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You've just presented a number talk to your students. Before you know it they are jumping up and down while wildly waving their hands and collectively squealing "I know! I know! I know! Pick me! Pick Me!" While you're thrilled that your students are so engaged, this may be a little more chaotic than you'd like. The answer? Silent hand signals for your daily number talk routine.

What are number talk silent hand signals?

Number talk silent hand signals give students a way to communicate without making a single peep.

Isn't that music to your ears?

Best of all, silent hand signals are easy to incorporate into your daily number talk routine.

Here are six silent hand signals that we use in our classrooms.

6 Silent Signals for Number Talks

1. I Have a Solution

After a problem is presented, students put a thumb up to show they have figured out a solution to the problem. Because this is done quietly and privately (not a hand in the air), it gives other students the processing time they need to solve the problem without pressure to solve it quickly.

Additionally, this hand signal helps with the pacing of the lesson. As soon as you see that most (or all) students have a solution to the problem, whole group sharing time can begin.

2. I Have Another Solution (or I Solved the Problem Using Another Strategy)

Fast finishers...you know the ones....the kids that have solved the problem before you've finished presenting it.

Well, this will stop them in their tracks.

These students need a challenge, and they'll get one.

Share with students that when they have an answer, put a thumb up and keep thinking.

Many number talks have multiple answers. Have students put an additional finger up for each additional answer they come up with.

For number talks with one answer, challenge students to solve the problem using a variety of strategies. Have them put up an additional finger for each strategy used to solve the problem.

You high-flyers will love these challenges.

Pinky promise!

3. We Have Both Shared

After most (or all) students have solved the number talk, it's time for them to share their thinking with a partner. It's important that each partner has adequate time to engage in mathematical conversations.

After each student in the partner pair has shared, the students face forward and place their fists in front of them to signal they are done sharing and are ready for the whole group share time.

4. Me Too (or I Agree)

Does a chorus of "Me too" make you want to crawl under your desk and cover your ears? We hear ya!

During the whole group share time, students listen to the answers and reasoning of their peers. If they got the same answer and/or used the same strategy, they show the "me too" signal by making a fist with their pinky and thumb sticking out, then moving their hand back and forth.

This strategy encourages students to listen attentively to their peers as they share solutions and strategies they used.

Additionally, it reduces the disappointment felt by some students because their answer was already shared.

5. I Solved the Problem Differently or I Have A Different Answer

During number talks, there will be many times when students either have a different answer than the one shared, or they used a different strategy to solve the problem.

The number talk silent hand signal for this is simply a raised hand.

The use of this hand signal encourages a variety of strategies to be shared.

After students share using the "Me Too" strategy, the teacher asks if anyone solved the problem differently or has a different answer.

Students take turns sharing their thinking.

One other perk of this hand signal - it reduces the urge for students to blurt out their disagreement while a student is sharing.

6. I Have Something To Say

Kids love to talk. This we know. So let them! In a controlled way of course.

As students listen to the explanations and reasoning of others, they may have a question, something to add, or a compliment to give.

The "I Have Something to Say" signal empowers students to share their ideas in a safe environment.

Do you need a year-long number talks program? Click on the picture below to check these out.

If you're looking for a yearlong set of number talks for your grade level (K-5) click here. These number talks are guaranteed to get your students conversing and engaging in mathematical conversations.

New to number talks? Click Here to read our blog " 5 Number Talks to Try This Week ".

Take Care and Happy Number Talking!

Team Primary Bliss Teaching

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6 Silent Hand Signals to Make Number Talks a Breeze - Primary Bliss Teaching

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Try these six number talks silent hand signals to bring calmness to your classroom during your number talk routine.

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Let's Dig Into the CRA Model - Concrete - Representational - Abstract by Don Mrozik | This newsletter was created with Smore, an online tool for creating beautiful newsletters for individual educators, schools and districts

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Skip the note of gratitude and go right to the Number Sense Routines now:

https://drive.google.com/drive/folders/1zybINWNfw7G_XYEAGwTTKiX4FZRYuRkM?usp=sharing

Thank you to the Steve Wyborneys, Amy Lucentas, Grace Kelemaniks, Sherry Parrishes, and Megan Frankes of the educational world who have shared their wealth of knowledge and resources and inspired these sets of Number Sense Routines. And thank you to my school district and colleagues who have supported and contributed to this project.

We have collected routines from several amazing open educational resources and have been inspired to create quite a few of our own to add to the collection. I am unable to tell you exactly how much time has been invested in this work (several hundred hours accrued over the past 3 years!), but I do know that every minute was time well spent selecting, editing, creating, organizing, and then fine-tuning again and again.... It's still not perfect, but the slide decks are finally in a place where we feel confident that they might be useful to others who are looking to build Number Sense with their students.

These slide decks contain 180 days of Number Sense Routines for PreK through Grade 5 (a unique set for each grade!). The files are organized into 20-day sets. The first 20 days have the same routine two days in a row with the first day designed for the teacher and the students to learn how to use the routine and then followed with a second day of the same routine (different prompt) to reinforce what was learned the day prior. Days 21-180 have nine routines randomly spread across the slides.

