Seeing the Math You Teach – Part 3

Hands-On Math, Professional Learning, Teaching and Learning

Math Makes Sense

By Katie Basham

I wish I had known that when I was a young, impressionable student. Maybe some kids knew it—maybe they even believed it—but I certainly didn’t. I was the kid counting on my fingers, skip-counting under my breath, and hoping no one would notice. I thought of these methods as “less-than” strategies and felt a pit in my stomach at the thought of others noticing me using them.

Everyone Can Be Good at Math

Also true. And I wish my teachers had known that, too. Maybe then I would’ve felt proud of my own strategies, instead of ashamed. Because here’s the truth: success doesn’t belong to a select few. It’s not limited. There’s enough of it for every single student to feel it, own it, and believe in it.

These two beliefs—math makes sense and everyone can be good at math—are the foundation of Seeing the Math You Teach. Success shouldn’t be reserved for an elite few. All students can succeed when they understand that math is really about visualizing and explaining abstract thinking. Because when students can see the math, they can understand it….and, when they understand it? It makes sense.

What a privilege we have as educators.
 It’s no longer our job to bestow knowledge from on high. Our mission is to create learning environments where students use concrete materials, build visual representations, and gradually progress to symbolic understanding. This book is designed to empower you—to help you see and understand the math you teach, so you can help your students do the same.

Take rounding, for example.
 I recently asked a respected educator how he would explain it. Here’s how the conversation went:

Me: “What does it mean to round a number?”
Him: “Five or more, raise the score. Four or less, let it rest.”
Me: “But what does that mean?”
Him: “It means if the number ends in a five…”

As you can see, we weren’t quite getting to the heart of it.

“Rounding” isn’t really something you do, like an action—it’s more about understanding where a number falls in relation to benchmarks or friendly numbers. In other words, rounding is determining which number a given value is closest to.

  • What number is nineteen closer to?
  • What number is 18.34 closer to?
  • THAT’S what rounding is.

Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.

The student who can show rounding on a number line isn’t just repeating a rhyme—they’re seeing the math. They’re understanding the concept. And that, at its core, is what this book is about.

We want you—and your students—to experience success.

Because: Math Makes Sense & Everyone Can Be Good at Math!

Interested in Workshops by Kim Rimbey?

Kim offers workshops on a variety of topics, including Building Thinking Classrooms, Ten-Frame Mathematics, Mastering Math Manipulatives, Small-Group Math Instruction, and See the Math You Teach: A New Vision for Math Teaching and Learning. View Kim’s PD Catalog to learn more.

 

In the Pursuit of Fluency Part 1

Professional Learning, Teaching and Learning

Math Fluency What is fluency? As a literacy specialist in my early career, I equated the word “fluency” with the ability to read with automaticity, accuracy, and prosody (expression using the patterns of rhythm and sound). Simply put, a fluent reader expresses herself fluidly and with expression. I understood fully that fluency is not the same as comprehension, and yet it contributes immensely to the ability to get to the higher levels of thinking required to comprehend well.

Several years later, as I shifted toward a mathematics specialization, I discovered that there is a mathematical equivalent to reading fluency. Much like reading fluency helps children succeed in higher levels of comprehension, mathematical fluency plays a foundational role in helping children succeed in problem solving. Therefore, just as students who struggle with reading fluency find comprehension tasks difficult, students who struggle with mathematical fluency find problem solving tasks difficult. For me, thinking about this in the form of an analogy helps illuminate the relationship:

Reading Fluency : Comprehension :: Mathematical Fluency : Problem Solving

So…what is mathematical fluency, and how can we develop it as a foundation that leads to stronger problem solving? This topic has many layers, and we’ll be revisiting it often over the next several weeks. For now, let’s take a look at what it is not.

Mathematical fluency cannot be reduced to memorization. This limited view of fluency puts undue pressure on students to focus on rote recall of facts. The mathematics community has agreed upon four qualities that comprise mathematical fluency:

  • Accuracy: finding the correct answer
  • Efficiency: using strategies or methods that allow for ease and flow while solving
  • Flexibility: knowing more than one approach to solve a problem and then selecting one that provides for efficiency
  • Appropriateness: knowing when to apply a particular procedure, strategy, or method

Students who are mathematically fluent are able to choose flexibly from a variety of methods and strategies to solve problems, and they solve them accurately and efficiently.

