Another way to support your student's understanding of quantity is by teaching him to count. Knowing how to count includes the following: knowing how to verbally count, understanding cardinality (that numbers represent quantities), and having the ability to interpret, represent, and compare quantities. Counting ability develops in the following developmental stages:
Before you learn about interventions that support one of these stages, you'll learn about tools you can use to support student mastery.
The intervention pages to come refer to the following tools: Counters, Drawing and Labeling Objects, Ten-Frames, Number Lines, Base-Ten Blocks, and Area Models. Use this page as a resource as you learn more about each of the tools in this activity.
Tool # 1: Counters
Counters are the concrete objects that students can manipulate in order to count or build quantities. Counters can come in many different forms: gummy bears, pebbles, buttons, Unifix cubes, and so on. When students are first learning to count, counters are the best tool to use, since the student can physically hold and move this object.
Tool # 2: Drawing and Labeling Concrete Representations of Objects
As students become better at counting or building groups of objects using counters, you can link the counters to visual representations of each quantity. Forbringer and Fuchs (2014, pp. 92-93) explain why and how to do so:
Sometimes students initially model a concept or procedure with counters and then are asked to perform the skill using visual or symbolic representation but without practice that explicitly connects the various forms of presentation. While normally achieving students may be able to successfully transition from one form of representation to another without the need for scaffolded support, students who have difficulty with number sense often struggle to connect the various forms of mathematical representation (Hecht, Vogi, & Torgesen, 2007), and so benefit when these connections are made explicit. For example, if students initially used M&Ms to count, they could use pencil, crayon, or chalk to draw a model of their M&Ms. Recording the written numeral on their drawing helps students connect the three-dimensional objects with pictorial and symbolic representations.
Drawing and labeling objects is a great tool to use if your students does not need to physically hold an object to count it.
Tool # 3: Drawing and Labeling Abstract Representations of Objects
As students become more fluent with drawing and labeling pictures of objects, they are able to represent objects through symbolic representations of the same objects. Using this tool, a student would draw a circle to represent each object in the story, instead of a picture of the object itself, knowing that each circle represents the actual object. Drawing and labeling abstract objects is faster than drawing and labeling more detailed pictures of objects.
Tool # 4: Ten-Frames
A ten-frame is a graphic organizer with ten empty spaces on it, over which a student can place concrete objects as he models a quantity. Ten-frames are a particularly useful tool for students who are learning about place value. Ten-frames help students quickly organize their counters into groups of ten when modeling larger numbers.
One of the best tools to help students connect three-dimensional concrete representation to two-dimensional visual representation is the ten-frame. A ten-frame consists of an empty 2 x 5 grid onto which students place counters...
When [students] are first learning to model numbers with frames, students should lay the frame horizontally so that there are five boxes in the top row. They begin by placing the first counter in the upper-left corner and progress from left to right across the top row, then move to the bottom row and continue placing counters from left to right, just as the eyes move when reading. Placing counters on the ten-frame in this set order helps students organize their counting and develop a mental model of each quantity.
Click here for a ten-frame template.
Tool # 5: Hundred Chart
Another graphic organizer that a student can use as he learns to count is a hundred chart. A hundred chart lists all the numbers from 1-100, in order, in rows of ten. This tool helps a student visualize each quantity (for example, 23 represents all of the space on the 100 chart up to the number 23). It can also help a student recognize symbolic representations of numbers as he learns how to count (e.g.,, the student can touch each number and count up to 31 to find out that 31 is written "31").
Tool # 6: Number Lines
A number line is another tool you can use to support your student's ability to represent a number visually or recognize symbolic representations of numbers. Teach your student that the distance from zero to a given number on the number line represents the size of that number. Or, teach the student to touch each number as he counts to find out what the symbolic representation of a certain number looks like.
Note: A common misunderstanding when using a number line is thinking that the number of ticks on the line represents the size of the number (when actually the quantity represented is the distance between the number and zero). Teach your student to focus on the spaces between ticks and not the ticks themselves, when identifying a quantity. You can do this by helping him build a number line, placing equal-sized units (e.g., blocks or squares) next to each other. Or, you might incorporate movement to draw your student's attention to the spaces. Forbringer and Fuchs (2014, pp. 99-100) write "...students can walk along a large number line taped on the floor, counting their steps as they move. The large muscle motion helps them focus on counting space. They can then progress to demonstrating bunny hops... on a small number line taped on the desk."
