“Poochie Poo” and Numbers Squared

Posted by & filed under Differentiated Instruction, Inspiration, Math Activities, Tabor Rotation.

“Creative thinking may mean simply the realization that there’s no particular virtue in doing things the way they always have been done.” -Rudolf Flesch

Today’s Inspiration Blog is also an Activity Blog. If you continue reading you’ll understand why…

Friday was an early release day for all three of my children, so I decided to make Glenna Tabor, educational consultant, take the day off. Glenna Tabor, a.k.a. Mommy, had great adventures with her children all day! We had a $25 gift card to spend at a bookstore, so computational proficiency was being used. Higher-order thinking,  such as evaluation, was also being exercised by all three as they determined which item was worth the price. After shopping, we went for a snack at our favorite restaurant. No mathematical calculations needed there. We know it’s a bargain—all the chips and salsa you can eat with at least 2 ice cream cones for dessert!

As we were resting that evening, we were petting our Heinz 57 dog. I had begun calling him “Poochie Poo” for the last week or so. I’m not really sure why he needed that name, but I like the alliteration. I decided that we should shorten it to P Squared. This shortening of names brought about the most incredible mathematical hour with my 4th grader.

My 9-year-old asked me why I was calling our dog P-Squared. I told her that “Poochie Poo” began with two P’s which was P times P or P squared. She put her hands on her hips and answered, “Mommy, I’m only in 4th grade and we haven’t studied squared yet. Can you show me?”

For me, her request was a rare one. I don’t know how many fellow educators experience this phenomenon, but our middle child doesn’t enjoy me teaching her anything. She knows I’m a teacher, but she’d much rather I not teach her.

We immediately pulled out rainbow centimeter cubes and centimeter grid paper. (If you don’t have those items, sugar cubes and any kind of paper will do.) I asked her to make me an array that was 3 by 3. She arranged the 9 cubes on the grid paper.

I asked her what geometric shape the 3 by 3 array looked like. She answered that it looked like a square. I asked her if she wanted to make another array out of “a number by the same number.” Being my diverse, hard to keep engaged learner, she said, “No, thanks.” That’s when I offered her the “magical math tool” (better known as needle-nose pliers) to use to make her new arrays. As it always did with every reluctant learner, she wanted to use the magical math tool. She busily set about creating two more arrays- a 2 by 2 and a 4 by 4. Each time she was able to observe that the array made a square.

As she worked, I wrote each of the equations on a sheet of paper and explained that the 2 at the top right was the symbol for “squared.” I asked her why she thought mathematicians called it that. Immediately, her eyes lit up and she shouted out, “It’s because it’s makes a square!”

I asked her what she thought would happen if we squared the number 6, 7, or 8. She wrote the equations on her paper and told me the answer. We even talked about square roots…but that exploration will have to come another night.

Not one to pass up the opportunity to differentiate based upon interest and her recently increased readiness for the concept of squared numbers; I asked her if she’d like to build a cubed number. Of course she wanted to, but only if she could use the magical math tool.

For the next 30 minutes she worked on building a 2 by 2 by 2 cube, 3 by 3 by 3 cube, and a 4 by 4 by 4 cube. I tried helping her keep the rainbow cubes straight after the first arrangement fell apart, but she didn’t want the assistance. She told me that she wouldn’t discover it on her own if I did it for her. How right she was! She created cubes, drew what she had created, and wrote the equations. She went from concrete-to-representational-to-abstract.

How did the learning session end? The abstractions were made; she could tell me what squaring and cubing a number meant. Then she began to clean up the cubes–that had spilled all over the floor. She spent the next 15 minutes picking them up with her magic mathematical tool—while “Poochie Poo” tried to eat them.

I will treasure that hour forever! The most reluctant learner in my house let me help her “turn on her math light.” I know she wouldn’t have done it if I had given her a brief explanation, written an example on a sheet of paper, and then handed her a workbook page to complete. I got her attention because I:

  • Connected the new mathematical concept to her world…her dog;
  • Used a concrete manipulation of objects to help her understand an abstract concept;
  • and, made the task of high interest by letting her use needle nose pliers to move the cubes.

