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 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.

 

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.

Keeping Math Routines Fun and Fresh!

Game Play

Do you experience points during the school year that send you into a mini-panic about where your kids currently are and where they need to be? For me, one of those moments tends to be Thanksgiving week. I suddenly realize that Winter Break is just around the corner, and with that comes the reality that the school year will be half over.

Over the past few years, I’ve developed a strategy for helping myself get past this brief moment of panic by focusing in on the math routines that are most influential for maximizing student learning. These are the routines I encourage teachers to use regularly…and we take a few moments to think about how they’re working. Are they being used consistently? Is there evidence of student learning? Do the routines need to be refreshed?

You may have your own go-to routines on which to reflect…and here are five of my current favorites:

Clothes Lines: This routine has been around for a while, and yet it never loses its appeal Basically, a student or a group of students is given a set of numbers or expressions which is written on folded paper or note cards to be properly placed on a “clothesline” – a piece of string or yarn that is stretched across the room to create a “hanging number line.” The number sets may include small numbers, large numbers, fractions, percents…you name it! After the numbers are placed, subsequent class discussions offer opportunities for students to clear up misconceptions and learn new insights from their peers.

The Daily Five: Used as a spiral review, the teacher places 5 problems on the board at the beginning of the math period. These generally include one arithmetic problem, one place value problem, one fraction/ratio problem, one story problem, and one problem related to the content currently being covered. As much as possible, teachers select operations and number sets that link strategically to the current content.

Number Talks: Number talks were developed for classroom teachers to engage students in “mental math” through grappling with carefully selected arithmetic problems. Teachers can use number talks regularly as introductions to the day’s mathematical practice, as “warm ups” for other lessons, or as stand-alone extended engagements with mathematical concepts. First, students solve the arithmetic problem mentally. Next they share their answers while the teacher records them on the board. Then they share their solution strategies with a peer using a “pair-share” format. Finally, two to four students share their solution strategies out loud as the teacher records their thoughts. To see how this works, check out these videos or these books.

Show Me: For this routine, students need manipulatives, paper, and writing utensils. The teacher calls out a quantity, and the students must represent that quantity as quickly as possible in as many different ways as possible. For example, if the teacher calls out, “74,” the students may use manipulatives, such as KP Ten-Frame Tiles, to show 7 tens and 4 ones, 6 tens and 14 one, 5 tens and 24 ones, etc. Another student may represent 74 on a number line. And still another student may represent 74 by drawing 7 boxes of markers and 4 extra markers. The goal is for everyone, as a class, to use as many different representations as possible. This can be done with small numbers, large numbers, fractions, decimals, etc.

Splat: Steve Wyborney, math coach extraordinaire, invented this routine, and it’s catching fire all over the place! If you go to his website (linked here), you can download 50 power point sets that help students work on number sense by figuring the quantities covered up by the “splats.” So simple, yet so profound!

 

In my experience, we’ve found that after these routines are in place for a while, they may need a refresher to keep them from getting stale. This is the perfect time of year for teachers to reflect on practice, refresh as needed, and perhaps introduce something new.

Please let us know how it goes as you reflect on the routines used at your school. Are they being used consistently? Is there evidence of student learning? Do the routines need to be refreshed? Our community is growing, and it’s in our shared experiences that we all get better. We’d love to hear from you!

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

Authentic Problem Solving is Fluid and Flexible

Game Play, Problem Solving, Professional Learning, Teaching and Learning

Years ago, I walked into a Weight Watchers meeting, shy, self-conscious, and unsure of what to expect. After signing my agreement and going through what would become my weekly ritual, I sat down to read through the stack of literature they gave to me. I read through many ideas, insights, and testimonials. Yet it was one very small side note that caught my eye. There, in an offset lavender box, were familiar words that resonated with me. “The Four Steps for Solving Your Weight Loss Problem.” Can you guess what those four steps were?

  • Understand your weight-loss problem.
  • Create a plan to address the problem.
  • Solve the problem by following your plan, adjusting as needed.
  • Reflect on your progress and check for new ways to address it.

Yes – these are the four steps I shared with you in last week’s blog post, So You Took Keywords Away – Now What?  These four steps, immortalized by George Polya in How to Solve It, are the same steps for solving mathematical problems. It turns out that these steps find utility in all of life for beyond math class! And, interestingly enough, we were encouraged to move back and forth among the steps, reflecting upon and revising our plans from one week to the next.

Fast forward to today…I had some great conversations with several of you last week after sharing the problem solving template I use (click here to download again). One such conversation centered on the notion that Polya’s problem solving steps are not isolated, sequential steps. Rather, they are stages in a process that is both fluid and flexible.

