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The CRA framework is an instructional strategy that stands for concrete, representational, and abstract; it is critical to helping students move through their learning of math concepts.

To fully understand the idea behind CRA, or concrete representational abstract, think about a small child learning to count. They may learn counting to 10 by memorizing a song. Then, these numbers are used to teach them to count. First, they may be counting blocks by pointing or moving the object. Then, they may be able to count dots on a page or something on paper that represents that same concept. Lastly, they may be able to then look at a picture with five dots and not need to count it, they have learned what five "looks like". This process takes place with all different math concepts and new learning.
Looking for more math intervention ideas?
This is part 3 of our Instructional Design series. We've covered how to unpack math standards and how to use questions to push students to think more critically. Make sure to grab our lesson planning freebie down below!

The CRA framework helps us to present this learning in a specific method to help tie all the connections together. There have been numerous studies measuring the effect of using the concrete representational abstract (CRA) sequence for students at risk of failure. You can read more here, here, and here.

This teaching method breaks down the concept (examples: multiplication, subtraction with regrouping, subtracting integers) in a methodical process in which students move from one phase to the next.

Sometimes CRA is referenced as a sequence, so moving from one phase of understanding to the next. But more so, it is called a framework, which shows that all three of the representations can be used simultaneously. If we were teaching solving equations, students would actually use the algebra tiles, draw a picture of the model, and then write it mathematically at the same time.

When students have a concrete understanding of a mathematical concept, they are utilizing manipulatives to demonstrate the math. For example, when my son was learning to count, I wanted him to physically touch or move the items he was counting. In doing this, he was developing one-to-one correspondence. He was internalizing that each item represents one.

When our middle school students are learning to solve equations, we want them using algebra tiles to physically remove manipulatives from both sides of the equation. This allows students to internalize that an equation must remain balanced.

There are many reasons why teachers might skip this crucial step:

Practice with concrete manipulatives takes time and energy. It is not easy to facilitate with 30+ students.

It can be expensive to purchase manipulatives.

It can be frustrating for students who seem to already grasp the concept.

Why it is beneficial:

Students are using their kinesthetic learning style.

Students are engaging with the content in a deeper way, and abstract concepts like "isolating the variable" are given meaning.

This foundation is formidable so that they can move to conceptual understanding and aren't just following a list of steps with no meaning.

Research shows that it is essential for students who have a shaky math foundation.

Representational

The representational part of the framework refers to students drawing a representation of those concrete materials.

In the case of my son above, that would be counting dots on a page or drawing eight tally marks to represent the number 8.

In the case of our middle schoolers and solving equations, this would look like drawing the algebra tiles, creating a key, and demonstrating how the process works by crossing off the tiles or grouping as they solve.

We see this quite a bit in middle school. From integer counters being sketched to a graph that represents a linear relationship, our standards include many representational aspects.
Why We Might Start Here:

This is a little easier to execute in a classroom full of students.

The standards include the representational language.

It also leaves some gaps for students who need to understand why you are crossing out the positive integer circles.

Math can be a very abstract subject. When you think about solving an equation and finding the value of x, why are we doing this? Why do we need to know this? The value of x is abstract. It is always changing. It doesn't actually mean anything without context.

I think this is where it is really difficult for students who struggle. When we don't provide a context for learning and for the application of the abstract, it just seems too confusing. I think it also warrants the phrase, "I don't get it." Many times students don't even know what to ask because the content means nothing to them.

This is when math becomes too procedural and you risk students understanding the process, going through the motions, and yet lacking number sense. When I was a student, I was taught solely in the abstract. It wasn't until I was a teacher trying to make sense of models before I had several AHA! moments. Why didn't I learn this when I was in school??

Best Practices

Shelley Grey says, "When you teach a math lesson, make it your goal to incorporate concrete, representational and abstract into the same lesson. This way you can be certain that you are differentiating for all your students, regardless of where they are in their understanding.

And it does naturally lend to differentiation - you are scaffolding, using different parts of the brain, and then eventually allowing students to work with a method they feel comfortable with.

You as the teacher can make sure to use different methods when you are tackling problems. Just because you taught the algorithm for multiplying fractions already doesn't mean you can't draw a model to solve at a later date. This ensures that students don't forget how to use the models themselves.

How do you teach difficult math concepts?

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The concrete representational abstract sequence (CRA) helps fill in gaps, teach difficult math concepts, & build a strong math foundation.

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Toy Theater is your source for free educational games specially designed for elementary-aged students. It's available on all desktop computers, tablets, and mobile devices, ensuring learning is never out of reach. If you are a fan of our site, we encourage you to share Toy Theater on your social media or website.

Explore Toy Theater

Here's a sneak peek into what Toy Theater has in store:

Math Games: Ranging from counting and numeral recognition to division and telling time, there's a vast assortment of interactive math games.

Language Arts: Students can work on spelling, letter recognition, and even practice reading with online books.

Art and Music : From teaching students to play simple songs on a keyboard to exploring symmetry via art games, the platform fosters creativity alongside learning.

Puzzles and Games: Ideal for early finishers or indoor recess, their online educational puzzles and games promise fun-filled learning.

Teacher Tools: Need a fun timer or a spinner? Or perhaps a simple way to evaluate basic math skills? Their virtual teacher tools have got you covered.

Whether you project Toy Theater games and interactive tools on your smartboard for the whole class to see or use them during small group work, you're sure to love them!

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Toy Theater | Fun Online Educational Games for Kids

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Make learning a blast! Toy Theater is chock full of interactive online art, reading & math games for kids. Designed with the needs of kindergarten through third grade classrooms in mind, these free games are trusted by teachers around the world to provide real educational value.

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working with a partner to form 2-digit numbers; cooperative activities dealing with concepts before, after, and between DIGITAL FORMAT (view & accomplish tasks on computer, ipad, or other hand held device with google)

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Original Tiling Task Card Packets include 20 unique task cards on card stock, 3 backdrop cards , tracking sheets, suggestions and answer sheets. These are contained in a printed manila envelope for convenient storing. A set of number tiles 0 - 9 is needed for each task card.

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How to Read the Next Generation Science Standards

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Backwards Planning for School Year Success Part 3

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School year curriculum mapping and planning for school year success is an essential part of being organized and making your teacher life easier. Part 3

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Place Value Pockets are amazing tools for developing strong number sense and place value understanding. This is a tool that will quickly become a daily routine in the elementary classroom. It is also an excellent tool for middle school students to practice divisibility rules and algebraic thinking. This product includes 30 sets of 0-9 digit cards in light pink cardstock and 30 place value pockets for ones, tens, hundreds and thousands in blue cardstock. Included are directions with games, questions and activities to facilitate the daily use of this tool. These place value pockets and digit cards are very powerful with Kim Sutton’s book, Place Value with Pizzazz.

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The Coherence Map shows the connections between Common Core State Standards for Mathematics.

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2023 Mathematics Framework

At its meeting on July 12, 2023, the State Board of Education (SBE) adopted the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve ( Mathematics Framework). The framework is important guidance designed to help educators align classroom teaching with California's rigorous math learning standards.

The Mathematics Framework reflects input from California parents, educators, students, and others who commented during the two 60-day field review periods that were held in 2021 and 2022.

California is committed to achieving excellence in math teaching and learning through curriculum and instructional approaches grounded in research and reflective of best practices across the globe. The Mathematics Framework provides guidance for mathematics learning for all students at all levels of math-including calculus-and ensures students have a wide variety of options including pursuing Science, Technology, Engineering, and Mathematics (STEM) in college and career.

Frequently Asked Questions
Information about the draftMathematics Framework.

Schedule of Significant Events for the Revision of the Mathematics Framework
At its meeting on July 13, 2022, SBE revised the Schedule of Significant Events (Timeline) for the Revision ofMathematics Framework. The Timeline was originally approved by the SBE on May 8, 2019, and was previously revised on September 11, 2019, May 7, 2020, July 14, 2021, and November 3, 2021.

A professionally-edited and designed PDF version of the framework will be posted at a later date. The content will not substantively change from the version posted below.

Curriculum Framework and Evaluation Criteria Committee

At their January 08, 2020, meeting the State Board of Education appointed 20 members to the Mathematics Curriculum Framework and Evaluation Criteria Committee (Mathematics CFCC) to assist in the development of the Mathematics Framework.

The Mathematics CFCC Meetings and Notices

Focus Groups

Pursuant to the California Code of Regulations, Title 5, Section 9511(c), the California Department of Education (CDE) convened four focus groups across California.

Listserv

Additional Resources

2013 Mathematics Framework

Assembly Bill 250 (Chapter 608, Statutes of 2011) called for the revision of the Mathematics Framework to align with the California Common Core State Standards for Mathematics.

Video: Overview of the Mathematics Framework (Dated 26-Feb-2015)
This video for teachers, administrators, and other educators provides an overview of the Mathematics Framework and how it supports implementation of the California Common Core State Standards for Mathematics.

Common Core State Standards for Mathematics (PDF; 3MB)
Links to the standards adopted by the California State Board of Education in August 2010 (Modified January 2013 Publication Version).

Common Core State Standards (CCSS) Initiative
The multi-state initiative released the CCSS on June 2, 2010. The initiative site includes information about the development and research behind the standards, appendices, the states that have adopted the CCSS, and frequently asked questions.

Appendix A
Designing High School Mathematics Courses Based on the CCSS.

2005 Mathematics Framework (Available on request)
The Framework was adopted by the State Board of Education on March 9, 2005.
*To order a copy of this framework please contact CDE Press.

Content Standards
Links to downloads of all Content Standards.

All Curriculum Frameworks
Links to downloads of all Curriculum Frameworks.

Questions: Curriculum Frameworks and Instructional Resources Division | CFIRD@cde.ca.gov | 916-319-0881

Last Reviewed: Friday, October 20, 2023

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Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

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The State Board of Education adopted the Science Framework on November 3, 2016. Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

Science

2016 Revision of the Science Framework

SB300 (Chapter 480, Statutes of 2013) calls for the revision of the Science Framework to align with the Next Generation Science Standards (NGSS).

California Science Framework

Science Framework For California Public Schools, Kindergarten Through Grade Twelve (2003).
*To order a copy of this framework please contactCDE Press

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The State Board of Education adopted the Science Framework on November 3, 2016. Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

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The widely adopted ISTE Standards provide a framework for educators to create high-impact, sustainable and scalable learning experiences for all students.

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Discover thousands of math skills covering pre-K to 12th grade, from counting to calculus, with infinite questions that adapt to each student's level.

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Unleash your creativity with the world’s best virtual manipulatives!

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Number Frames by The Math Learning Center

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Number Frames help students structure numbers to five, ten, twenty, and one hundred. Students use the frames to count, represent, compare, and compute with numbers in a particular range.

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Number Line helps students visualize number sequences and demonstrate strategies for counting, comparing, adding, subtracting, multiplying, and dividing.

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Sine of the Times | The Web Sketchpad and Math Education Blog

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Interactive Array Explorations | Sine of the Times

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The Web Sketchpad and Math Education Blog

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Create your own jeopardy template online, without PowerPoint. Or browse the pre-made templates created by other people!

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The DoK Levels added to questions will be reflected in reports. Teachers can easily track and show growth of students understanding in an individual assignment from start to end.

What do individual DoK Levels help you assess?

Level 1: Recall - Focus here is on specific facts, information, details, and procedures e.g specific definitions, details, applies simple steps.

Level 2: Skills and Concepts - Processing of conceptual knowledge, comparing and contrasting, extracting main points, interpreting facts etc. More content knowledge required here than Level 1. Actions require more than one step.

Level 3: Strategic Thinking - Focus is on reasoning based on deep knowledge: which include understanding and rationalizing thoughts, manifesting them into achievable steps. E.g Drawing conclusions from Examples, citing evidence to support a logical argument, concept based explanations.

