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Monday, April 12, 2010

Time and Again

Ask any teacher what the most challenging part of their job is and they'll say: time. How often have you thought: wouldn't it be great if we... Only to realize that you couldn't spare the time?

Anything worth doing is worth doing right. But our society is a just 'do it' society. As teachers we are expected to 'cover' so many concepts within the year, that there is not enough time to teach. Students (and teachers too for that matter) learn better, understand more fully, and remember the tings that they spend time on, do hands on investigations, and play around with. But how often do you have time to give to one concept before you have to move on to another?

I am both blessed and challenged: I don't have to cover the curriculum. I do extra support for students who need a little more time or practice. However, I, too, run out of time. I see the students only one day out of 3 (some only one day out of 6), and, though I initiate some great in-class project learning activities, there have been many times when I, too, need to move on and leave things unfinished.

I recently did two activities that I think were very successful: for those who managed to complete them. I did an activity with my 7th graders on the golden rectangle and the golden ratio.

They got to choose one of the following:

Concrete Random:

Deign and conduct an experiment or survey to determine if more people prefer the golden ratio or not.

Reflect: “Why do we or don’t we prefer the golden ratio?” Be prepared to present your findings and opinions.

Abstract Random:

Work with a partner or a small group to find examples of golden ratios by:
Measuring your height and arm span, etc.

Can you find any other relationships between body measurements that are golden ratios?
Measuring things that you find in the room, such as cinder blocks, tables, bulletin boards, doorways, etc.

Speculate, discuss, then illustrate “How does the golden ratio affect our lives?”

Concrete Sequential:

Construct a puzzle or poster of different sized rectangles by coloring, cutting, and using the rectangles found on an 8 by 11 inch sheet of paper.

Indicate whether or not they are “golden rectangles” (having a ratio of 1.67).

Be prepared to prove your findings and explain how you did it.

Abstract Sequential:

Gather examples of golden rectangles through historic photos or artifacts online.
Organize your findings into a table or chart.

Develop a hypothesis as to “Why this might be so?”

I also did a cubing activity with my 6th graders on prisms and pyramids. They got to choose which dice they wanted to construct, cut out the net, tape it, then roll. They then choose between the two activities listed. I thought it was very cohesive to have them use a net to create a 3D object when they were studying prisms and pyramids.

Cubing: Prisms and Pyramids
Summarize the similarities and differences of “prisms” and “pyramids”.
Group photos:
“Prisms” l “Pyramids”

Organize in a poster: Prisms’ Nets
Pyramids’ Nets...
Explain / demonstrate which nets are “prisms” and which are “pyramids”.

Speculate: Why weren’t the pyramids in Egypt prisms?

Give examples of “prisms” and “pyramids” in real life.
Discuss the similarities and differences between “prisms” and “pyramids”.
Imagine you helped construct the ancient pyramids.

Tell us...
Demonstrate the similarities and differences between “prisms” and “pyramids”...

Brainstorm all of the uses you can for prisms and for pyramids.
Illustrate the connection between nets of “prisms” and “pyramids”.

How are they similar?
How are they different?
Write step-by-step directions for building prisms and pyramids.

How are your directions similar?
How are they different?

I was pleased with the work and the thinking that these activities brought forth. However, some students couldn't complete them within one class period and the next time I saw them, 3 days hence, we had to review and practice to prepare for a quiz. Hopefully we'll find time to return to these activities, to learn, to think, to speculate, and to grow. Hopefully we will find/ make time for learning.

Saturday, January 30, 2010

"The Goals of Differentiation" by Carol Ann Tomlinson

In Giving Students Ownership of Learning (Nov. 2008, volume 66, no. 3, pages 26-30), Carol Ann Tomlinson states, "How can I create a real learner? As teachers address this question, they need to consider four elements that help students take charge of their own learning and thus take charge of their lives: trust, fit, voice, and awareness."

"Trust begins when students believe that the teacher is on their side... (and) believes in their capacity to succeed..."

"Fit suggests that we ask students to do only what they are ready to do." Challenging them just enough, so that they strive and succeed. "Fit also requires that what we ask students to learn connects with what they care about." It's relevant to their lives today.

