Instructional Routines

1) Notice/Wonder

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?" 

• The 5 minute video Ever Wonder What They'd Notice by Annie Fetter is an inspiring introduction to the routine,
• Begin to Problem-Solve with I Notice, I Wonder is brief 2-page overview of the routine created by of the Math Forum. 
• NCTM's What is Notice Wonder? Collection
• What's Going On In This Graph is a collection of graphs and charts gathered by the NY Times to use with Notice/Wonder. 
• Explore the Multiplicity Lab's Image of the Week (and extensive archive) for photos ripe for noticing and wondering. 

2) Dot Talks

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. 

3) SPLAT!

Splat! is an interactive number sense strategy that is accessible to all students, that can be used to develop a visual approach to solving equations. The materials and routine were developed by Steven Wyborney. 

For more ideas on using SPLAT! with adult students, watch Some Numbers Walk into a Language Class: Splat!

There are 22 dots total.

How many are under each Splat?

4) Fraction Talks

A Fraction Talk is a discussion-rich routine that invites multiple ways of identifying fractions. They are similar to number talks, but instead of drawing out strategies through mental calculations, students evaluate images like this one and asking questions like, "What fraction of this figure is yellow?"

Nat Banting created the Fraction Talks website, which has a huge collection of images to inspire discussion and exploration, as well as suggestions for using this routine in your class. 

MATH FOR LOVE has some great Fraction Talk images using pattern blocks. 

5) Number Talks

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 FOR LOVE has a nice collection of resources for using number talks.

7) Test Talks 

A Test Talk is an instructional routine developed by Sarah Lonberg-Lew designed to (1) help students apply and extend their learning in the context of standardized test questions, (2) Solidify learning and act as a formative assessment, (3) Empower students to think flexibly and creatively in test situations, and (4) Build students’ capacity to make sense of problems and persevere in solving them 

Test Talks help create and nurture a classroom culture that values flexible thinking and conceptual understanding and that explicitly prepares students to apply their learning in the context of standardized tests.

To help you get started with Test Talks to your class, Sarah created a packet called Will This Be On The Test?: Inspiration For Tackling Standardized Test Questions Conceptually and Creatively, published by the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center. This packet has 22 separate test-like questions that can be used for this routine, including full explanations of multiple strategies for each question.

The test-like questions in this packet can also be found in the ongoing blog series Will This Be on the Test? 

8) 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?

9) Estimation 180

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:

• Read Number Sense: Helping Adult Students Close the Gap 
• Watch Some Numbers Walk into a Language Class: Estimation 180

How long is I Got You (I Feel Good)?

10) 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?

There is a WODB Generator on Polypad that uses prime factor circles, fraction circles, dice, dots, equations, rectangular prisms, and more. Just click on the Generate! button to produce a new WODB. 

With fraction circles, you can label the pieces as fractions, decimals, percentages, or not at all (as seen above). You can also remove the unit fraction dividers (the white lines). 

With the factor circles, you can label the products (as seen above), and/or the prime factors around the circle. 

11) 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  

(To see the entire sequence with the graph above, read Slow Reveal Graphs: A Quick Intro for ABE/HSE Teachers)

12) Numberless Word Problems

Many students in adult education struggle with word problems. Because of their prior educational experiences often emphasized memorization and tricks over sense-making, they sometimes throw numbers together without thinking about the mathematics in the situation. Similar to slow reveal graphs, numberless word problems involve removing the layers of a word problem and then adding the layers back, one at a time, giving students space and time for sense-making and discussion.

Brian Bushart's website, NUMBERLESS WORD PROBLEMS has ready-to-use and ready-to-customize numberless word problems, as well as suggestions for getting started using this instructional routine with your students. 

14) Always, Sometimes, Never

The basic structure of the Always, Sometimes, Never routine is, students are given a series of mathematical statements to sort into one of three categories: "always true," "sometimes true," or "never true." Students, usually in groups, have to defend their answers and come up with examples and counterexamples. 

A good way to get started with Always, Sometimes, Never is this brief blog post by Fawn Nguyen describing using the routine with her students. The image here shows the work of some of her students.

In her post, Fawn shares another collection of statements developed by Swan and Ridgway. 

Consider a few statements that Fawn shared with her students. Which column would you place them in?