Level 4: Extended Thinking - Focus is on applying complex reasoning, experimental design, and planning over an extended period of time. It requires extended investigation (research) and time to process information and problem solving. Students are required to make multiple connections, relate ideas to specific content area(s), and select an approach from among many as a solution.

Download the PDF Depth of Knowledge Question Stems.

To learn more about the DoK framework, you can refer to Brenda Matthews' presentation:
Brenda Mathew's DoK framework
Add Depth of Knowledge Levels:

STEP 1: Select "Options" then "Add DoK"

STEP 2: Select the DoK Level you want

View a student's proficiency by DoK Level in your Student Reports [Team Plan]

This will give you real-time metrics on how your students are engaging with texts and how they are performing by DoK Levels.

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The DoK Levels added to questions will be reflected in reports. Teachers can easily track and show growth of students understanding in an individual assignment from start to end. What do individual DoK Levels help you assess? Level 1: Recall - Focus here is on specific facts, information, details, and procedures e.g specific definitions, details, applies simple steps.

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Common Core Instruction: Exploring Dividing Decimals to the Hundredths

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For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is "a mile wide and an inch deep."

These new standards build on the best of high-quality math standards from states across the country. They also draw on the most important international models for mathematical practice, as well as research and input from numerous sources, including state departments of education, scholars, assessment developers, professional organizations, educators, parents and students, and members of the public.

The math standards provide clarity and specificity rather than broad general statements. They endeavor to follow the design envisioned by William Schmidt and Richard Houang (2002), by not only stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value and the laws of arithmetic to structure those ideas.

In addition, the "sequence of topics and performances" that is outlined in a body of math standards must respect what is already known about how students learn. As Confrey (2007) points out, developing "sequenced obstacles and challenges for students...absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise." Therefore, the development of the standards began with research-based learning progressions detailing what is known today about how students' mathematical knowledge, skill, and understanding develop over time. The knowledge and skills students need to be prepared for mathematics in college, career, and life are woven throughout the mathematics standards. They do not include separate Anchor Standards like those used in the ELA/literacy standards.

The Common Core concentrates on a clear set of math skills and concepts. Students will learn concepts in a more organized way both during the school year and across grades. The standards encourage students to solve real-world problems.

Understanding Mathematics

These standards define what students should understand and be able to do in their study of mathematics. But asking a student to understand something also means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student's mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

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For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country.

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The 8 Standards for Mathematical Practice

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California’s newly revised Mathematics Framework highlights the importance of using the Standards for Mathematical Practice as part of high-quality instructional strategies.

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The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

The progressions can explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. They would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks. Progressions documents also provide a transmission mechanism between mathematics education research and standards.

For more information on the project, please visit the project website.

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Progressions Documents for the Common Core State Standards for Mathematics

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The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

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There are so many challenges we're facing with teaching remotely or in a hybrid model. One issue is that students might not have access to the colorful and fun manipulatives you are used to teaching with. But chances are, students will have at least some of the at-home math manipulatives below. Stay flexible with the manipulatives kids choose to use and invite them to explore the math learning around them.

1. Beans, cereal, or dried pasta

Little kids can use dried goods as counters, or to practice estimation or measurement. Challenge older students to find the angles in dried pasta, like the activity above!

2. Dice

Not all kids will have access to dice (no harm in borrowing from a favorite board game), but if they do, there are so many fun ways to use them for math learning.

3. Playing cards

We love using cards to teach evens and odds, or to play a fun math card game.

4. Coins

Students can obviously use coins to practice money skills, but they also make great counters or weights for other units.

5. Beads

String beads on pipe cleaners to use them as counters, or place them in egg cartons as impromptu ten-frames.

6. Buttons

A jar full of loose buttons isn't as common as it once was, but if your students have access to them, they can be a fun way to support all kinds of learning. Check out our favorite ways to use buttons for learning here.

7. Loose parts

Source: @adventschool

In early childhood, educators use the term "loose parts" to describe any group of items that could be used for learning, such as seashells, egg cartons, blocks, Barbie shoes ... the list goes on. When it comes to at-home math manipulatives, older kids can adopt a "loose parts" philosophy too. Here are some of our favorite ways to use loose parts for learning.

8. Paper plates

Students can use paper plates to explore fractions, fact families, and more. Check out some of our favorite ways to use paper plates here.

9. LEGO bricks

LEGO bricks are an amazing manipulative in and out of the classroom and can be used to teach almost any math skill you can think of. Here are some of our favorite ways to teach math with LEGO.

10. Play-Doh

Play-Doh is great for tactile and sensory learners! Use it to practice fractions, model 3D shapes, and more. Check out more ideas for using Play-Doh here.

11. Craft sticks

Pair craft sticks (or regular sticks from outside!) with Play-Doh to build geometric shapes. See more ideas for using craft sticks here.

12. Yarn

Younger students can use yarn to make numbers and practice non-standard measurement. Older kids might try making one of these geometric line designs.

13. Cardboard or other recyclables

Have students hit the recycling bin to create their own math manipulatives from cardboard, bottle tops, or other recyclables. Here's our full list of ways to teach with cardboard.

14. Pumpkins, apples, or other produce

The refrigerator is another place to turn for math learning. Check out ways to teach math with pumpkins and apples.
What at-home math manipulatives have worked for you? Come and share in our WeAreTeachers HELPLINE group on Facebook. Plus, the best websites for teaching math.

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14 Easy Math Manipulatives Kids Can Use for At-Home Learning

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There are so many challenges we're facing with teaching remotely or in a hybrid model. One issue is that students might not have access to the colorful and fun manipulatives you are used to teaching with. But chances are, students will have at least some of the at-home math manipulatives below.

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Math is all around us. We can see it in nature and in our homes. We use math skills every day, even if we don't realize it. And you can help your students see math in all of its forms with fun and engaging math manipulatives.

These hands-on objects and activities enhance your math lessons, giving students a concrete way to practice and play with math concepts. Sometimes it can be difficult for children to grasp the abstract concepts of adding, subtracting and multiplying numbers. But math manipulatives can help them actually see what the numbers are doing.

Common math manipulatives include blocks of all kinds, whiteboards, and interesting online materials. Even if you don't have access to these resources right now, you can easily find creative alternatives for your classroom, especially once you understand what math manipulatives are and how they're used.
This article will give you a strong grasp of what math manipulatives are, the key benefits of using them, and our top 10 favorite math manipulatives for the classroom.

What are math manipulatives?

Math manipulatives are objects, charts, and activities that engage learners while helping them develop their math skills.
They can assist in teaching lessons such as graphing, decimals, geometry, arithmetic, and more.

Using math manipulatives during a lesson brings hands-on learning to your classroom. And this can appeal to a multitude of different learners.

As you know, not every student learns in the same way. Math manipulatives allow students to work with math in the way that's best for them. By working independently with math manipulatives, a student is better able to work at their own pace and spend more time on concepts or problems that are particularly difficult for them.

Many math concepts are conceptual and hard for students to visualize on paper. By practicing math processes with math manipulatives, learners can see exactly how the process affects the numbers and how problems are solved.

There are many places to buy math manipulatives or manipulative kits, including online or in school supply stores. But you don't need to buy an expensive, specialized kit to bring manipulatives to your classroom. Especially for younger grades, any toy collection or set of blocks will work.

There are also many printable options available online. We're talking loads of variations and choices, some geared towards homeschooling families, math centers or traditional classrooms. The digital choices are endless.

With all these options, it can be overwhelming and difficult to know where to start. Here's our list of the ten most common and versatile math manipulatives to get you started.

10 Math manipulatives to implement in your classroom

1. Dominoes

Dominoes are much more than just a fun game. They're also one of the best foundational math manipulatives. Dominoes can be used to teach a variety of math skills - like addition, multiplication, division, and subtraction. And they're particularly helpful in younger grade levels.

They're easy for even the youngest students to grasp and move, and they can help math learning feel like a game.

Dominoes work well for both individual and small group problem-solving. They also allow you to take a quick trip around the room, gauging student understanding at a glance.

During your math manipulative time, encourage your students to create and solve their own math problems by arranging the dominoes in different ways. There are loads of math games you can play with dominoes. If you need some ideas, a quick Google search can pull up enough ideas for the entire school year.

2. Pattern blocks

Pattern blocks are a math manipulative perfect for geometry lessons. These blocks are made of either plastic or wood and come in a variety of shapes - like triangles, squares, rectangles, and trapezoids.

Your students can experiment with how these shapes fit together to make new shapes or images. These blocks often come with pattern cards. These can guide your student in how to make a picture, or encourage them to be creative and create their own pattern.

Pattern blocks are also great for basic math learning. Lessons in fractions, counting and patterns can all be enhanced with pattern blocks. With so many diverse applications, many educators feel these blocks are an important and necessary component of their teaching toolkit - especially in younger grade levels.

3. Online math games

Regardless of your classroom setup - traditional, virtual, homeschooling or hybrid - digital math manipulatives are a great option! They allow students to engage with math in a unique and fun way. And there's a huge variety of online math games online, targeting whichever math skills you're currently teaching.

Online math games can even support student learning over the summer months or during school breaks. The fun nature of online games keeps students excited and learning at home - because they don't feel like homework.

Looking for a math game for your class, try Prodigy Math!

A lot of my higher level students are really motivated by Prodigy, but the part that I like the best is when my students can work cooperatively. My lower level math students can play with my higher level ones and they both are engaged in their math work and having fun. It's a great supplementary tool to help keep math fun and engaging, and it works.

William Stenross

4th Grade Classroom Teacher

4. Cuisenaire rods

Cuisenaire rods are rectangular rods of various lengths. They're often made of wood, and each length corresponds to a different color. For example, all red rods are 10 units long, while all yellow rods are 5 units long.

These colorful manipulatives can help bring the otherwise boring number line to life. Students can visualize the distance between numbers, and see the relationship numbers have with each other. How many times does 5 go into 10? Grab the number rods and find out!

Cuisenaire rods are a math favorite, mostly due to their versatility. They can be used to teach everything from basic number sense to fractions and patterns. Whether you teach Pre-K or 8th grade, you can find a use for Cuisenaire rods.

5. Two color counters

Another popular manipulative in math classrooms is two-color counters. Color counters are simple plastic circles with one color on each side. You'll see them most often used to demonstrate fractions, basic addition, subtraction and counting skills.

Two color counter's simplistic nature means they're most popular in lower grade levels. And even better, two-color counters are often magnetic, making them the perfect choice for whiteboard activities and math games. Use them to demonstrate lessons when you're front of class, or divide students into small groups so they can experiment and learn.

6. Fraction tiles & fraction circles

Fraction tiles and fraction circles are two different math manipulatives with a similar purpose.

Fractions can be particularly difficult for students to understand and work with. These manipulatives can help students deepen their conceptual understanding of fractions with two different math visuals - making them a perfect pair.

Fraction tiles are often flat, rectangular pieces of plastic or cardboard made of different colors and varying lengths. They usually have the corresponding fraction printed on them so kids can quickly look and compare. The varying lengths allow students to experiment with how fractions work together to create not just whole numbers, but also different fractions within the whole.

Fraction circles are very similar, but instead of rectangles, they use parts of a circle. This is the manipulative version of the pie or pizza example. Students can use differently sized pieces to understand how fractions can add up to 1.

Bonus - both of these math manipulatives are easy to DIY with paper print outs if you're on a time or budget restraint.

See how to make your own fraction tiles below!

7. Rekenreks counting frames

Rekenreks counting frames are often one of the first math manipulatives a student will use. In fact, many use similar bead counters before they ever come to school.

