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 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.

 

In Pursuit of Fluency Part 5: Math Fact Fluency

Game Play, Math, Teaching and Learning

Math Fact Fluency

In my experience, when discussing mathematical fluency, math-fact fluency is by far the most frequently referenced topic. After all, success with more complex computation, including work with fractions and algebra, depends on math- fact fluency. Interestingly, teachers at every grade level I work with, all the way up through high school, voice their frustration that the grade level(s) before send them students who are not math-fact fluent. As a former kindergarten teacher, I used to joke that we might as well drill all the way down to kindergarten, where it all began, and blame kindergarten teachers for this problematic epidemic.

Epidemic? Perhaps this is a bit extreme, but it makes the point. So many students pass through math class year after year without math-fact fluency, and this truly is problematic. Lack of math-fact fluency bogs down the short-term memory and inhibits success with topics such as whole number and decimal operations, fractions, algebra, and problem solving.

I personally experienced what it felt like to finally break free of this debilitating stumbling block. As a math specialist, I knew good and well that I was still a finger-counter when it came to addition and subtraction facts. I had simply never committed them to memory. However, as I worked with first- and second-grade students to use mental strategies rather than counting by ones, I finally decided I needed to take my own advice. So, as a teacher with a mathematics degree and ten years of teaching experience, I set out to commit these facts to memory. It was fascinating to experience this as an adult and an educator, noticing that the process initially slowed me down. However, once automaticity set in, I was amazed not only at my speed, but at the clarity that accompanied it. My brain was free to contemplate the larger issues with the problem at hand rather than having to continuously downshift into counting.

Ever since my experience of conquering my lack of math-fact fluency, I have advocated that we do whatever we can to help students address this issue.  That said, I personally agree with experts such as Jo Boaler, who point out that traditional methods such as timed tests and whole-class games such as Around the World perpetuate math anxiety and the notion that being good at math means possessing fast recall. Therefore, we need other ways of promoting math-fact fluency.

So if we’re to abandon the methods that failed me and oh-so-many others as children, what are we left with? My go-to for developing and maintaining math fluency centers on providing five to ten minutes of DAILY fluency-building activities, primarily game-play and other fun activities. Here are a few of my favorites…each requires nothing more than a deck of cards.

Fishing for Tens (or any number): Students play Go Fish using the typical rules, removing the tens and face cards beforehand. However, rather than asking for the same number they are holding in their hands, they ask for a number that pairs to make a ten. For example, if I am holding a 3, I will ask for a 7. Once I have a pair that adds to ten, I place it in front of me. The first player to run out of cards wins the game. This game can be played for other totals by taking removing cards from the deck. For example, if you want students to “fish for nines,” remove the nines, tens, and face cards from the deck. Now they simply ask for a number that pairs with one of their cards to make a nine.

Concentration: Students play Concentration using the typical rules, removing the tens and face cards beforehand. Rather than flipping two cards in hopes of making a match, students flip two cards in an attempt of making a ten (2 and 8, 3 and 7, 5 and 5, etc.). As with Fishing for Tens, this game can be played for other totals by removing cards from the deck. For example, if you want students to make pairs that total nine, remove the nines, tens, and face cards from the deck.

Double War: This game can be played for addition or multiplication. Players each flip over two cards and state the sum or product. The highest sum/product wins the round. The winning player takes all four cards. The game is over when all cards are used up. The champion is the player with more cards at the end.

Salute: This game requires three students and can be played for addition or multiplication. All face cards should be removed from the deck beforehand. Shuffle the cards and divide them into two equal piles, face-down. One student acts as the “judge” while the other two are the players. The two players each pick up one card from their respective piles and place them on their foreheads, face-out (it is important that each player can see the other player’s card but not his/her own). The judge adds/multiplies the two numbers together and announces the sum/product. The players race to figure out their own card values. The first player to correctly call out his/her own card number wins that round and collects both cards. The winner is the player with more cards at the end.

These four card games are but a few that can be used to reinforce math-fact fluency. A simple Google search will turn up hundreds of other options. The point is that daily fact practice using fun and motivating interactions need not center on methods that do more harm than good. By providing students with daily opportunities to enjoy number play and develop their math-fact fluency, we instill in them a confidence, an enthusiasm, and a level of success that so many adults missed out on as children.

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

 

In Pursuit of Fluency Part 6: Operations Beyond Basic Facts

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

Operations Beyond Basic FactsBecause we have been talking a lot about fluency in the past few blog posts, it suddenly occurred to me the other day that I can see a surprising connection between our efforts to help students develop fluency and my experiences becoming a successful high school sprinter. Let me explain.

