Hands-On Math, KP® Ten-Frame Tiles, Math, Problem Solving, Professional Learning, Teaching and Learning
Because 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:
- 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.
- We honed and customized our individual races by putting together our own well-practiced skills.
- 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.
- 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).
- 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.
- 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.
- 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).
- 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).
- 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.
- 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.
- 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.
Hands-On Math, KP® Ten-Frame Tiles, Math
Summertime… a time to relax, a time to learn, a time to catch up, a time to dream and to plan and to prepare for next year. It’s a beautiful thing that every school year is a fresh start with a brand-new group of students. And summertime gives us a chance to reflect on what went right in the past year as we anticipate the upcoming one.
I recently received a profound reflection from a teacher about her experience using manipulatives to help her students understand grouping and ungrouping.
Although I had been using Ten-Frame Tiles in my classroom, a colleague suggested I just teach my students the robotic mechanics of regrouping, teaching them to “go next door to the tens, knock on the door, and ask to ‘borrow’ a ten.” He mentioned that teaching them the concept just got too complicated and mucked up the mechanics of the automatic actions the kids were supposed to go through every time they needed to “borrow.”
This went against my better judgment, though, because in the 80’s I had previously taught my kids to regroup using flip cards, beans, straws, and bundles of tens that they would unbundle, and cards they would flip to represent each step of ungrouping and regrouping. Teaching the concepts this way had made sense to my kids then, and they had learned how to ungroup and regroup with a firm understanding. That day, after he advised me to jump in and teach the kids to “go next door and knock on the door and ‘borrow’ a ten,” I did just that, and we had a fun lesson knocking and asking and laughing about all of it.
But, the next day, I gave the kids more problems that required them to ungroup and regroup, and, sure enough, not only could they not do the problems, there was a strong air of unrest and dissatisfaction in the room. The kids acted like the world was not a friendly place for them to be in, and they were disgruntled and downright grumpy. I didn’t know this tension was coming from their place of confusion with the math. I brought out the tiles again the next day, and we spent the whole morning learning grouping and ungrouping with the tiles.
Time really did fly by. After two hours had passed, and the kids were back in a place of true understanding, one of the kids saw the clock and said, “Wow, we have been doing math for two hours.” Not one of the students had complained or asked about the time during our lesson or even noticed how much time had passed.
But after that two hours, I had happy kids again, and not only was there an air of true satisfaction in the room, but the kids were clearly happy—peaceful, content and what seemed to me with a sense of being one with their world again. They acted as if I had turned on the lights or opened a door to a better place for them.
I asked them, “Now, is this a planet you want to live on?” (I had mentioned earlier that a planet where everyone keeps losing ten dollar bills was not a place I wanted to live!) and they immediately agreed, “Yes!” It was a place they knew they wanted to be, and it was a place that finally made sense! They have been on this planet happily ever since!
While I had known that this teacher was enthusiastic about Ten-Frame Tiles, what struck me about her reflection was her understanding of the KIND of difference these manipulatives make. To her, her students’ use of the tiles contributed directly to their emotional well-being, a perspective not typically taken into consideration or perhaps even recognized.
This reflection reminds us that children want to make sense of the math they are learning. When they do, they find a world of satisfaction, and, when they don’t, the results may have implications far more significant than low test scores. And both outcomes can have life-long implications!
We would love to hear from you! As you reflect on your work last year, what one or two things went well and are worth saying out loud? Please let us celebrate with you by leaving your thoughts in the comments box below.
Hands-On Math, KP® Ten-Frame Tiles, Math, Teaching and Learning
We’ve all done it…teaching rules without meaning. For me it was rounding rules. I had found the perfect song. The teacher next door used a baseball metaphor. Our colleague down the hall used the Rounding Mountain. We were all trying help our students remember the rounding rule: “If the digit to the right is 0-4, round down. If the digit to the right is 5-9, round up.” And in each case, of course, we failed to develop any understanding whatsoever.
That’s because rounding is really about “closer to” thinking. The dictionary states that to round is to “alter a number to one less exact and more convenient for calculations.” In other words, rounding is about finding the closer benchmark number for estimation purposes. The benchmark number doesn’t even have to be a multiple of 10!
Here are three ways to develop “closest to” thinking:
1. Use a concrete manipulative such as KP Ten-Frame Tiles. If your students are rounding 68 to the nearest ten, for example, ask them to place 6 tens and 8 extra ones on a large ten-frame. Ask them if the small ten-frame is closer to full or closer to empty? Since only two tiles are missing, they can easily see that the ten-frame is closer to full, and, therefore, closer to 70 than to 60.
