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).
Once a student is able to count, interpret, represent, and compare quantities up to 30, he is ready to apply the same skills to larger whole numbers. Building fluency with numbers greater than 30 requires developing one's understanding of place value and one's ability to apply this understanding to identify and represent numbers. If your student is just developing his understanding of whole numbers above 30, you will want to have him identify and build the quantity using concrete objects (e.g., base ten blocks). If your student has mastered identifying or building numbers above 30 using concrete objects, you can teach him to identify and build numbers using abstract representations (such as drawing base 10 blocks or writing numerals). 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 align 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 above 30, which will support his understanding of place value, you should start by explicitly teaching the skill. This sounds like:
In order to support a student's ability to verbally count numbers above 30, the following interventions can be used:
Activity A: Rote Counting (by ones above 30)
If your student is unable to count from above 30, teach him Rote Counting (above 30). In this strategy, a teacher helps a student practice counting by repeated practice. A teacher may also consider counting in a chant or song form to help the student remember the numbers. If your student has no trouble counting to a certain number (such as 33), but then gets stuck (at 34), have him start counting slightly below the number in which he gets stuck (such as 20 or 30). This will help him be able to count enough numbers to keep the pattern in mind without having to repeat all the numbers he already knows.
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 100! Ready?
Teacher and student choral count to 100.
Teacher: Nice work counting to 100! Now, you'll count to 100 on your own.
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 Hundred Chart (by ones above 30)
Another strategy to use if your student is struggling to count whole numbers above 30 is Counting with Number Lines or Hundred Charts. This strategy mirrors the former strategy, but gives the student a visual to use as he says each number. When teaching counting with one of these tools, the teacher should refer to the tool as she counts with the student, pointing to each number as she counts.
Counting with Number Lines or Hundred Charts in Action (above 30)
Teacher: We are going to count up to 100! As we say each number, I will point to it on the number line. Ready?
Teacher points to a number on the number line and choral reads with student up to 100.
How Do Hundred Charts Teach Place Value and Skip Counting? (n.d.). ThoughtCo. https://www.thoughtco.com/hundred-charts-place-value-and-multiplication-3110499
Activity C: Skip Counting with Number Lines or Hundred Charts (above 30)
Once your student gains fluency with counting whole numbers above 30, it's important to teach him how to skip count with numbers that he will use as he learns about place value. For example, when a student learns about place value with base ten blocks, he must be fluent in counting by 2s, 5s, 10s and 100s. Skip counting by these numbers will support a student's ability to count base ten blocks efficiently. Therefore, this intervention sets up students to be able to develop this skill. In order to help your student develop this skill, teach him Skip Counting with Number Lines or Hundreds Charts (above 30). When teaching a student to skip count, the teacher can visually support his understanding with a number line or hundred chart, pointing to each number as they count.
Skip Counting with Number Lines or Hundred Chart in Action
Teacher: We are going to count by 10s up to 100! We will skip count, pointing to each number on the hundreds chart.
Teacher (pointing to each number as she and students read it on the hundreds chart): 10! 20! 30! 40! 50! 60! 70! 80! 90! 100!
How Do Hundred Charts Teach Place Value and Skip Counting? (n.d.). ThoughtCo. https://www.thoughtco.com/hundred-charts-place-value-and-multiplication-3110499
Activity D: Skip Counting On by 10s with a Number Line
If your student is able to count by tens, and to count by ones through 30, but struggles to skip count numbers that aren't whole tens, teach him Skip Counting on by 10s with a Number Line (Clements & Sarama, 2009). In this intervention, a teacher uses a number line to help a skip count by 10s after starting at a given number (such as 23).
Skip Counting On by 10s with a Number Line in Action
Teacher: I'm going to show you how to skip count after I've counted on. Let's say I've counted up to 23. Now, what if I wanted to skip count by tens from there? I can use my number line to count as I go. Ready? 23! 33! 43! 53! Great! Now, let's count up to 14, and then we'll skip count by 10s.
Student (having counted on to 14): 14! 24! 34! (Teacher points to number line as student reads each number).
How Do Hundred Charts Teach Place Value and Skip Counting? (n.d.). ThoughtCo. https://www.thoughtco.com/hundred-charts-place-value-and-multiplication-3110499
In order to support a student's ability to understand cardinality and interpret a quantity above 30, you'll want to intervene to support your student's understanding of place value, or that the position of numbers indicates their value.
