Teaching Math as a Life-Long Tool

Hands-On Math, KP® Ten-Frame Tiles, Math Content

I don’t know about you…but whenever I get on an airplane, the first thing I usually do is put in my headphones to subtly let everyone around me know that I intend to keep to myself during the flight. However, on those occasions when I strike up a conversation with the person next to me, it doesn’t take long for them to find out that I work with math teachers. And then is happens…I get their math autobiography. And it’s rarely a happy story!

Far too many adults had bad experiences with school math and have carried that negative perception into adulthood. We cannot afford for this to continue!

Way too many adults in the U.S. are eager to tell their horror stories about word problems and fractions and formulas and how math just never made sense. Far too many adults had bad experiences with school math and have carried that negative perception into adulthood. We cannot afford for this to continue! We cannot afford for the next generation of students to go into adulthood without the tool set that everyone recognizes as critical to future career success. We owe it to our students to prepare them for school and for life. They don’t all have to love math – but they all deserve to be comfortable and competent with the logic and thinking skills that mathematics provides.

So, what are we going to do to break this generational cycle of fear of mathematics and the bragging rights that come with it?

For starters, all across the United States, educators are using powerful mathematics standards that aim high for student thinking and understanding. These standards, designed by the best and the brightest mathematics educators, mathematicians, and practicing teachers and leaders, reflect what we know to be in the best interest of our students.

Here’s something you need to know about these standards – they’re really nothing new! Great math teaching and learning always has always been and always will embody the content of these standards. These standards include what math educators and teachers have known for decades about what our students need to know and understand to be successful with mathematics…not just in school, but in life.

To take things further, KP Mathematics offers tools and strategies that help bring state standards to life! Using revolutionary manipulatives that focus on the basic principles and structures of mathematics, students gain conceptual understanding that leads to procedural fluency. The rigor found in the new state math standards is met using these simple tools and the strategies they encompass. Check out our previous posts to learn more about what these revolutionary tools can do for the students in your life!

Are Your Mathematics Games C, P, or A?

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

Serious Play #5

Imagine this…you walk into a noisy, bustling classroom where students are spread out in small groups at tables, on the floor, at the counter. They’re everywhere, really. They’re huddled around game boards and decks of cards, playing mathematics games that engage them in mathematical thinking, discourse, and strategy.

As you reflect on what you’ve witnessed, what did you notice? What math concepts were being reinforced? Were all the students playing the same games, or was there a variety? Did you see evidence of different levels of thinking?

After several years of using mathematics games in my own classroom, I noticed that that most of my games centered on arithmetic and math-fact practice, almost always at the abstract level (e.g., using the symbols of mathematics). It was at this time that I started looking for games that included representations other than numerals. Could I find games that incorporated concrete manipulatives? Were there games out there that incorporated pictures, diagrams, or other graphic representations? It turns out that there are plenty of options if you know what you’re looking for! These days, when selecting great mathematics games, I look for a balance in concepts, DOK (Depth of Knowledge), thinking strategies, and representations.

For today’s post, let’s look at representations that promote thinking at different levels. Below you will find three games, each addressing one of the C-P-A levels: concrete – pictorial – abstract. Notice that they all exhibit the characteristics described in last week’s blog post that make mathematics games fun.

1) Mathematics Games That Use Concrete Representations

Since students often manipulate objects when exploring a concept, games in this category typically include a game tboard or concrete materials.  Race to 100 provides a concrete way for students to focus on grouping (and ungrouping) tens as they build numbers to 100.

Click here to download Race to 100.

When playing this game, students work as partners rather than as competitors. As they repeatedly group ten ones into one ten, the “concreteness” of the game promotes visualization of place value.

2) Mathematics Games That Use Pictorial Representations

Ruler Races involves the use of a 12-inch ruler marked in fractional parts of inches. Students are encouraged to think of the ruler as a number line marked in fractional parts. They physically identify different points on the fractional number line as they explore flexible ways to think of fractions and fraction addition.

Click here to read the directions for Ruler Races.

Although one may think of a ruler as a concrete object, it is not mathematically concrete. It really is a pictorial representation of the 1-inch units (or 1-cm, or whatever unit is being used) used to measure objects. Some teachers even use rulers as number lines, which is the case in this game.

3) Mathematics Games that Use Abstract Symbols

Salute has been a favorite in my classroom for years! Students practice inverse operations (addition/subtraction OR multiplication/division) during game play. Students alternate between finding sums/differences and related missing addends/factors.

Click here to read the directions for Salute.

This game uses the numbers (abstract representations) on the playing cards to generate new problems for each round, and the students use mental math strategies to determine the missing values.

So…think about the games you use in your classroom. Do they offer students the opportunity to engage in mathematics beyond math-fact practice? Do your mathematics games include various levels of representations? How might you incorporate more concrete and pictorial games into your repertoire? Please share your thoughts in the comments box, below.

Next Steps for Teachers: Set up the games in this blog post (and others) and provide them for students to play. Also, take inventory of the games you already have in your classroom – do they provide concrete and pictorial opportunities as well as abstract practice?

Next Steps for Leaders: Lead your teachers in a conversation about this post. Have them list the mathematics games available in their classrooms and categorize them by C-P-A. Then walk through the classrooms to see how students are responding to the math games that use different levels of representation.

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

Are Base Ten Blocks Road Blocks? (Part 1)

Hands-On Math, KP® Ten-Frame Tiles

Spoiler Alert: If you haven’t already read John Van de Walle’s Teaching Student-Centered Mathematics, you’re going to want to get a copy right away!!!