My daughter is one of the students I keep in mind every time I encourage others to differentiate instruction in mathematics. She’s one of the reasons why I believe in systematically structuring meaningful mathematical learning experiences for all students. So many of our students need to be engaged, to be connected to the information, and to have fun while they are learning-Tabor Rotation does all of that!

Side note- The next morning I found a sheet of paper with some equations written on them. Next to it were three bags that were filled evenly with cubes and two left over. On her own volition, she had counted organized the cubes mathematically. Amazing! Problem-solving strategies are used when a learner encounters a real problem!

“There is no doubt that creativity is the most important human resource of all. Without creativity, there would be no progress, and we would be forever repeating the same patterns.” -Edward de Bono

The Sounds in the Silence

Posted by & filed under Math Activities.

“I have often regretted my speech, never my silence.” -Xenocrates

“I have nothing to say / and I am saying it / and that is poetry / as I needed it” John Cage

John Cage’s most famous musical composition is called 4’33”.
It consists of the pianist going to the piano, and not hitting any keys for four minutes and thirty-three seconds.  If the listener truly listens, the sounds of the audience and the environment around the pianist fill the silence.

What happens when you are not speaking, guiding, or directing? That is the composition of your instruction…the activity when you are silent…

Differentiating Means “Shaking it Up”

Posted by & filed under Differentiated Instruction, Mathematics.

“You have your way. I have my way. As for the right way, the correct way, and the only way, it does not exist.” -Friedrich Nietzsche

There are many, many different (and very long definitions) for the term Differentiating Instruction. The simplest one I’ve ever used is:

“Differentiating instruction means regularly “shaking up” what’s going on in your classroom.”

There are several ways to base the differentiating of instruction in a classroom. It can be based on the learner’s interests, learning style, readiness level, or a combination. Stewart and Felicetti (1992) define learning styles as those “educational conditions under which a student is most likely to learn.” Theories of learning styles are not really concerned with “what” learners learn, but rather “how” they prefer to learn. Research also shows that it is more important to match the presentation of information with the subject matter and then provide opportunities for various learning styles to understand the content.

That means “shaking up” how you present information, how the students process the information being learned, and how the students demonstrate their understanding of the concepts.

What made me think about learning styles today? Well, my husband started reading me something he had written this morning. About half-way through the first paragraph his voice started to sound like Charlie Brown’s parents, “WA, WA, WA.”

It’s not that I wasn’t interested (I really was, Honey!) but I am a visual learner and he was reading it to me while he was looking at it. Because I know I’m a visual learner, I asked if I could look at it, too, while he read it. Immediately, he had much more of my brain attending to what he was trying to say.

Let me give you another example of my need to visualize. One of my friends left the room when I was talking to her. I stopped talking.

She immediately shouted back, “I can still hear you. Keep talking!”

I replied, “I know you can hear me, but I want to be able to see you hearing me!”

She came back in the room and the conversation continued-with both of us laughing about my need to “see” a person when I was talking to them.

Ok, here’s my last example of being a visual, tactile learner. I bought a new digital camera a few years ago. My best friend learned how to use it first. She was very nice. She put in the batteries, set up the memory card, and took everything out of the box. She even took pictures with it to make sure it worked and could be downloaded to the computer.

But…I hadn’t had a tactile experience with the new camera. Her giving me a verbal explanation of my new camera and how to use it should have been helpful, but it wasn’t! I thanked her many times, then took the

Do you know your own learning style? You might want to find out, because the way you teach is usually the way you learn.  Knowing your own learning style will help you recognize pedagogical tendencies and make you more aware of the need to differentiate activities based upon all the learning styles that are represented in your classroom.

Helping your students discover their individual learning styles will help them develop strategies for areas in which they might need help and capitalize in ways they are strong. Here are just a few resources available online. LdPride.net has some good explanations for learning styles. They also offer a free learning styles test.

<a href=”ldpride.net”>Free Learning Styles Test</a>

Learning-styles-online.com offers a free online learning styles inventory that could also be taken by your and your students. The results are shown in graphic form—great for a visual learner like me!

http://www.learning-styles-online.com/inventory/

I wish more of my teachers had known my learning style. I really didn’t understand the information that was shared about mathematics when the teacher was writing on the board and explaining it with her back to me. I have never developed a deep understanding of the concepts I explore until I Have a concrete, meaningful, tactile experience with them!

