KP® Ten-Frame Tiles, Professional Learning, Teaching and Learning
A while back, I was talking with a fourth-grade teacher at my school. She questioned, “What did the third-grade teachers do last year? I just started my fraction unit this morning, and none of the kids knows anything about fractions!” Rather than explaining what I already knew (since I was the one who taught the third-grade fraction unit the previous yea!), I simply asked her to bring her class to my math lab after lunch.
When they arrived, I had several manipulatives out on the tables – representations they had used in third grade. Before diving in, though, I simply asked the students to sing a song with me. I started the tune, which was about numerators and denominators, and almost everyone joined in. The teacher, now standing in the back of the classroom, threw her hands up and simply smiled.
You see, far too often, teachers begin mathematics units where they think they should start rather than tapping into where others left off. In this case, my colleague simply began the fraction unit as her textbook directed rather than talking with the previous year’s teachers to find out where they had left off. Had she simply inquired, she would have known which manipulatives, which visuals, which vocabulary, and which instructional strategies to use in order to tap into students’ prior knowledge.
Whether you’re a teacher, a coach, or a site leader, facilitating conversations among colleagues within and across grade levels is critical to connected learning. As discussed in our previous blog post, Math “Rules” That Expire, taking time for such conversations is critical. In the case of common visuals, here are three ways to ensuring students use visuals to “see” how inter-connected mathematics truly is…
- Connections within grade levels: Invite teachers within a grade level to map out the math concepts to be taught throughout the school year. Under each concept, have them list the manipulatives, diagrams, and other visuals to be used. Finally, ask them to look for opportunities to use common visuals across the year. The more overlaps that exist, the greater the opportunity for children to see mathematics as a system rather than isolated skills.
- Connections across grade levels: Once each grade level has compiled a year-long list of visuals and mapped out the connections, ask them to engage in cross-grade conversation about how visuals might be used to connect the mathematics from one year to the next. The more connections they make, the better students will be able to tap into prior knowledge and use that prior knowledge to learn new ideas.
- Connections using common visuals: The list below includes some of the common visuals that help students see mathematical connections…
- Manipulatives: pattern blocks, base-ten manipulatives such as KP Ten-Frame Tiles, unifix cubes, two-color counters, Cuisenaire rods, fraction bars
- Diagrams: number lines, bar models/tape diagrams, number bonds, ten frames, place-value charts, arrays
- Other visuals: hundred charts, multiplication charts (may be used for skip counting, multiplication facts, arrays, equivalent fractions, etc.), spreadsheets, Excel graphs
We owe it to our students to make these connections explicit. And we owe it to our teachers to ensure they have time and space to discuss and internalize these ideas.
Please share your thoughts on the visuals most likely to help students make connections. Do you have experiences with any of the listed visuals? Do you have other favorites you would like to recommend?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
One more thing: A great example of common representations is KP Ten-Frame Tiles. They are so superior to base-ten blocks! Check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks.
KP® Ten-Frame Tiles, Math, Professional Learning, Teaching and Learning
“Multiplication makes bigger and division makes smaller.” “You can’t subtract a larger number from a smaller number.” “Improper fractions should always be written as mixed numbers.” “You always divide the larger number by the smaller number.”
We’ve all heard these statements…and I’ll bet we’ve all said them at one time or another. And every time we say these, we further perpetuate misconceptions that are difficult, if not impossible, to un-teach in future years.
Far too often, teachers inadvertently teach mathematics in an overgeneralized and imprecise way. We teach tricks that promote nothing more than memorization. The result is that our students misunderstand the very ideas we are trying to illuminate. How many teachers have had to contradict their colleagues from previous grade levels because what was taught was so painfully limited in scope?
So…how can we stop this practice in its tracks and work together to teach mathematics correctly and coherently? Here are a few ideas to get you started.
- Read Up! First, I’d like to introduce you to two recently-published articles that drive this point home: 13 Rules That Expire and 12 Math Rules That Expire in the Middle Grades. These two articles simply and eloquently outline overgeneralized, commonly-accepted, inadequate strategies we need to abolish from our vernacular. Seriously – check them out – you’ll likely find one or two or more to work on.
- Talk to Your Colleagues: Professional discourse provides foundational and at-your-fingertips opportunities for growth. After perusing one or both of the articles above, talk to your colleagues. Last month, I had a group of 40 teachers and leaders read and discuss these articles, and their conversations were robust.
- Create School-Wide Agreements: Take some time during grade-level and staff-wide meetings to discuss these ideas. Then come up with a list of five to ten school-wide agreements, identifying the concepts, vocabulary, procedures, etc. that everyone, or no one, will use on campus. This will make for a lively conversation!
I’m in the process of working through these steps in my own school district right now. Won’t you join me on this journey to abolish rules that expire by engaging in professional discourse and by creating school-wide agreements?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Preview for Next Week: When selecting visuals and manipulatives to use within and across grade levels, choose wisely. Focus on ones that allow for connections throughout the grades. A great example is KP Ten-Frame Tiles. They are so superior to base-ten blocks! More on this topic next week…for now, though, check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks from the get-go.
Professional Learning, Teaching and Learning
“Many hands make light work,” claimed John Heywood more than 600 years ago. And we still find this to be true today. Especially in the world of education, when we join forces to serve a collective goal, our work becomes magnified and exponentially impactful.