These simple arithmetic frames combine the concepts of the number line, individual counters, and base-ten models to create a useful learning tool. Students move the beads from one end to the other to practice counting or assist with basic addition and subtraction problems.

Younger students, like those in Pre-K and elementary school, benefit the most from Rekenrek counting frames. Once students have mastered counting and single-digit math problems, they're ready to move on to more diverse manipulatives.

8. Bear counters

Bear counters are another great option for younger students.

These fun and colorful plastic bears will keep your little learners excited to play with math. The bears are perfect for teaching the foundational math skills of counting and sorting. They can also be paired with a scale to teach measurement and weight.

If you don't have bear counters in your classroom, look for other small toys that can serve the same purpose. It's helpful for them to be similar in type, but that's not a necessity. Students can practice counting zoo animals, cotton balls, or erasers just the same.

9. Place value blocks or base ten blocks

Place value blocks, also called base-ten blocks, are used in classrooms from Pre-K all the way to 8th grade. These little plastic blocks connect to each other, making them a fun option for teaching a variety math concepts - like addition, subtraction, fractions, measurement, counting and more.

Most sets are colorful and can help students visualize different parts of an equation. They're perfect for exploring place value as well. Once students move up to solving double-digit problems, different colors can be used to represent the tens and even hundreds place.

Bonus tip: You can also use this place value chart as another in-class supplement!

10. Geoboards

Looking for something besides blocks for your geometry activities in class? Geoboards are the answer.

Geoboards are small wooden or plastic pegboards that use string or rubber bands to create a variety of shapes. They're a great hands-on activity for practicing shape recognition and names.

Geoboards can double as a fun brain break for young students. Pre-K and elementary students are often still working on the fine motor skills needed to draw shapes and write numbers. And manipulating the rubber bands or strings can be a welcome departure from the frustration of writing.

Like pattern blocks, you can have students use these in a structured way by making specific shapes. Or give them the freedom to create whichever shapes they'd like. You're sure to end up with some fantastic, mathematical art!

See how one teacher used a geoboard to explain basic shapes to her first grade students below!

Physical vs. virtual math manipulatives

Classrooms today don't look the same as they did twenty years ago, or even 3 years ago. The number of students in virtual or hybrid classrooms has increased dramatically. So, when you can't physically hand your students math manipulatives, what are your options?

There's actually quite a few! In the last few years, many free, online programs offering virtual math manipulatives have popped up. These may not be manipulatives in the traditional sense, but online games stretch the brain using a variety of learning models. They help students nail down complicated concepts in much the same way as the manipulatives mentioned above.

And it's not just virtual classrooms that can benefit from these tools. Homeschooling rates are on the rise. But many parents may not have the budget to purchase a large variety of math manipulatives. Online options provide a budget-friendly way to teach math at home or in the classroom.

It's easy to see how virtual manipulatives are beneficial for both students and teachers. Teachers now have a wide variety of effective teaching tools at their fingertips. And learners can gain a deeper understanding of math concepts on a platform they already enjoy. That means less frustration and more opportunity. Let's dive deep into a few more benefits of math manipulatives, whether virtual or physical.

3 Key benefits of using math manipulatives in the classroom

Increase interactive learning

In a classroom of 15-30 students, no two students are alike. They all learn in unique ways. And trying to make one lesson fit all those students is impossible.

For students who are struggling with math, hands-on learning activities with manipulatives can be a huge help. Not everyone can make sense of math on paper alone. Hands-on activities give learners a deeper understanding of how numbers work. And this understanding will continue to help them as they progress through school.

Students need to have a full understanding of basic concepts. Math builds on itself, so have a strong foundation will only help them as they gain more knowledge and advanced skills.

Also, manipulatives keep math learning exciting. Students grow bored of always using the same learning method. There's little to love about lectures and equations on a whiteboard. Math manipulatives can help you diversify your teaching style. You'll reach more students by using more methods, and that means more engagement and more learning. Awesome!

Develop problem-solving skills

Math manipulatives help students develop more complex problem-solving skills. When a student fully understands a math concept, they're better able to apply that concept to real-world problems. We want to teach students how to apply math skills across all of life, not just to next week's test.

Also, math manipulatives allow for a variety of problem solving methods. Students often interpret and think about problems in different ways. And that means they'll approach the solution differently too. Math manipulatives give your students the freedom to explore multiple paths to a solution. And if it doesn't work, it's easier for them to see why.

Build confidence in each math lesson

All of this problem solving with math manipulatives can give your students a major confidence boost! Activities with math manipulatives allow students to take the reins of their learning process. And because many math manipulatives are self-correcting, they can work individually on their problems. When they reach the right answer, all on their own, they'll feel an amazing sense of pride and accomplishment.

When students are able to understand and apply mathematical concepts, they're filled with that I did it feeling. Mastering the steps of a complicated math process gives students the confidence and encouragement to keep moving forward.

Students can then translate this deeper conceptual understanding to tests, quizzes, and standardized tests. They won't need a last minute cram session to succeed. They can stress less knowing they have the knowledge to tackle a variety of math and life problems.

Whether physical or virtual, math manipulatives can create a ripple effect that helps students throughout their entire educational journey.

Develop your student's math skills in a fun and engaging way

Math manipulatives are fantastic tools for helping your students understand and engage in their math lessons. And there are so many math manipulatives to choose from - something to fit every lesson and each student's unique needs.

Even if you don't have physical math manipulatives at your disposal, you can easily find virtual and online options that are just as helpful.

And some of them come at no-cost to teachers and schools! 👇

A no-cost, virtual math manipulative that's fun!

When you're exploring virtual options for math manipulatives, be sure to check out Prodigy Math. Prodigy's online math game is a stress-free way to help students master standards-aligned math skills. It's safe, fun and engaging, keeping your students excited and ready for their next lesson.

And with a free teacher account you can:

Set up engaging math practice in just a few clicks

Easily find out any learning gaps your students may have

Track how your students are progressing through your local or national curriculum

Sign up for free

Math is all around us. We can see it in nature and in our homes. We use math skills every day, even if we don't realize it. And you can help your students see math in all of its forms with fun and engaging math manipulatives.

These hands-on objects and activities enhance your math lessons, giving students a concrete way to practice and play with math concepts. Sometimes it can be difficult for children to grasp the abstract concepts of adding, subtracting and multiplying numbers. But math manipulatives can help them actually see what the numbers are doing.

Common math manipulatives include blocks of all kinds, whiteboards, and interesting online materials. Even if you don't have access to these resources right now, you can easily find creative alternatives for your classroom, especially once you understand what math manipulatives are and how they're used.
This article will give you a strong grasp of what math manipulatives are, the key benefits of using them, and our top 10 favorite math manipulatives for the classroom.

What are math manipulatives?

Math manipulatives are objects, charts, and activities that engage learners while helping them develop their math skills.
They can assist in teaching lessons such as graphing, decimals, geometry, arithmetic, and more.

Using math manipulatives during a lesson brings hands-on learning to your classroom. And this can appeal to a multitude of different learners.

As you know, not every student learns in the same way. Math manipulatives allow students to work with math in the way that's best for them. By working independently with math manipulatives, a student is better able to work at their own pace and spend more time on concepts or problems that are particularly difficult for them.

Many math concepts are conceptual and hard for students to visualize on paper. By practicing math processes with math manipulatives, learners can see exactly how the process affects the numbers and how problems are solved.

There are many places to buy math manipulatives or manipulative kits, including online or in school supply stores. But you don't need to buy an expensive, specialized kit to bring manipulatives to your classroom. Especially for younger grades, any toy collection or set of blocks will work.

There are also many printable options available online. We're talking loads of variations and choices, some geared towards homeschooling families, math centers or traditional classrooms. The digital choices are endless.

With all these options, it can be overwhelming and difficult to know where to start. Here's our list of the ten most common and versatile math manipulatives to get you started.

10 Math manipulatives to implement in your classroom

1. Dominoes

Dominoes are much more than just a fun game. They're also one of the best foundational math manipulatives. Dominoes can be used to teach a variety of math skills - like addition, multiplication, division, and subtraction. And they're particularly helpful in younger grade levels.

They're easy for even the youngest students to grasp and move, and they can help math learning feel like a game.

Dominoes work well for both individual and small group problem-solving. They also allow you to take a quick trip around the room, gauging student understanding at a glance.

During your math manipulative time, encourage your students to create and solve their own math problems by arranging the dominoes in different ways. There are loads of math games you can play with dominoes. If you need some ideas, a quick Google search can pull up enough ideas for the entire school year.

2. Pattern blocks

Pattern blocks are a math manipulative perfect for geometry lessons. These blocks are made of either plastic or wood and come in a variety of shapes - like triangles, squares, rectangles, and trapezoids.

Your students can experiment with how these shapes fit together to make new shapes or images. These blocks often come with pattern cards. These can guide your student in how to make a picture, or encourage them to be creative and create their own pattern.

Pattern blocks are also great for basic math learning. Lessons in fractions, counting and patterns can all be enhanced with pattern blocks. With so many diverse applications, many educators feel these blocks are an important and necessary component of their teaching toolkit - especially in younger grade levels.

3. Online math games

Regardless of your classroom setup - traditional, virtual, homeschooling or hybrid - digital math manipulatives are a great option! They allow students to engage with math in a unique and fun way. And there's a huge variety of online math games online, targeting whichever math skills you're currently teaching.

Online math games can even support student learning over the summer months or during school breaks. The fun nature of online games keeps students excited and learning at home - because they don't feel like homework.

Looking for a math game for your class, try Prodigy Math!

A lot of my higher level students are really motivated by Prodigy, but the part that I like the best is when my students can work cooperatively. My lower level math students can play with my higher level ones and they both are engaged in their math work and having fun. It's a great supplementary tool to help keep math fun and engaging, and it works.

William Stenross

4th Grade Classroom Teacher

4. Cuisenaire rods

Cuisenaire rods are rectangular rods of various lengths. They're often made of wood, and each length corresponds to a different color. For example, all red rods are 10 units long, while all yellow rods are 5 units long.

These colorful manipulatives can help bring the otherwise boring number line to life. Students can visualize the distance between numbers, and see the relationship numbers have with each other. How many times does 5 go into 10? Grab the number rods and find out!

Cuisenaire rods are a math favorite, mostly due to their versatility. They can be used to teach everything from basic number sense to fractions and patterns. Whether you teach Pre-K or 8th grade, you can find a use for Cuisenaire rods.

5. Two color counters

Another popular manipulative in math classrooms is two-color counters. Color counters are simple plastic circles with one color on each side. You'll see them most often used to demonstrate fractions, basic addition, subtraction and counting skills.

Two color counter's simplistic nature means they're most popular in lower grade levels. And even better, two-color counters are often magnetic, making them the perfect choice for whiteboard activities and math games. Use them to demonstrate lessons when you're front of class, or divide students into small groups so they can experiment and learn.

6. Fraction tiles & fraction circles

Fraction tiles and fraction circles are two different math manipulatives with a similar purpose.

Fractions can be particularly difficult for students to understand and work with. These manipulatives can help students deepen their conceptual understanding of fractions with two different math visuals - making them a perfect pair.

Fraction tiles are often flat, rectangular pieces of plastic or cardboard made of different colors and varying lengths. They usually have the corresponding fraction printed on them so kids can quickly look and compare. The varying lengths allow students to experiment with how fractions work together to create not just whole numbers, but also different fractions within the whole.

Fraction circles are very similar, but instead of rectangles, they use parts of a circle. This is the manipulative version of the pie or pizza example. Students can use differently sized pieces to understand how fractions can add up to 1.

Bonus - both of these math manipulatives are easy to DIY with paper print outs if you're on a time or budget restraint.

See how to make your own fraction tiles below!