Activity E: Counting Jar (above 30)
If your student can rote count to 30, can skip count by 5s and 10s, and is able to count objects to at least 30, one activity to teach him is 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 (above 30) in Action
In the video below, Emily Art models how to use skip counting to count quantities efficiently.
As you watch, consider: How does this strategy help a student learn when skip counting is more efficient than counting by ones?
Counting Jar – KIPP DC Math. (n.d.). Retrieved February 18, 2023, from https://kippdcmath.com/counting-jar/
Activity F: Count Ten Frames
Another strategy to use to help your student more accurately count sets of objects above 30 is Ten Frames. In this intervention, the student places single counters on ten frames. This tool can be used with a student who is developing his understanding of place value because it helps a student quickly organize his counters into groups of ten when modeling larger numbers.
Count Ten Frames in Action
Teacher: This is a ten frame. It helps organize the counters into groups, which makes it easy for us to count them. Each of these rows is 5 counters, which means if the ten frame is full, there are 10. Watch me count these objects using my ten frame. 10, 1, 2 , 3, 4. 14. Now, you try.
Activity G: Count Base Ten Blocks
Once a student has developed skills for counting a quantity above 30 and has a grasp on counting cubes (which are single units, or objects), he is ready to use base ten blocks to count by ones, tens, and eventually hundreds to identify how many there are in a set. In this strategy, a student learns that single units can be represented by base ten blocks, 10 single units can be represented by a 10 rod, and 100 single units (or ten 10 rods) can be represented by a 100 block. In this intervention, the teacher models how to count by 10s and 1s to identify the number shown using base ten blocks, and then asks the student to do the same.
Count Base Ten Blocks in Action
First, watch the short video in which Emily shows the student how a 10-rod is the same as 10 ones. Then, read the script below to solidify your understanding of how to build on this understanding.
As you watch, consider: How do the prompts that Emily uses help the student interpret base ten blocks?
Teacher: Base Ten Blocks are sometimes used to represent bigger numbers. I have a single cube that's worth one. Then I have this rod, which is bigger. How many single cubes do you think there are in this rod? Watch what I can do to figure it out. (Teacher lines up 10 cubes and then puts a 10 rod next to them). What do you notice about these cubes and this 10 rod when I put them next to each other?
Student: They are the same length?
Teacher: Exactly. So the rod is worth ten. When we see 10 rods, we can start by skip counting them by 10s, and then count the single cubes, or ones, after. (Teacher models how to count 4 rods and 6 ones to identify the total amount in the set) 46! Now, you can try. Can you count these cubes? (Teacher lays out 3 ten rods and 2 single cubes.)
Student: 10, 20, 30, 1, 2... 32 total!
Base-Ten Blocks | Lesson plan | Education.com. (n.d.). Www.education.com. https://www.education.com/lesson-plan/el-support-lesson-base-ten-blocks/
In order to support a student's ability to represent (show or build) a quantity above 30, you'll want to use the following interventions:
Activity H: Organize Counters
If your student is able to count counters, but struggles to build numbers using counters, teach her to Organize Counters into smaller groups of objects and then skip count to keep track of how many she has in all. In this strategy, the teacher tells the student a number (such as 52) and then asks that she build that number with counters, reminding her to build and count in groups.
Organize Counters in Action
Teacher: Here is a box of counters. I am going to teach you how to build numbers by organizing these counters into groups. Watch as I build 52. First, I will organize my counters in groups of 5. (Teacher makes groups of 5, and counts as she makes them). 5, 10, 15, 20, 25, 30, 35, 40, 45. Okay, so I need one more group of 5. (Teacher makes one more group of 5.) 50. Now, I know that if I add another 5, I'll have more than 52, so now I'm going to count by ones: 50, 51, 52... Now, I'll recount all of my groups and singles to make sure I accurately represented 52. (Teacher recounts all groups and singles.)
Teacher, T. A. (2020, August 18). How to Organize Math Manipulatives Like a Boss. The Average Teacher. https://www.theaverageteacher.com/how-to-organize-math-manipulatives/
Activity I: Represent Ten Frames
If your student is able to count with ten frames, but struggles to build numbers above 30, teach him to Represent Ten Frames. In this intervention, the student builds a number using a ten frame. As you use this tool, teach the student to fill up each ten-frame one at a time to model the value of the digit in the tens place, and to then partially fill the last ten frame to model the value of the digit in the ones place. This strategy will help your student see two-digit numbers as composed of tens and ones.