Voice is evident when we ask students to assess us as teachers. When...student feedback was sought, shared, and acted on, it was clear that students understood the power of their voice." Another way that teachers encourage students' voice is through giving students the opportunity to choose.

When "students ...(become) metacognitively aware... They understand how to capitalize on their learning strengths and how to compensate for their weaknesses. (They) reflect on their work... Academic awareness builds academic success."

"Differentiated instruction is...concerned with developing ... student efficacy and ownership of learning. (Helping students to) identify their own identity as learners."

This was a great article. In my work with middle school students who need extra support with math, I strive to do all of this. I have a sign in my classroom. The words "I Can't" with a line drawn through them. I don't allow my students to say that they can't do something. Saying "I can't closes a door. Saying I need help with... opens the door and allows the possibility that they can.

It has been proven that positive thinking strengthens. Athletes don't say i can't before the big game or event. They visualize themselves succeeding, winning, and then they do it.

I believe in my students. I've come to believe that everyone can learn math. Some need longer, different strategies, materials, and supports, but everyone can succeed. Not only can they do as well as the other students, when they're with me, they can do anything. I often ask more of them than their teachers do. I tell them that I am there to support and help them. We can do it together. Together we strive to use every minute we have to become the best that we can be.

Or most of the students do so.

I remind them that anything that I ask them to do, is for their own good: I am on their side all the way. And I won't give up--even when they want to. I keep pushing-- encouraging, and affirming my belief that they can do it. It may take hard work, but if they commit themselves to it, I will be right there with them working just as hard.

I'm on of those teachers who is usually the first one in and often one of the last to leave each day. I try to model for the students that hard work pays off.

I ask the students to set goals every time we meet. I ask them what they want to practice, then I support them in their endeavors. I have the best job there is-- where else can one say that the student drives the instruction? I demonstrate for the group when asked, work one on one when they prefer. I'm there to give feedback--not just whether they are doing it correctly or not, but what I think they understand and what they seem confused about. What they are doing well, as well as where they need to improve.

I take copious notes daily and try to get know my students as learners. I hope that they realize how much they all mean to me, and how much I'm pulling for their success.

Several times I have asked my students for feedback. Sometimes it's in the form of an exit query: Was that helpful? or Do you understand it a little better now? And I really want to know.

I have asked students to assess my websites (I've created 2 over the years.) and tried to use some of their suggestions. Some of what they would like to see is beyond my capabilities at this time. I always ask them to let me know if there is a concept or skill that I have not included, or if they prefer videos, games, or practice quizes online. I update and change my website accordingly.

In the past I have led students through activities that highlight what type of math learner they are: visual, number-oriented, or verbal, and whether they are more often a global thinker--always focused on the big idea, or whether they tend to prefer step-by-step procedures. The forest or the trees.

In this way we can encourage them to become more metacognitively aware. I always ask them what they were thinking. So that I can understand where they are coming from. I believe that teaching is a conversation between the teacher and the student. I cannot teach if I cannot listen, observe, and ask questions.

I ask the students to reflect upon their work. I ask them how they feel about a concept or skill, a quiz or a test. Do they think they understand? Do they think they did well? I worry more about a student who continually says that they don't need to practice, then they take a test and fail, than I do about a student who is aware when they don't understand and knows when they were not adequately prepared.

Years ago I tried to engage some difficult to reach students. I was a science teacher at that time, so I considered how the concepts and skills we were learning related to the careers they might pursue in the future. I called it "Taking Ownership of their Learning". When we were studying fossils, I buried bones in aquariums filled with dirt, gave the students toothbrushes and didn't tell them what they might find. I tried to make it as close to an archetologist's dig as possible. It worked. They were engaged, intrigued, motivated, and totally focused.

Carol Ann Tomlinson has long been a favorite author of mine. I recommend that you read this article in its entirety.

How do you differentiate? Please send me your ideas and suggestions.