To better understand the purpose and process for using these routines, please take time to read the information provided on the first several slides of the first slide deck at each grade level. We are very proud of the work we've done and the results we've seen from using these routines regularly over the last 3 years with our students across our school district.

I am happy to answer questions you may have about using these routines, and I'd love to hear about YOUR successes if you decide to try them :) These Routines have become our passion here in my district, and we'd love to talk about helping your district get started using them, too. We've supported several districts with implementation through professional development trainings and have presented at a few major conferences, including the California Mathematics Council conference in Palm Springs. We'll also be presenting at NCTM Baltimore on December 1, 2022.

I'm just an email away.
Dawn Caine HelloMrsCaine@gmail.com

You can ACCESS and DOWNLOAD the FILES HERE:
https://drive.google.com/drive/folders/1zybINWNfw7G_XYEAGwTTKiX4FZRYuRkM?usp=sharing

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Join me on my learning journey - each month I share math strategies and ideas designed to build reasoning and conceptual understanding.

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Grade 1 Number Talks/Strings Routine Quarter 1

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This google slide show is for use with grade 1 students as a routine to build number sense. Strings of addition expressions are used to build connections and as a springboard for discussion. Each slide has a target addition strategy, though other strategies are encouraged as well.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Cherilee Wilkinson

Grade 1 Clothesline Routine Quarter 1

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This google slide presentation describes a numeracy routine for grade 1 use during the first quarter. The routine involves sequencing various representations of numbers on a clothesline. Printable materials are included.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Cherilee Wilkinson

Grade 1 Same But Different Routine Quarter One

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This google slide show will be used by first grade teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Students then engage in discourse to tell how the two pictures, numbers, expressions, or equations are the same and how they are different.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Cherilee Wilkinson

Grade 1 Quick Images Routine Quarter 1

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This google slide show will be used by first grade teachers with their students as a numeracy routine. Students will use a google slide presentation to subitize, compose, and decompose numbers presented as quick dot images.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Cherilee Wilkinson

Grade 1 Splat Routine Quarter 1

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This google slide show will be used by grade 1 teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Focus is on composing and decomposing numbers as well as discussion of addition and subtraction strategies. Steve Wyborney is the original creator.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Cherilee Wilkinson

Grade 1 Counting Jar Routine Quarter One

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This google slide show will be used by first grade teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Multiple opportunities to count objects exist within each slide's photo. The slides build upon the counting collections strategy, which involves multiple ways to count collections of physical objects.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Cherilee Wilkinson

Grade 2 Splat Routine Quarter 1

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This google slide show will be used by grade 2 teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Focus is on composing and decomposing numbers as well as discussion of addition and subtraction strategies. Steve Wyborney is the original creator.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Sarah Martinsen

Grade 2 Number String Routine Quarter 1

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This google slide show is for use with grade 2 students as a routine to build number sense. Strings of expressions are used to build connections and as a springboard for discussion. Each slide has a target addition strategy, though other strategies are encouraged as well.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Sarah Martinsen

Grade 2 Clothesline Routine Quarter 1

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This google slide presentation describes a numeracy routine for grade 2 use during the first quarter. The routine involves sequencing various representations of numbers on a clothesline. Printable materials are included.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Sarah Martinsen

Grade 2 Estimation Routine Quarter 1

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This google slide show will be used by second grade teachers with their students as a numeracy routine. Directions for the teacher are found here. For quarter one, students will estimate using physical objects in a jar. Students should be given time to notice and wonder about each jar of items.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Sarah Martinsen

Grade 2 Which One Doesn't Belong Routine Quarter 1

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This google slide show will be used by second grade teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Students should then provide reasoning for how each number or picture is different from the others. Teachers should focus on connections and precise vocabulary.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Sarah Martinsen

Grade 3 Hundreds Chart Mazes Quarter 1

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This google slide show organizes hundreds charts number puzzles in a sequence by weeks. Puzzles are online and linked to this slide show. Teachers can use these with students to promote number sense, addition, subtraction, and place value.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Jessica Hudson

Grade 3 Who Am I? Number Routine Quarter 1

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This google slide show organizes "Who Am I?" number puzzles in a sequence by weeks. Puzzles are online and linked to this slide show. Teachers can use these with students to promote number sense and precise mathematical vocabulary.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Jessica Hudson

Grade 3 Splat Routine Quarter 1

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This google slide presentation is a collection of Steve Wyborney's SPLAT! routine, arranged in a sequence by weeks for use with grade 3. Students will work on subitizing, finding missing addends, and creating equations to represent their thinking.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Jessica Hudson

Grade 3 Estimation Routine Quarter 1

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This google slide show organizes estimation exercises in a sequence by weeks. Teachers use the slide show with students to promote finding structure and estimating.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Jessica Hudson

Grade 4 Hundreds Charts Routine Quarter 1

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This google slide show organizes place value exercises in a sequence by weeks. Teachers use the slide show with students to promote finding structure and adding and subtracting ones and tens.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Stacy Ottman

Grade 4 Estimation Routine Quarter 1

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This google slide show organizes estimation exercises in a sequence by weeks. Teachers use the slide show with students to promote finding structure and estimating.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Stacy Ottman

Grade 4 Who Am I Routine Quarter 1

(View Complete Item Description)

This google slide show organizes "Who Am I?" number puzzles in a sequence by weeks. Puzzles are online and linked to this slide show. Teachers can use these with students to promote number sense and precise mathematical vocabulary.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Stacy Ottman

Grade 4 Would You Rather Routine Quarter 1

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This google slide show organizes "Would You Rather?" questions in a sequence by weeks. Teachers can use these with students to promote number sense and precise mathematical vocabulary. Students will construct viable arguments and discuss their choices mathematically.