Mathematical fluency is not limited to knowing and using basic facts. Rather, it is much broader, involving concepts such as number sense, math facts, multi-digit operations, fraction operations, problem solving — and the list goes on (see below for more on this). Children need opportunities to develop fluency with the concepts on which they are focused at any given time. Students at the elementary level should have opportunities to develop fluency in at least four areas:

  • Number Sense: thinking about numbers and number relationships fluidly and flexibly.
  • Math Fact Fluency: adding numbers through 10+10 and the related subtraction facts; multiplying numbers through 10×10 and the related division facts.
  • Operational Fluency: using strategies to add, subtract, multiply, and divide whole numbers, decimals, fractions, and integers.
  • Problem Solving Fluency: selecting from and applying a variety of strategies and methods with the goal of solving a contextual or mathematical problem.

Mathematical fluency does not just happen spontaneously. Fluency develops over time as students are given multiple opportunities to focus on mathematical relationships and manipulations, often as a result of engaging in deliberately designed and sequenced experiences. Here are a few examples to get you started (more to follow in the coming posts):

  • Number Sense: use ten frames to help students understand relationships among numbers, both large and small. Ten frames needn’t be restricted to numbers within ten – they can be used to explore multi-digit numbers, decimals, and integers, as well! At KP Mathematics, we call this the “infinite ten frame,” and I’ll share more during this fluency series.
  • Math Fact Fluency: use card and dice games to allow students to practice their math facts in ways that are enjoyable and build confidence. For example, Double War is a simple card game where pairs of students each flip over two cards and either add or multiply the two numbers. Each student calls out his sum or product. The student with the greater (or lesser) sum or product takes all four cards. Repeat until all cards are used.
  • Operational Fluency: implement daily Number Talks with your students. This robust-yet-simple daily routine helps students develop number sense and mental math skills while focusing on developing operational fluency.
  • Problem Solving Fluency: engage students in daily problem solving, giving them tasks that require them to use strategies other than standard algorithms. For example, engage young children in solving a division problem before teaching them “how” to divide. Observe how they approach the process. Provide them with tools and manipulatives to guide their thinking. Resist the temptation to jump in – let them struggle. They will amaze you!!!

If you’re wanting more ideas in each of these categories, tune in for the next few weeks as we unwrap each category and share specific classroom-tested ways to develop fluency with your students.

For now, let’s carry on this conversation. What is your understanding of math fluency? What do you do in your classroom to develop fluency among your students? Please leave your comments in the boxes below.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.

 

The Equal Sign Means Relationship, Not Action

Math, Math Content, Math Discourse, Math Standards, Professional Learning, Taking Care of Teachers, Teaching and Learning

Equal Sign Means Relationship

Blog post #1 in the series, Lies We Tell Our Students

The Story

A while back, I was working in a fifth-grade classroom, and as a warm-up, I put the following equation on the board:

8 + 4 = 1 + 11

I then asked the students, “Is this statement true or false?” After waiting several seconds, I asked them to raise their hands to indicate their votes. At that point, two students voted for true, and the other 22 students voted that the statement was false.

Being the good teacher that I am, I asked the students to discuss their thinking, believing that those two students who said it was true would quickly convince the others. However, when I brought the class back together and once again asked my questions, 24 hands went up stating that the answer was false.

“Yikes!” I said, “What’s going on here? This statement is actually true. Can you tell me what you’re thinking?”

“Mrs. Rimbey,” started one of those original two students, ”8 + 4 does not equal 1.”

The Lie

I’ve since come to learn that this is not an isolated case. Many students across the grade levels believe that the equal sign is a symbol that indicates the action of finding the answer, when, in fact, it actually signals a relationship between the expressions on each side. In the vast majority of classrooms, students develop this misconception because virtually every equation they see in the early grades follows the same format: 3 + 3 = 6; 4 x 4 = 16; 20 – 13 = 7; and so forth. The “answer” appears last.

The Truth

So, what do we do about this? We can be sure to provide students with multiple daily opportunities to see equations written in different formats, pointing out that the equal sign indicates an equal relationship, not the action of finding the answer. Provide examples with the “answer first” (e.g., 6 = 2 + 4), with “nothing to do” (e.g., 1000 = 1000), or with “no answer” (e.g., 32 + 42 = 20 + 5).

