In the Pursuit of Fluency Part 1

Professional Learning, Teaching and Learning

Math Fluency What is fluency? As a literacy specialist in my early career, I equated the word “fluency” with the ability to read with automaticity, accuracy, and prosody (expression using the patterns of rhythm and sound). Simply put, a fluent reader expresses herself fluidly and with expression. I understood fully that fluency is not the same as comprehension, and yet it contributes immensely to the ability to get to the higher levels of thinking required to comprehend well.

Several years later, as I shifted toward a mathematics specialization, I discovered that there is a mathematical equivalent to reading fluency. Much like reading fluency helps children succeed in higher levels of comprehension, mathematical fluency plays a foundational role in helping children succeed in problem solving. Therefore, just as students who struggle with reading fluency find comprehension tasks difficult, students who struggle with mathematical fluency find problem solving tasks difficult. For me, thinking about this in the form of an analogy helps illuminate the relationship:

Reading Fluency : Comprehension :: Mathematical Fluency : Problem Solving

So…what is mathematical fluency, and how can we develop it as a foundation that leads to stronger problem solving? This topic has many layers, and we’ll be revisiting it often over the next several weeks. For now, let’s take a look at what it is not.

Mathematical fluency cannot be reduced to memorization. This limited view of fluency puts undue pressure on students to focus on rote recall of facts. The mathematics community has agreed upon four qualities that comprise mathematical fluency:

  • Accuracy: finding the correct answer
  • Efficiency: using strategies or methods that allow for ease and flow while solving
  • Flexibility: knowing more than one approach to solve a problem and then selecting one that provides for efficiency
  • Appropriateness: knowing when to apply a particular procedure, strategy, or method

Students who are mathematically fluent are able to choose flexibly from a variety of methods and strategies to solve problems, and they solve them accurately and efficiently.

Mathematical fluency is not limited to knowing and using basic facts. Rather, it is much broader, involving concepts such as number sense, math facts, multi-digit operations, fraction operations, problem solving — and the list goes on (see below for more on this). Children need opportunities to develop fluency with the concepts on which they are focused at any given time. Students at the elementary level should have opportunities to develop fluency in at least four areas:

  • Number Sense: thinking about numbers and number relationships fluidly and flexibly.
  • Math Fact Fluency: adding numbers through 10+10 and the related subtraction facts; multiplying numbers through 10×10 and the related division facts.
  • Operational Fluency: using strategies to add, subtract, multiply, and divide whole numbers, decimals, fractions, and integers.
  • Problem Solving Fluency: selecting from and applying a variety of strategies and methods with the goal of solving a contextual or mathematical problem.

Mathematical fluency does not just happen spontaneously. Fluency develops over time as students are given multiple opportunities to focus on mathematical relationships and manipulations, often as a result of engaging in deliberately designed and sequenced experiences. Here are a few examples to get you started (more to follow in the coming posts):

  • Number Sense: use ten frames to help students understand relationships among numbers, both large and small. Ten frames needn’t be restricted to numbers within ten – they can be used to explore multi-digit numbers, decimals, and integers, as well! At KP Mathematics, we call this the “infinite ten frame,” and I’ll share more during this fluency series.
  • Math Fact Fluency: use card and dice games to allow students to practice their math facts in ways that are enjoyable and build confidence. For example, Double War is a simple card game where pairs of students each flip over two cards and either add or multiply the two numbers. Each student calls out his sum or product. The student with the greater (or lesser) sum or product takes all four cards. Repeat until all cards are used.
  • Operational Fluency: implement daily Number Talks with your students. This robust-yet-simple daily routine helps students develop number sense and mental math skills while focusing on developing operational fluency.
  • Problem Solving Fluency: engage students in daily problem solving, giving them tasks that require them to use strategies other than standard algorithms. For example, engage young children in solving a division problem before teaching them “how” to divide. Observe how they approach the process. Provide them with tools and manipulatives to guide their thinking. Resist the temptation to jump in – let them struggle. They will amaze you!!!

If you’re wanting more ideas in each of these categories, tune in for the next few weeks as we unwrap each category and share specific classroom-tested ways to develop fluency with your students.

For now, let’s carry on this conversation. What is your understanding of math fluency? What do you do in your classroom to develop fluency among your students? Please leave your comments in the boxes below.