Sunday, January 24, 2010

"What's Math Got to Do with It?" by Jo Boaler

Jo Boaer has written a book about "Helping Children Learn to Love Their Most Hated Subject...". In the first chapter s tells us about a math class she once attended. "The students were given problems that interested and challenged them... As the students filed out of the room at the end of class one of the boys sighed, 'I love this class.' His friend agreed."

"Unfortunately, very few math classes are like (this)... (and) Far too many students in America hate math and for many it is a source of anxiety and fear." It's not just students who hate math, "...In 2005, an Associated Press--America Online (AOL) news poll showed that ...twice as many people hated math (when in school) as any other subject."

However, recent "...trends suggest... (that while) math is widely hated, ...the mathematics of life, work, and leisure is intriguing and much more enjoyable." And so, "Our task is to ...get (students) excited about math..." by introducing them to the beauty, intrigue, and fun that math can be.

I had a great experience at work this past week. I've been helping my sixth graders to understand and work with fractions. They're good kids and work hard, but some of them are still having difficulty with some of the basics such as finding common denominators and simplifying fractions.

Why is this so hard? Haven't they compared 2/8 to 1/4 and seen that it is the same? Haven't they had experiences working with fraction circles? Fraction bars? Visualizing pizzas and pans of brownies?
Magnetic Fraction Circles
images copied from:
Hands-On Math Tiles

I think it's because we don't always emphasize the importance of the pieces being exactly the same size. Face it, it's difficult dividing circles be hand and arriving at equal sized pieces. How many times do we say, "Now, imagine that these are the same size..." ? When we use rectangles or circles to illustrate fractions, do we always draw them each the exact same size? How can we say that we are comparing fractions, if the whole is a different size?

There is a sequence to all learning: hands-on, illustrations, verbal, then symbolic. After we've had experience working with the actual manipulatives, we draw them, then we are able to visualize them and mentally manipulate them. Can you do this easily with circles or rectangles? I can't.

For this reason, I like to use pattern blocks when working with fractions. For those of you who don't know about pattern blocks, they are, for the most part, regular polygons: a yellow hexagon, a red trapezoid, a green equilateral triangle, and a blue rhombus.

image copied from:

  • 6 green equilateral triangles = 1 yellow hexagon
  • 3 blue rhombuses = 1 yellow hexagon
  • 2 red trapezoids = 1 yellow hexagon
They've created two new polygons that make these blocks even more versatile: a brown right trapezoid that is exactly half the size of the regular red trapezoid, and a purple right triangle that is exactly half the size of the green equilateral triangle. So now you also have:
Overhead Fraction Pattern Blocks
image copied from: Didax
  • purple right triangles = 1 yellow hexagon
  • 4 brown right trapezoids = 1 yellow hexagon
In this way, you can represent 1/2's (red trapezoid), 1/3's (blue rhombus), 1/4's (brown right trapezoid), 1/6's (green equilateral triangles), 1/12's (purple right triangles), and, of course, 1 whole (yellow hexagon).

When you move from working with these blocks to drawing them, you know the relationship between the triangle and the hexagon; the rhombus and the hexagon; the triangles and the rhombus; etc. You've felt this relationship.

Then there's the color. I can visualize these blocks much easier than I can rectangles or circles, because they are color coded. Mention the yellow pattern block and not only do I see the yellow hexagon in my mind, I can see all of the other blocks as well. Thus I can visualize and mentally manipulate these blocks.

Not only that, but I can change the size of the whole simply by combining or decomposing the blocks. One time the whole might be a hexagon and a red trapezoid. I call this my fish cake. I call it a cake so that the students will begin to see it as a whole and no longer a group of blocks.

Or the whole might be a hexagon minus a green equilateral triangle. This I call my rabbit cake, because it looks like there are 2 ears.

The whole might be a smaller block, such as the blue rhombus. How does this change all of the relationships?

A lot depends on what the whole is. A LOT depends on what the whole IS. I am continually asking what is the whole? Or stating, this blue rhombus; or hexagon and green triangle; or 2 green triangles, this is the whole cake, so...?

Students begin to really reason, to think, to make sense of fractions and wholes, as well as mixed numbers. They use the blocks, solve difficult problems, then I show them how what they just did with the blocks might be represented by numbers and operational signs. There is an "AH-HA!" moment. THEY GET IT! THEY REALLY GET IT!