• Max gets a pay raise of 30%.  Jim gets a pay raise of 25%.  So Max gets the bigger pay raise.
• When you cut a piece off a shape, you reduce its area and perimeter
• The square root of a number is less than the number.
• 2(3 + s) = 6 + 2s

15) Solve Me Mobiles

Solve Me Mobiles allows students to develop algebraic thinking through balance puzzles like the one to the right. The site offers a collection of puzzles and allows teachers (or students) to create their own. 

16) Clothesline Math

Clotheslines are interactive number lines. Students place (and move) cards on a clothesline where they think the number belongs. It is a great activity to show proportional reasoning, precision, equivalency between numbers, and magnitude. The clothesline draws out student sense-making around relationships between numbers. It can be used across a range of math content, from building students’ number sense to algebra.

Read What is Clothesline Math? from the Summer 2024 issue of the Math Practitioner. 

Kristen Acosta's clothesline math cards can be used to explore topics like fractions, decimals, percents, integers, money and expressions. Patricia Helmuth's Numeracy Adventure Day materials has a clothesline activity with many number representations.

In this blogpost - The Intersection of Fraction Talks & Clothesline Math: Formative Assessment - Jenna Laib writes about a clothesline math activity with fractions. Her description includes her planning, photos of students work, and the connections she made during the class discussion. 

17) Can You See It?

I learned this routine from Dee Crescitelli. It is a great way to strengthen connections between visual representations and quantities. It allows you to assess students in a puzzle-like atmosphere that lends itself to building a culture of collective knowledge building. 

It is a simple set up. Provide a visual like the one to the right and ask if folks can see the quantities in the image. Try it for yourself. Can you find all of those quantities in those colored circles?

18) What math do you see?

So many students have internalized an invisible barrier at the door to our math class, and all the math they do in their lives everyday gets devalued and checked at the door. If pressed, students will say most real-world math happens while shopping. Giving students an image and asking them to talk about the math they see is a great way to open up our ideas of what math is, where it happens, and why it is important. The question gives an opportunity for students to value the math they do and see in their lives and in their communities. The images here are just a few examples to get you started. We'd love to collect and share any images you use with your students! Send yours to mark.trushkowsky@cuny.edu or post them on Twitter using the hashtag #ANNMath. The Multiplicity Lab's Image of the Week (and their extensive archive) is full of images you can use to pose this question.

19) SAME or DIFFERENT?

Similar to Notice/Wonder, Same or Different is a simple pair of questions to draw out student observations and mathematical argument.   

Same or Different is a good introduction to this routine created by Brian Bushart. It has a collection of images and brief videos that teachers can use, but don't feel limited to what is there! 

What's the same? What's different?

21) Unit Chats

For unit chats, teachers share an image with different kinds of objects to count, arranged in arrays or some other structure and ask students a simple question - “How many?”. Give students some time to see how many of which objects they see. Then after they have had 20 - 60 seconds to look, ask students what they see. As they share, open the floor for students to explain they different ways of counting what they saw and the different things they saw to count in the image. 

You can find a great introduction to unit chats, along with some compelling images at the Math for Love UNIT CHAT page.

How many?

22) Convince me that...

"Convince me that..." is an easy to implement instructional routine that can be used with any math content. The routine draws out student visualizations and lots of language in a way that shifts the goals from finding solutions to communicating ideas. The basic structure is you find a word problem or problem, remove the question, give students the answer and then ask them to convince you that it's true. Here's a collection of over 100 examples of "Convince me that..." prompts, or you can create your own. 

23) CONTEMPLATE then CALCULATE (CtC)

This instructional routine has 4 main steps.

1. Teachers flash an image for about 1.5 seconds. Students then share what they noticed with a partner and then as a whole class. 
2. The teacher reshares the image and students find a calculation shortcut, working on their own.
3. Students share and study the strategies that come out. 
4. The final step is to reflect on the learning and look for something that can help them as they are working on other problems in the future. 

The New Visions for Public Schools has a great collection of CtC resources to bring this routine into your classroom including a collection of tasks, videos, an introduction to the routine, a guide for how to annotate student strategies, slide templates and lesson plan templates.

24) CONNECTING REPRESENTATIONS

This instruction routine has 4 steps:

1. Connect Representations. In this step, students are given 5 different representations. Their first task is to find the two pairs that match.
2. Share and Study Connections
3. Create a matching Representation for the one that was left.
4. Reflection 

Similar to Contemplate then Calculate, the New Visions for Public Schools has a great collection of Connecting Representation tasks, videos, an introduction to the routine, a guide for how to annotate student strategies, slide templates and lesson plan templates. 