Material Type: Activity/Lab

Authors: Kelly Schaefer, Stacy Ottman

Clothesline Numeracy Routine Kindergarten Quarter 1

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This google slide presentation describes a numeracy routine for kindergarten use during the first quarter. The routine involves sequencing various representations of numbers on a clothesline. Printable materials are included.

Material Type: Activity/Lab, Teaching/Learning Strategy

Authors: Kelly Schaefer, Katie Campbell

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Riverdale Numeracy Routines Group Resources

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This google slide show will be used by first grade teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Multiple opportunities to count objects exist within each slide's photo. The slides build upon the counting collections strategy, which involves multiple ways to count collections of physical objects.

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NUMBER SENSE ROUTINES: DESIGNED TO BE USED AS CLASS STARTERS TO ENGAGE YOUR STUDENTS, FOSTER MATHEMATICAL DISCOURSE, AND CREATE CLASSROOM COMMUNITY. CLICK HERE TO LEARN ABOUT THE DIFFERENT TYPES OF ROUTINES.

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The content in this High Yield Routine Slideshow has been organized by standard to assist with finding routines that can be used to strategically expose students to content for standards 3.6 Money, 3.7 Measurement, 3.9 Time, 3.10 Temperature, 3.11, 3.12, 3.13 Geometry and 3.14 Probability.

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Math routines are embedded in each week’s activities and provide students an opportunity to develop number sense by connecting critical concepts through discourse. When students are provided an…

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Discover 3 effective number sense routines that you can use instantly in your classroom. These fun math games and activities are ideal for your math block.

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Let's Talk Math! Implementing Math Talk in the Classroom

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Math manipulatives and hands-on concrete resources have always been acceptable in pre-k, kindergarten, and 1st grade classrooms as a method of teaching mathematical concepts, but only recently have they been considered relevant in upper elementary and middle school math.

The adoption of math manipulatives or concrete resources is driven by the move towards a math mastery approach in schools.

Using manipulatives to help teach elementary math

At first, children need hands-on and visual hooks to understand new math concepts. This is true whether they are simply learning to count or exploring patterns in algebra.

It's also why counting cubes, making 'cookies' out of playdough and using plastic coins to pay for these 'cookies' have long been key ways to teach concepts to younger students in early elementary math.

In upper elementary, we decrease the chances of children realizing the concepts for themselves if we stop using math manipulatives at this point. Essentially, we could be denying children the most powerful type of learning!

With mathematics itself being abstract, concrete math manipulatives provide the learner a 'window' in, to make sense of the problem at hand by touching them, playing with them, exploring the patterns and relationships which make a huge difference between understanding for depth or just for procedure.

Research has shown that teaching with physical manipulatives has a significant positive impact on learning in Maths (Carbonneau, K.J., Marley, S.C. & Selig, J.P. 2013). A summary of their main findings can be accessed here.

Certainly, the Asian-style model of mathematics teaching strongly advocates using manipulatives in the classroom, with the concrete representational abstract approach they've popularized.

But before you think, "Great - let's just get out some practical resources each lesson and all students will succeed!", there is more! You see, it's also about how students use the equipment and learning resources and relate what they are doing practically to the problem you've asked them to solve!

Guide To Hands On Manipulatives

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Hands-on resources are a tool

It is fundamental that children understand that what they have in their hands is a tool to help them make sense of the underpinning ideas of math, and not as an implement to depend on to work out the math each time.

If the latter happens, then children become dependent on the equipment and the manipulative then becomes a 'crutch' to rely on.

How not to teach math with manipulatives

My own experiences draws me back to a lesson I observed where the teacher, with the utmost of good intentions, set about teaching her 2nd grade class all about subtraction with decomposition, with the use of base ten blocks.

Fantastic you may think! So did I. Until I realized that she had gone about it in a fashion where she was demonstrating and modeling how to decompose the numbers by just telling them what to do as opposed to letting the class find out for themselves.

It then became apparent that many children in the class were simply mimicking the procedure of decomposition they had seen their teacher do, without any real understanding of what was going on.

They followed the 'steps' and although came to the correct solution with the blocks, could not then relate this back to the algorithm that was subsequently taught later on. Just because practical equipment is being used, if not used correctly, it will actually have a detrimental effect.

Mastery involves variation

As you know, the key to mastery is the ability to use and apply while making sense of mathematics across topics, one step towards achieving this is through variation - variation of the math problem and math activity posed, but also variation of the manipulatives used.

By this I mean there is not just one practical resource used for a particular type of question. For instance, when teaching number and place value, you could use bead strings, base ten blocks, place value counters...the list goes on!