Represent Ten Frames in Action
Teacher: We are going to use ten frames to build numbers. This is a blank ten frame, because it doesn't yet have any objects in it. We are going to learn to build and count numbers using this ten frame. Watch as I do it first: I am going to build 14. So, I know that one ten frame is 10, so I will draw a circle in each of these squares to give me 10. (Teacher fills out first ten frame). Now, I need 4 ones, so I will fill in 4 squares on my next ten frame. 1, 2, 3, 4. Now, I'll recount to see if I represented 14 accurately. 10, 11, 12, 13, 14.
Forbringer, L., & Fuchs, W. (2014). RtI in Math: Evidence-Based Interventions for Struggling Students. Hoboken, NJ: Routledge Ltd
Activity J: Represent Using Base Ten Blocks
If your student is able to count base ten blocks, but struggles to build numbers above 30, teach him to Represent Base Ten Blocks. In this strategy, a teacher models how to build a given number using base ten blocks, and then gives the student time to practice this strategy.
Forbringer and Fuchs (2014, p. 103) state that place value mats can also support a student's ability to Represent Base Ten Blocks:
To further solidify their understanding, students can arrange the blocks on a place-value mat, a paper or plastic mat divided into columns, with each column labeled with a different place value.
Place-value mats help students transition from concrete blocks to abstract numbers because the mats provide a two-dimensional graphic representation that forces learners to organize the blocks in the same order they will use when writing numbers. Initially, students place actual blocks on the mats. When they are ready to progress to the visual representation level, they can simply draw the blocks on the mat. Explicitly linking the blocks on the mat with the abstract symbols in both expanded notation and standard form will help students understand place value within the base-ten system.
Click here to download a place value mat.
Represent Base Ten Blocks in ActionTeacher: Base Ten Blocks help us count bigger numbers. Watch what I do as I build the number 42. (Teacher counts by tens as she pulls out each 10 rod.). 10, 20, 30, 40... Okay, now I need to switch from counting by tens to counting by ones, so I'll take single cubes and count on until I get to 42. 41, 42... Now, I'll recount my base ten blocks. 10, 20, 30, 40, 41, 42... 42 blocks! Now, you try to build 31. Remember, you'll want to count your ten rods before your ones.
Forbringer, L., & Fuchs, W. (2014). RtI in Math: Evidence-Based Interventions for Struggling Students. Hoboken, NJ: Routledge Ltd
Represent Using Base Ten Blocks (and a Place Value Mat) in Action
Activity K: Build It
If your student no longer requires concrete models to understand place value, you can teach him to represent and compare quantities through a strategy called Build It. In this strategy, a student refines his understanding of place value by associating the position that each digit is in with that digit's magnitude. You can reinforce this understanding by giving your student cards with single digits on them and having him use these cards to build numbers. You might ask your student to manipulate the digits to build and record multiple numbers and then compare the sizes of the numbers he builds. Once he can do this, you might ask him to use his digits to build the smallest or largest numbers he can and then ask him to defend his answer.
Build It in Action
Watch as Ama Awotwi introduces an activity called Build It. How does this intervention reinforce understanding of place value?
Note: We recommend using the term greatest or largest in place of the word "highest."
How to Teach Number Representations. (n.d.). Www.understood.org. https://www.understood.org/en/articles/number-representations-an-evidence-based-math-strategy
In order to support a student's ability to compare quantities above 30, the following intervention could be used:
Activity L: Build It & Compare It
If your student understands place value but struggles to draw comparisons between different numbers, teach him to Build It and Compare It. This strategy is a variation of Build It in which the student uses digit cards to make a number. In this intervention, both the teacher and the student use digit cards to build numbers, and then the student must compare the two numbers and identify which is greater.
Build It & Compare in Action
As you watch, how does this intervention facilitate a student's ability to compare numbers above 30?
Note: We recommend using the term greatest or largest in place of the word "highest."
Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for using counting whole numbers above 30.
Teacher (giving the student base ten blocks that make 117 total): "How many blocks are
there total?"
Student (counting the blocks as she moves them): "100, 10, 1, 2, 3, 4, 5, 6, 7... There are 7
blocks total!"
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 blocks again from the beginning? This time, remember to count on." |
| Medium Scaffold | Back it Up. If a student is struggling, back up your process. Put all the blocks back together, and ask the student to remember to count on as he moves them. |