By way of example, please humor me as I tell my own story. This past year, my school district adopted new mathematics resources. Because we had varied perspectives for what mathematics teaching and learning should look like, I brought together diverse teams on each campus to work and learn together. Each team had one representative from each grade band, K-2, 3-5, and 6-8, as well as an instructional coach, a SPED teacher, and an administrator.
Initially, during phase one, each team had the responsibility to represent their grade bands, taking inventory of what was working, what materials were needed, and what PD was requested. This one-way communication allowed them to take ownership of the needs of their staff.
However, there was a second intention, phase two, if you will, behind the formation of these teams: equipping the members as teacher leaders, both individually and corporately. Here’s the plan I implemented for evolving the teams from phase one to phase two…
- Monthly Meetings: During both phases one and two, I met with each team about every four to six weeks. During phase one, I gave the team assignments to talk with everyone they represented about their needs: materials, professional development, strategies and methods, etc. During phase two, I am now giving them things to take back to their teams: number talk strategies, games, manipulatives strategies, word problem methods, etc. This helps their colleagues begin to see them as the experts.
- Summer Symposium: The best way I’ve found to transition from phase one to phase two is to provide a common experience for the teams. This year, I offered a Summer Math Team Symposium where the teams came together to examine mathematics content across the grade-levels, pedagogical strategies, and leadership techniques. They created vision statements for what mathematics teaching and learning will look like on their campuses. And they planned for how to share important ideas with their colleagues.
- Teacher Leadership: An amazing (and deliberate) by-product of this effort is the empowerment of teachers to see themselves as leaders on their campuses. It’s thrilling to hear teachers confess how their discomfort with mathematics has been overshadowed by the summer experience and how excited they are to support their colleagues. In just the past month, I’ve seen the teams blossom as they presented mini math sessions for their colleagues, offered vision-casting seminars for their staffs, and tutored their colleagues in areas of discomfort. This effort is resulting in teachers taking on new challenges and supporting their colleagues unlike anything I’ve witnessed before.
So…are you interested in creating or taking part in such an effort…or at least in hearing more about how you might do something like this within your own sphere of influence? You don’t have to be a site or district leader to get this started. Sometimes the best efforts begin within the teacher ranks. We would love to hear your thoughts on building camaraderie amongst the math teachers on your staff.
And for those of you following Peggy’s journey with KP Ten-Frame Tiles, please click here for her latest entry. You may find great ideas in here to take back to your colleagues to inspire new conversations about the teaching of mathematics.
As always, we would love to hear your thoughts regarding these ideas or your own journey. Please continue the conversation in the comments section below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
PS – If you’ll be in Phoenix this Wednesday morning, August 29, please join us for our FREE leadership event taking place in the downtown area. You can learn more by clicking here.
Math, Professional Learning, Teaching and Learning
I get asked this question a lot. Everywhere I go, teachers are looking for how to make the best use of their math time.
First, let me answer by saying there is no silver bullet. There are many effective ways to structure your math block. The structure I share below reveals what I’ve personally found to be most effective for me and those I work with. Whether you’re teaching kindergarten or middle school math, these components really make a difference!
1. Daily fluency practice (5-10 minutes): In my classroom, this usually includes interactive card and dice games as well as solitaire activities for those students who don’t like competition. Fluency practice may focus on basic math facts, place value, fractions, proportions and ratios, or properties of operations. This time is used for pretty much anything I want my students to be fluent with, with an emphasis on mental math and reasoning.
2.Mental Math/Problem of the Day (5-10 minutes): I love to begin the content portion of the day with a word problem that pushes students to focus on math concepts with which they aren’t yet fluent. For example, I’ll present a division problem involving dollars and cents prior to students learning the standard algorithm for decimal division. This pushes them to use a variety of strategies.
Another great use of this time is to engage the students in mental math work. Of course, my favorite is Sherry Parrish’s Number Talks.
3. Whole Group Instruction (10-30 minutes): Note that “whole group instruction” does not necessarily translate into “direct instruction.” This time may be used
- to build anticipation.
- to review prerequisite skills.
- to “set up” an inquiry-based exploration and let students do the bulk of the talking.
- to demonstrate new learning stations.
The point is that learning is a social endeavor, and I want to use the whole-group time to build the notion that we are all in this together.
4. Small Group Work & Independent Practice (20-30 minutes): Students work in small groups (and sometimes independently). Throughout the week, they rotate through four or five activities that focus on the content at hand using a variety of learning modes. For example, if we’re studying fraction multiplication, I may have
- one center that uses pattern blocks,
- another with a number line activity,
- a third with a fractions game,
- a fourth with independent work, and
- a fifth with interactive notebooks.
The students rotate to a different station each day. During this time, I pull small groups of students, a couple from each rotation, to work with me on interventions or enrichment, depending on their needs. I personally promote the Guided Math approach – here are some great resources.
5. Closure (5 minutes): Think formative assessment. What can you ask the students to say or do that will indicate that they met the day’s objective? This may include a ticket-out-the-door, a response board solution, a simple reflection, or a quick series of hands-up-multiple-choice questions. Whatever you do, don’t skip your closure. THEY should be assessing their progress – immediate feedback is hugely effective in helping students retain learning.
So…does anything here resonate with you? How do you structure your day? How might you incorporate one or more of these ideas into your work with students? Please join the conversation by leaving your comments below. I love hearing from you!
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.