Let the students choose the resources for greater depth

If we are encouraging children to become independent learners, they should then also have the choice of which manipulative they think would best suit the task and then be able to reflect on and explain what they did.

To encourage greater depth and conceptual understanding, could the student use different manipulatives to prove the same answer?

The Best Math Equipment List For Elementary

Here are some ideas of math manipulatives for your classroom:

Base ten blocks

Geoboards

Fraction Tiles

Fraction Circles

Playing cards

Dominoes

Rekenreks

Two-color counters

Ten frame

Snap cubes

Many of these math tools can be expensive to purchase for your classroom, but don't worry. There are plenty of 'do it yourself' options and virtual manipulatives, such as these from Didax, which can be displayed on a whiteboard, to support your classroom teaching.

Looking for more math mastery resources?

Do you have students who need extra support in math?
Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor.

Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state's standards and assessments, plus you'll receive regular reports every step of the way.

Personalized one-on-one math tutoring programs are available for:
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Why not learn more about how it works?

The content in this article was originally written by math specialist & consultant Wendy Liu and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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How Math Manipulatives Transform Elementary Math Lessons

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Math manipulatives and concrete resources aren't just for kindergarten. Find out how they can boost elementary student progress too!

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The Concrete Representational Abstract approach is now an essential tool in teaching math at elementary school, so here we explain what it is, why its use is so widespread, what misconceptions there may be around using concrete resources throughout a child's primary math education, and how best to use the CRA approach yourself in math lessons.
In this blog, we will be focusing specifically on how the CRA approach can be used to support the teaching and learning of the four written calculation methods.

What is the Concrete Representational Abstract approach in Math?

The Concrete Representational Abstract (CRA) approach is a system of learning that uses physical and visual aids to build a child's understanding of abstract topics.

Decimal Place Value Chart

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Students are introduced to a new mathematical concept through the use of concrete resources (e.g. fruit, base ten blocks, fraction bars, etc). When they are comfortable solving problems with physical aids, they are given problems with pictures - usually visual representations of the concrete objects they were using.

Then they are asked to solve problems where they only have the abstract, i.e. numbers or other symbols. Building these steps across a lesson can help students better understand the relationship between numbers and the real world, and therefore helps secure their understanding of the mathematical concept they are learning.

Origins of Concrete Representational Abstract Approach

Looking at the origins of the CRA approach, we need to go back to the teaching methods of the 1960s, when American psychologist Jerome Bruner proposed this approach as a means of scaffolding learning.

He believed the abstract nature of learning (which is especially true in math) to be a 'mystery' to many children. It, therefore, needs to be scaffolded by the use of effective representations and math manipulatives.

He found that when students used the CRA approach in math, they were able to build on each stage towards a greater mathematical understanding of the concepts being learned, which in turn led to information and knowledge being internalized to a greater degree.

Many teachers mistakenly believe the CRA approach to have been a method imported from Singapore.

In actual fact, the Singapore Math curriculum has been heavily influenced by a combination of Bruner's ideas about learning and recommendations from the 1982 Cockcroft Report (a report by the HMI in England, which suggested that computational skills should be related to practical situations and applied to problems).

Why use the Concrete Representational Abstract approach in Math?

Students achieve a much deeper understanding if they don't have to resort to rote learning and are able to solve problems without having to memorize.

When teaching reading to young children, we accept that children need to have seen what the word is to understand it. Putting together the letters c- a- t would be meaningless and abstract if children had no idea what a cat was or had never seen a picture.

People often don't think of this when it comes to math but to children, many mathematical concepts can be equally meaningless without a concrete resource or picture to go with it. This applies equally to mathematics teaching at elementary school.

What is a 'Concrete' representation in the CRA approach?

As part of the CRA approach, new concepts are introduced through the use of physical objects or practical equipment. These can be physically handled, enabling children to explore different mathematical concepts. These are sometimes referred to as math manipulatives and can include ordinary household items such as straws or dice, or specific mathematical resources such as base ten blocks or ten frames.

The abstract nature of math can be confusing for children, but through the use of concrete materials, they are able to 'see' and make sense of what is actually happening.

Previously, there has been the misconception that concrete resources are only for learners who find math difficult. In fact, concrete resources can be used in a great variety of ways at every level. All children, regardless of ability, benefit from the use of practical resources in ensuring understanding goes beyond the learning of a procedure.

Practical resources promote reasoning and discussion, enabling children to articulate and explain a concept. Teachers are also able to observe the children in order to gain a greater understanding of where misconceptions lie and to establish the depth of their understanding.

How can we portray the "representational" part of the CRA approach?

Once children are confident with a concept using concrete resources, they progress to drawing visual representations or quick sketches of the objects. By doing this, they are no longer manipulating the physical resources but still benefit from the visual support the resources provide.

Some teachers choose to leave the representational stage (also known as semi-concrete) out, but it is key to ensuring that children can make the link between a concrete resource and abstract notation. Without it, children can find actually visualizing a problem difficult.

One of the most common methods used in the representational stage is the bar model, often used in more complex multi-step problem solving.
Read more: What is a bar model

What is an 'Abstract' representation in the CRA approach?