7. Rekenreks counting frames

Rekenreks counting frames are often one of the first math manipulatives a student will use. In fact, many use similar bead counters before they ever come to school.

These simple arithmetic frames combine the concepts of the number line, individual counters, and base-ten models to create a useful learning tool. Students move the beads from one end to the other to practice counting or assist with basic addition and subtraction problems.

Younger students, like those in Pre-K and elementary school, benefit the most from Rekenrek counting frames. Once students have mastered counting and single-digit math problems, they're ready to move on to more diverse manipulatives.

8. Bear counters

Bear counters are another great option for younger students.

These fun and colorful plastic bears will keep your little learners excited to play with math. The bears are perfect for teaching the foundational math skills of counting and sorting. They can also be paired with a scale to teach measurement and weight.

If you don't have bear counters in your classroom, look for other small toys that can serve the same purpose. It's helpful for them to be similar in type, but that's not a necessity. Students can practice counting zoo animals, cotton balls, or erasers just the same.

9. Place value blocks or base ten blocks

Place value blocks, also called base-ten blocks, are used in classrooms from Pre-K all the way to 8th grade. These little plastic blocks connect to each other, making them a fun option for teaching a variety math concepts - like addition, subtraction, fractions, measurement, counting and more.

Most sets are colorful and can help students visualize different parts of an equation. They're perfect for exploring place value as well. Once students move up to solving double-digit problems, different colors can be used to represent the tens and even hundreds place.

Bonus tip: You can also use this place value chart as another in-class supplement!

10. Geoboards

Looking for something besides blocks for your geometry activities in class? Geoboards are the answer.

Geoboards are small wooden or plastic pegboards that use string or rubber bands to create a variety of shapes. They're a great hands-on activity for practicing shape recognition and names.

Geoboards can double as a fun brain break for young students. Pre-K and elementary students are often still working on the fine motor skills needed to draw shapes and write numbers. And manipulating the rubber bands or strings can be a welcome departure from the frustration of writing.

Like pattern blocks, you can have students use these in a structured way by making specific shapes. Or give them the freedom to create whichever shapes they'd like. You're sure to end up with some fantastic, mathematical art!

See how one teacher used a geoboard to explain basic shapes to her first grade students below!

Physical vs. virtual math manipulatives

Classrooms today don't look the same as they did twenty years ago, or even 3 years ago. The number of students in virtual or hybrid classrooms has increased dramatically. So, when you can't physically hand your students math manipulatives, what are your options?

There's actually quite a few! In the last few years, many free, online programs offering virtual math manipulatives have popped up. These may not be manipulatives in the traditional sense, but online games stretch the brain using a variety of learning models. They help students nail down complicated concepts in much the same way as the manipulatives mentioned above.

And it's not just virtual classrooms that can benefit from these tools. Homeschooling rates are on the rise. But many parents may not have the budget to purchase a large variety of math manipulatives. Online options provide a budget-friendly way to teach math at home or in the classroom.

It's easy to see how virtual manipulatives are beneficial for both students and teachers. Teachers now have a wide variety of effective teaching tools at their fingertips. And learners can gain a deeper understanding of math concepts on a platform they already enjoy. That means less frustration and more opportunity. Let's dive deep into a few more benefits of math manipulatives, whether virtual or physical.

3 Key benefits of using math manipulatives in the classroom

Increase interactive learning

In a classroom of 15-30 students, no two students are alike. They all learn in unique ways. And trying to make one lesson fit all those students is impossible.

For students who are struggling with math, hands-on learning activities with manipulatives can be a huge help. Not everyone can make sense of math on paper alone. Hands-on activities give learners a deeper understanding of how numbers work. And this understanding will continue to help them as they progress through school.

Students need to have a full understanding of basic concepts. Math builds on itself, so have a strong foundation will only help them as they gain more knowledge and advanced skills.

Also, manipulatives keep math learning exciting. Students grow bored of always using the same learning method. There's little to love about lectures and equations on a whiteboard. Math manipulatives can help you diversify your teaching style. You'll reach more students by using more methods, and that means more engagement and more learning. Awesome!

Develop problem-solving skills

Math manipulatives help students develop more complex problem-solving skills. When a student fully understands a math concept, they're better able to apply that concept to real-world problems. We want to teach students how to apply math skills across all of life, not just to next week's test.

Also, math manipulatives allow for a variety of problem solving methods. Students often interpret and think about problems in different ways. And that means they'll approach the solution differently too. Math manipulatives give your students the freedom to explore multiple paths to a solution. And if it doesn't work, it's easier for them to see why.

Build confidence in each math lesson

All of this problem solving with math manipulatives can give your students a major confidence boost! Activities with math manipulatives allow students to take the reins of their learning process. And because many math manipulatives are self-correcting, they can work individually on their problems. When they reach the right answer, all on their own, they'll feel an amazing sense of pride and accomplishment.

When students are able to understand and apply mathematical concepts, they're filled with that I did it feeling. Mastering the steps of a complicated math process gives students the confidence and encouragement to keep moving forward.

Students can then translate this deeper conceptual understanding to tests, quizzes, and standardized tests. They won't need a last minute cram session to succeed. They can stress less knowing they have the knowledge to tackle a variety of math and life problems.

Whether physical or virtual, math manipulatives can create a ripple effect that helps students throughout their entire educational journey.

Develop your student's math skills in a fun and engaging way

Math manipulatives are fantastic tools for helping your students understand and engage in their math lessons. And there are so many math manipulatives to choose from - something to fit every lesson and each student's unique needs.

Even if you don't have physical math manipulatives at your disposal, you can easily find virtual and online options that are just as helpful.

And some of them come at no-cost to teachers and schools! 👇

A no-cost, virtual math manipulative that's fun!

When you're exploring virtual options for math manipulatives, be sure to check out Prodigy Math. Prodigy's online math game is a stress-free way to help students master standards-aligned math skills. It's safe, fun and engaging, keeping your students excited and ready for their next lesson.

And with a free teacher account you can:

Set up engaging math practice in just a few clicks

Easily find out any learning gaps your students may have

Track how your students are progressing through your local or national curriculum

Sign up for free

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Math Manipulatives: How Can They Be Used to Enhance the Classroom Experience?

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Math is all around us. We can see it in nature and in our homes. We use math skills every day, even if we don't realize it. And you can help your students see math in all of its forms with fun and engaging math manipulatives.

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How To Use Manipulatives In Math for Elementary Students

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Demonstrate key math concepts in your classroom using our new virtual manipulatives!

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Mental Math - visualize adding and subtracting 10s

Common Core Connections - Grades K to 2

Add and subtract with 100.
Count to 120, starting at any number less than 120.
Given a two-digit number, mentally find 10 more or 10 less than the number.
Subtract multiples of 10.

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Interactive Hundreds Chart | Math Playground

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Interactive Hundreds Chart at Math Playground.com! Explore addition, subtraction, patterns, and more.

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.
Watch the video to learn how sharing works.

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.

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Unfortunately, not everyone loves math. Some may find it boring, others difficult! By using fun math apps in your math lessons you can make everyone discover how interesting math really is.

In this blog post, I will show you some of the most interesting math apps you can use in your digital classroom.

These math apps help teachers make their lessons more interactive and more interesting. Nowadays, there are apps for students of all ages. Hopefully, these math tools for teachers will make students like math more by teaching them to do so in a funny way or by helping them with problems.

Most of these apps are free. As a teacher, the only thing you need to do is to make sure your students download and install these apps on their smartphone or 1-to-1 device.

20+ math apps for math teachers

GeoGebra is a dynamic mathematical software program. It is made for every level of education, from beginner to expert. This app combines geometry, algebra, spreadsheets, graphs, statistics and analysis, calculus in one user-friendly package. The community of GeoGebra is expanding very fast and has millions of users worldwide. They are the number one provider of dynamic mathematical software all over the world.

Geometry Pad offers a fun way to learn geometry and practice important constructions. It's your personal assistant in learning geometry. Students can easily present their geometric constructions, take measurements, use the compass and experiment with a lot of different geometric shapes.

This app can be compared to the math tool GeoGebra above, and is designed for usage on iPads and Tablets specifically. Today's students know very well how to use an iPad or tablet; perhaps teachers can learn something from them too?

To learn math, and to take the frustration out of math and to bring more peace to a student's (and teacher's) life, use the Photomath app. This app will help you understand mathematical problems with content to improve your math skills. Every month they solve and explain more than 1 million math problems. By scanning your math problem, this app will help you instantly with solving it. You can do this by using the camera on your mobile device. After it found out what your math problem is, the app gives you a step-by-step explanation. Photomath explains you the calculation steps with animations just like a teacher would do, in real life!

The Khan Academy app uses instructional math videos, practice exercises, and has a personalized learning dashboard so you can study on your own pace in and out of the classroom. Besides mathematics, they also offer science, computer programming, (art) history, economics and more. All of the content they offer is free because they want to provide a free world-class education for anyone and anywhere. Khan Academy is a non-profit organisation that works with a community of volunteers and sponsors.

Shapes 3D is an AR app to teach geometry. You can create prisms, pyramids, solids of revolution and Platonic solids. Pick an easy figure to start with, and slowly build up until you reach the most complex figures. This tool wants to improve the teacher's capabilities and provide possibilities to show things you cannot show with physical tools or within the classroom.

This math app is designed for children aged 3 years and older. It helps students to pick up the four basic concepts of math easily: subtraction, addition, division and multiplication. To give kids a good practice you can find different exercises and games in this app.

This free math app is available on almost any device. It offers adaptive practice, workbooks, quizzes, tests and several simulations. The app is divided into different sections, so depending on your level you can handle a certain section or not. Because it's available on almost every device, it's easy to connect with anytime, anywhere.

This is a math app for students from elementary to middle school. Because it's student focused, students are more motivated to use this app. It offers more than 3000 math problems which students can try again and again until they found the right solution. It used to only be a web-based app, but nowadays you can use it on any device.

Cuethink is an innovative application for students in grades 2-12. On the one hand, it tries to engage students who are already doing well, and on the other, it supports students who have math difficulties. They want students to see challenges as opportunities with a growing mindset.

Teachers can give students math tasks that come from the so-called "problem bank". The tasks are linked to a certain level so that each student has tasks at their level. Students use a process to create and present their solutions. Cuethink is a community where students can work together in virtual groups. Thanks to these groups, they can learn from each other, both from their success as errors.

Fluidmath is one of the first to create this kind of teaching and learning tool. The tool is created for use on Pen-Centric platforms such as tablets and whiteboards. Students and teachers can create, solve, graph and animate math and physics problems in their own handwriting in a very easy way. Teachers can create dynamic instructional materials for the classrooms while students can understand concepts in math. You can see Fluidmath as a smart piece of math paper.

The founder of Rocket Math says that students get motivated by seeing their own learning and success. This is why he invented Rocket Math. This math app offers a way to teach students math while having fun, doing their best and doing it on a quick tempo. Rocket math offers a "worksheet program" and an online game. The worksheet program can be printed. The printouts are 'one minute tests' where students have to work in pairs and try to solve the sheet as fast as possible while enhancing each other's mistakes.

In the online game, students need to answer questions while using the knowledge they have with numbers in order to pass missions. The further they get into the tool the more difficult the missions become. The online game gives immediate feedback so the intervention of the teacher is not longer necessary.

The Desmos tool is great for teaching about graphs. They offer a free graphing calculator that can be used by students all over the world for free. It is even accessible for visually impaired students. In addition to the calculator, they also offer more than one hundred digital activities such as small math games.

Students can do these activities or can even create their own math ideas. These ideas can be shared with each other. Not only students can make their own activities and ideas, teachers can also do this thanks to the activity builder. This way, teachers can offer the right material to fit the needs of the students.