Tool # 7: Base-Ten Blocks & Place Value Organization Mats
Base-ten blocks are physical counters that are scaled to represent ones, tens, hundreds, and thousands. There are four types of base ten blocks, including a cube thousand (representing 100 units), a flat hundred (representing 100 units), a 10-rod (representing 10 units), and a single cube (representing one unit). These blocks allow for more efficient modeling of multi-digit whole numbers. To use base-ten blocks, a student will represent a given number using the lowest number of blocks. For example, if a student wanted to represent the number 23, he would use two 10-rods and 3 single cubes (instead of counting out 23 individual cubes). Using base ten blocks allows a student to quickly glance at the blocks and understand how many there are.
Tool # 8: Area Models
There are numerous concrete objects that students can manipulate in order to identify or build fractional parts of a whole. These include: fraction tiles, pattern blocks, legos, or strips of paper. Each of these tools can be used to visually model the number of equal parts a whole is divided into (as represented by the denominator). The student can then identify the number of equal parts represented by a particular fraction (as reflected in the numerator).
When a student first learns to count, he learns to count whole numbers from 0 to 30. Knowing how to count whole numbers from 0-30 includes the following: knowing how to verbally count, understanding cardinality (that numbers represent quantities), and having the ability to interpret, represent, and compare quantities. As you teach these skills, you should start with concrete representations of numbers and move to more abstract representations of numbers. For example, if your student is just developing his understanding of whole numbers, you will want to have him identify and build the quantity using concrete objects (e.g., a group of 23 buttons or blocks). If he's mastered identifying or building the number using concrete objects, you can teach him to identify and build the number using pictures (e.g., drawing 23 buttons or blocks) or symbols (e.g., writing 23). This page includes intervention strategies that you can use to support your students in this area. As you read, consider which of these interventions best aligns with your student's strengths and needs in the whole-learner domains.
Explicit Instruction
If you are intervening to support your student's ability to count from 0 to 30, you should start by explicitly teaching the skill. This sounds like:
In order to support a student's knowledge of the sequence of numbers from 0-30, the following interventions could be used:
Activity A: Rote Counting
If your student is unable to count from 0-30 because he's skipping numbers or getting stuck at a certain point in the counting sequence, teach him Rote Counting. In this strategy, a teacher helps a student learn the sequence of numbers from 0 to 30 through repeated practice. A teacher may also consider teaching the student a number chant or song to help the student remember the sequence of numbers.
Note: Another strategy to use when teaching rote counting is to have the student clap his hands as you say each number. This multisensory approach allows the student to feel the "beat" of each number as he counts.
Rote Counting in Action
Teacher: We are going to count up to 10! As we say each number, I will point to it on the number line. Ready?
Teacher points to number on the number line, as she and student count: 1! 2! 3! 4! 5! 6! 7! 8! 9! 10!
Williams, M. (n.d.). What Is Rote Counting? (And How To Teach It). Early Impact Learning. https://earlyimpactlearning.com/what-is-rote-counting-what-it-is-and-how-to-teach-it/
Activity B: Counting with Number Lines or Hundreds Chart
Another strategy to use if your student is struggling to count whole numbers to 30 is Counting with Number Lines or Hundreds Chart. This strategy mirrors the former strategy, but gives the student a visual cue that matches each number as he says it. When teaching counting with one of these tools, the teacher should point to each number as she counts with the student.
Counting with Number Lines or Hundreds Charts in Action
Teacher: We are going to count up to 10! As we say each number, I will point to it on the number line. Ready?
Teacher points to number on the number line, as she and student count: 1! 2! 3! 4! 5! 6! 7! 8! 9! 10!
Activity C: Skip Counting with Number Lines or Hundreds Charts
Once your student gains fluency with counting ones up to 30, you can start to intervene to support his ability to skip count. This intervention teaches students to count by a multiple. When teaching a student to skip count, the teacher can visually support his understanding with a number line or hundreds chart, by jumping over or skipping certain numbers and just pointing to benchmark numbers as she counts.
Note: When teaching a student to skip count, start with teaching 5 or 10, then 2, and then you can go on to teach students to count by 3s, 4s, etc.
Skip Counting with Number Lines or Hundreds Chart in Action
Teacher: We are going to count by 5s up to 20! As we say each number, I will point to it on the number line. Ready?
Teacher points to number on the number line, as she and student count: 5! 10! 15! 20!