Maybe this blog will spur you to think about the instruction in your room and encourage you to “shake it up” some!

“Everyone who enjoys thinks that the principal thing to the tree is the fruit, but in point of fact the principal thing to it is the seed. Herein lies the difference between them that create and them that enjoy.” -Friedrich Nietzsche

Solving Problems = Balloons = Fun

Posted by & filed under Inspiration, Math Activities, Problem Solving, Thinking.

“It takes as much energy to wish as it does to plan.”     -Eleanor Roosevelt

I was working with a school a few weeks ago as we planned for the remainder of the year. The things we identified, at the beginning of the planning process, were the most challenging topics or concepts that the teachers were going to teach in mathematics. The concept that appeared on every grade level’s list was problem solving. Especially challenging was making a decision about which strategy or operation to use.

I have to admit…when I was a student I didn’t really care for word problems. It’s not that I didn’t know how to calculate elapsed time. I just didn’t care when Mr. Jones would arrive at his house if he left the airport 3 hours earlier and drove 55 miles an hour.  When my students rolled their eyes at the problems they were required to solve I wanted to roll my eyes, too!

One of things I do to decrease the number of eye-rolling exercises for my students is give them problems that are fun and engaging. I try to do this at least once a week—especially during test season. It keeps the thinking energy at a higher level!

My favorite is “The Balloon Building Task.”

The Balloon Task

Have a camera ready. Then, give this task to your students and watch what happens. After observing the results from the first balloon tower building, have your students reflect with their group and create a better plan based on their observations of all of the Balloon Towers. Give the task to them a few days later. You’ll be amazed at their application of what they’ve learned from experimentation, meaningful problem solving, and decision making that is done in a systematic way. Be careful! You just might become addicted to the thinking energy in your room!

Note: This is a great task to do with a faculty. The laughter and the team building that occurs will be worth the $5 an administrator, or staff developer, spends on balloons and tape!

“We know what happens to people who stay in the middle of the road. They get run over.” -Aneurin Bevan

A Purple Cow and Algebra

Posted by & filed under Math Activities, Math Games, Mathematics, Thinking.

“As long as algebra is taught in school, there will be prayer in school. ” -Cokie Roberts

“It is hard to convince a high-school student that he will encounter a lot of problems more difficult than those of algebra and geometry.” -Edgar Watson Howe

I woke up this morning and looked at the stuffed animal my parents brought back from Germany a few years ago. It’s a purple cow. I know, you’re probably saying the purple cow poem in your head. I love the cow, not because it’s purple, but because it has been a perfect tool to introduce algebraic thinking to many groups of students. When you squeeze its stomach it moos 3 times.

Here’s how I do it. Before I begin the lesson, I place the cow in a large, brown shopping bag with the top folded down. I call this The Mystery Bag. The Mystery Bag The children use deductive reasoning to guess what is in the bag. I can only answer questions that can be answered with yes or no. I only allow 10 questions, so the students have to work in groups to develop the questions that will best guide them to what is in the bag.

After the students have guessed what is in the bag, I squeeze the cow. The cow moos 3 times per squeeze. I ask the students to think about how many moos the cow will make if I squeeze it 3 times. I have counters, pencil and paper, and calculators available to each group.

After I have given each group enough time to think about it, I record their responses on the board. I also ask them to share with the whole group how they thought about the answer. Sharing their thinking is very important because it gives the students the chance to hear how others think. It also is a time to “think about your thinking.”

I ask the students if there is a way to write about the squeezes and the moos. The students come up with many ways to do this. After recording each of these on the board, I ask them if there is an shortened way to write 1 squeeze = 3 moos. The students eventually determine that 1s = 3m is the shortest way to record it.

I then write 3s = _m and 5s = _m on the board. I ask the groups to think about what number would go in each blank and why. After giving think time, I ask several students to share how they determined the answer.
I then write _s = 18m and ask them to determine how what will go in the blank. Within just a few minutes the children have been able to determine the answer.

The activity continues with the students generating problems, writing them on the front of a folder, placing the answer inside, and passing them to the next group to solve.