BookWidgets offers a wide range of different widgets or exercises you can create yourself for your students. This way, depending on the level of your students, you can make widgets that are very easy or very difficult. You choose the content you put in.

BookWidgets has different widgets, developed for the youngest students to teach the basics of mathematics. For example, there is the ' Arithmetic ' widget where students learn to add, subtract, multiply, divide, ...

You can also use the ' Active Plot ' widget. In this widget students learn how different functions look on a graph. They also see what happens to the graph when they change the parameters.

Then, there is the ' Randomness ' widget, if you want to teach your student's probability. Here you can make anything random from numbers to symbols.

To make the lesson more enjoyable, you can use the ' Bingo ' widget. There are many different ways to play bingo in a math lesson.

In the ' Crossword ' widget the questions are mathematical calculation the students have to solve.

Another fun way to learn mathematics is through a pair matching game where students have to find the right pairs, you can put in mathematical formulas or fractions and their matching images.

If students find it difficult to remember mathematical formulas, you can use the 'Flash Card' widget. Here, they can first practice and then use the practice mode to have as many flash cards as possible correctly as quickly as possible.

Finally, you can also use the 'Quiz' widget to test your students on their mathematics knowledge.

The possibilities here are endless, as you put in the content yourself.

Mathalicious looks at mathematics from a different perspective. To teach mathematics to students, they don't use old-fashioned, boring mathematic. Instead, they learn mathematics by using real-life subjects such as sports, food, economics, games, ...

For example, a subject of the lessons can be "Spinning Your Wheels: Why do tires appear to spin backwards in some car commercials?" Students will learn to convert speeds, calculate the circumference of a tire, convert between the speed of a car and the rotational speed of its wheels, ...

Thanks to these different lessons, students will not immediately see this as mathematics, although it is actually pure mathematics. This way, they will link mathematics to daily things which will make mathematics much more fun and valuable for them.

This Pokémon based online math game is created for students of grades 1 to 8. Students have to create their own avatar and can adjust as they progress through the game. Because it's a fun game, students will also play this game at home so they double their math practice time.

Teachers can see, thanks to reports, which skills students have been working on, how much time they have spent playing the game and what they found difficult. The biggest benefit of this math app is that students can play it anywhere where they have an internet connection.

This online mathematics program is designed for students from 7 until 18 years old. Students have to solve questions. When they succeed, the questions become harder. The Mathspace app remembers the students' past performance and thanks to this, it can decide the difficulty level of the questions.

Students can also give feedback on their progress. The app takes over the role of the teacher by helping the student when they encounter difficulties. The support is offered by video tutorials and tips that help the students solve the problem.

The application MathBoard is created for all school-ageing students. The younger the students, the easier the exercises. In the beginning students get exercises where they have to add and subtract. As they grow older, you go to the next step: divide and multiply.

Students can set their level themselves in the app. To prevent guessing, the answer is shown in multiple choices style and a scratchboard where students can solve the problem by hand. When it becomes too difficult, they can consult the MathBoard's Problem Solver who gives them hints and tips to solve the problem.

Virtual Nerd can be used in every math class. The only thing you need is a device with an internet connection. Virtual Nerd is made for students of all ages. On their website, you can find all kinds of videos explaining all kinds of mathematical problems. In these videos, they explain step by step how to solve these problems. In fact, this is a virtual teacher. This can be an easy tool for teachers to let students solve mathematical exercises. If students don't understand the explanation of their teacher, they may understand it through the explanation of a Virtual Nerd video.

Mangahigh is a game-based learning tool. They want schools and teachers to use more games in their classrooms, especially within math lessons. This is a platform children understand and like to use instead of the boring books they normally have to study from. When playing the game, students don't feel the pressure they have when answering in front of a full class. Instead, they build up more confidence which is very good for them.

If children really like something they have just learned in class, they will be more likely to continue this at home. So the barriers between school and home life will disappear.

With Brainly, students can ask their math (and other) questions to a community of people. This community-based app is a kind of forum where everyone comes together and helps each other with school-related questions. No matter what your question is, the people from this community will help you.

This app is made for questions from the youngest students to students from colleges. Sometimes, you need to wait a bit until you get your answer, but this is because it is a free app based on the help of a community.

CanFigureIt Geometry is a math platform where students can work on geometric proofs with the help of a guide or on their own. They can choose to work forwards or backwards. Work forward with the provided information (called 'gifts'), or work backwards from the goal on a step-by-step approach. If a student is stuck solving geometric proofs they get hints. When a student reaches the final proof, a great pop-up shows up with a congratulations message.

This platform covers a lot of different types of geometric proofs. You can try to prove simple facts about lines to difficult proofs on circles, triangles and so on. This tool allows students to learn for themselves and how to prove something rather than simply copying it from the teacher.

With Pattern Shapes, students can explore all kinds of geometric figures. Students can create things themselves by combining different shapes with each other. This way, students discover all geometric forms and geometric relations that exist such as angles, symmetry, and more. This is an ideal tool for students who are new to geometric shapes. With this tool, they learn the geometric shapes that exist in a playful way at their own pace.

The Mathway app is much more than just a calculator app. With this app, students learn how to solve a mathematical problem step-by-step, because mathematics is much more than just being able to give the correct answer. Students can easily fill in their math problem or take a picture with their camera and the app will explain step-by-step how to find the right solution. The app has a large range of topics from basic math to statistics, algebra, and so on.

It's a very useful study aid, except for the two dangers: lazy students will only look for the solution and won't learn how to solve it step-by-step, and this app is unfortunately not free.

Maths Chase is a free site where you can quickly test your times tables, additions, subtractions and more. The site is a very simple game but kids will find it a really fun way to learn arithmetics.

Conclusion

That's it! These were more than 20 math apps for math teachers you can certainly use if you want to teach your students something new or if you want them to learn something at their own pace. If my mathematics teacher had used some of these tools, I (and most of my friends) would certainly have liked mathematics much more!

Make sure to share this post with your fellow math teacher colleagues as it will help them teaching mathematices in a more interactive way.

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How these 20+ math tools can make your lessons more exciting

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Unfortunately, not everyone loves math. Some may find it boring, others difficult! By using fun math apps in your math lessons you can make everyone discover how interesting math really is. In this blog post, I will show you some of the most interesting math apps you can use in your digital classroom.

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Chapter 9: Developing Basic Fact Fluency

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Chapter 9: Developing Basic Fact Fluency

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Chapter 9_ Developing Basic Fact Fluency

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Chapter 10: Developing Whole-Number Place-Value Concepts

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Chapter 10: Developing Whole-Number Place-Value Concepts

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Chapter 10_ Developing Whole-Number Place-Value Concepts

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Chapter 11: Strategies for Addition & Subtraction Computation

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Chapter 11: Strategies for Addition & Subtraction Computation

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Chapter 11_ Developing Strategies for +_- Computation

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Chapter 12: Strategies for Multiplication & Division Computation

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Chapter 12: Strategies for Multiplication & Division Computation

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Chapter 12_ Developing Straategies for Mult. & Divi, Computation

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Free Teacher Training

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Complimentary data entry when you sign up for a school/district license. Simply send us your spreadsheet.

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Free Teacher Training If you sign-up for a school membership, we'll be happy to provide your teachers with a free introductory training via video conference. Free Data Entry Complimentary data entry when you sign up for a school/district license. Simply send us your spreadsheet.

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Picture books about math are the perfect cross-curricular tool to introduce your kids to complicated concepts and mathematical thinking. We chose some of our favorite authors and stories, both new and old, to add to your classroom library and inspire your students' mathematical minds.

Our picks for the best books about math are listed by grade level and math concept so you can easily find the right selection for your class.

1.by Mac Barnett and Jon Klassen

A sassy story about sneaky Triangle trying to play a trick on his friend Square that turns into a shape showdown. Kids will be mesmerized by this whimsically illustrated tale.

Math concepts: Shapes, relative size

Best for: Grades K-4

2. Place Value by David Adler and Edward Miller

Love the story about monkeys jumping on the bed? You'll adore this version that tells the story of the monkeys at Banana Cafe making one very large banana cupcake.

Math concept: Place value

Best for: Grades 1-5

Also by this author: Fraction Fun, Mystery Math: A First Book of Algebra, How Tall ,How Short, How Far Away, Shape Up!: Fun with Triangles and Other Polygons, Perimeter, Area and Volume: A Monster Book of Dimensions, Millions, Billions and Trillions: Understanding Big Numbers, and more.

Mr. and Mrs. Comfort are having a family reunion and need to arrange the tables so that everyone in the family can enjoy their fabulous spaghetti and meatballs. A witty tale by math guru Marilyn Burns.

Math concepts: Area and perimeter

Best for: PreK-Grade 3

Also by this author: The Greedy Triangle, Math for Smarty Pants, The $1.00 Word Riddle Book, This Book is About Time, and more.

Brave knights, a strong ruler, and a dilemma that needs to be solved using math. This book about math is great for introducing geometry concepts in a fun and accessible way.

Math concepts: radius, diameter, circumference

Best for: Grades 2-7

Also by this author: The Sir Cumference series has more than 10 books that explore math concepts from algebra and geometry to measurement and fractions.

Delightful illustrations and a dreamy young girl pondering the vastness of the universe make a very difficult concept approachable and interesting in this book about math.

Math concept: Limitless numbers and infinity

Best for: Grades K-5

Watch as this busy construction crew creates impressive structures by grouping bricks by twos, fives, and tens. Kids will love the catchy repetitive verse and charming illustrations.

Math concept: Counting in quantities

Best for: PreK-Grade 2

Ethan wakes up one morning with a strange cat on his head and the only way it will agree to get it off is if Ethan agrees to play a game of probability with it. This kooky book about math will entertain your students as it teaches a difficult concept.

Math concept: Probability

Best for: Grades 2-5

Also by this author: Fractions in Disguise: A Math Adventure

Kids will love this fun story with delightful cartoon illustrations that teach the importance of zero as a placeholder in our number system. Without Zero the Hero, we wouldn't be able to count beyond the number nine!

Math concept: Concept of zero

Best for: Grades 1-5

A beautifully illustrated book about math that shows how flips, slides, and turns can create amazing symmetrical images.

Math concept: Geometry, symmetry

Best for: Grades 1-3

A playful book with silly characters and perfect rhyming that teaches not just the basics of telling time, but the tricky concept of relative time.

Math Concept: Time

Best for: Grades 1-4

Also by this author: Cleary has written a whole series of books called Math is CATegorical, including The Mission of Addition, The Action of Subtraction, A Dollar, a Penny, How Much and How Manym, A Fraction's Goal- Parts of a Whole, and more.

Mouse and her friends want to play a game of tug-of-war but have a hard time making the sides fair. This charming story teaches the concept of balance and equality.

Math concept: Equality

Best for: Kindergarten-Grade 3

Two more rhyming tales with colorful, busy illustrations that kids will love. In each of these books about math, a boy wakes up one day and finds that things are kookily out of order. Great as a read-aloud or for independent readers who will love hunting through the detailed pictures to find the odd or even items.

Math concepts: Odd and even

Best for: Kindergarten-Grade 3

Your students will love quirky Henry and his wild imagination as he uses his dog Frank as a unit of measure. Henry thinks about real life objects and how they can take up specific spaces. This story will get kids thinking creatively about numbers and many other math concepts.

Math concepts: Problem-solving, measurement, estimation, and calculatio n.

Best for: PreK-Grade 3

Colorful fish bob, glide, and swish through the ocean until POP, a shark appears! Kids not only love hearing and seeing this story, it will get them interested in learning and creating their own patterns.

Math concepts: Patterns and variations of pattern types

Best for: Kindergarten-Grade 2

16. A Remainder of One by Eleanor Pinczes and Bonnie MacKain

The 25th Beetle Army Corp is trying to solve the problem of how to divide their troops into equal groups for the parade march without leaving any stragglers. A fun, rhyming book about math that will get kids thinking about how numbers are put together and pulled apart.