I just had to share all of this with you because it made my day, my week, maybe even my month, when a student said recently, "I LIKE FRACTIONS!" This student said this with surprise, delight, and joy. Wow, Math Was Fun Today! Isn't that a hoot?

What do you do to make fractions; decimals; or math in general fun? Please write to me and let me know. --Lauren

For more fun with pattern blocks go to:

Monday, January 18, 2010

Allowing Students to Make Sense of Math

I recently went to a seminar entitled, Manipulatives and Modeling Make Math Meaningful through MACS (Mathematics and Computer Science Collaborative at Bridgewater State College). This particular session dealt with "Number Sense" and was presented by Nancy Anderson. Am I glad that I went!

Nancy Anderson was great. She asked us to estimate the quotient for a problem such as 3 1/6 divided by 1/3. We very naturally decomposed the mixed number into a whole number and a fraction. None of us made it into an improper fraction. It made more sense to think how many 1/3's would be in 1: 3. So how many would be in 3: 9. How many 1/3's would be in 1/6? Well it takes 2 1/6's to make 1/3, so it must be half. So the answer would be 9 1/2. We divided mixed numbers by fractions without pen or paper! We made sense of it by using the distributive property.

I was reminded of that night recently when reading an article in Mathematics Teaching in Middle School, Oct. 2009 edition. Susan Taber writes, in Capitalizing on the Unexpected, "If ...the teacher invites various students to describe and explain the differing ways they combined the blocks to represent adding the two numbers, the students will...develop a more complete understanding of the meaning of what is represented by each of the digits and the relationship among units, tens, and hundreds. An Additional benefit is that ...(it) allows the class to discuss the associative property of addition. ...They ...learn that numbers can be decomposed and composed in various ways."

The article goes on to describe work with fraction circles. It states that, "Using a set of fraction pieces with denominators of 2, 4, 6, 8, and 12 provides many more opportunities for students to explore equivalent fractions than does using pattern blocks, because the pattern blocks can represent only halves, thirds, and sixths."

This is no longer true. You can now purchase an additional set of pattern blocks which includes right trapezoids (half the size of the regular trapezoid) and right triangles (half the size of the equilateral triangle). Thus you can now work with fourths and twelves as well. So, in actuality, you are only missing 1/5's, 1/7's, 1/9's, and 1/10's.

The best manipulative I know for 1/10's is money. Beyond that cuisenaire rods work well, as there are ten differing sizes, ranging from one to ten.

We did some great work with cuisenaire rods at that seminar. We not only explored proportions and percents using an activity from Math Matters, but Nancy encouraged us to write our own lessons, with guiding questions to take back to our students.

This is the power of good teaching. This is the power of encouraging students to use manipulatives, mental math, and various strategies to solve problems, rather than the traditional algorithm. Then, be ready to take advantage of the unexpected. Explore the properties of numbers and operations the way a scientist would: does it work now? Now? Does it still work? Why does it work?

The article asks, "How do you decide how many students' solutions should be presented during the class discussion?" This is very telling. You don't want to overload the students with alternative solutions, but you do want to explore similarities and differences in strategies, and why each works, or doesn't. Keep in mind the big idea: what do you most wish the students to understand at this point? Which examples of student work might best highlight and illuminate this concept?

In summary, students learn best when allowed to play around, explore, and test ideas: both mentally and through hands-on manipulatives. Our job is to encourage and guide this exploration. Please write to me and let me know what your favorite manipulatives are and how you use them. --Lauren

Sunday, January 10, 2010

Game Playing

Everyone knows that games are more fun. More fun than work. More fun than practice. But the great thing is, great games are work. Great games are practice. So, the question is, how do we find those great games?

In Sharon we use two curricula in elementary and middle school: Everyday Math and Prentice Hall. On their website, Everyday Math says, "The curriculum has a wide variety of fact practice games. Because children find these games much more engaging than standard drill exercises, they are willing and eager to spend more time practicing their basic facts."