When I was in high school, I loved running the quarter mile, and I made it to the state track meet every year. The mile relay was my absolute favorite event on the track – four young women working tirelessly to perfect our individual races in order to combine our skills to claim the final prize of the meet. We each played an important part in creating that final success:

  1. We worked on our form and rigorously built our individual skills (using the starting blocks, perfecting our just-right paces, engaging in distance running, sprinting shorter distances, etc.), all the while receiving immediate and continuous feedback from our coach.
  2. We honed and customized our individual races by putting together our own well-practiced skills.
  3. And then we took that last, important step — we combined our efforts into a longer, collaborative race, blending the traits of our individual races into a more complex and challenging final event.

So here’s the connection to math fluency…Just as my coaches helped me and my teammates achieve success through a prescribed sequence of skill-building activities, we, as teachers and coaches, can use the same strategies to help our students build the skills necessary to achieve fluency.

  1. I equate students’ development of number sense and math-fact fluency with the smaller, individual skills we sprinters developed (our sprints, form drills, and the like).
  2. Then, students’ fluency with number sense and math facts become the building blocks to fluency with multi-digit operations, just as we sprinters individually put together our running skills to design our full races.
  3. Eventually, operational fluency supports problem solving (the “more complex and challenging event” that is the goal of fluency) by enabling students to invest their thinking in the problem-solving process without getting stuck in the mire of operations.

So here we are, having developed number sense and math fact fluency with our students, laying the groundwork for them to perform more complex operations with multi-digit base-ten numbers (and fractions). Ensuring that students achieve operational fluency is critical…and how we get there can be a challenge. Below are the five steps I have found to be most important in helping students gain operational fluency.

  1. Continue to reinforce conceptual understanding. Using tools such as KP Ten-Frame Tiles to reinforce and describe what’s happening in the procedures will continue to support procedural understanding (Akin & Rimbey, 2017).
  2. Understand that speed and fluency are not synonymous. Fluency combines accuracy, efficiency, flexibility, and appropriateness. An over-emphasis on speed alone increases math anxiety (Boaler, 2018).
  3. Support fluency by giving relevant, immediate feedback. If you simply give students arithmetic worksheets with problems to practice over and over, they will not thrive because there is no opportunity for them to receive necessary and meaningful feedback.
  4. Build fluency in a fun and motivating atmosphere (Boaler, 2018). Provide opportunities for students to work on their skills through interactive games and online activities where meaningful feedback is possible. Something as simple as having students work side-by-side with models and with white boards, one “acting out” while the other records the operation, can make these practice sessions more meaningful.
  5. Make math fluency meaningful. All along the way, provide students with authentic opportunities to experience application of their math skills so they can see first-hand how fluency is an asset (Akin & Rimbey, 2017).

Had my mile-relay team neglected to rigorously build our individual skills, had we not had the opportunity to practice together toward a cohesive outcome, had we not focused on our common goal, we would not have been successful in seeing our vision become reality.

So it is with math fluency – students must actively build their competence with number sense and math fact fluency, put these skills together as they practice toward cohesive strategies for operations, and apply these skills as they successfully engage in deep and meaningful problem solving.

What are your thoughts about building math fluency for operations? What strategies have you found successful? Please 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.

 

The Best Math Tools For Your Classroom

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

I hear these questions all the time — “What math tools should be available in my classroom?” “Which manipulatives purchase would give me the best bang for my buck?” “What math tools should I buy for my teachers so their students will be successful?”

My first thoughts probably go without saying:

  1. When the manipulatives stay on the shelf or in the closet, they don’t get used. They are not a good investment.
  2. If students don’t engage with the manipulatives in meaningful ways, the manipulative will not be a successful tool.
  3. If the teacher is ill-equipped to help the students use the manipulatives effectively, the impact is likely to be negligible.

My “tier one list” of math tools will help teachers stock their classrooms. The list identifies what I recommend be present in every elementary classroom.

Base-Ten Manipulatives as Math Tools

The most conventional version of base-ten manipulatives are Base-Ten Blocks. These pre-grouped blocks can often hinder students’ understanding of arithmetic because they undo place value understanding (see our previous journal entries). Instead, I prefer groupable manipulatives that help students develop a concrete understanding for the “nesting” nature of our base-ten system. By “nesting” I mean where one ten is concurrently 10 ones, one hundred is concurrently 10 tens and 100 ones, etc. KP Ten-Frame Tiles facilitate developing understanding of numbers within ten and building with tens for both whole numbers and decimals. Click here to check them out!

Counters as Math Tools

My preference is either clear colored disks or two-color counters. I prefer the two-color counters because they transition so nicely from whole numbers to integer operations. I’m not a huge fan of all the “cute” counters with which we stock our primary classrooms, such as teddy bear, dinosaur, or people counters. It’s far more meaningful for students to “pretend” that the simple counters are these other objects. Such “pretending” involves substitutionary thinking that can lay the groundwork for using variables as substitutes for numbers later on.