Math concept: Grouping, division

Best for: Preschool-Grade 3
What picture books have you used in your math lessons? Come and share in our WeAreTeachers HELPLINE group on Facebook. Plus, Why It's Important to Honor All Math Strategies.

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16 Picture Books About Math to Inspire Curious Kids - We Are Teachers

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Picture books about math are the perfect cross-curricular tool to introduce your kids to complicated concepts and mathematical thinking. We chose some of our favorite authors and stories, both new and old, to add to your classroom library and inspire your students' mathematical minds.

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Erikson Institute Early Math Collaborative

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One sure fire way to warm up children's attitude towards mathematics is to build math activities and lessons out of the books they all love to listen to and read.

Children explore the Big Ideas about foundational mathematical concepts as they interact with math in everyday life. When reading and listening to books, mathematical situations come to life in new and fanciful ways. And, we're not just talking about books that are explicitly mathematical, since math is often central to the problem and resolution of our favorite stories. So many questions that drive children, such as "Is it fair?", "Who is bigger/faster/stronger?", "What will happen next?" are shared by the characters they read about.

But, don't forget to take the math out of the book! Play with the stories. Act them out and innovate. Ask the what-if questions. Create your own version. This is where mathematical thinking gets creative.

We have collected many lists and activity ideas related to some of the best children's books we know of, all infused with math concepts. We've outlined links below, organized under the early math topics of our website. Use books to bring math to life today!

Where's the Math? Books, Games, and Routines to Spark Children's Thinking

Make math learning both meaningful and fun by building on children's natural curiosity to help them grow into confident problem solvers and investigators of math concepts.
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Best Childrens Books for Early Math Learning | Early Math Collaborative

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One sure fire way to warm up children's attitude towards mathematics is to build math activities and lessons out of the books they all love to listen to and read. Children explore the Big Ideas about foundational mathematical concepts as they interact with math in everyday life.

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6th Grade Literature List

Related Links

See advice from Everyday Mathematics teachers on working with parents, using technology in the classroom, pacing, and more.

The UChicago STEM Education offers strategic planning services for schools that want to strengthen their Pre-K-6 mathematics programs.

McGraw-Hill Education's website features supplemental materials, games, assessment and planning tools, technical support, and more.

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6th Grade Literature List - Everyday Mathematics

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See advice from Everyday Mathematics teachers on working with parents, using technology in the classroom, pacing, and more. The UChicago STEM Education offers strategic planning services for schools that want to strengthen their Pre-K-6 mathematics programs. McGraw-Hill Education's website features supplemental materials, games, assessment and planning tools, technical support, and more.

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6th Grade Literature List

Related Links

See advice from Everyday Mathematics teachers on working with parents, using technology in the classroom, pacing, and more.

The UChicago STEM Education offers strategic planning services for schools that want to strengthen their Pre-K-6 mathematics programs.

McGraw-Hill Education's website features supplemental materials, games, assessment and planning tools, technical support, and more.

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6th Grade Literature List - Everyday Mathematics

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See advice from Everyday Mathematics teachers on working with parents, using technology in the classroom, pacing, and more. The UChicago STEM Education offers strategic planning services for schools that want to strengthen their Pre-K-6 mathematics programs. McGraw-Hill Education's website features supplemental materials, games, assessment and planning tools, technical support, and more.

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Children's books often contain valuable lessons that can help young students begin to think like mathematicians.

monkeybusinessimages / iStock

Many students and teachers view math as a subject for numbers and computation, instead of one that benefits from discussion and interpretation. Based on our experience as children's literature and mathematics teacher-educators, we've found that providing the context to mathematical problems through literature supports students' learning-children's books can be used to integrate math and literacy and to provide context for math.

For example, the books When Sophie Thinks She Can't, a story about a little girl who discovers the importance of struggle, and , a story about the history of the color blue, are great examples of books that integrate mathematical thinking into storytelling.

USING LITERATURE TO SUPPORT MATHEMATICAL THINKING

1. Selecting books to support mathematical thinking: First, find books that have a worthy math problem to discuss or a story to connect to a math problem in your curriculum. Context is one of the most important parts of choosing a book. If the problem is not challenging or accessible enough for students, think about ways to revise the problem. We advocate for low-threshold, high-ceiling problems, which allow access for all students with their previous knowledge and build more knowledge and challenge all students.

When Sophie Thinks She Can't, by Molly Bang, poses a worthwhile math task, asking Sophie and her friends to form as many different rectangles with 12 tiles as they can. Because she can't figure out the solution to the problem, her teacher says, "You can't figure it out yet," which encourages her to keep trying. This book is a perfect example of a mathematical problem that is easily seen in the context of the story and encourages the notion of the power of yet as a problem-solving technique.

In many books, the math problem is not as easily seen. As a teacher, you may not think that most books have some type of mathematical focus within them, but upon close examination, you can discover that many of them can be part of your math curriculum. For example, Blue, written by Nana Ekua Brew-Hammond and illustrated by Daniel Minter, can be connected to an examination of patterns and shapes in textile designs. Working with different shades of blue papers, students could be asked to create patterns by folding the papers and study the patterns that they created mathematically. Studying the relationships between the number of folds and partitions in the final shape would take this task to a higher ceiling.

1. Whose story is this? How might it relate to students' mathematical stories?

2. Through what point of view is this story told?

3. What mathematical concepts shape the story?

4. How are mathematical tasks and understandings established?

2. Critically examining literature: Teachers should spend time critically examining their texts; as Maria José Botelho and Masha Kabakow Rudman advise, teachers should read beyond the words in the text and make connections between local, sociopolitical, personal, and mathematical content. Teachers can ask questions like the following:

The first two questions help students relate to the issues, topics, and questions they may have about the story. The last two questions help students think like a mathematician. For example, Blue is told from a historical perspective and touches on classism throughout time. By examining the book closely, teachers can pull out the patterns in the illustrations and ask students to create mathematical patterns on their own-this is shown throughout this book and particularly on the endpapers.

This picture book provides so many historical background understandings of who cultivated the materials to make the color blue and who had access to textiles made from blue dyes. By leading the students through the questions above, teachers will help them begin to see math in all things and begin to think like mathematicians.

3. Designing a lesson to integrate literacy and mathematics: It's beneficial to think ahead about questions for both disciplines that can help students extend and guide their thinking; in previous work, we developed a full lesson with content-specific questions. Asking students to analyze or interpret their peers' mathematical work and making connections to the text would also extend students' thinking. After coming up with their own set of rectangles with 12 tiles, students can compare their rectangles with the rectangles created by their peers or by Sophie and her friends. This can lead to a discussion of how similar or different all of the rectangles are.

Circle Dogs, written by Kevin Henkes, illustrated by Dan Yaccarino. Exploring shapes through a day in the life of two dachshunds. Perfect read-aloud for a geometry unit! Pair this with Hot Dog (below) to help your students make text-to-text connections.

Bean Thirteen, written and illustrated by Matthew McElligott. This book explores the concept of division with the tribulations that the two characters encounter as they work their way through a problem. A great way to engage children in discussions of trying to work through problem-solving.

, written and illustrated by DougSalati. This 2023 Caldecott-winning book explores a day in the life of a dog as readers are taken on an adventure throughout an urban setting to find solitude for the owner and the dog in a noisy city. A wonderful way to look at mapping.

A Land of Books: Dreams of Young Mexihcah Word Painters, written and illustrated by Duncan Tonatiuh. This story focuses on the creation of books by the Indigenous people of Mexico, exploring their knowledge, culture, and history. Readers can dig deeply into the illustrations and see how precise measurement, geometry, and special relationships of mathematical concepts enhance the telling of this story.

We hope these ideas and books will support you as you provide meaningful conversations with your students by integrating mathematical understandings through children's literature.

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Using Stories to Support Mathematical Thinking in Young Students

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Children's books often contain valuable math lessons that can help young students begin to think like mathematicians.

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Chapter 14: Developing Fraction Concepts

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Chapter 14: Developing Fraction Concepts

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Chapter 14_ Developing Fraction Concepts

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Chapter 15: Developing Fraction Operations

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Chapter 15: Developing Fraction Operations

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Chapter 15_ Developing Fraction Operations

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Integrating ELD Standards into K12 Math and Science

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There are a lot of misconceptions out there about Guided Math. I've seen many things labeled "Guided Math" that truly have me scratching my head. With so much (mis)information out there, I thought I'd take some time to help break down what Guided Math *IS* and what Guided Math *IS NOT*. Let's dive in!

Guided Math IS NOT:

Guided Math is NOT teaching the same lesson to students in small groups.

While teaching content in small groups is important and allows you more face time with students, Guided Math is NOT teaching the same lesson multiple times to different groups.

Why not? Guided Math focuses on what Vygotsky called a student's Zone of Proximal Development, which is individualized to each student. The model of teaching the same content to different groups doesn't allow the teachers to meet students at their instructional level. Instead, the teacher is still "teaching to the middle," but in a smaller group setting.

Guided Math is NOT students rotating through the same centers

Centers are a great way to give students time to practice skills independently! Especially while you are also trying to pull those small math groups. However, Guided Math is not setting up weekly centers (or something similar) where all students complete the same centers throughout the week.

Why not? Guided Math focuses on differentiating student work, and ensuring that students are working on skills at their instructional level. Just like with reading, the most meaningful gains come when students are pushed *just slightly* above their independent level. This can't happen when we aren't strategic about the centers we choose, and instead have everyone rotate through the same activities.

Guided Math is NOT Teaching a Mini Lesson and checking in with students individually

Providing common math experiences is awesome, so mini-lessons and Number Talks are very much a part of Guided Math. However, Guided Math is not teaching mini-lessons and then following up with individual students as they engage in independent work/centers.

Why not? The mini-lesson is a way to establish a community of mathematicians, and allow students to hear from one another, practice mathematical discourse, and engage in common experiences around math topics. But, the mini-lesson is not the only direct instruction that takes place in a Guided Math model.

Guided Math *has* to include targeted small group instruction beyond the mini-lesson to ensure that we are meeting students where they are in their understanding of a concept.

SO... What IS Guided Math?

Chances are, if you are familiar with Guided Reading, you have a great foundation for understanding Guided Math. Why? Both models follow very similar blueprints. They are both based in meeting students at their instructional level via targeted small group instruction, followed up with practicing those same skills at independent/partner/small group centers.

You might be asking, okay... then what does Guided Math look like? Here are some key components:

Use Assessments to Find Instructional Levels:

Just like with reading, Guided Math utilizes assessments (formal and informal) to guide instruction (hence "GUIDED" math). Those assessments guide targeted small group instruction as well as the selection of centers and independent work students engage with while not working directly with a teacher.

Create Flexible Learning Groups

Flexible grouping is also a key component of Guided Math. It uses assessments to regularly switch groupings in order to work within a student's zone of proximal development. We know student growth is not linear. This is why groupings are constantly changing as students gain and make meaning of new knowledge that pushes them forward.

Teach Targeted Lessons to Small Groups

Within the flexible groups mentioned above, teachers are working to push students at their instructional level within a concept. Each group should be engaging in a targeted lesson that is meant to bring students forward from where they are in their understanding.

Students Work Independently at Targeted Centers

While the teacher is pulling small groups, the rest of the class is engaged in targeted center work. That means that each group will be working through different centers that extend the learning they worked on within their small group time.

I know that sounds like a lot of extra work, but it is actually much less! You will most often be using the same materials for each group, just at different times to reflect their current understanding.