At the end of the lesson we talk about the fact that we have just “thought like a high schooler.” They are THRILLED with themselves and are encouraged to go home and tell someone they were doing algebra.

You can study for years about algebraic thinking. Every article you read about the topic will include a bibliography with more articles about it. My primary goal with elementary level students and teachers is to remove the intimidation of algebra. Hopefully, after reading this post, you heart won’t start beating faster thinking about x and y. Remember, it’s really just a squeeze and three moos!

“The most difficult thing is to just start the ball rolling. Once it starts, it’s actually more difficult to stop it.” -Butch Lovelace

By the way, Gelet Burgess, author of “The Purple Cow,” hopes to be remembered by all his other works. He also says he wished he’d never written the one that all of us can quote…

Readiness Grouping in Mathematics

Posted by & filed under Differentiated Instruction, Mathematics, Tabor Rotation.

“I have been impressed with the urgency of doing. Knowing is not enough; we must apply. Being willing is not enough; we must do.” -Leonardo da Vinci

A group of teachers once told me that using small groups in mathematics didn’t meet the needs of all their students. How true!!! It depends on how a teacher effectively uses differentiated strategies throughout a unit of study and throughout a week. Being willing to use small groups in mathematics, at stations or centers, is a brave first step for many teachers. This should be applauded and encouraged!

Once teachers and schools have begun to release themselves from strictly using whole-group instruction, they are ready to sophisticate their readiness small groups. A math specialist, who is trying to help her district better understand Tabor Rotation, asked me what Days 4 & 5 of Tabor Rotation might look like.
Day 4 of Tabor Rotation begins the same as any other day. The teacher guides the students in a constructivist manner in a Vocabulary Review for 5-10 minutes. The emphasis is on the words the students need to attach meaning to and place into long-term memory. Next is the whole-group mini-lesson when the students are working with partners in an interactive way on one of the mathematical concepts for the week.

The math block continues with a brief explanation of The Application Menu of Options. Using The Application Menu of Options is a great way to tier assignments and empower the students in making choices about which task most interest them. After the brief explanation, the teacher gives out the menu and has the students place it in their Tabor Rotation Folder. The students may be working independently or with a partner.
Some teachers find it more convenient to use a standardized formula for creating The Application Menu. An example of this was created by a specialist and is being used by a primary school in south Texas. If you’re interested in a copy, let me know and I’ll upload it to the Free Resources Page of glennatabor.com.

Other teachers use a menu that is based on the concepts being learned but may vary in assignment type. The top row is simplest, middle is average, and the bottom row of assignments is challenging.

While the students are working on their menus, the teacher calls the readiness groups–beginning with the Above-Level students. Remember that the readiness groups were based on the information gathered by the teacher during pre-assessment and via clipboard cruising throughout the first part of the week. You may want to use a simple sheet like the Tabor Rotation: Differentiated Readiness Grouping Sheet when you’re clipboard cruising.

All of this on-going assessment is based on what the students do and don’t know about the concepts being studied that week (not on results from a placement test given at the beginning of the year). The above-level students will receive qualitatively different and challenging instruction. They will not be given quantitatively more work with the skills and concepts they have proven to have mastered. The teacher will give the students work to respectfully challenge them but will not grade the work. Instead there will be sharing with each other and in the Mathematician’s Circle on Friday.

After meeting with the above-level group for approximately 10 minutes, the teacher sends the students back to work on the Application Menu. Then the teacher calls 1/2 of the on-level students to her whisper readiness grouping. This is the group of students who is rarely worked with in a classroom setting in mathematics. If the teacher builds on their interest and learning styles, while sophisticating the socratic dialogue, the students will become the highly able–EVERY TIME!!!

The next readiness group who is called is 1/2 of the at-promise students (again, just for the skills and concepts for the week). These students receive instruction that is engaging and “hands-on.” It needs to be different from what was previously done in the week, because those activities didn’t develop mastery.

“Advancement only comes with habitually doing more than you are asked.” -Gary Ryan Blair

The last group of students to be called on Day 4 is the group of students who may need to have “pieces of the puzzle completed for them.” Maybe this is the group of 4th grade students who need 3rd grade skill instruction. It doesn’t make sense to reteach these skills to the entire class, so the teacher teaches them to the group of students who need it.