Several years ago, I had the amazing opportunity to meet and work with John Van de Walle for a brief time. Subsequently, as I read the original version of his Teaching Student-Centered Mathematics, I was awed by the compilation of so many ideas I had read, heard, and discovered during my years as a classroom teacher and math coach. Honestly, I couldn’t put the book down. It brought together so many incredibly important ideas I had discovered over the years, all in one place!!!!

Perhaps one of the most impactful messages in the book was the reference to groupable vs. pre-grouped base-ten manipulatives. It was only a two-page spread, but it sent me on a journey that has forever changed my professional life.

It turns out that pre-grouped physical models, including the most-commonly-used base ten blocks, can obstruct student understanding rather than support it. I had heard some of my favorite researchers, such as Constance Kamii and Pat Thompson, make similar claims in their talks, and it was in this moment that their words came to life for me.

You see, in this very brief section, the authors stated that the potential is great for children to use pre-grouped manipulatives to demonstrate operations “without reflecting on the ten-to-one relationships or without really understanding what they are doing – this is especially true if children have not had adequate experience working with groupable models.” The added cognitive load increases dramatically when students must think about the structure of the manipulative in addition to representing numbers and operations (more on this next week).

The authors suggest that we use objects that can be grouped into tens, such as popsicle sticks, beans, or unifix cubes. They go on to suggest that an organizing structure such as the ten frame might further help students internalize the grouping-by-tens structure.

After reading this, my head started to spin with possibilities. What if we could somehow use ten frames in every place in our base-ten numbers to represent quantities, both large and small. How might this help children understand and internalize numbers and operations???

Well…that journey continues to this day. Peggy Akin and I joined forces a decade ago when we created Ten Frame Tiles, a groupable manipulative that helps children view mathematics in entirely new ways. This tool has the potential to transform the teaching of elementary mathematics.

Interested in learning more? Check out Peggy’s series of journal entries, Beyond Base-Ten Blocks: The Search for a Better Solution. Each entry visualizes the contrasting ways base ten blocks and Ten-Frame Tiles address a common base ten standard and then speculates on the contrasting impacts on children’s learning — and, ultimately, on the way children think about mathematics.

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

After reading this post, please share your thoughts in the comments box below. Have you faced frustration with base-ten blocks the way I did? Have you gone looking for alternatives? Have you found any that worked?

Are Base Ten Blocks Road Blocks? (Part 2)

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

Becoming a teacher-turned-inventor never crossed my mind earlier in my career. However, since founding KP Mathematics, this is the path I’ve traveled. KP Ten-Frame Tiles, an embodiment of the base-ten number system, came to life as we searched together to find a solution to the problems presented with pre-grouped manipulatives such as base-ten blocks.

As physical and pictorial models, base-ten blocks do an adequate job of illustrating place-value-based quantities. However, everything falls apart when using base-ten blocks to demonstrate the action of grouping and ungrouping tens and magnitudes of ten.

In her article, Choosing the Right Tool, Stacy Boote explains that base-ten blocks have a static form that renders them less useful when demonstrating operations such as division due to the need to group and ungroup. In contrast, when used to demonstrate the area model of multiplication, they will suffice because the action of grouping occurs after the model has done its job.

“When using the area model, exchanges between place values do not occur in the creation of the rectangle or when finding the areas of the four inner rectangles. Renaming occurs at the end of the process, when calculating partial products. Therefore, using materials that can be easily manipulated is not an area model requirement” (Boote, p. 479).

This regrouping issue that exists for long division also appears when representing the operations of addition and subtraction. The need to manipulate the materials for exchanges between place values is an important process. And when using base-ten blocks, the trades that must take place for grouping and ungrouping (aka, carrying and borrowing) obstruct student understanding rather than support it.

Using groupable manipulatives, such as KP Ten-Frame Tiles, craft sticks, or unifix cubes, requires less cognitive load because the actions children demonstrate are relevant to the mathematics they are learning. When students are “regrouping,” they literally group and ungroup rather than “trade,” which is the only option with pre-grouped manipulatives such as base-ten blocks.

So, you may ask, how do we resolve this issue? The answer is simple…use groupable manipulatives when the mathematics being modeled requires the action of grouping and ungrouping. KP Ten-Frame Tiles, craft sticks, or unifix cubes are a few options. It’s okay to use pictures of base-ten blocks to help kids visualize quantities, especially given this is the most common model used in testing situations. But when students need to manipulate to represent the movement between places, groupable models are the way to go.

If you’d like to learn more, I recommend reading the groupable vs. pre-grouped manipulatives section in Teaching Student-Centered Mathematics, or the article linked above, Choosing the Right Tool. And take a look at Peggy’s series of journal entries, Beyond Base Ten Blocks.

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

After reading this post, please join in on the conversation! What have been your experiences with pre-grouped manipulatives such as base-ten blocks? What have been your successes and challenges? We always love to hear from you!

In the Pursuit of Fluency Part 2: Number Sense

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

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

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

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

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

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

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

Numbers within 10

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

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

Place value

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

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

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

Number Relationships

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

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

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

Fractions

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

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

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

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

 

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

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

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

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

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

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

Building Numbers Within Ten (Early Number Fluency)

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

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

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

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

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

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

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

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

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

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

Building the First Ten

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

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

Building Numbers With Tens

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

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