2. Use a visual representation such as a number line. If your students are rounding 368 to the nearest hundred, for example, ask them to plot 400 on one side of the number line and 300 on the other side of the number line. Then ask them to mark the halfway point, 350. Finally, ask them to mark 368. Is 368 closer to 300 or closer to 400, using 350 as the halfway point? NOTE: If you’re familiar with “Clothesline Math,” this same visual can be achieved by having students place folded paper strips on a long piece of clothesline rope or string.
3. Write the “closer to” numbers above and below. Simply ask the students to write the possible rounded numbers on above and below the number at hand. If your students are rounding 2.34 to the nearest hundredth, for example, ask them to write 2.3 above and 2.4 below. Then ask them if 2.34 is closer to3 or to 2.4 (note: they may go back to using a number line for help).
Lesson learned – these are now my go-to strategies rather than teaching my favorite rounding song. Know that kids might find these a bit tricky at first, but honestly, they’ll catch on by the second day if you really get them thinking about “closer to” numbers and halfway points.
How about you? What are your thoughts about teaching thinking strategies rather than memorized rules? Please share your stories and comments below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, KP® Ten-Frame Tiles, Teaching and Learning
When was the last time you tried to teach a young child to count coins? If you’re like me, you probably found this to be a daunting task, as many math and non-math related issues surface.
As a child, I remember making a deal with my little sister: I offered to trade my four big nickels for her three little dimes. She thought I was being so generous…and we made the trade right there in the backseat of our Ford Pinto.
This goes to show that value-counting coins is a complex skill. It includes
- memorizing coin names and values,
- understanding the difference between counting the number of coins and “counting” their combined values,
- understanding the proportional relationships between the values of various coins,
- skip counting by different denominations,
- adding mentally,
- seeing coin values as fractions as a whole,
- and the list goes on.
It turns out that our traditional method, using coins as the teaching tool, just doesn’t cut it. Coins are concrete objects, but there is nothing about them that represents their proportional values. Today, I’d like to make a suggestion – let’s, instead, use the ten-frame to teach money!
Using the Ten-Frame to Represent Coin Values
The ten-frame, a visual organizer for numbers 0-10, can be extended and partitioned to represent multiple ten-frames and fractional parts. As you can see below, we can partition and/or combine ten-frames to represent coin values – 1, 5, 10, 25, 50, and 100.
Using the largest ten frame to represent $1.00, students can see that $1.00 is composed of 100 small squares – we’ll call those pennies. They can also see that there are 10 small ten frames – we’ll call those dimes. Then it’s a matter of partitioning and combining ten-frames to find the nickels, quarters, and half-dollars.
From Ten Frames to Money Boards
This visual model, pictured below as a KP Money Board and Coin-Value Cards, helps students see the proportional relationships between the various coin values. It also helps them explore the fractional values of the coins.
KP Dollar Board & Coin Cards for download
Using Money Boards and Coin-Value Cards for Problem Solving
Money Boards and Coin-Value Cards also provide a foundation for students to conceptualize a number of important money-related skills:
- Finding coin equivalences for $1.00 (4 quarters, 10 dimes, 20 nickels, 100 pennies)
- Comparing relative values of coins (e.g., 1 dime = 2 nickels; 1 nickel = 5 pennies)
- Value-counting a set of mixed coins
- Making change from $1.00
- Counting past $1.00 by counting in groups of $1.00
- Solving money-based word problems and story problems
One Final Thought – Bills Before Coins
At KP, we take the stance that since whole numbers are mastered before fractions, value-counting bills (whole numbers) should be taught before value-counting coins (fractions of bills). Several years ago, I conducted an action-research project to see if students would learn to value-count coins faster if they first learned to value-count bills. After all, the bills are uniform in size and clearly indicate their values. Guess what I found? Yep – the class that learned to value-count bills first subsequently learned to value-count coins in about half the time it took the class that began with coins. Furthermore, we used $1, $10 and $100 bills to connect to place-value concepts rather than pennies, dimes, and dollars. This was a huge eye-opener to me and my colleagues. Might you give it a try?
We hope you’ll consider using the KP Money Boards and Coin-Value Cards as you plan for your upcoming money units and problem-solving units. Click here for a downloadable version of the KP Money Board and Coin-Value Cards: KP Dollar Board & Coin Cards for download.
What do you think of these ideas? Do they resonate with you? Are they new for you, or have you thought of something similar in the past? If so, will you share with our readers what have you tried and how it has worked? Please join the conversation by leaving your comments below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