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

 

The Equal Sign Means Relationship, Not Action

Math, Math Content, Math Discourse, Math Standards, Professional Learning, Taking Care of Teachers, Teaching and Learning

Equal Sign Means Relationship

Blog post #1 in the series, Lies We Tell Our Students

The Story

A while back, I was working in a fifth-grade classroom, and as a warm-up, I put the following equation on the board:

8 + 4 = 1 + 11

I then asked the students, “Is this statement true or false?” After waiting several seconds, I asked them to raise their hands to indicate their votes. At that point, two students voted for true, and the other 22 students voted that the statement was false.

Being the good teacher that I am, I asked the students to discuss their thinking, believing that those two students who said it was true would quickly convince the others. However, when I brought the class back together and once again asked my questions, 24 hands went up stating that the answer was false.

“Yikes!” I said, “What’s going on here? This statement is actually true. Can you tell me what you’re thinking?”

“Mrs. Rimbey,” started one of those original two students, ”8 + 4 does not equal 1.”

The Lie

I’ve since come to learn that this is not an isolated case. Many students across the grade levels believe that the equal sign is a symbol that indicates the action of finding the answer, when, in fact, it actually signals a relationship between the expressions on each side. In the vast majority of classrooms, students develop this misconception because virtually every equation they see in the early grades follows the same format: 3 + 3 = 6; 4 x 4 = 16; 20 – 13 = 7; and so forth. The “answer” appears last.

The Truth

So, what do we do about this? We can be sure to provide students with multiple daily opportunities to see equations written in different formats, pointing out that the equal sign indicates an equal relationship, not the action of finding the answer. Provide examples with the “answer first” (e.g., 6 = 2 + 4), with “nothing to do” (e.g., 1000 = 1000), or with “no answer” (e.g., 32 + 42 = 20 + 5).

The Series

This blog post is the first of a series that addresses the lies we tell our students. It’s important to note that these lies are not deliberate untruths. Rather, they are rules, procedures, mnemonics, and tricks we teach kids in an effort to make math easier for them. Sometimes, the lies come in the form of omission, such as the example of the equal sign shared above, where we don’t provide enough varied examples. Sometimes, they come in the form of half-truths, such as addition and multiplication, which always make bigger, which is only true with natural numbers (not including 0).

In Summary

To sum up this week’s Lies We Tell Our Students:

  • The inadvertent lie: The equal sign indicates the action of finding the answer.
  • The truth: The equal sign indicates relationship, not action.

As always, we love hearing from you! Do you think your students fully understand the meaning of the equal sign? Have you tried any of these (or other) strategies with your students? Please share your thoughts in the comments box below.

Next Steps for Teachers: Use a variety of equation structures with students on a daily basis (see examples above). Consider making a set of 4-6 “true-false cards” each day to show the class. Simply write one equation on each card, using a variety of formats, making some true and some false. Then ask the students whether each is true or false and to justify their answers.

Next Steps for Leaders: Discuss the meaning of the equal sign with grade levels, focusing on the notion that the equal sign indicates a relationship, not the action of finding the answer. Although this concept typically appears in the K-1 math standards, this misconception continues to exist in middle school and beyond. Discuss ways to avoid this misconception in each grade level, and then share their ideas across grade levels.

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.

 

In Pursuit of Fluency Part 6: Operations Beyond Basic Facts

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

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

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

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

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

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

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

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

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

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

What are your thoughts about building math fluency for operations? What strategies have you found successful? Please share your thoughts in the comments box below.

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

 

Grading Practices That Impact Learning

Problem Solving, Professional Learning, Teaching and Learning

Last week, I shared ways to shift homework practices to maximize student learning and understanding. This week, I turn my attention to grading practices. Just as with homework, when students receive their projects or tests back with grades written at the top, they simply look at the grade, make a judgement about whether or not they are good at math or if their teachers like them, and pretty much disregard any other comments on the paper.

We, as teachers, know that reflecting on work and learning from mistakes is where real learning takes place. However, even when we ask students to reflect on how they did, that score at the top of the page supersedes anything else on the page that could lead to a learning experience.

So…what can we do to push students to focus on the learning rather than on the grades? Here are a few ideas you might try…

  • Grading Practices 1: Give students the opportunity to share everything they know. As teachers, we hold the power to choose which facts and details we deem most important to include in an assignment or test. However, our students have so much more knowledge to share than we have anticipated with our questions. Why not include a blank page at the end that asks students to tell us everything they know about the topic at hand that wasn’t asked? The example below is from a science class, and is easily transferrable to math class!