The Pearson website goes on better: they have teamed up with Tabula Digita to create video games involving math. "Research shows that when kids learn in an engaging, motivating environment with research-based, standards-aligned curriculum, their test scores soar," said Mike Evans, Pearson's senior vice president for mathematics. "What better way to engage young learners than with the video games that they love customized to teach the critical math concepts that they will need to be successful in our 21st century economy"


When looking for online games, you will find that there is a wealth of resources. However, just because a game is good and effective, it may still not be what you need. If you truly wish to provide games so that the students may practice every concept and skill, then you will have to create your own.

I have found the place to do just that: Here you will find hundreds of games created by math teachers from around the world. You can sign up for a 30 day free trial, but if you wish to keep your games available, A one-year individual subscription to Quia costs only $49." Not only can you use the games created by others, you can copy them to your account then make any revisions, changes, or modifications that you wish, then save it as a new game.

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In this way you can tailor each game to meet the needs of your students. Need a game to practice division with remainders? Can't find one? Make it! Need a game for identifying properties such as the associative or distributive? No luck? Make it! Anything and everything is now possible!And, once you have made a game, it will always be available: you can even share it with others on the web, adding to the wealth.

How great is that?!

They have 16 options available, you just make up the questions and answers and --Voila!-- you have a game!
  • Create 16 types of games and learning activities (as well as many formats for quizes):
  • Matching game
  • Flash cards
  • Concentration game
  • Word search puzzle
  • Battleship
  • Challenge board
  • Columns activity
  • Hangman game
  • Jumbled words
  • Ordered list activity
  • Picture perfect activity
  • Pop-ups
  • Rags-to-riches game
  • Scavenger hunt
  • Cloze activity
  • Patterns
  • Create quizzes with 10 types of questions:
  • Multiple choice
  • True-false
  • Pop-up
  • Multiple correct
  • Fill-in
  • Initial answer
  • Short answer
  • Essay
  • Matching
  • Ordering

I really recommend that you try it out. Let me know what you think, and, if you know of a better website or software program to make your own games, please let me know.

Sunday, December 20, 2009

What's My Rule?

I used the 'What's My Question?' format to have the sixth graders practice writing rules (or as Everyday Math calls them, 'general cases') for various patterns.

I first demonstrated how to write a rule: (the constant difference/ or change) times the variable plus (or minus) the constant/ what doesn't change. Example: the perimeter of a triangle is 3. When 2 triangles are placed side by side so that one side touches, the perimeter is now 4. Three triangles side by side have a perimeter of 5. etc. So the change is +1. So 1T +2 (that doesn't change)=the total perimeter.
The "What's My Question?' format worked great! I left the previous examples up and reviewed the basics in between each pattern. The students had partners to help them out. They were very engaged, motivated, and excited. Writing one rule after another like this was very powerful for many of them. I could see the light in their eyes, and they're growing confidence.

Try it!

Cooperation vs. Competition

When choosing a game format to practice math skills, we often have to choose between cooperation and competition.

Cooperation encourages risk-taking, promotes self esteem, and positive interactions among students.

Competition increases motivation, promotes self esteem (with the students' abilities are similar and neither one always wins or loses), and encourages positive interactions among the students.

I have found the perfect game format: it's called 'What's My Question?' and it involves both cooperation and competition. Students work with partners competing against another team. It can be used for any and all math skills/ concepts. The only requirement is that the students be capable enough to not need the teacher's support and input. It works best when you have homogeneously mixed teams so that the stronger students can teach their peers.

To begin, draw 12 to 20 squares on a black or white board, and number them. The teams take turns throwing something at the board to determine which question is theirs. If the team that threw gets it correct, the square is theirs and they may mark it accordingly. If they miss it, the other team gets to try for it. Should they have the correct answer, the square is theirs.

The first team to get 3 in a row (with 12 squares), or 4 in a row (with 20 squares) wins the game.

I have created questions based upon the skills they are currently working on, written on index cards, or used worksheets with problems. They sometimes like the worksheet format because they get to see the question before they throw and can try to pick the questions that they feel most comfortable answering. The card format works well if you want to pick the level or difficulty or type of question based upon the team members.

Try it out and let me know what you think.