Built on Strong Structures and Routines

Just like the Guided Reading model, Guided Math is only made possible by creating strong structures and routines. This means that before any instruction truly begins, students need to have internalized the expectations or how to engage in learning within a whole group, small group, and independent center, AND how to transition between those settings.

It takes front loading and consistency, but it is SO WORTH the effort! Why? Because when you are able to work with a small group of learners and provide powerful lessons that push their understanding ALL WHILE the rest of your group is engaged in meaningful, independent learning, there is NOTHING BETTER!

Are you looking to learn EVEN more about making Guided Math work in your classroom?

Be sure to get on the waitlist for my online professional development course called Guided Math Academy.

Guided Math Academy is an interactive six-week course for teachers who want to learn how to teach ALL their students WITHOUT adding hours of prep! It includes a step-by-step blueprint, Q&A sessions, and incredible bonuses to help you AND your students.

This course opens every June. Be sure to get on the waitlist so you don't miss out!

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There are a lot of misconceptions about Guided Math. I'm taking some time to break down what Guided Math *IS* and what Guided Math *IS NOT*.

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Guided Math Small Groups in First Grade - Happy Hearts in 1st

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Guided Math Small Groups have been such a game changer in my 1st grade classroom! I LOVE the opportunity for differentiation and all of the light bulb moments! 💡💗 Small…

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Guided Math with Dr. Nicki - What is Guided Math?

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Update 2022: For a recent article on how to talk about math, click here.
Think about the questions that you ask in your math classroom. Can they be answered with a simple "yes" or "no," or do they open a door for students to really share their knowledge in a way that highlights their true understanding and uncovers their misunderstandings? Asking better questions can open new doors for students, helping to promote mathematical thinking and encouraging classroom discourse. Such questions help students:

Work together to make sense of mathematics.

Rely more on themselves to determine whether something is mathematically correct.

Learn to reason mathematically.

Evaluate their own processes and engage in productive peer interaction.

Discover and seek help with problems in their comprehension.

Learn to conjecture, invent and solve problems.

Learn to connect mathematics, its ideas and its applications.

Focus on the mathematical skills embedded within activities.

Below are 100 questions from mathematics expert Dr. Gladis Kersaint to help you address these core areas and promote mathematical thinking and discourse in the classroom. Want these questions visible in your classroom? Curriculum Associates has released an infographic that you can print and have at your desk or in your class for quick access! Here's a preview:

For more, please download Dr. Kersaint's new whitepaper:Orchestrating Mathematical Discourse to Enhance Student Learning.
Dr. Gladis Kersaint
Help students work together to make sense of mathematics

1. What strategy did you use?

2. Do you agree?

3. Do you disagree?

4. Would you ask the rest of the class that question?

5. Could you share your method with the class?

6. What part of what he said do you understand?

7. Would someone like to share ___?

8. Can you convince the rest of us that that makes sense?

9. What do others think about what [student] said?

10. Can someone retell or restate [student]'s explanation?

11. Did you work together? In what way?

12. Would anyone like to add to this?

13. Have you discussed this with your group? With others?

14. Did anyone get a different answer?

15. Where would you go for help?

16. Did everybody get a fair chance to talk, to use the manipulatives, or to be recorded?

17. How could you help another student without telling the answer?

18. How would you explain ___ to someone who missed class today?

Refer questions raised by students back to the class.

Help students rely more on themselves to determine whether something is mathematically correct

1. Is this a reasonable answer?

2. Does that make sense?

3. Why do you think that? Why is that true?

4. Can you draw a picture or make a model to show that?

5. How did you reach that conclusion?

6. Does anyone want to revise his or her answer?

7. How were you sure your answer was right?

Help students learn to reason mathematically

1. How did you begin to think about this problem?

2. What is another way you could solve this problem?

3. How could you prove that?

4. Can you explain how your answer is different from or the same as [student]'s?

5. Let's see if we can break it down. What would the parts be?

6. Can you explain this part more specifically?

7. Does that always work?

8. Is that true for all cases?

9. How did you organize your information? Your thinking?

Help students evaluate their own processes and engage in productive peer interaction

1. What do you need to do next?

2. What have you accomplished?

3. What are your strengths and weaknesses?

4. Was your group participation appropriate and helpful?

Help students with problem comprehension

1. What is this problem about? What can you tell me about it?

2. Do you need to define or set limits for the problem?

3. How would you interpret that?

4. Would you please reword that in simpler terms?

5. Is there something that can be eliminated or that is missing?

6. Would you please explain that in your own words?

7. What assumptions do you have to make?

8. What do you know about this part?

9. Which words were most important? Why?

Help students learn to conjecture, invent and solve problems

1. What would happen if ___? What if not?

2. Do you see a pattern?

3. What are some possibilities here?

4. Where could you find the information you need?

5. How would you check your steps or your answer?

6. What did not work?

7. How is your solution method the same as or different from [student]'s?

8. Other than retracing your steps, how can you determine if your answers are appropriate?

9. What decision do you think he or she should make?

10. How did you organize the information? Do you have a record?

11. How could you solve this using (tables, trees, lists, diagrams, etc.)?

12. What have you tried? What steps did you take?

13. How would it look if you used these materials?

14. How would you draw a diagram or make a sketch to solve the problem?

15. Is there another possible answer? If so, explain.

16. How would you research that?

17. Is there anything you've overlooked?

18. How did you think about the problem?

19. What was your estimate or prediction?

20. How confident are you in your answer?

21. What else would you like to know?

22. What do you think comes next?

23. Is the solution reasonable, considering the context?

24. Did you have a system? Explain it.

25. Did you have a strategy? Explain it.

26. Did you have a design? Explain it.

Help students learn to connect mathematics, its ideas and its application

1. What is the relationship of this to that?

2. Have we ever solved a problem like this before?

3. What uses of mathematics did you find in the newspaper last night?

4. What is the same?

5. What is different?

6. Did you use skills or build on concepts that were not necessarily mathematical?

7. Which skills or concepts did you use?

8. What ideas have we explored before that were useful in solving this problem?

9. Is there a pattern?

10. Where else would this strategy be useful?

11. How does this relate to ___?

12. Is there a general rule?

13. Is there a real-life situation where this could be used?

14. How would your method work with other problems?

15. What other problem does this seem to lead to?

Help students persevere

1. Have you tried making a guess?

2. What else have you tried?

3. Would another recording method work as well or better?

4. Is there another way to (draw, explain, say) that?

5. Give me another related problem. Is there an easier problem?

6. How would you explain what you know right now?

Help students focus on the mathematics from activities

1. What was one thing you learned (or two, or more)?

2. Where would this problem fit on our mathematics chart?

3. How many kinds of mathematics were used in this investigation?

4. What were the mathematical ideas in this problem?

5. What is the mathematically different about these two situations?

6. What are the variables in this problem? What stays constant?

Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. In the next and final blog in this series, we will dive into the specific strategies that teachers can use to foster meaningful conversations about what students are thinking, doing and learning.

This blog is part of a three-post series on the importance of mathematical discourse from Curriculum Associates and Dr. Gladis Kersaint, the author of the recently published whitepaper Orchestrating Mathematical Discourse to Enhance Student Learning. Download your free copy .
For more on mathematical discourse and Curriculum Associates, check out:
Dr. Gladis Kersaint is a Professor of Mathematics Education at the University of Connecticut. Stay in-the-know with all things EdTech and innovations in learning by signing up to receive the weekly Smart Update. This post includes mentions of a Getting Smart partner. For a full list of partners, affiliate organizations and all other disclosures please see our Partner page

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Talking Math: 100 Questions That Help Promote Mathematical Discourse

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By Gladis Kersaint - 100 questions from a mathematics expert to help promote mathematical thinking and discourse in the classroom.

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Administrators can create custom assessment windows to measure students' knowledge levels, better informing high-level decision-making. Plus, teachers can get reliable insights between those assessment windows in just 10 minutes a week.

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Administrators can create custom assessment windows to measure students' knowledge levels, better informing high-level decision-making. Plus, teachers can get reliable insights between those assessment windows in just 10 minutes a week.

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Why do higher order thinking questions matter for science? In education, we know that want to move away from questions and instruction that simply focuses on correct answers. Instead, we want to develop critical thinkers by posing a variety of questions that take topics of study and learning to a deeper level. Especially if you use the Next Generation Science Standards, you'll want to emphasize the higher levels of bloom's taxonomy by prompting students to engage in arguments, construct explanations, and analyze and interpret data.

Of course, I think it's important to point out that there is a place for all types of questions when it comes to teach key concepts and new ideas. Students must be able to remember vocabulary and understand the main ideas before they are able to move on to more provocative questions, especially for an english language learner. When it comes to classroom instruction, lower level questions should not be considered a bad thing and may be useful for student engagement. Personally, I've always found, for better or worse, that the majority of my students are a lot more engaged when they know how to get the right answer. There are, naturally, some students who thrive on difficult questions. Differentiation has always been the most difficult part of teaching science for me, so perhaps if you find a way to strike that perfect balance, you can let me know!

Why ask questions?

To extend pupils' thinking from the concrete to the more abstract.

To check prior knowledge and understanding of key ideas.

To clarify misconceptions.

To challenge pupils to apply the key ideas to a range of observations and findings.

To lead pupils through a sequence which establishes their understanding of certain enquiries.

To promote the use of thinking skills, for example, reasoning, evaluation.

To develop a systematic approach to scientific enquiry and problem solving.

To promote pupils' thinking about what they have learned and how they have learned.

Source: https://stemedhub.org/resources/760/download/

Strategies for for improving questioning

Plan your Questions Ahead of Time

Great teaching rarely happens without any prior preparation! It's important to take time before your lesson to think about the questions you plan to ask. I like to incorporate my questions into my presentation if I'm using one, or make a short list that I can keep on my clipboard while I teach.

Wait time

Are you constantly getting the same kids raising their hands in class? Giving wait time is an often neglected solution to this problem! Not every student has an answer prepared instantly, so giving wait time can bring out more from students that you may have underestimated.

Tell students Big Questions in Advance

I like to use higher order questions on my assessments because I think a good question can really get to the heart of what students know and understand. If the question isn't Google-able, there really isn't any harm in giving it to the students beforehand so that they can do their research and plan their response. I also find that this seriously helps alleviate testing anxiety, which affects some of my students pretty severely.

Provide Talking Stems

It's very possible that your students don't actually feel equipped to answer higher order thinking questions. I know my own students have often been traumatized into thinking they can't really speak unless they "know the answer." As a result, classroom discussion suffers and kids with really good ideas rarely speak up.

To remedy this problem, consider providing talking stems. Accountable Talk is a great way to give kids the tools they need to participate in class discussions. I've known teachers who post these talking stems on desks or on the wall. If your students struggle with getting off track, you can also make it a requirement that students must begin their comment with one of the stems.
Take a look at some of the talking stems for science below:

"I can relate to _____ because..."

"I disagree with _____ because... "

"To recap, .... and the consensus is..."

"I agree with what was said about _____ because..."

"I think..."

"Can you clarify what you mean when you said...?"

"What's your evidence to prove that...?"

"I think that was a great point because in the text it says..."

"I found it interesting when..."

"I would like to add..."

"I understand your opinion, but would like to add a different viewpoint..."

"When you mentioned ____it made me think about..."

What is a higher level thinking question?

Higher-order questions cannot be answered by repeating prior learning verbatim. These questions foster critical thinking skills and often require students to look at topics in different ways. They may require students to incorporate new information with a big idea that they've already studied and determine the best way to integrate the two. Often, these questions don't have a right or wrong answer but require students to provide evidence for their claims. They also refer to those closer to the top of the Bloom's taxonomy hierarchy.Good questions are critical for helping students learn and deeply understand the wonders of science.