The Series

This blog post is the first of a series that addresses the lies we tell our students. It’s important to note that these lies are not deliberate untruths. Rather, they are rules, procedures, mnemonics, and tricks we teach kids in an effort to make math easier for them. Sometimes, the lies come in the form of omission, such as the example of the equal sign shared above, where we don’t provide enough varied examples. Sometimes, they come in the form of half-truths, such as addition and multiplication, which always make bigger, which is only true with natural numbers (not including 0).

In Summary

To sum up this week’s Lies We Tell Our Students:

  • The inadvertent lie: The equal sign indicates the action of finding the answer.
  • The truth: The equal sign indicates relationship, not action.

As always, we love hearing from you! Do you think your students fully understand the meaning of the equal sign? Have you tried any of these (or other) strategies with your students? Please share your thoughts in the comments box below.

Next Steps for Teachers: Use a variety of equation structures with students on a daily basis (see examples above). Consider making a set of 4-6 “true-false cards” each day to show the class. Simply write one equation on each card, using a variety of formats, making some true and some false. Then ask the students whether each is true or false and to justify their answers.

Next Steps for Leaders: Discuss the meaning of the equal sign with grade levels, focusing on the notion that the equal sign indicates a relationship, not the action of finding the answer. Although this concept typically appears in the K-1 math standards, this misconception continues to exist in middle school and beyond. Discuss ways to avoid this misconception in each grade level, and then share their ideas across grade levels.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.

In the Pursuit of Fluency Part 2: Number Sense

KP® Ten-Frame Tiles, Math, Teaching and Learning

Number SenseWhen you think about the term “math fluency,” what comes to mind? My guess is that the first thing, and perhaps the only thing, that comes to mind is math fact fluency. After all, it is extremely important that our students know their math facts within 10+10 and 10×10, from memory, before they can find success with higher levels of mathematics, isn’t that right?

Few would disagree with this statement. Myself included. And yet, as educators, we must understand that math fact fluency is only one part of “math fluency.”

Most recently, the National Council of Teachers of Mathematics (NCTM) has defined mathematical fluency as accuracy, efficiency, flexibility, and appropriately working with a variety of foundational mathematical ideas. Yes, this includes math facts, and it also includes other math concepts such as quantity, place value, multi-digit operations, fractions, decimals, percents, equations, functions…and the lists goes on. And, of course, as one progresses into higher levels of math, new fluencies become necessary for success. So, yes, math fact fluency is foundational, important, and necessary, but not exclusive.

Therefore, today we focus on number sense fluency. Number sense “refers to a child’s fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparisons.” (Gerston & Chard, 1999, quoted in Hornigold, 2017). Number sense develops over time as students engage in activities that facilitate playing with numbers, learning to see relationships within and among numbers and becoming increasingly flexible in the ways in which they think about numbers. And as children develop number sense, they should develop accurate, efficient, flexible, and appropriate ways of thinking about numbers, thus making “number sense fluency” an important building block.

Successful math students develop number sense fluency within various number ranges – numbers within 10, 100, and 1,000,000; place value; decimals; fractions; percents; ratios; rational numbers; irrational numbers; and so on. And becoming fluent with number sense concepts continues beyond the early grades as mathematical concepts become increasingly sophisticated.

For today, let’s take a look at some practical ways to develop number sense fluency in the elementary years, beginning with numbers within 10 and continuing through fractions.

Numbers within 10

Young children develop a sense of quantity and comparative quantities through a variety of activities. For example, when working with the number “5,” a child might count out 5 objects and then decompose the total into “1 and 4” or “3 and 2.” This task can be made visual by using two colors of tiles, for example, or by placing counters on a ten frame.

Speaking of ten frames, the use of  ten-frames can help a young child develop a sense of “ten-ness” (a component of number sense). Placing 7 counters on a ten frame, for example, can help him/her see the relationship between 7 and 10: “7 is 3 less than 10”, and “7 is 2 more than 5” are two observations the child might make.

Place value

A critical factor in understanding number and operations with very small and very large numbers is knowing how numbers are constructed. Finding various ways to compose and decompose whole numbers and decimals, especially around the idea of tens and magnitudes of ten, plays a vital role in building place value understanding.

