive Which One Doesn't Belong.
For a variation of the WODB routine, try How Does It Belong? on Desmos.
What makes each of these numbers different from the other three?
Slow Reveal Graphs
Slow Reveal Graphs refers to a process for using graphs with students that promotes sense-making. You start with a graph and peel back the layers until you have a numberless graph that every student can make observations about. Then you add back the layers, step by step, encouraging students to make sense of the graph and engage with the story of the graph.
Slow Reveal Graphs is a collection of slow reveal graph sequences organized by type of graph as well as by context. The site is curated by Jenna Laib.
The CUNY Adult Literacy Program has a growing collection of Slow Reveal Graphs in SS and Science.
In this brilliant article from the Math Practitioner (Summer 2022 issue), Tim Berrigan, NY explains how he uses slow reveal graphs in his classroom and why the routine is so effective - Slow Reveal Graphs: A Powerful Instructional Routine