How Do Hundred Charts Teach Place Value and Skip Counting? (n.d.). ThoughtCo. https://www.thoughtco.com/hundred-charts-place-value-and-multiplication-3110499
Once a student learns how to recite numbers in the appropriate sequence, he can apply this understanding to count sets of objects. In order to do so, the student must understand cardinality or the concept that each set consists of a certain number of elements. The student must then be able to accurately count the number of objects in that set. In order to support your student's ability in this area, the following interventions can be used:
Activity D: The Last One I Said
If your student is struggling with understanding cardinality, to the extent that he doesn't understand that the last number he counted is the number of objects in the set, teach him The Last One I Said. This intervention helps a student conceptualize that each number he says aloud (or counts) represents an object. Students who struggle with this concept may count a set of cubes (such as 5 cubes), and when they are asked how many they have, they may say a different number (such as 2 cubes). In order to help a student understand that the last number he says represents the number of cubes he has, a teacher and a student count objects (fingers, cubes, or counters), and then they practice with different techniques until the student understands the meaning of counting.
The Last One I Said in Action
Teacher: In a minute, I am going to show you a trick. First, I want to ask you a question: How many fingers do I have?
Student: Five. Everyone has 5 fingers.
Teacher: Okay, I'm going to prove to you that I have 5 fingers. I will count them! Ready: 1, 2, 3, 4, 5! (As teacher counts each finger, she sticks it in the air). 5 fingers, you were right! Wait, let me count them again. 1, 2, 3, 4, 5. Yep, the last number I said was 5, so that means I must have 5. The last number I say is always the number I have in all. Now, you try!
Ferlazzo, L. (2021, February 21). Ten Ways to Use Retrieval Practice in the Classroom (Opinion). Education Week. Ten Ways to Use Retrieval Practice in the Classroom (Opinion) (relay.edu)
Activity E: Cubes in the Box
If your student must count a set of objects over and over to remember how many there are in the set, he needs additional support with understanding cardinality. If this is the case, teach him Cubes in a Box (Clements & Sarama, 2009). This intervention teaches a student that the quantity of cubes, once counted, stays the same, even if the cubes aren't visible. In this strategy, the teacher has the student count a small set of cubes, and then puts them in a box and closes the lid. Then, she asks the child how many cubes are in the box. Once the student gives an answer, the teacher dumps out the box and, together, the teacher and student count the cubes again to confirm.
Cubes in the Box in Action
Teacher: Count these cubes for me.
Student: 1, 2, 3, 4, 5, 6. Six cubes.
Teacher: You said 6 cubes? Okay. Count them again to make sure. (Student counts 6 cubes again). Okay, now I'm going to put them in a box. (Teacher puts cubes in a box.) How many are there now?
Student: I don't know. I can't see them.
Teacher: Well, how many did we have before? (Teacher dumps out cubes, and student counts 6. Then, teacher puts the cubes back in the box.) So, I didn't add or take any cubes away, so we must have the same number as we had just a moment ago... So, how many do I have in the box?
Student: You still have 6.
Clements, D.H. & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.
Activity F: Count and Move (Given Cubes)
If your student can rote count with some accuracy (count up to 15 or 20), but is struggling to count objects using one-to-one correspondence, teach Count and Move (Clements & Sarama, 2009). Students who would benefit from this strategy make counting errors because they skip objects when they are counting or count the same object more than once. In this intervention, the teacher gives a student a certain number of objects and has him move each one as he counts them, helping him understand that he should say one word for each object.
Count and Move (Given Cubes) in Action
Teacher: Here are some cubes. I'd like you to count each one and move it as you count it. That way, you'll know that you've already counted it. Ready?
Student: 1, 2, 3, 4, 5, 6, 7. (Student skips over a cube)
Teacher: It looks like you missed one. Try again, and count them one by one. Make sure to move the cubes over to the other side, so you know you've already counted them. You'll know you're done when you've moved and counted all of the cubes.
Student, counting and moving all 10 cubes: 10! There are 10! See- I don't have any left.
Clements, D.H. & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.
Activity G: Count and Label
Once your student is able to count and move concrete objects, he will begin to count sets of objects on a page. In this case, the student can't move each object as he counts it and may, once again, begin to skip or double count objects in the set. If this is the case, teach the student Count and Label. In this strategy, a teacher gives a student a picture of a set of objects, and the student has to count and label each one with a number as he counts. When the student has finished counting and labeling the whole set, he should be taught to go back and count again to double-check that he labeled the objects in sequence.