To help make this notion visible, KP Mathematics has created the notion of the “infinite ten frame,” a unique way of helping children solidify place value understanding. Ten frames, primarily used as a structure for helping young children understand relationships among quantities within ten, have been around for decades. With KP Ten-Frame Tiles, students can build “nesting” ten frames that extend the ten-frame representation beyond ten. Students place 10 ones onto a small ten frame. When the small ten frame is filled, they place a cover on top to represent 1 ten. Then, they can collect the tens on a large ten frame. When the large ten frame  is filled with 10 tens, they cover to represent 1 hundred. One more iteration creates 1000. At that point, the process continues with drawings. Click here to see a video example.

Students can use this same “nesting” process to explore decimal fractions, as well. The large ten frame becomes a unit of “1.” Students lift the cover to view 10 tenths. Then, they remove the tenths-covers to reveal 100 hundredths. This powerful representation does wonders for helping students carry out, visualize and internalize decimal fraction concepts and operations.

Number Relationships

Understanding the relationships among numbers is an important building block of number sense fluency. For example, a teacher might present four numbers to her class:  23, 20, 15, 25, and ask them to select the number that is not like the others. Of course, this particular exercise allows for multiple responses:

  • 23 is the only prime number.
  • 20 is the only even number.
  • 15 is the only number not in the twenties.
  • 25 is the only square number.

As you can see, this brief encounter might spark a lively conversation as students begin justifying their responses and realizing that there is more than one correct response. (Thank you to Mike Askew for this great example.)

Fractions

Students should develop a deep sense of fractions just as they do with whole numbers and decimals. By drawing pictures, placing fractions on number lines, folding and cutting paper, and using mental strategies to decompose fractions, students gain a deeper understanding of what fractions represent. For example, knowing that ¾ is composed of ¼ and ½ helps a child understand that ¾ is greater than ½ and less than 1 whole. Such comparisons support a much more robust sense of number than when students use an equivalent fractions strategy to compare. Knowing that 1/3 is less than ½ and that 3/5 is greater than ½ provides another solid way to compare 1/3 and 3/5 rather than finding common denominators. Knowing that 7/8 is closer to 1 whole than 5/6 because the “missing piece” is smaller provides another line of logic that creates a robust sense of number.

As mentioned before, number sense fluency develops over time and requires numerous carefully-crafted opportunities through which students make progress. This post, lengthy as it is, provides only a few examples. Most commercial textbooks neglect to offer the types and quantities of experiences children need in order to develop number sense fluency. We owe it to our students to build the foundations they need for success.

What activities have you done to develop number sense at your grade level(s)? What else would you like to know? Please take a moment to share your thoughts in the comments box below.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.

 

In Pursuit of Fluency Part 3: Fluency with Early Number Sense

Game Play, Hands-On Math, KP® Ten-Frame Tiles, Professional Learning, Teaching and Learning

Early Number SenseWhen Josie, a young athlete, had the goal of making her middle-school basketball team, she spent time preparing for tryouts. She knew she needed to learn to dribble, shoot, guard, throw, and jump with ease – all moves necessary to be a successful basketball player. Josie worked with her brother for hours each week, focusing on drills to help her master each move until it became effortless. Whenever she and her brother play-practiced games, her improvement exceeded her expectations. Finally, it was time for tryouts. Josie succeeded with flying colors and made the team. Because she had already mastered every single move, she never had to practice or drill again. Right??? Wrong! Our young athlete had to continue drilling to refine her skills and to maintain her proficiency.

The same is true with mathematics. Students must spend time in repetitive yet motivating activities that develop foundational skill fluency and competence so they are able to perform increasingly complex tasks. Then, after achieving fluency, they must continue to practice to maintain the fluency and competence they worked so hard to attain.

In Part 2 of this Fluency Series, we took a broad look at number sense and why it’s important for students to be fluent with these concepts. Internalizing numerical relationships requires fluency. When students become accurate, efficient, and flexible with numbers and operations, they are better equipped to problem solve and to use mathematics to describe the world around them. Otherwise, their working memory becomes bogged down with unsophisticated thinking and basic counting to the point that they cannot hold onto the rich and useful mathematical ideas that help with rich tasks.

Here, in Part 3 of this Fluency Series, we take a look at what it means to be fluent in early number sense.

Building Numbers Within Ten (Early Number Fluency)

The 0-10 number concepts developed in pre-K, kindergarten, and 1st grade lay the foundation for understanding our entire number system (more on this in part 4). Thus, it is critical that our youngest children become fluent in the language of building numbers.