Once children have a secure understanding of the concept through the use of concrete resources and visual images, they are then able to move on to the abstract stage. Here, children are using abstract symbols to model problems - usually numerals. To be able to access this stage effectively, children need access to the previous two stages alongside it.

For the most effective learning to take place, children need to constantly go back and forth between each of the stages. This ensures concepts are reinforced and understood.

How to teach using the Concrete Representational Abstract method in elementary school

A common misconception with the CRA model is that you teach the concrete, then the representational, and finally the abstract. But all stages should be taught simultaneously whenever a new concept is introduced and when the teacher wants to build further on the concept.

When concrete resources, visual representations and abstract recordings are all used within the same activity, it ensures students are able to make strong links between each stage.

If you're concerned about differentiating effectively using the CRA approach, have a look at our differentiation strategies guide for ideas to get you started.

Using 'The Four Operations' to model how CRA works

In the following section, we will be looking at the 'four operations' and how the CRA approach can be used at different stages of teaching them. In particular, we will examine how the 3 parts of the CRA approach should be intertwined rather than taught as 3 separate things.

As this blog is to share ideas rather than say how the calculation methods should be taught, we will only cover the four operations briefly.

Teaching addition using the Concrete Representational Abstract approach

In the early stages of learning column addition, it is helpful for children to use familiar objects. For example, straws or lollipop sticks can be bundled into groups of ten and used individually to represent the tens and ones.

Once children are familiar with making 2-digit numbers using these resources, they can set the resources out on a baseboard to represent the two numbers in a column addition calculation.

Initially, children complete calculations where the units do not add to more than 9, before progressing to calculations involving exchanging/regrouping. Alongside the concrete resources, children can annotate the baseboard to show the digits being used, which helps to build a link towards the abstract formal method.

Once children are confident using the concrete resources they can then record them representationally, again recording the digits alongside to ensure links are constantly being made between the concrete, representational and abstract levels.

The next step is for children to progress to using more formal mathematical equipment. Base ten blocks should be introduced alongside the straws, to enable children to see what is the same and what is different. As confidence grows using the base ten blocks, children can be introduced to the hundreds column for column addition, adding together 3-digit and 2-digit numbers

Alongside the concrete resources children should be recording the numbers on the baseboard, and again have the opportunity to record representational representations.

Once children are completely secure with the value of digits and the base ten nature of our number system, base ten blocks can be replaced with place value counters. These help children as they progress towards the abstract, as unlike the base ten blocks, they are all the same size.

These should be introduced in the same way as the other resources, with children making use of a baseboard without regrouping initially, then progressing to calculations, which do involve regrouping.

As with the other equipment, children should have the opportunity to record the digits alongside the concrete resources and to progress to recording representationally once they are secure. The place value counters can be used to introduce children to larger numbers, calculating column addition involving the thousands and then the ten thousands column.

Read also: How To Teach Addition For Interventions In Upper Elementary

Teaching subtraction using the Concrete Representational Abstract

The method for teaching column subtraction is very similar to the method for column addition. Children should start by using familiar objects (such as straws) to make the 2-digit numbers, set out on a baseboard as column subtraction.

To begin with, ensure the ones being subtracted don't exceed those in the first number. Once children are confident with this concept, they can progress to calculations, which require exchanging. As with addition, the digits should be recorded alongside the concrete resources to ensure links are being built between the concrete and abstract.

Once secure with using the concrete resources, children should have the opportunity to record representationally, again recording the digits alongside.

As with addition, children should eventually progress to using formal mathematical equipment, such as base ten blocks.

These should be introduced alongside the straws so students will make the link between the two resource types. Digits are noted down alongside the concrete resources and once secure in their understanding children can record the base ten blocks representationally, to ensure links are built between the concrete and abstract.

Once secure with the value of the digits using base ten blocks, children progress to using place value counters.

Again, the counters enable children to work concretely with larger numbers, as well as bridge the gap from the use of base ten blocks to the abstract. Along with the counters, children should be recording the digits and they should have the opportunity to record representationally once confident with the method using concrete resources.

Read also: How To Teach Subtraction For Interventions In Upper Elementary

Teaching multiplication using the Concrete Representational Abstract approach

The grid method is an important step in the teaching of multiplication, as it helps children to understand the concept of partitioning to multiply each digit separately.

Students can begin by drawing out the grid and representing the number being multiplied concretely. For example, 23 x 3 can be shown using straws, setting out 2 tens and 3 ones three times. This way, children can actually see what is happening when they multiply the tens and the ones.

As with addition and subtraction, children should be recording the digits alongside the concrete apparatus, and recording representationally once they are confident with the concrete resources.

Children are then able to progress to representing the numbers in a grid, using place value counters.

Once children are confident using the counters, they can again record them representationally, ensuring they are writing the digits alongside both the concrete apparatus and the visual representations.

Progressing to the expanded method and then the short method of column multiplication is much easier for children if these are introduced alongside the grid method, to enable them to see the link.

Read also: How to Teach Multiplication Interventions in Elementary

Teaching division using the Concrete Representational Abstract (CRA) Approach

As children work towards the standard algorithm for division (long division), it is important they understand what is meant by both division as grouping and division as sharing.