Count and Label in Action
Teacher (pointing to a worksheet with 17 trees drawn on them): Count and label each tree with a number. Remember, you want to start from 1.
Student (counting the first tree): 1... (Student writes a 1.) 2... (Student writes a 2.) 3... (Student writes a 3 and continues to do this for all 17 trees.)
Teacher: I see that you are done. Now, go back and recount all of the trees, making sure you counted and labeled them correctly.
Student: 1, 2, 3, 4, 5, 6, 7, 9. Ooops! That should be 8. I'll change it to 8, and then start counting from the beginning.
Nguyen, H., Laski, E., Thomson, D., Bronson, M., & Casey, B. (2017). Playful Math = Engaged Learning More Than Counting: Learning to Label Quantities in Preschool. http://www.bclearninglab.bc.edu/downloads/Nguyen_Laski_Thomson_Bronson_Casey_2017.pdf#:~:text=Tell%20children%20the%20number%20labels%20for%20Example%3A%20%E2%80%9CSee%2C
Activity H: Touchmath
Once your student is able to count sets of numbers, you'll want to teach him to interpret symbolic representations of numbers. For example, the student needs to learn that the numeral 2 represents a set of two objects. If your student is struggling to interpret numbers, teach her TouchMath. TouchMath is a research-supported multi-sensory approach to teaching students to associate numeric symbols with quantities.
Each numeral from 1 through 9 has "TouchPoints" that align to the digit's value.
As students count the TouchPoints, they associate numerals with real values. They learn that a numeral is more than a mark on a page; a written numeral represents a quantity. Students count aloud as they touch the single TouchPoints once and double TouchPoints twice. This multi-sensory approach supports mastery of single-digit numbers.
Touchmath in Action
Teacher: Count each TouchPoint in 2, beginning by saying the number.
Student (using his finger to point to each TouchPoint as he counts): 2.... 1... (Touching the point at the top of the 2). 2... (Touching the point at the end of the 2).
Activity I: Counting Jar
If your student can rote count to 10 and has some understanding of what skip counting means, one activity to teach him is called Counting Jar. In this intervention, the teacher models how to use group counters and then count them. This strategy supports a student's ability to start seeing sets of numbers as made up of smaller groups, which will help him count more efficiently.
Counting Jar in Action
In the video below, Emily Art models how to use skip counting to count quantities efficiently. Note: In the video below, the student is working with a quantity above 30. As you watch, consider what this strategy might look like if you were working with a quantity below 30.
As you watch, consider: How does this strategy support a student's ability to count a large quantity efficiently?
Counting Jar – KIPP DC Math. (n.d.). Retrieved February 18, 2023, from https://kippdcmath.com/counting-jar/
Once a student knows how to count sets of objects of a certain size, she can learn to build sets of objects as well. In order to support a student's ability to represent a quantity from 0-30 (showing or building the quantity), the following interventions can be used:
Activity J: Count and Move (Representing Cubes)
If your student can count and move a certain number of objects, but has trouble representing a quantity (for example, he can count 10 cubes if he's given them, but he can't take a large group of cubes and count 10 of them out), teach Count and Move (Representing Blocks) (Clements & Sarama, 2009). Students who would benefit from this strategy make counting errors because they struggle to count the cubes and keep track of how many they have so far. In this intervention, the teacher gives the student a large pile of cubes and asks him to count out a certain number (such as 10). However, to do this, a student must attend to two actions simultaneously: the student must move and count the cube at the same time. Therefore, after the student counts and moves the cubes, the teacher should direct him to go back and recount the cubes to check his work.
Count and Move (Representing Cubes) in Action
Teacher: Here are a whole bunch of cubes. I'd like you to count out 10 of them. When you do this, move each cube away from the big pile as you count it. That way, you'll know that you've already counted it. Ready?
Student: 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10. (When student moves cubes, he counts 8 twice)
Teacher: Now, count your pile.
Student (counting all 11 cubes): Oops, I counted 11, not 10.
Teacher: Now, take your 11, and count 10 from there, moving each as you count.
Clements, D.H. & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.
Activity K: Represent and Label
Once a student can represent a set using concrete objects, you'll want to teach him how to represent a set of objects using visual representations of those objectives. If your student is struggling to represent numbers from 0 to 30 on paper, you can teach him to Represent and Label. In this strategy, a teacher gives the student a paper and asks him to draw a certain number of objects, counting and labeling each object as he draws it. When the student has drawn the entire set, he should go back and count again to make sure that he has not made a mistake in his counting.