Connecting Cubes as Math Tools

These have been around for years. They can be used for counting, simple arithmetic, and non-standard units for linear and area measurement.

Inch Tiles, Inch Cubes, and CM Cubes as Math Tools

All of these are great for counting, early arithmetic, and transitioning to standard measurement units. I also use the 1” tiles for introducing the area model for early multiplication.

Measurement Instruments

You’ll want to be sure your students have access to rulers, balance scales, and liquid volume pouring containers. Of course, using non-standard units is a must prior to introducing students to standard units and the tools that measure them.

Pattern Blocks

Pattern blocks are incredibly useful from kindergarten through middle school. They can be used meaningfully for geometry, algebraic thinking, whole number work, or fraction concepts, 

Proportional Coin Cards

It’s common for us to think of plastic (or real) coins as math manipulatives. However, there is nothing conceptual about coins that helps students see the proportional relationships among their values. Therefore, KP Mathematics has created a set of proportional coin cards, based on ten frames.

These coin cards help students develop an understanding of these relationships. Download a free set from our website by clicking here.

Of course, there are many, many other things out there. My “tier two list” includes things like geometric solids, unmarked fraction bars, Cuisenaire rods, and attribute blocks. However, most of the items on my “tier two list” can easily be reproduced with paper, scissors, and tape.

Now I’m wondering what’s at the top of your manipulatives list. How do you stock your math tools in your classroom? Please let us know by responding in the comments box below.

We look forward to continuing the conversation!

Mathematical Motivation Through the Holidays

Game Play, Math, Problem Solving, Teaching and Learning

As the holidays approach, and with them an extended break between semesters, motivation wanes, not only among students but also within the teaching ranks. Classroom teaching can be a challenge – trying to keep students on-target with their learning goals while acknowledging the distracting festivities all around.

As a classroom teacher, I sometimes found myself frustrated by the ways in which my students got caught up in the holiday cheer. After a few years, I realized that it wasn’t only the students who were pulled away from the learning at hand; I suddenly realized that I was contributing to the tension between learning and celebrating. I was eager to get out the door to get more shopping done, to get ready for yet another party, to be released from the pressures of work so I could engage in more holiday play. I was contributing to the frenzied atmosphere in my classroom because I was distracted by my own desire to be on vacation. I was tired, overwhelmed, and a bit burned out — and my students knew it.

At that moment, I realized I needed to do two things: have fun with teaching during the holiday season and attend to some self-care.

Having Fun with Teaching and Learning

First, I made a list of the standards to be covered during the weeks between Thanksgiving Break and Winter Break. I looked at the tasks and problems included in my district-adopted textbook. And then I contemplated ways I could redesign the tasks to include holiday flair while meeting the objective.

  1. Shopping Task: Given that shopping is a big activity this time of year, I like to set students up with a math activity that involves buying gifts for others. I give them a budget, a list of people, and a time limit for purchasing gifts for each person. This activity may be as open-ended as letting them shop online or as limited as giving them a specific scenario such as this one from NCTM. You might even take your students on a field trip to the mall. When I did this, I made arrangements ahead of time with 5 stores, all of whom agreed to ring up the purchases and give my students mock receipts (no money changed hands). The students worked in groups to buy gifts for a family of 5. They had to plan ahead as there was a minimum for each family member in order to qualify (they couldn’t just buy a deck of cards for the last person because they had cut it too close). The team that came closest to $100 without going over “won.” Afterwards, each team received $25 (earned from a math-related fund raiser we did before Thanksgiving) to shop for a child on the Angel Tree at the mall.
  2. Coordinate Geometry: My 5th-graders are usually working on coordinate geometry around December, so I like to replace some of the textbook tasks with graphing tasks that result in creating holiday pictures on the coordinate grid. The students love this so much that they ended up creating their own coordinate grid puzzles for one another.
  3. Game Play: Designating a set period of time for game play provides a fun way to review concepts already covered during the year. I like to set out some marshmallows and chocolate chips for the kids to munch on as they play games that have become favorites throughout the semester such as One, Two, Switcheroo or Go Fish.

There are so many ways to connect holiday fun to the math standards you’re currently teaching. Give it a try…and let us know how it goes!

Attending to Self-Care

I’m saving this one for next week. In the meantime, take some time for yourself!!!

We wish you the best as you navigate these weeks between Thanksgiving and Christmas. Focus on making things fun for your students – and you’ll have more fun, too! And be sure to take care of yourself in the process.

We would love to hear from you. What are you going to do to make math fun for your students during the holidays?

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