“When you do the common things in life in an uncommon way, you will command the attention of the world.” – George Washington Carver

Note that some of the students in the first at-promise group may also be in this one. It is not necessary for these students to complete all of the Application Menu of Options since it’s more important for them to have small-group time with the teacher.

Now, for Day 5:

The day begins the same way as Day 4 with vocabulary instruction and WGML (Whole-Group Mini-Lesson) with a different partner. Next, the students take their Application Menu of Options out of their Tabor Rotation Folder and begin to work. The teacher calls the first readiness group of the day that is comprised of the other 1/2 of the on-level students. She meets with them for 10 minutes. She might play one of the games from the Games Station that the students really liked and make parts of it more challenging.

The second group the teacher meets with on Fridays is the other half of the at-promise learners. Again, making the instruction different on the skills and concepts for the week.

The last group met with on Friday is the group of students who need what is called “front-loading.” If you have studied Accelerated Learning and the work of Max Thompson, you are familiar with this incredible tool for changing the lives of students. The tendency for the at-promise learner is to remediate and “fix” their problems. Instead, the teacher should be looking for the opportunity to teach them the concepts that will be taught the next week.

This provides the students with the chance to be a step ahead instead of trying to catch up. It builds confidence and positive efficacy for the students. And, as research as shown, it WORKS!!!

More information about Clipboard Cruising, Anchoring Activities, and grouping can be found in some of my previous blog posts.

I probably answered more than you thought you needed, but readiness grouping in mathematics cannot be skipped or skimped. Teachers who are doing what is best for all of their students, do it all the time and find ways to use differentiated, small-group guided instruction in mathematics. They aren’t just trying…

“Do, or do not. There is no ‘try’.” -Yoda (‘The Empire Strikes Back’)

Hopefully, the information contained in this blog and on this website will help all of you on your journey deeper into differentiating instruction in mathematics!

“Good people do not need laws to tell them to act responsibly, while bad people will find a way around the laws.” -Plato

Do it Anyway!

Posted by & filed under Inspiration.

mother teresas & children

“Do not wait for leaders; do it alone, person to person.” -Mother Teresa

When someone asks me what schools I’ve worked with during my career as an educational consultant, my answer is always, “the ones where you wouldn’t drive at night.” My heart is pulled to the teachers and students in these schools. For most of these children, a good education is their only ticket out of a challenging situation.

I admire the work that is accomplished by all of those who keep coming back day after day to do what is best for children. Today’s Inspiration Blog comes from the thoughts of one of the most unselfish, giving souls I know. I hope her words mean as much to you as they do to me…

“At the end of life we will not be judged by how many diplomas we have received, how much money we have made, how many great things we have done.

We will be judged by ‘I was hungry and you gave me to eat, I was naked and you clothed me, I was homeless and you took me in.’

Hungry not only for bread — but hungry for love. Naked not only for clothing — but naked for human dignity and respect. Homeless not only for want of a room of bricks — but homeless because of rejection.”

Mother Teresa
Catholic Missionary, Nobel Peace Prize Recipient

“People are unreasonable, illogical and self-centered.
Love them anyway.
If you do good, people will accuse you of selfish, ulterior motives. Do good anyway.
If you are successful, you will win false friends and true enemies. Succeed anyway.
The good you do today will be forgotten tomorrow.
Do good anyway.
Honesty and frankness make you vulnerable.
Be honest and frank anyway.
The biggest person with the biggest ideas can be shot down by the smallest person with the smallest mind.
Think big anyway.
What you spend years building may be destroyed overnight. Build anyway.
People really need help but may attack if you help them.
Help people anyway.
Give the world the best you have and you might get kicked in the teeth.
Give the world the best you’ve got anyway.” -Mother Teresa

The Right Tools for the Job

Posted by & filed under Math Activities, Math Games, Mathematics, Tabor Rotation.

“Man must shape his tools lest they shape him.”     -Arthur Miller

“An architect’s most useful tools are an eraser at the drafting board, and a wrecking bar at the site.”     -Frank Lloyd Wright

Many of you have counted down the days till you received a much deserved spring break. I could hear the cheers up and down my street today as children made their way home with no homework written all over their faces. My own children are looking forward to “screen time” during the week and the chance to stay up past their school bedtime.