Concrete resources are invaluable for representing this concept. Students need to understand how numbers can be partitioned and that each digit can be divided by both grouping and sharing.

Once confident using concrete resources (such as bundles of ten and individual straws, or base ten blocks), children can record them representationally, before progressing to more formal short division.

As children work towards understanding long division, concrete resources can be used to help them understand that 2-digit numbers can be partitioned and divided by both sharing and grouping. This can be through the use of bundles of ten straws and individual straws or base ten blocks to represent the tens and ones.

As with the other operations, it's important that children are recording the digits alongside the concrete resources and are having the opportunity to draw visual representations. As children grow in confidence and once they are ready to progress to larger numbers, place value counters can replace the base ten blocks.

Read also: How To Teach Students Division For Interventions In Upper Elementary

The Concrete Representational Abstract Approach is here to stay

It may have taken many years for CRA to reach the level of popularity it has today, but it is definitely here to stay.
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I have seen firsthand how successful it can be when children have the opportunity to work in this way and I love the fact that children are now starting to have the conceptual understanding in math that I never had as a child.

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The content in this article was originally written by assistant headteacher Emma Johnson and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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Concrete Representational Abstract: What It Is And How To Use It

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A guide to The Concrete Representational Abstract (CRA) approach and how to use it in your elementary math classroom.

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How To Effectively Implement CRA Math Lessons - The Math Spot

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In an effort to implement CRA math lessons, you used hands-on materials all last week in math. Your students were absolute rock stars. This week you were sure

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What is CRA you ask? It stands for concrete, representational, and abstract, and it's a research based instructional sequence that results in deeper understanding of mathematics concepts.

Concrete learning is hands on. It's using manipulatives to make meaning of a new concept.

Representational is showing that same concept using pictures.

Abstract is representing a concept using symbols.

Now that I've set the stage, let's talk about place value. I talked with both 2nd and 3rd grade teachers today, and both grade levels are starting the year with place value. In 2nd grade, they will use groupable manipulatives, linking cubes, to model numbers with tens and ones (a review from 1st grade). Van de Walle recommends groupable manipulatives prior to using traditional base-10 blocks, because they can physically be joined together and broken apart. Traditional base-10 blocks are actually a little more abstract because, for example, you can't break the tens rod apart into ones-you have to trade it for ones. 2nd grade will then transition from the linking cubes to base-10 blocks as they extend their learning to hundreds.

In 3rd grade, my good math buddy Jeremy wanted a place value mat that the kids could use to work with base-10 blocks (concrete) and that also had representations of each place value (representational), so I whipped up this PV mat for him. Note that it prints on 11 x 17 so the columns fit the base-10 blocks. Of course, you can scale it down to print it on letter-sized paper, but the columns won't fit the manipulatives. I love how he wanted the ten-frame for the ones! Great bridge to prior learning. Click to grab yours and read on for suggested uses and another freebie.

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Place Value, CRA Style - Math Coach's Corner

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What is CRA you ask? It stands for concrete, representational, and abstract, and it's a research based instructional sequence that results in deeper understanding of mathematics concepts. Concrete learning is hands on. It's using manipulatives to make meaning of a new concept. Representational is showing that same concept using pictures.

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Let's start off with the number line, and as you can see this is not a number line. But it's the precursor to a number line. Kids in kindergarten are working with physical objects, and we start to get them to line them up because it becomes a number line later in the grades. So, kindergarten and first grade actually should not be using a number line.

There was wording in the original Common Core Standards that talked about, it specifically called it out, number lines should not be used before second grade. Now there is no wording in a lot of standards that says that, but I want you to check and see because I can almost bet that there is no mention of number lines in your standards until second grade.

Until then, kids should be using actual objects to model, and then we can turn this into a representational phase using what's known as a number path. Kids still need to see the actual objects, which a number path has those rectangles that kids could still see five things when they are showing five, or whatever it might be.

As they move into number lines, number lines become more abstract. It's a length view of a model that kids aren't really seeing five things, it's the space from zero to five that's supposed to represent five, and it becomes really abstract for our young kiddos.

But number lines are very important and an essential model to allow kids to be able to visualize some cool mathematics, and eventually, they get to the point where they're able to work fluently in a number line. The image above shows an example of an open number line because it does not have all of the tick marks that a number line typically has.

I want to just share a few types of problems and have you think about how it makes a difference if we can model this on a number line. I'm not going to go through all the way things that you can model using a number line because it would take forever. I just want to give you a little glimpse into some of the things that are difficult as kids progress up into their mathematics that, if they can visualize these types of tasks on a number line, it can make the mathematics so much easier.

In the grades before they actually get to this actual content, it's helpful that they are using number lines in lots of different mathematical concepts. Here's a typical one, where kids have to order fractions from least to greatest. and they will be given crazy fractions. Are we really expecting kids to find common denominators on this to be able to compare them? No.

If you can get kids to visualize where these fractions would fall on a number line, that automatically puts them in order from least to greatest. Now I'm not going to go through and show you all the answers to this.

I just want you to take a second and work on it yourself. Think about how much does that make a difference on this kind of a problem. We're not jumping to the abstract algorithm of find common denominators and then you can order fractions. Let kids use a representation to help them solve that problem.