Subitizing. Instantly recognizing quantities (subitizing) and their relationships to other quantities plays a major role in children’s building early number fluency. With practice, children can become accurate, efficient, and flexible in identifying quantities at a glance.

My favorite way to get started is using five-frames and ten-frames. By using these structures, students recognize quantities in relationship to 5 and 10. For example, they see that 4 as one less than 5, 7 as 2 more than 5, and 9 as one less than 10. The ten-frame provides a powerful way for children to visualize quantities without having to count every time.

In addition to using the ten frame structure, children might also use unstructured sets of counters, pattern blocks, etc. for subitizing opportunities. When a set of 3-10 counters is revealed, students might use spatial skills (I see five tiles because there are 3 on the top and 2 on the bottom) or color (I see 7 blocks because I see 3 yellow, 2 red, and 3 blue) to help them recognize quantity.

Subitizing activities are often done with a whole group, but these activities can also be done with children  in pairs and small groups. One child plays the “teacher” who places a specified number of objects  (no more than ten) in a cup and “spills” the counters for a partner to subitize. The important part of this “game” is that the students have the opportunity for repeated practice to build fluency.

For those of you working with older students, note that this “game” works with students up to 5th grade . It reminds them to use their visualization skills to identify quantities. (We’ll extend this idea further in Part 4 of this series).

Composing Numbers Within 10. Young students should also be fluent in identifying various ways to compose and decompose numbers within 10: identifying number pairs that compose a specified quantity (e.g., 5 can be composed of 1+4, 2+3, and 0+5) as well as multiple addends (2+1+2 = 5). While they are not necessarily composing and decomposing symbolically, they are developing those skills by using a variety of objects and drawings. The subitizing activities listed above can also be used for composing numbers within ten using ten-frames, pattern blocks, counters, etc.

Composing 10 With Number Pairs. As mentioned in the previous post, the idea of making a ten lays a strong foundation for later work. Students should be able to name the number pairs that make ten (1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5) as well as name the missing number when one addend is named (e.g., what goes with 7 to make ten?).

A great game for developing fluency with composing 10 with number pairs is Making Tens. Two students work together to make as many tens as possible. They need only a six-sided number cube, several ten-frames, and objects to place on the ten frames.

  1. Player A rolls a die and places that many counters on a ten frame.
  2. Player B announces how many more are needed to make a ten and then rolls the die.
  3. Player B counts out the “rolled” number of counters into his/her hand.
  4. Player B fills the current ten-frame, if possible. If there are extra counters, s/he places them on a new ten frame.
  5. Player A then announces how many are needed to fill the new ten frame and repeats the process.
  6. The players repeat steps 2-5 until all the ten frames are filled.

Building the First Ten

Young students must also go beyond making tens to making one group of ten. Making the shift from ten ones to one group of ten is a huge developmental jump! Understanding that one group of ten and ten individual ones concurrently exist proves to be difficult for most young children. It’s important to provide students with many opportunities to build fluency with making one ten from ten ones.

A simple way to do so is to play the Making Tens game described above, only this time providing a cover for each ten frame. KP Ten-Frame Tiles work great for this, but you can also use paper ten frames and blank paper to serve this purpose. Once students fill a ten frame with ten counters, they should place a cover on the ten frame to represent one ten. They can lift the cover and see that there are still ten ones inside and then replace the cover to represent one group of ten.

Building Numbers With Tens

In the next blog post, we’ll examine ways in which these activities can be “grown up” for use with multi-digit whole numbers and with decimal fractions. You’ll be amazed at how these simple activities can be transformed for use in the middle-grades.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.

 

In Pursuit of Fluency Part 4: Number Sense in the Middle Grades

Hands-On Math, KP® Ten-Frame Tiles, Math, Teaching and Learning

Place ValueWelcome to Part 4 of our Fluency Series! In Part 3, we discussed fluency with early number concepts such as subitizing, composing numbers within ten, finding number pairs that total ten, and making the first ten. Giving students opportunities to engage in these types of activities over time helps them internalize a strong sense of number and maintain fluency. Far too often, we leave these early number experiences behind without realizing how robust they might be if we grow them up a bit for older students.