I always spend the first few days of spring break recovering and doing at least one day of absolutely nothing. By the end of the break I had to open the books and start to plan again. If some of you choose to do a little bit of planning for games and activities in mathematics, this blog is meant to help.

The following work mat can be used with just about any concrete object or counter and placed in the Tabor Rotation Manipulative Station. I can never have too many work mats on file. Hope this one helps!

AB Pattern Work Mat

“Do not wait; the time will never be “just right.” Start where you stand, and work with whatever tools you may have at your command, and better tools will be found as you go along.”     -George Herbert

Reading and Writing…in Math!

Posted by & filed under Differentiated Instruction, Mathematics, Tabor Rotation.

“You write in order to change the world, knowing perfectly well that you probably can’t, but also knowing that literature is indispensable to the world… The world changes according to the way people see it, and if you alter, even by a millimeter, the way … people look at reality, then you can change it.” -James Arthur Baldwin


Most teachers make a mental connection to Tabor Rotation when I ask them to think about how schools instruct in a balanced literacy program. Students’ brains have to make a connection to what they are learning in any subject. This is a well-known and well-emphasized fact in literacy instruction, but it is not always of primary importance in mathematics- it should be!

In the book Educating Everybody’s Children (Robert Cole, Editor), they say that all learners, especially the diverse learner, needs the following in math:

  • Relate math to real-world experiences of their own.
  • Work cooperatively to solve problems.
  • Explore concepts with hands-on materials.
  • Use calculators and computers.
  • Construct one’s own mathematical knowledge.
  • Write and talk about math.

All teachers have to do is use the same comprehension strategies they are teaching in literacy when teaching math. Students need to be encouraged and given plenty of opportunities to write and talk about math, connect it to their own world, work cooperatively, and explore concepts in a hands-on way. Tabor Rotation assists the teacher in facilitating that.

Going right along with using the comprehension strategies that all good readers use, in every subject and every occasion, is writing in order to better understand what you have just experienced. Writing in math should be a daily activity. In Tabor Rotation, students should carry their Math Journals with them to each group or activity and take notes on the left side of the book. On the next page, the right side, they turn their notes into explanations, illustrations, and descriptions. Every person in the room is writing during this time…including all the adults who were assisting. The most important model in the room and the person with the most influence is the teacher. If he writes, then the students will realize the importance of it. If a student is an emerging writer, then he can record what he has learned by drawing.

Some people get caught up in the fact that they can’t afford composition books or math journals. If you have any kind of paper that can be fastened together, your students have a math journal! Begin by modeling, scaffold support as they write, and share what is written from the Mathematician’s Circle on Friday.

Every state math test is also a reading and writing test, too. So, read and write in math on a daily basis and you won”t have to sweat when test time comes!!!

“The discipline of the writer is to learn to be still and listen to what his subject has to tell him.” -Rachel Louise Carson

Here’s to the Crazy Ones

Posted by & filed under Inspiration.

“Remember when you see a man at the top of a mountain, he didn’t fall there.”
– unknown

This morning my son asked me what I was doing today. I showed him my “to do” list. At the top was written, “write my inspiration blog.” He asked me how many people read it. I told him that I didn’t know how many read it, but if one person did and like it–it was a success.

Today’s blog is late for all kinds of reasons, but it’s still going to be written. I gave my word that I would blog three times a week (every Monday, Wednesday, and Friday), so I do. My family thought I was crazy tonight for heading up to my office to my computer, but they also know that if I say I’m going to do something–I do it. I hope there are others out there who still give their word and keep it. I hope that some of them are the teachers my children will have one day.

“Here’s to the crazy ones. The misfits. The rebels. The trouble-makers. The round heads in the square holes. The ones who see things differently. They’re not fond of rules, and they have no respect for the status-quo. You can quote them, disagree with them, glorify, or vilify them. But the only thing you can’t do is ignore them. Because they change things. They push the human race forward. And while some may see them as the crazy ones, we see genius. Because the people who are crazy enough to think they can change the world, are the ones who do.”     -Jack Kerouac, from his book, On The Road

My mechanic told me, “I couldn’t repair your brakes, so I made your horn louder.”