  • Grading Practices 2: Highlight mistakes, but don’t give a grade. When grading an assignment or test, simply highlight the mistakes each student made. Do not assign point values, and do not put a score at the top of the page. At first, your students will ask, “What did I get?” or “How much is this one worth?” Resist, resist, resist! Do not go there.

  • Grading Practices 3: Give students time to reflect upon and discuss mistakes. Rather than going over the assignment or test as a class, ask students to reflect on their on their own work. They may discuss and compare answers with their friends, but in the end, everyone must learn from his/her own mistakes if learning is to take place. As teacher Leah Alcala says, “Learning from mistakes is really what learning is.” Click here to see Leah and her students in action.
  • Grading Practices 4: Share “my favorite mistakes” before passing out papers. After highlighting mistakes as described above, go through the papers again to look for trends and teachable opportunities. Select 3-5 of your favorite mistakes to discuss with the whole class prior to passing out their papers. This helps to glorify mistakes, communicating that mistakes are among the best learning opportunities we have.

I’m hoping that between last week’s homework post and this week’s grading post, you’re seeing ways in which teachers might shift their practice to maximize student learning without adding to the plethora of things they already do every single day. Simple tweaks in our practices can bring about tremendous shifts in student learning.

Please let us know how it goes by sharing your thoughts and stories in the comments box below.

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

Math Homework – Helpful or Harmful?

Math, Professional Learning, Teaching and Learning

I remember math class as a child. As soon as we walked in the door, we would get our homework out and grade it as a class, usually trading papers to keep us honest. The teacher would read the answers, and we would simply mark each item that was incorrect, total the number of correct answers, and write that number at the top. And then we would give the papers back to the original owners.

As a good math student, I remember both the pride and fear associated with this practice. I was proud when my paper came back with 100% written at the top. However, anything less than 100% resulted in silent shame, the certain knowledge that everyone in the class realized I was a fraud, clearly unworthy of the reputation I had built. (And I felt even worse for my friends who struggled in math, knowing the humiliation they felt when they received their papers back.)

More importantly, we were unaware that our scores were not the feedback we needed. With no opportunity to understand why each answer was correct or incorrect and to self-correct, as needed,  we simply looked at our score, proud with a good score and ashamed with a bad score. End-of-story.

Let’s take it a step further – even now, math homework often includes practicing the same skill 20-30 times. Although students need to develop automaticity, if they already know how to do long division, for example, they need only practice a few times before the practice becomes tedious. And, of course, if they don’t already know how, then practicing 20-30 times incorrectly is only going to reinforce bad habits.

If the point of homework is to see if students “get it,” then 5 practice problems is enough. If kids get it after 5 problems, and you ask them to do another 35, you’re just being mean. And if kids don’t get it after 5 problems, and you ask them to do another 35, you’re really being mean.    – Matt Cwalina, Discovery Education

So, what can we do to improve students’ homework experiences?

The 2-4-2 Homework Policy

Steve Leinwand, the author of  Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement and Sensible Mathematics: A Guide for School Leaders, suggests a new way of thinking about homework. Rather than assigning 20-30 problems, assign students 8 well-crafted, engaging problems using the 2-4-2 homework policy.

  • 2 problems on the new skill (This is usually enough to determine understanding and, at the same time, avoids practicing and reinforcing) mistakes);
  • 4 cumulative review problems roughly drawn from content developed the day before, the week before, the month before (think about being strategic with prerequisite skills for the current content); and
  • 2 problems that require showing work or including explanation that supports problem solving, reasoning, and justification.

A Feedback Process That Promotes Learning

And then comes the most important part…feedback time. At the beginning of math class the next day, rather than asking students to trade papers to “grade,” the teacher posts the answers to these eight problems on the board. S/he provides students with five minutes to review their work in pairs or triads with particular attention to the last two problems. After five minutes of small-group discussion, s/he leads a group discussion of any problems that are still causing trouble. And here’s the kicker: homework is only to be recorded as completed.

 

This practice sure would have been helpful to me as a young scholar.  As a matter of fact, as a middle-school student, I would meet with two of my friends before school to “compare answers.” For me, this self-created practice was especially about knowing that I know – going into class confident of my ability to do and understand the math. What I didn’t know then was that, someday, this would be the exact practice I would encourage in my own students.

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 using the comments box below! What homework practices do you currently employ? How might you reframe your practices to make them meaningful and useful for students?

Interested in learning more about KP Ten-Frame Tiles? Please check out Peggy’s latest Beyond Base Ten Blocks journal entry, Manipulatives First.