Here's another one that I personally hated in school because it was taught that you have to pull them out, you have to create the equation, solve for the missing part, and it just seemed so abstract to me.

Here's the problem: I got a dress on sale for 40% off and I paid $36 for the dress. What was the original price? Now there's a little bit of layering in here. Kids have to realize that that $36 was really 60% of the original price because it was 40% off. So, it's a little tricky in there.

But, if they can model this on a number line, that, if 100% was the full price of the dress, and what I paid was only 60%, would this help them have an avenue to be able to solve that problem without having to jump to the abstract, create the equation, solve and do the procedure?

It sure helped me. This was a game changer. When I saw finding percentages on a double number line like this, it totally changed my life, seriously. I love doing problems like this now because I can see the relationships that I could never see before. And that's really what working in the CRA continuum, I guess, is supposed to do for kids. It helps build these relationships and connections that kids often don't see when they're just working in the abstract area.

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How To Use Representation In The CRA Method - The Recovering Traditionalist

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Curious about how to use representation in the CRA method in your math classroom? Here are two that you must be using when using the CRA method.

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Do you do Number Talks in your first grade classroom?

We do! And we can honestly say that Number Talks are the BEST way we have found to increase our students' computational fluency and their ability to think about numbers flexibly.

This simple addition to our math warm up time has truly transformed mathematics in our classrooms. Pinky swear!

However...if the thought of implementing daily number talks gives you hives....we've got you covered!

If you have an interactive whiteboard, you literally just display a number talk each day and let the conversations begin. If not, just print a number talk out each day and you are ready to roll.

Step 1: Display Your Daily Number Talk & Give Individual Think Time to Solve

The first step is to display the daily Number Talk and give students individual think time to solve the problem.

This step is critical as we want all learners to have the time to access this learning opportunity. All too often the quick thinkers blurt out answers before others even have time to think and process. Please do not allow this to happen or the value of this powerful activity will be greatly diminished. Silence is golden during this portion of the math talk activity.

Step 2: Have Students Communicate Using Silent Hand Signals

Keep this activity running smoothly and calmly by having students communicate with you using silent hand signals. For this portion of the lesson, students will be signaling that they have an answer or that they have more than one answer to the problem.

Why silent signals? Again, we don't want to disrupt students' process time with discussion yet. During this portion of the lesson we want to check in to find out where each student is in their thinking process. Do they have an answer? Two answers? Do they have a question? Do you have students not showing silent hand signals that need more process time?

Guess What? You receive these adorable posters FREE with our First Grade YEARLONG Number Talks BUNDLE! Here is a blog we have on

Step 3: Partner Share Answers & Strategies Used

When you see that all students have an answer, ask your students to partner share their answer and how they got it. What strategies did they use?

We like to assign partners. One is Peanut Butter and one is Jelly. Then we alternate who shares first.

The important thing is that both students have the opportunity to not only share the answer they got, but also share how they got the answer. Sharing the how is critical. Learning and understanding go deeper when the how is shared. Mathematical concepts are better internalized through the process of sharing.

Step 4: Have a Student Share Answer & Strategy Used (Record)

Call on a student to share their answer and how they arrived at the answer. What strategies did they use? Record the students' thinking. (We like to also record the student's name.)

Then ask students to share who got the same answer and solved it in the same way. Students will use their silent signal for "me too" to communicate if they did.

Oftentimes several students will arrive at the same answer and use the same strategy. This is the time for those students to silently share that information with you instead of yelling out me too.

Step 5: Have Other Students Respond Through Silent Hand Signals & Share Thinking (Record)

Ask students who got the same answer, but solved it in a different way; or who got a different answer. (Many of our number talks are open ended and have multiple answers.)

Again, have students respond using silent hand signals.

Call on students to share both their answer and how they arrived at the answer. Record.

Through this sharing process students are able to learn from their peers, which is very powerful! They may learn new strategies to try or more effective strategies to use. They may also recognize that they have misconceptions and ask clarifying questions to resolve gaps in their understanding.

And there you have it.... Number Talks MADE EASY! If you are not currently doing Number Talks in your classroom, we highly encourage you to give them a try!

Here's What Teachers Are Saying About Our #1 Selling Number Talks Resource!

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"Oh my goodness, this resource is fantastic. I've never done number talks before, so I'm learning right along with the kids - and it's amazing to hear their explanations and conversations each day! They are learning so many different strategies for looking at and thinking about numbers. These activities have enough repetition that the kids (and I) know what to do, but each month they "ramp up" a bit and the kids just love it! Even my least engaged students get involved in figuring out and sharing their answers. I can't thank you enough for such an amazing product!" - Amanda K.

"Thank you so much for this product! It is very comprehensive and saves me loads of time. My students LOVE the interaction with the math, are learning to talk about their problem-solving processes and think critically and talk about the strategies others may use to solve problems! I love the directions and the plans! This is an awesome product." -Tressa

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5 Easy Steps to Teach Number Talks in First Grade - Primary Bliss Teaching

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Are you ready to take on teaching number talks in your first grade clasroom. If so, hop on over to our blog to get started on the right foot.