Represent and Label in Action
Teacher (point to a worksheet): Draw and label 17 trees. Remember, you want to start from 1. (Student counts and labels 17 trees).
Teacher: I can see that you are done. Now, can you go back and recount your trees, making sure you have 17?
Student goes back and recounts: 1, 2, 3, 4, 5, 6, 7, 8, 8. Oops! I counted 7 twice. I'll fix that, and then recount from the beginning.
Activity L: One More!
If your student is able to represent a small set of objects (such as quantities up to 5), but then gets stuck or miscounts when asked to represent larger numbers, teach One More (Clements & Sarama, 2009)! In this intervention, the teacher has the student count two objects. Then, the teacher has the student add one and asks him "How many now?" The teacher directs the student to count on to answer. Then, the teacher continues to add another counter, until they have counted to 10. Then, the teacher asks the student to count on from a different starting amount (such as 5 instead of 1). When the student is able to do this activity easily, the teacher can add an additional layer of challenge by adding 2 or 3 more, instead of just one more.
One More in Action
Teacher: Here is one counter. How many are there if I add one more (teacher moves counter next to other counter)?
Student: Two.
Teacher: Great! Now, I'll add another counter. Now, how many are there?
Student: Three. (Teacher and student continue to do this until they hit 10).
Teacher: Okay, now we'll try something new. I'm going to put 5 counters here. How many? (5.) Now, I'm going to ask you to count on as I add more. (Teacher adds two more counters.) How many?
Student: 5, 6, 7. 7!
Clements, D.H. & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.
In order to support a student's ability to compare the size of two quantities from 0-30, the following intervention can be used.
Activity M: Counting Towers
If your student is struggling to compare the sizes of sets of objects (especially sets greater than 10), teach him Counting Towers (Clements & Sarama, 2009). In this intervention, the student uses counters to make one tower as high as he can. Then, the student makes another tower (or compares his tower to a tower that another student has constructed). The student then needs to guess which tower has more, based on the relative height of the towers, and estimate how many counters are in each tower. The student should then use the towers to draw a conclusion about the relative size of the quantities.
Counting Towers in Action
Teacher: Make a tower using as many counters as you'd like. While you do that, I'm going to make my own tower. Then, you are going to tell me which tower you think has more counters in it! (Student and teacher build towers.) Okay, which tower has more counters in it?
Student: Mine!
Teacher: How do you know?
Student: Because my tower is taller!
Teacher: Can you estimate how many counters are in each tower?
Student: Well, I used 25, and mine's just a bit taller than yours, so I think you used 20.
Teacher: I actually used 21. So which number is bigger, 25 or 21?
Student: 25.
Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for counting whole numbers 0-30.
Teacher (giving the student 7 cubes): "Count and move these cubes."
Student (counting and moving each cube): "1, 2, 3." (Student stops and then counts
other cubes, starting at 1.) "1, 2, 3, 4... 4 cubes."
In such a case, what might you do?
When you are planning your lessons, you should anticipate that your student will make errors throughout. Here are a series of prompts that you can use to respond to errors. Keep in mind that all students are different, and that some students might respond better to some types of feedback than to others.
| Level of Support | Description of Scaffold | Script |
|---|---|---|
| Smallest Scaffold | Try Again! In order to get him into the habit of checking his work, remind a student to recount from the beginning. | "Can you count your cubes again from the beginning?" |
| Medium Scaffold | Back it Up. If a student is struggling, back up your process. Put all the cubes back together, and ask the student to remember to count each one as he moves it. | "When you just counted, you started from 1, and moved some cubes. Then, you started from one again and moved the rest, This time, start from one and keep counting until you count all the cubes in the pile." |
| Highest Scaffold | Model. If the student continues to struggle, model how you would count and move the cubes, and then recount to check your answer. | "I can see that you are stuck. I'll show you how I count and move the cubes. 1, 2, 3, 4, 5, 6, 7. Now, I'll recount to check that I counted correctly. 1, 2, 3, 4, 5, 6, 7." |
If your student struggles to meet your objective, there are various techniques that you might try in order to adjust the activity so as best to meet your student's needs.
| Activity | Description of Strategy | Script |
|---|---|---|
| All Activities | Choral Count. If a student struggles when counting numbers, count with him. | "I'll count with you!" |