This week, we’re going to take a look at similar number concepts, only with larger and smaller numbers. By no means is this a comprehensive survey of everything middle-grade students need to know for number sense. Rather, I’ve included a few ideas to make the case for how important number sense is and how simple and fun it can be. For this post, we’ll focus on base-ten numbers and save non-decimal fractions for another time.

Subitizing with Ten Frames – Tens, Hundreds and Thousands

 Subitizing is the instant recognition of quantity without counting. Quite often, students use structures or groupings to help them organize quantities into smaller chunks. For young children, we often use a ten frame, colored counters, or colorful blocks to help them easily chunk quantities into smaller amounts so they can see relationships. For older students, subitizing can be a great way to help them recognize patterns in the number system and apply their understandings of number relationships and properties.

Young children recognize quantities on a ten frame, using 5 and 10 as benchmark numbers. For example, they can see that 7 is two more than 5 or that 9 is one less than 10. Older students benefit from seeing “tens” on a large ten frame and then stating the quantity that is represented. For example, they can see that 70 is two tens more than 50 or that 90 is one ten less than 100. Next, the experience can be repeated by placing “hundreds” on a large ten frame so students can see that 700 is two hundreds more than 500 or that 900 is one hundred less than 1000. Click here to view slides that illustrate this idea.

Make a Ten:

Make a Hundred:

Make a Thousand:

 

Recognizing Quantities Within Complex Figures

Grace Kelemanik introduces an advanced form of subitizing in her Contemplate Then Calculate routine. You can find it on her website, Fostering Math Practices. Basically, the teacher presents students with a configuration of dots, and the student is asked to determine the quantity without counting. Students may use principles such as compensation, chunking, symmetry, and other ideas to help them determine the quantity. Students can then represent their thinking mathematically, connecting to symbolic representations. A sample of one of these configurations appears below.

One Possible Representation: (5×3) + (4×2)

Building Decimal Fractions with “Tens”

How does one build decimal-fraction sense? Primarily by connecting base-ten ideas with fraction ideas. The digits to the right of the decimal point represent a fraction of a whole. However, we often neglect to help students make the connection between fraction and decimal notations. Here are a few suggestions:

  • Help students see that the ones place is the point of origin of a base-ten number, not the decimal point. There is not a “oneths” place to the right of the decimal point. The decimal point simply identifies the ones place as the unit. Tens, hundreds, thousands, etc. appear to the left of the ones place, and tenths, hundredths, thousandths, etc. appear to the right of the ones place. In other words, magnitudes of ten emerge from the ones place, going in both directions.
  • Ask students, on occasion, to write decimal numbers in fraction form. For example, 2.25 can be written as 2 25/100. This reminds students that a number that has digits on both sides of the decimal point is actually a mixed number. This also helps the students remember that decimal fractions are fractions written in base-ten notation.
  • Provide students with opportunities to group and ungroup tenths and hundredths in the same way they group and ungroup tens and hundreds. My favorite way to do this is with KP Ten-Frame Tiles. Students identify the blue-covered tile as 1 whole; then, they can “unpack” to see that there are 10 tenths and 100 hundredths inside the whole. Students can then visualize the concept behind the fraction language as well as see the magnitude of hundredths in comparison to tenths and wholes. Students do this with whole numbers in the early grades, so doing this with decimal fractions in the upper grades is extremely helpful in allowing students to see all base-ten numbers as part of a system.

Number Relationships

I shared this idea in Part 2 of our Fluency Series, and this is a great place to repeat it. Understanding the relationships among numbers is an important building block for fluency with number concepts. For example, a teacher might present four numbers to her class:  23, 20, 15, 25, and ask them to select the number that is not like the others. Of course, this particular exercise allows for multiple responses:

  • 23 is the only prime number.
  • 20 is the only even number.
  • 15 is the only number not in the twenties.
  • 25 is the only square number.

As you can see, this brief encounter might spark a lively conversation as students begin justifying their responses and realizing that there is more than one correct response. (Thank you to Mike Askew for this great example.)

In this post, we barely scratched the surface on ways to build fluency with number concepts in the middle grades. The point is that students need frequent and specific opportunities to engage in activities that develop number sense to the point of fluency. Automaticity and flexibility with thinking about number provides a vital foundation for success in higher levels of mathematics. Isn’t it worth taking 5 minutes per day to help develop and maintain this fluency? I think so!

Let us know what your thoughts about these and other activities for developing fluency with number concepts in the middle grades in the comments section below.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.