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Creating a Learning Environment for Number Talks

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One important factor for successful number talks is having a learning environment where students feel safe to participate and try out new ideas and ways of thin

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by Tina Jennings, Guest Author

Since the introduction of the Common Core Standards for Mathematical Practice, teaching math has shifted its emphasis from the rote memorization of equations and facts to one of conceptual understanding. We're no longer asking students to remember algorithms they may forget as soon as the year is over. Instead, we want our children to be able to explain their thinking, justify their solutions, and recognize that their peers may have alternative solutions as well.
Sentence Stems
In order to put kids on the path to becoming lifelong math learners, we must set up norms for math conversations that are deep-diving, accountable, and low-risk. Sentence stems and silent gestures are two techniques for streamlining math conversations in your classroom.

I agree with what _____ said and would like to add on...

I respectfully disagree with __ and think that...

I saw how to solve this in another way...

I'm confused. Can you help me with this part?

I didn't understand you. Can you explain it in another way?

Sentence stems serve as a great starting point for creating norms during your classroom math conversations. They provide students with the word structure they need to state their own solution, add on to another student's thinking, disagree with someone's thinking, or ask for help. This encourages an atmosphere of respect where all ideas are heard and valued. A list of sentence stems I teach includes:
Silent Hand Gestures
The book Intentional Talk: How To Structure and Lead Productive Math Conversations by Allison Hintz and Elham Kazemi is another excellent resource for sentence stems. These leaders in the field of elementary mathematics provide actionable tips, titled "Math Talk Moves," for helping students communicate. These include "revoicing" or restating what another student said, reasoning, adding on, revising thinking, and encouraging "wait time" that allows students to reflect on a problem or discussion before participating. Teaching and practicing communication skills in these explicit ways streamlines discussions and normalizes engagement.
No Opting Out
Using hand signals is another way to establish a supportive math environment. Teach your students how to give "silent support" by raising their arms in the air when another person is talking, to signal they like what they hear. Alternatively, they can use thumbs up or thumbs down to connect with another student's idea without distracting the class. Giving agency to students allows them to feel comfortable and open to sharing with one another. Furthermore, using silent hand signals as a norm during math conversations supports those more kinesthetic learners in the classroom.

"No opt out" is at the center of strong math conversations. This motto, borrowed from behavior management guru Doug Lemov, sets an expectation that everyone in the class participates in these discussions. Explain that everyone is an equal part of the class team, and must contribute. To that end, it's important to also stress that there are many ways that kids can participate, including the sentence stems - even repeating or asking for help - and using silent gestures.
Create a Low-Risk Environment
While this policy has led to some difficult moments in the past where a student stares at me like a deer in headlights, over time I've learned that holding students accountable actually makes it easier for them to participate. Posting silent signal charts and referring to them frequently gives puzzled pupils a menu of options. You'll find that enforcing the rule leads to class-wide engagement becoming the norm, establishing a community where all are ready to speak. For more on "no opt out" and a myriad of other invaluable behavior management strategies, check out Doug Lemov's website.

Be aware of your language. As the leaders of the classroom, the words we use hold a lot of weight. We want to name the qualities in our students that they can use to become conceptual thinkers. Substitute phrases like "perfect," "great job," or "you got it," for "way to persevere," "what effort you've made," and "I can see tremendous thinking."

Focus on the quality of curiosity instead of getting the right answer. After a student answers a question or a problem, ask probing questions in a friendly manner, and be an active listener. This will encourage your class to persist in explaining the thought process behind their solutions.

Model the benefits of taking risks. Risk-taking and an acceptance of failure are critical aspects of a safe space. To model this, start with a math problem. Incorrectly solve the problem to start a class discussion. Choose two volunteers to solve the problem correctly, and discuss how they did it. Express excitement when students are taking risks, and limit your reactions when students make errors.

Create a behavior management system that reinforces risk-taking. Offer rewards for those who support their peers, and give appropriate consequences for those who tease, are rude, or act inappropriately.

It's essential to create an environment where students feel safe to express their ideas. There are several ways this can be done, the majority of which stem directly from your behavior in the classroom.

What's so gratifying about this conceptual shift in today's math standards is that it modifies our end trajectory. Instead of focusing solely on "getting the right answer," we get to foster the vital qualities of risk, exploration, innovation, and growth. With these norms, we can create a community of thinkers by modeling risk, normalizing engagement, and creating a standard of accountability to reach this end.
Tina Jennings, M.A. is a second grade teacher at Roosevelt Elementary School in Santa Monica, CA. She received her undergraduate degree in 2006 from The State University of New York at Purchase with a major in art history. In 2010, Tina began her teaching career at Aspire Public Schools, a charter school system working in low-income communities. She was a founding teacher at Aspire Firestone Academy while simultaneously earning her Master's Degree in Curriculum and Instruction from the University of the Pacific. Since then she has taught second grade and loves working in a close-knit community so near to her home. In addition to teaching, Tina is a freelance writer.
Find more teaching resources at education.com.

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Math Talks: Setting up Norms for Deep-Diving Math Conversations

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Since the introduction of the Common Core Standards for Mathematical Practice, teaching math has shifted its emphasis from the rote memorization of equations and facts to one of conceptual...