Here are a few thoughts on ways to encourage and demonstrate flexible, fluid problem solving with your students:

  1. UNDERSTAND the Problem: This step has two parts. And you’ll likely revisit this step several times throughout the process as a way to monitor and adjust your progress.
    1. Understand the Story: Read the problem for understanding. Cover up the question at first, and relate to the context. Who is the problem about? What is happening? Where is it going? What story is being told?
    2. Understand the Question: Read the question to discover where the story goes next. What is being asked? Given this direction, what information in the story is useful? What are the quantitative relationships that emerge based on what the question is asking?
  2. PLAN a Course of Action: As with the Understand stage you’ll likely revisit your plan several times throughout the solution process, revising it as needed. Therefore, although you may be tempted to create a definitive plan, it’s likely that you will benefit more by coming up with something loose that you can revise as you go. The more flexible you are with your plan, the more likely you will be able to engage with the problem realistically, especially if you get stuck or need to back-track a bit. And remember that good plans includes both thinking strategies and a representations (see last week’s post for more on this).
  3. SOLVE by Implementing Your Plan: You’ll want to be flexible and fluid during this step of the process! Dive right in, knowing that you’ll be revisiting the Understand and Plan stages a few times to ensure that you’re on the right track. It’s okay to revise your plan once you gain new insights and notice new nuances within the problem itself.
  4. CHECK Your Solution and Reflect on Its Accuracy: This stage is so much more than using inverse operations to check for arithmetic accuracy. Did you really understand the address the original question? Does the solution match the original context? Does it answer the question? Did the plan work? Did both the thinking strategies and the representations help get to an accurate solution? Was the solution process messy or elegant? Might there have been a different plan that may have been more efficient or revealing? Does the final solution make sense? Does your solution need tweaking? Do you need to go back refine your process a bit more?

As you can see, problem solving is not at all linear, with siloed steps that happen in isolation. But, rather, problem solving is both fluid and flexible, allowing for movement in all directions throughout the process. Just as with real-life problems such as weight loss, job searches, or financial decision-making, moving back-and-forth among these steps allows you dive in, refine your understanding, revise your process. And even when you get to the check/reflection stage, you’re not necessarily finished.

Let us know what you think – we would love to hear your stories! Do you have any real-life examples of when this problem-solving process could be helpful? Have you had the chance to see students engage with problem solving that revealed flexibility and fluidity?

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PS – Beyond Math Blocks Teacher Institute is just around the corner – October 25-26 in Phoenix, AZ. My colleagues and I are super-excited about what we have in store for you. Have you signed up yet???? Click here to learn more.

PPS – Check out Peggy’s latest Beyond Base-Ten Blocks: A Search for a Better Solution journal entry and learn more about how base ten blocks may be problematic as manipulatives representing number and operations in our elementary classrooms.

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

Writing Rough Drafts in Math Class?

Game Play

When do “rough draft” and “math task” ever land in the same sentence? Honestly, when I first heard these two put together, it took me a couple of minutes to figure it out. After all, writing a rough draft is something I work on with my students during the ELA block, not during math class. However, as I learned while attending Dr. Amanda Jansen’s NCTM session a couple of days ago, rough drafts belong in math class, as well.

Here’s how it works:

  1. First, set up norms to help students understand the value of rough draft thinking. Click here for a creative example you may glean from.
  2. Introduce a mathematical task to your students, giving them a few minutes to individually think about what the task is asking them to do and how they might respond. If you need a good task, check out the free tasks on the Illustrative Mathematics website.
  3. Provide individual work time for students to attempt to represent their thinking while working solo. Help students understand that the focus is much more about the process and on their representations rather than on the “right answer.” Of course, students may use manipulatives and visuals throughout the process.
  4. Ask students to discuss their thinking so far with a partner or small group. IMPORTANT: Be sure you ask students to stop and share while they are still in the middle of their work so they can participate in “rough draft sharing” long before they’re finished.
  5. After the first-round discussion, students go back to their work, making revisions as warranted. Beforehand, be sure students understand that revision is not “fixing something that’s wrong.” It’s adding, extending, and reorganizing one’s work with new insights.
  6. Repeat steps four and five, above, a couple of times until students have had the opportunity to fully develop their ideas and get feedback from their partners.

As you can see, rough-draft thinking in math class very much resembles the iterative process of rough-draft writing during the ELA block. When students realize that they don’t have to have their ideas completely fleshed out prior to sharing with others, they learn to lean into one another in a collaborative effort to move forward.

This sounds magical – I can’t wait to try it out this week. How about you?

To learn more, check out this article by Dr. Jansen and her colleagues.

 

I would love to hear your thoughts on rough-draft thinking. How do you think it would play out in your classroom? Have you tried anything like this before? Do you have any refinements to suggest?

Please join in on the conversation by leaving your comments below.