25) Push & Support Cards

Push and Support cards are tools that allow teachers to dial up and dial down problems, keeping every student in a mixed-level math class engaged in high-cognitive level math tasks.  

26) Wrong Answers Only

For this routine, present your students with a problem. Their task is to come up with an incorrect answer and to share their rationale for how they know the answer is incorrect. 

Read more about how Zuzka uses this routine at the TERC Adult Numeracy Center blog post, Wrong Answers Only.

27) My Favorite No

My Favorite No is a great assessment activity that turns students’ mistakes into collective opportunities for learning. It can be done with any math topic or content. It takes very little time, so teachers can do it often and weave it into the daily routine of their class. This 6 min video provides an overview of the routine. 

For some ideas about how to use it with adult students, read My Favorite No: A Great Way to Celebrate Student Mistakes

28) Clarify, Critique, Correct

Clarify, Critique, Correct is a math language routine where teachers give students a piece of mathematical writing (that is not their own) to analyze, reflect on, and develop. The intent is to prompt student reflection with an incorrect, incomplete, or ambiguous written argument or explanation, and for students to improve upon the written work by correcting errors and clarifying meaning. We learned about Clarify, Critique, Correct from Illustrative Math's collection of Math Language Routines.

The goal of the routine is for students to discuss and understand why the mistake happened (more than just having them correct mistakes). For example, you might give students a fictional student statement like:

And ask them to discuss in pairs/groups:

Clarify: Why do you think this student wrote this?
Critique: What changes do you suggest? Why?
Correct: Improve the response.

29) Information Gap

During an Information Gap routine, students conduct a dialog in a specific way. One partner gets a Problem Card with a math question that doesn’t have enough information, and the other partner gets a Data Card with information relevant to the problem card. Students ask each other questions like “What information do you need?” and are expected to explain what they will do with the information. 

We learned about Info Gap from Illustrative Math's collection of Math Language Routines. Info Gap cards can be found throughout their free curriculum. Read a teacher describe why she likes the routine Info Gap Cards: The Hidden Gem. 

30) Going on a Number Picnic

Going on a Number Picnic is a math variation of a classic game "Going on a picnic." In the original version of the game, a player comes up with a rule in their head (ex. you can only bring things that have the letter L in them) and then says something like, "I'm going on a picnic and I'm bringing lettuce, lemons, and chanterelles but I'm not bringing mushrooms, spinach, or sandwiches." Then the other platers need to try to guess the rule by asking if they can bring certain items to the picnic. Can I bring apples? - Yes! Can I bring pears? - No! 

The number picnic version was developed by Carol Cashion, an adult education teacher in NYC. In Carol's version, you come up with a group/category of numbers (ex Multiples of 6) and say something like, "I'm going on a number picnic and I'm bringing 36, 48, and 54 but I'm not bringing 27, 33, or 63." Then students try to figure out the rule by asking if they can bring certain numbers to the picnic. Can I bring 66? - Yes! Can I bring 34? No!

There are all kinds of rules you can use for Going on a math picnic - you can make them as challenging or accessible as you like. For example, the rule could be: perfect squares, equivalent fractions, prime numbers, factors of 72, numbers who digits add up to 8, etc  

It can help make two columns on the board (or a sheet of paper) - Yes and No to keep track of the guesses. 

You can use Going on a Number Picnic to introduce a topic and you can also use it to review a topic. Once you've done it a few times together as a class, it's also fun to have students come up with their own rules and play in pairs.

31) Menu Math

Menu Math is a new instructional routine developed by Nat Banting.

A Menu Math task is a collection of 6-10 constraints that appear as an unordered list. For each task, there is a type of mathematical object that needs to be designed to satisfy these constraints. That object could be a certain type of function, an expression, a shape, a number, etc. Each constraint must be satisfied at least once, and students try to complete this goal using as few mathematical objects as possible. 

The structure provides a low entry point (teachers might ask students to design objects that satisfy one or two constraints), and a high ceiling (as students analyze which constraints pair well together and which cannot pair together).

Here is an example of what Menu Math looks like:

Nat Banting's collection of Menu Math Tasks 

Chad Williams created a collection of Menu Math specific to K-5 grade math content - MENU MATH ELEMENTARY COLLECTION

32) Number of the Day

The basic routine of Number of the Day is simple:

Give the students a number to consider each day. Students think of different ways to make that number. As you develop different number concepts in class, encourage your students to incorporate these concepts into the different ways they develop the number.