[C]lassroom environments should at the same time be learnercentered, knowledgecentered, assessmentcentered, and communitycentered…The instruction described is learnercentered in that it draws out and builds on student thinking. It is also knowledgecentered in that it focuses simultaneously on the conceptual understanding and procedural knowledge of a topic, which students must master to be proficient, and the learning paths that can lead from existing to more advanced understanding. It is assessmentcentered in that there are frequent opportunities for students to reveal their thinking on a topic so the teacher can shape instruction in response to their learning, and students can be made aware of their own progress. And it is communitycentered in that the norms of the classroom community value student ideas, encourage productive interchange, and promote collaborative thinking.
– Secondary Lenses on Learning: Team Leadership for Mathematics in Middle and High Schools, 2009
– Secondary Lenses on Learning: Team Leadership for Mathematics in Middle and High Schools, 2009
Problem Solving as SenseMaking
As this video demonstrates key word approaches fail students over the long term. That's because solving problems requires thinking. Good mathematicians make sense of problems before they solve it. How do we get students to learn to do this? Teaching students questioning stems to help them understand a problem before they solve it is a better strategy than the key word approach (which essentially takes the thinking out of it). Try integrating this problemsolving questioning sequence whole class, small group, and one on one.


Andy Warhol Math Investigations

I took these photos at the Andy Warhol exhibit at the Chicago Art Institute in Fall 2019. These pieces provide rich opportunities for exploring multiplication concepts and strategies.
Use them for a whole class or small group number talk. How many? How do you know? What are easy/friendly ways to calculate? Use them at a center. Beyond calculating totals, students can use materials to make their own "multiplicative" art pieces. Work with other subject area teachers to create an inquiry unit around math in art. Students can research an artist, identify ways the artist uses math in their work, and create original pieces for a math inspired art show. 
Photo Collection: At the Store
Stores provide an excellent context for exploring mathematics. What math questions do these photos inspire? Use these photographs at a center or during collaborative work time to anchor mathematical investigations. Better yet, take students on a walking trip so that students can capture their own math images and write/solve their own math investigations
Intermediate grade students can use these pictures to:
Middle school students can use these pictures to:


OnLine Resources: Number Sense and Problem Solving Tasks
Math Talk for Slides
This site has easy to access slides of some of my favorite number sense routines ready for classroom use. Many of the resources listed below are included in slide version.
Algebra mobiles
A variety of cool number and visual puzzles that build algebraic reasoning and number sense.
Steve Wyborney Splat, Estimysteries, Cube conversations
This site has a number of ready to use routines that develop number sense, reasoning, and math joy in the classroom K8.
Which One Doesn’t Belong
These are quick look cards where students use reasoning to determine and defend which one doesn’t belong there are many ways to reason “correctly”. They have shape, number, and graph based cards. Can work at many levels KHS.
Estimation 180
A collection of picturebased contexts to spark reasoning and problemsolving around estimation and number sense. Pictures are organized to build upon one another.
Number Talk images
This site has a variety of prompts (mostly photos) where kids answer the question "How many?" and share how they figured it out.
Inside Mathematics
This site has lots of great resources for teachers. Check out the "inside problem solving" tab for open ended problems by grade level. The "assessments" tab has MARS performance assessment tasks by grade level available for download.
You Cubed
Jo Boaler's Mathematical Mindsets is a must read book for any practitioner. This site houses her resources to support teaching for a positive math mindset. Under "Tasks" you can find open ended problems by domain and gradelevel as well as her her week of inspirational math(s) which is a great way to kick off the school year. There are ideas, classroom resources, and online PD workshops available as well.
Illustrative Mathematics
Download common core aligned math tasks by gradelevel here.
Also check out their K8 curriculum you have to register to see it
Nrich Maths
Huge site with tons of resources, I find it easiest to search a topic “Fractions” and go from there. Has PreK, Elementary, and Secondary tasks and resources sorted by level and challenge.
Open Middle
These open ended and challenging tasks can be searched by grade level or standard. They also provide links to google slide versions of all tasks.
Dan Meyer’s blog
This former math teacher has a huge collection of tasks and posts to support meaningful learning great for MS/HS teachers click on “Archives” to launch list of tasks.
Visual Patterns
Has a huge collection of visual growing patterns focus question of “What is the equation?”
101 Questions
Tons of picture prompts that have kids generate questions. “What is the first question that comes to mind?” Pushes for mathematizing the world and encourages thinking
Would you rather math
This site contains real life scenarios that ask kids to pick and justify a choice in a “would you rather?” format. For example buy one and get 2nd 50% off vs. buy 2 and get 3rd 50% off.
Clothesline math
http://www.estimation180.com/clothesline.html
Using a flexible number line (on a clothesline) to support numerical reasoning. These sites offer a collection of tools and cards and explain what clotheslines are and how to use.
Desmos
Desmos has engaging, sense making digital activities for students. Teachers can use Desmos as a curriculum or assign by task. Make sure to check out the "getting started" tab under resources.
University of Waterloo
You can sign up for a subscription to a problem of the week by grade level (grades 3 and above). There are also archives of past problems and solutions by grade level
Mindset kit
Provides resources for supporting growth mindset. Includes resources for parents.
This site has easy to access slides of some of my favorite number sense routines ready for classroom use. Many of the resources listed below are included in slide version.
Algebra mobiles
A variety of cool number and visual puzzles that build algebraic reasoning and number sense.
Steve Wyborney Splat, Estimysteries, Cube conversations
This site has a number of ready to use routines that develop number sense, reasoning, and math joy in the classroom K8.
Which One Doesn’t Belong
These are quick look cards where students use reasoning to determine and defend which one doesn’t belong there are many ways to reason “correctly”. They have shape, number, and graph based cards. Can work at many levels KHS.
Estimation 180
A collection of picturebased contexts to spark reasoning and problemsolving around estimation and number sense. Pictures are organized to build upon one another.
Number Talk images
This site has a variety of prompts (mostly photos) where kids answer the question "How many?" and share how they figured it out.
Inside Mathematics
This site has lots of great resources for teachers. Check out the "inside problem solving" tab for open ended problems by grade level. The "assessments" tab has MARS performance assessment tasks by grade level available for download.
You Cubed
Jo Boaler's Mathematical Mindsets is a must read book for any practitioner. This site houses her resources to support teaching for a positive math mindset. Under "Tasks" you can find open ended problems by domain and gradelevel as well as her her week of inspirational math(s) which is a great way to kick off the school year. There are ideas, classroom resources, and online PD workshops available as well.
Illustrative Mathematics
Download common core aligned math tasks by gradelevel here.
Also check out their K8 curriculum you have to register to see it
Nrich Maths
Huge site with tons of resources, I find it easiest to search a topic “Fractions” and go from there. Has PreK, Elementary, and Secondary tasks and resources sorted by level and challenge.
Open Middle
These open ended and challenging tasks can be searched by grade level or standard. They also provide links to google slide versions of all tasks.
Dan Meyer’s blog
This former math teacher has a huge collection of tasks and posts to support meaningful learning great for MS/HS teachers click on “Archives” to launch list of tasks.
Visual Patterns
Has a huge collection of visual growing patterns focus question of “What is the equation?”
101 Questions
Tons of picture prompts that have kids generate questions. “What is the first question that comes to mind?” Pushes for mathematizing the world and encourages thinking
Would you rather math
This site contains real life scenarios that ask kids to pick and justify a choice in a “would you rather?” format. For example buy one and get 2nd 50% off vs. buy 2 and get 3rd 50% off.
Clothesline math
http://www.estimation180.com/clothesline.html
Using a flexible number line (on a clothesline) to support numerical reasoning. These sites offer a collection of tools and cards and explain what clotheslines are and how to use.
Desmos
Desmos has engaging, sense making digital activities for students. Teachers can use Desmos as a curriculum or assign by task. Make sure to check out the "getting started" tab under resources.
University of Waterloo
You can sign up for a subscription to a problem of the week by grade level (grades 3 and above). There are also archives of past problems and solutions by grade level
Mindset kit
Provides resources for supporting growth mindset. Includes resources for parents.