On the other hand, lower order questions, refer to those near the bottom of the hierarchy. These questions typically have a right or wrong answer. They require students to match or identify vocabulary. Students can often answer these questions by copying content from their notes, textbook, or by using Google.

Of course, both types of question have a place in educational settings and classroom discussions!

What are so me examples of higher order thinking questions?

Remember, rigor does not mean more - it means better. We really want to challenge our students to think critically in science. You may also notice that the majority of the NGSS standards really push students to those levels of higher order thinking. Most teachers would agree, though, that you've got to get the lower levels down pat before you can move up.

Take a look at the table below to read a little more about the various levels of Bloom's taxonomy, plus some examples that you could use in your science class.

Remember

Recall basic facts and basic concepts

define, duplicate, list, repeat, state

Which organelle is responsible for producing proteins?

What is the chemical formula for photosynthesis?

List the steps of the engineering design process.

Can you name...?

What is...?

Understand

Explain ideas of concepts

classify, describe, explain, identify, select

Classify the following list of terms as biotic or abiotic.

How can you use the idea of forces to explain why a boat floats in water?

Pick the one that doesn't belong: elastic energy, gravitational energy, mechanical energy.

Can you explain why...?

Can you write in your own words?

Write a brief outline.

What was the main idea of...?

Apply

Use information in new situations

execute, solve, use, demonstrate, interpret, sketch

Using the graph, what can you conclude about the connection between ticket price and concert attendance?

Do you know of another instance where...?

Ilustrate...

Analyze

Draw connections among ideas

differentiate, compare, contrast, examine, organize

What is the difference between an independent and dependent variable?

Analyze and interpret data on the properties of substances before and after the substances interact to determine
if a chemical reaction has occurred.

If ... happened, then what would be the outcome?

How is... similar to...?

Can you distinguish between...?

What was the problem with...?

Why did... changes occur?

Evaluate

Justify a stand or decision

argue, defend, support critique,

Support an argument that differences in the apparent brightness of the sun compared to other stars is due to their
relative distances from Earth.

Use argument supported by evidence for how the body is a system of interacting subsystems composed of
groups of cells.

Construct an argument supported by empirical evidence that changes to physical or biological components of an
ecosystem affect populations.

Evaluate competing design solutions for maintaining biodiversity and ecosystem services.

Create

Produce new or original work

design, assemble, construct, develop, investigate

Develop a model using an example to describe ways the geosphere, biosphere, hydrosphere, and/or atmosphere
interact.

Undertake a design project to construct, test, and modify a device that either releases or absorbs thermal energy
by chemical processes.

Plan an investigation to determine the relationships among the energy transferred, the type of matter, the mass,
and the change in the average kinetic energy of the particles as measured by the temperature of the sample.

What would happen if...?

Can you see a possible solution to...?

Get my Question Stems Resource for Any Core Subject!

If you're interested in getting a little more help in writing solid higher order thinking questions for science or any other core subject, take a look at the question stems resources listed below.

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Higher Order Thinking Questions for Science - Laney Lee

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Need examples and suggestions for higher order thinking questions for science? Get useful resources and ideas after the jump!

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To learn more about the standards, click here. To learn more about the NGSS Appendices,click here.

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Over 10 percent of students in the United States-more than 4.8 million kids-are English language learners (ELLs), and the number is on the rise. Though these students do not learn differently than their native-English-speaking peers, they do have particular educational needs.

To learn about these needs-and best practices for addressing them-I interviewed a range of educators and observers, including Larry Ferlazzo, an educator and author of The ESL/ELL Teacher's Survival Guide; longtime teachers of English as a second language (ESL) Emily Francis and Tan Huynh; and the journalist Helen Thorpe, who spent a year observing a teacher who works with ELLs.

The group emphasized that the strategies listed here, which include both big-picture mindsets and nitty-gritty teaching tactics, can be incorporated into all classrooms, benefiting both native English speakers and ELLs.

1. Cultivate Relationships and Be Culturally Responsive

No surprise here. A successful classroom, our educators agreed, is one in which students feel known, appreciated, and comfortable taking emotional and intellectual risks. That requires intentional planning and consistent messaging by the teacher.

Emily Francis, an ESL teacher in Concord, North Carolina, makes clear that she wants her students to "embrace their culture and their language as a foundation of who they are" and to consider their acquisition of a new culture and language "not as subtractive, but as additive." To help support students who may never have attended school before or may be coping with migration-based trauma, Francis emphasizes that little things make a big difference. "The first thing that I need to think about is, how is my student feeling in my classroom?" she says. "Are they sitting next to a buddy they can ask a question in their home language? Do they feel comfortable tapping me on the shoulder if they have to go to the bathroom?"

But creating a supportive environment is also about cultivating an appreciation of diversity-it's critical that both the curriculum and the classroom environment honor and reflect the lives of the students. For instance, Francis makes certain that her classroom library is reflective of her students' diverse backgrounds and identities.

ESL teacher Katie Toppel, who teaches near Portland, Oregon, champions the value of home visits to get to know students' families and takes care to incorporate the particulars of students' lives, such as pets' names and favorite sports, into lessons. Students are most engaged when they feel a personal connection to a lesson or unit, a connection that's created in part by a teacher's investment in culturally competent relationships.

2. Teach Language Skills Across the Curriculum

English language learners should not be learning the fundamentals of English in isolation; they should be applying their developing language skills to rich academic content in all subjects.

"It is key that in mainstream classrooms, teachers understand their role as language teachers," says Valentina Gonzalez, a district leader in Katy, Texas, suggesting that all teachers should be aware of the specialized idiom they work within. "If we teach math, then we teach the language of mathematics. If we teach science, then we teach the language of science." Math teachers, in other words, should take the time to teach the unfamiliar vocabulary of mathematics- add, subtract, calculator, solve -concurrently with the teaching of math skills.

3. Emphasize Productive Language

The educators I interviewed agreed that productive language skills-hard-to-master dimensions of language fluency like speaking and writing-should be front and center from day one, even if students feel hesitant about them.

Beginning ELLs often develop receptive language skills like listening and reading first. Educators who are unaware of the typical path to fluency may believe that students who can follow verbal or written directions will be able to produce oral or written language, but that's usually not the case.

To support reluctant speakers, Tan Huynh, an educator who blogs at Empowering ELLs, suggests using sentence frames. "For example, when a science teacher wants ELLs to produce a hypothesis, they might offer the sentence, 'If _____ was added, then _____ because _____.' This sentence frame provides clues that empower ELLs to sound and think like scientists," Huynh says.

Andrea Honigsfeld, a professor of education at Molloy College in Rockville, New York, suggests that all lessons touch on every letter of the acronym SWIRL, which stands for Speak, Write, Interact, Read, Listen. The approach intentionally privileges productive language skills "from the beginning," she says.

4. Speak Slowly-and Increase Your Wait Time

Easier said than done. Still, many of the teachers I spoke to said this simple change is vital. You can record yourself speaking in class to measure your cadence, and adjust.

"Adding in an extra three to five seconds after we pose a question offers all students time to think," explains Gonzalez. "However, for English learners, it also gives time to translate, process their thinking, translate back into English, and develop the courage to answer. If we call on students too quickly, many of our students will stop thinking about the answers-or trying to answer at all."

That means undoing some of our own habits, according to Larry Ferlazzo, a high school ESL teacher in Sacramento, California. "Researchers have found that typically most teachers give one to two seconds between asking a question and expecting a student response," he says. "The same researchers have shown that if you wait three to five seconds, the quality of responses is astronomically greater."

5. Differentiate-and Use Multiple Modalities

All kids learn better when they engage with material in multiple ways: Lessons that involve writing, speaking, drawing, and listening, for example, give students four opportunities to deepen their understanding of the work. For ELLs, those additional engagements also provide a little breathing room so they can work through the language barrier.

Helen Thorpe, a journalist who spent a year observing Eddie Williams's classroom in Denver in order to write a book about immigrants called The Newcomers, marveled at Williams's method of differentiation. "Eddie would enunciate very clearly in English multiple times, he would write it on the board, and he would go to the projector to give a visual. And, finally, he'd be asking the kids to verbalize aspects of the lesson themselves," she says, tracing Williams's efforts to move his students from receptive to productive language fluency.

Toppel uses a strategy called QSSSA to scaffold classroom discussions with ELLs. The letters stand for Question (the teacher poses a question, then gives ample think time); Signal (a designated motion like a thumbs-up, a finger on the nose-something that lets students indicate that they're ready to answer); Stem (the teacher provides a sentence starter for the question-for example, for "What is your favorite place to read a book?" the stem might be, "My favorite place to read a book is _____"); Share; and Assess.

6. Incorporate Students' Native Languages-and Don't Be Afraid of Technology

Bilingualism is the goal, of course, not replacement.

Ferlazzo uses a strategy called "preview, view, review," which leverages a student's native language skills as a foundation for learning the new language. He introduces a topic and encourages students to preview it in materials in their home language (often using multilingual videos and other online resources); he then teaches the topic in English, and then has students review the information in their home language.

Ferlazzo also encourages teachers to embrace technologies that students find helpful, like Google Translate. He stresses that the app can be a useful way for students to translate words quickly-but warns that it can quickly "become a dependency" if it's used as more than a handy dictionary.

Thorpe agrees that Translate can take pressure off teachers, who sometimes find that they "just stand at the front of the room and say things over and over. And if the kids didn't know what those words are, they're out of luck." Translation technology means you can say it once (or twice) and let the students use the technology to translate key words and phrases.

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6 Essential Strategies for Teaching English Language Learners

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We interviewed educators with decades of experience in teaching ELLs and tapped a network of experts and observers to find the strategies that work.

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Support Reading Comprehension with Activity Worksheets for Organizing Information

Science A-Z Graphic Organizers support content-area learning and reading comprehension as students read fiction and nonfiction science texts. The Graphic Organizer collections are arranged by primary (grades K-2) and intermediate (grades 3-6) as well as by nonfiction and fiction. Select appropriate sheets for informational texts or literary texts to support students who are learning at different developmental levels and reading for various purposes.

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Science A-Z Graphic Organizers support content-area learning and reading comprehension as students read fiction and nonfiction science texts. The Graphic Organizer collections are arranged by primary (grades K-2) and intermediate (grades 3-6) as well as by nonfiction and fiction.

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Find Science resources | TPT

Importance of learning science

Learning science is particularly important for students as it equips them with the essential knowledge and skills to make sense of the world around them. Science education is not just about memorizing facts. It fosters critical thinking, inquiry, and problem-solving abilities. These skills are not only useful in school but also in everyday life beyond the classroom.

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Learning science offers many benefits to students both in school and beyond. In fact, it helps students:

Question the world around them and seek evidence-based explanations

Cultivate their ability to observe, experiment, and analyze data

Nurture a sense of curiosity, awe, and wonder about the natural world

Understand natural phenomena and the laws that govern the universe

With a solid foundation in science, teachers can help empower students to think critically, solve problems, make informed decisions, and engage with the world around them in meaningful ways.

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One of the best parts about science class is all of the possibilities for hands-on, collaborative learning! Think: labs, experiments, inquiry-based activities, and design challenges. These activities encourage students to explore scientific concepts by observing, asking questions, and conducting investigations. Additionally, with small group centers, students can engage in group experiments, discussions, and projects that require them to collaborate, share ideas, and solve problems together.

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There are many different types of science resources sold by Sellers on TPT. Some popular science lessons include biology, earth sciences, chemistry, environment, engineering, and anatomy.

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Students learn best when they're engaged! Sprinkle a little fun into your science lessons and make them feel like real scientists by investigating everyday objects, exploring the world outside the classroom, and doing hands-on and collaborative experiments and design challenges.

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Browse free science graphic organizers on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources.

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Printable graph paper mystery pictures. Plot the points on the graph paper and connect to reveal a special picture.