What routines will support mathematical thinking and reasoning?
"As teachers from multiple grades levels plan and consistently implement common routines, students will experience a more coherent and better-articulated mathematical experience. In addition, the use of common models such as number lines and Venn diagrams within the routines will improve students' ability to accurately and strategically use these models. (Barnett, Combs, McCoy, High-Yield Routines Grades K-8, 2013, p 3)
- establish effective lesson models, such as “before/during/after”
- ask students to monitor their thinking with prompts, such as “What do I know? What do I need to know? How do I find out?”
- ask students to reflect on their learning, collectively and individually with prompts, such as “What have we/I learned? What questions do we/I still have?”
- establish routines, that foster the core/curricular competencies in relation to content, such as:
- math games
- number talks
- counting collections
- parallel questions & open questions (Marion Small)
Math Routines: Description, Links, and Key Resources
Instructional Math Routines
Primary: Jessica Shumway is the author of Number Sense Routines for K - 3 teachers. This resource provides many different 5 to 10 minute routines, which will assist students in developing number sense in ways that engage all learners in the classroom.
K - 8: Ann McCoy, Emily Combs, and Joann Barneet are the authors of High-Yield Routines for Grades K - 8. In this book, seven easily implemented mathematical routines are shared. They can be used effectively at a variety of grade levels and with a variety of mathematical content.
Instructional Math Routines - Quick Guides with Links
Teacher Education by Design has a variety of Numeracy Routines with one page information sheets, example videos, and planning guides. You will need to register to access these but registration is free. Some of these routines include Counting Collections, Choral Counting, True/False Equations, Three-Act Tasks, Quick Images, Number Strings, etc.
Number Talks are a great way to create a dialogue with students on mathematical topics. The format of Number Talks helps to create a community of respectful math learners, develops computational fluency.
Janice Novakowski has written several blogposts on how she has using Number Talks in classrooms at various grade levels.
Jennifer Barker has information about how to begin, the learning involved, guiding questions, as well as video examples on her website.
Click here to view Number Talks in action in a Grades 6/7 classroom in Surrey, with educator Alex Sabell.
Additionally, click here to view Jonathan Vervaet explains how Number Talks had a profound impact on his Grades 4/5 students learning.
Number Talks with Quick Images
A great place to begin is to use quick images to elicit a number talk.
An excellent website that images of dots, arrays, etc. is NTimages. It is written in both French and English.
This PPT created by Ryan Dent has many examples of arrays.
Jo Boaler has a video that shows how dot images can be used in classrooms for number talks.
Which One Doesn't Belong
The idea of comparing groups of items looking at similarities and differences has been explored by many math educators. Christopher Danielson, the author of the blogs Overthinking my Teaching and Talking Math With Your Kids has taken this idea to new heights. Inspired by Sesame Street's "One of these things in not like the others", Danielson wrote the book Which One Doesn't Belong. In this routine students are asked to look at all four images in each of the quadrants and asked to share their reasoning as to "Which One Doesn't Belong and Why?" The great thing about this routine is that there are no wrong answers, as long as the student's reasoning is true. The focus is not on the answer, but on the student being able to communicate his/her reasoning/justification of their choice.
"Which One Doesn't Belong" website
Counting Collections are just as described; they are collections of items that students count. For more on how to implement this routine and the learning involved, please click on one of the links below.
Kassia Wedekind shares information about conferring with students during Counting Collections.
Videos, information, planning templates: TEDD.org
Same / Different
This routine, also known by some as "Alike or Different" is a routine in which the teacher presents two numbers, shapes, or objects to be compared. The teacher carefully selects what will be compared to focus student thinking on a desired mathematical concept. This routine is powerful because it fosters so many curricular competencies, including communication, reasoning, constructing arguments, etc in relation to mathematical ideas. Brian Bushart, with the support of several other math educators have created an outstanding website with examples that have been categorized into galleries based on math concepts.
Chris Shore authors the Clothesline Math website. The Clothesline is an interactive number line that makes the facilitation of class discourse on number sense much more efficient and effective. The Clothesline is dynamic, meaning that the “benchmark” numbers may be adjusted when needed, as well as the values that are placed on the line.
Other Clothesline enthusiasts
Andrew Stadel has some excellent videos and resources!
Jennifer Barker had described this routine and has editable clothesline cards on her site.
Kristin Gray has written about some experiences she has had in intermediate classrooms.
Janice Novakowski has shared some examples of using this routine in early primary classrooms.
Example of a Kindergarten Clothesline
Example of a clothesline activity in a math 9 class (October 2017)
Students placed cards on a clothesline over three rounds:
- exponents with integer basess
- exponents with variable bases
Sample: Intentions and cards for a Math 9 exponents clothesline activity.
The educators at the Math Forum, Annie Fetter and Max Ray have shared the power of using Notice Wonder as a routine to tap into children's curiosity and show them the importance of asking questions - "What do you notice? What do you wonder?" Check out the one-pager to the right to learn more about what this could look like in your classroom and the learning involved.
Annie Fetter's IGNITE talk "Every Wonder What They'd Notice"
Suggestions for Notice Wonder and Recording Template
Choral counting is basically exactly as it sounds. As a class, everyone counts out loud together in unison. In her book Number Sense Routines, Jessica Shumway suggests Choral Counting, as a great strategy teachers can use to develop their students counting abilities and overall number sense.
Kristen Gray, author of MathMinds website, and a math educator in Delware, has created some outstanding videos showing what Choral Counting looks like at various grade levels. Here is another video Kristen recommends of a Grade Three class.
Tell Me Everything
In this routine, the teacher selects a number. He/she then asks the students to tell them everything they know about the number. Some teachers have students record this information in math journals or on white boards. Depending on the number the teacher selects and how he/she records the students' thinking, different mathematical concepts can be highlighted. As well, when the students are asked to share what they know about a particular number they are using the curricular competency communicating and representing. Students can demonstrate their understanding of the number through concrete, pictorial, and abstract representations. This could involve using patterns, place value, computation, money, etc. Therefore the content learning is open and provides choice for the students.
Why play math games?
Links to math games